I. 
Exact
circle division by five —
how to draw the perfect five point star 
II. 
What
Natural (whole) numbers divide a circle exactly? Meet
the perfect allstar families 
III. 
Construct
the real and the virtual shapes —
and take them to 3D 

"Give
me a ring and I will make it sing"


Geometric
construction of the pentagram, pentacle, pentagon, and the five
pointed star —
and spice it up here and there with three kinds of ten pointed stars
How
to
 sketch
a five pointed star on a clock template, for kids [yes, pick a
better shade of pink]. Tilt and rotate and double them
 draw
stars of pentagrams and pentagons
using a compass and straightedge (two geometric methods)
 construct
a pentacle directly inside your own circle
or with your
own sides length (pointtopoint dimension). What you pick —
large or small —
will be a rational and therefore an exact measure
 tell
the diff between a pentacle and a pentagram and why there are
several pentacles
 determine
the inside and outside angles of a pentagram
 find
the golden proportions that are
inherent in a penta~ star and grow them, too
 make
a fivepoint star indirectly via pentagon tiling —
and vice versa
 create
fancy
five pointed stars plus a ten pointed hyperstar from a
halfsquare rectangle (in a new tab)
 double
a pentagon in one circle for a decagon
 get
the only harmonious ten pointed star of a decagram
by always skipping 6 points
 design
5 point stars as art forms —
inherit the energies if you stick to nature
 combine
the up and down pointing pentacles into a brand new hyperstar.
On five circles. You'll be in the micro but this is huge
What
does it mean if a pentagram is
 encircled
once or twice —
a star is on a single circle or inbetween of two concentric circles
 built
from interlocking rings
 taken
as a shape of a human body
 upside
down. Devil's in details
Symmetry
in a five pointed star — there are several
kinds of symmetry and .. ..
A
perfect three
pointed
star can be had on a circle or semicircle —
and it's doubled for a regular sixpoint star or a hexagon already.
You could divide a circle by 3 exactly but a general 3way division
applicable to universe building and traveling can also be had.
Hexagon and hexagram stars are behind viruses.
A
perfect square is a four pointed star
An
eight pointed star and a twelve pointed
star share the same construction. Both stars
are perfect and both take you beyond counting as well
16
and 24 pointed stars also share the same construction. Just
as the eight and 12 pointed stars, they are in the 3:2 musical ratio
and because the growth of stars in this ratio is unbounded we have a
separate page on that (in a new tab)
How
many perfect stars
can you make with geometry? With arithmetic?
Number 36 is (abundantly) divisible by 2, 3, 4, 6, 9, 12, and 18 –
yet, you cannot make a perfect 36 pointed star
A
comment on a seven pointed star
(heptagon). Hit a snag in a circle, but ..
Pentagon
Pyramid. Fusion of
numbers in three dimensions is good but it is not about averaging
The
four sided Great Pyramid construction is via
the golden proportion
and has its own page (in a new tab)

A
five pointed star just for kids. If you can tell time you can sketch
this and other stars by hand. Oh, use the free template below at first.
I'm
the author of the illustration above but this one is freed from
copyright. (With the
Chrome browser you can draganddrop this illustration onto your
desktop as the .gif file.)
Have
a look at some pics from the stone circle at Avebury in Wiltshire,
UK.
Draw
a star, any perfect star. Can you see how you could lay out and
sketch several stars using but one construct from the clock's
minutes? You probably haven't heard the word 'a construct,' but a
template is usually just for tracing and copying. Can you join the
points by skipping some? Odd and even number of points makes a big
difference. Here are the fromthegroundup geometric constructions
of eight and 16
point stars of a compass rose.
[Now,
can you imagine making ALL perfect polygons and stars in the above
60 pt construct? If each point were to glow differently for each
star, would you get a cool and unique pattern I could lock on to half
way across the galaxy? Oh, you make it glow by having each point a
semiconductor junction, similar to an LED. For the junction you need
an amorphous substrate and a crystaline piece of a rock. Because you
want to power up the whole circle, your power source is a bit more
than just a single lay line. Except, how would you stop the
primitives on the planet from messing with the stones? Yeah, it is
all sacred feminine kind of stuff. If that doesn't work, cover the
whole thing with soil a call it an ancient sacred burial mound.] 

Intro
Historically,
geometric drawing of a pentagram star was considered a secret. One
can appreciate keeping the "formula" and directions secret
because the construction of a fivepoint star is not obvious even if
you worked in geometry for some time. Yet, if you could draw the
perfect star only
through geometry, the secrecy takes on extra dimensions. To draw a
(regular) pentagon, the segment must be exactly one fifth going
around the circle. 
Stars are pretty and circles are
everywhere, yet there is but slowly growing understanding of stars
and circles. (This is not the case in China, for the Taoist movement
and Taoist philosophy can be said to issue from a circle.) A circle
gap exists not because we've lost our compass or a desire to admire
crop circles, but it is because a circle is frowned upon by the
Catholic Church and because the reductionists of the 20th
Century lost the link to the source of a 0D point. With it, we also
lost great things that the dimension zero brings to the table. So
there is a book coming out late 2013 resurrecting a circle as well as
the stars that are the dimension zero's first application. Oh, 99% of
all moving energy in the universe is in spin and orbits. Not only
that. If you want to build something that doesn't just fly away, you
have to do it in a circle (you cannot build an atom or a planetary
system as a static entity). Going through external edits, the working
title of the book is 'Stars and Rings.' 

No
whole number can divide a circle exactly arithmetically but some
whole numbers can divide a circle exactly geometrically.
This is the first differentiator between geometry and arithmetic.
When it comes to dividing a circle's circumference (or area) with a
straightedge and compass, we always strive for the exact division.
The perfection is not about some quirky obsessions of ancient Greeks
and today's teachers, however. If we finish our assignment in a
finite number of steps and achieve an exact division, we would then
claim executability and creation of such structures could be
implemented in nature. Computer's arithmetic gives us precise —
but not exact — answers when working with incommensurable
(irrational and transcendental) numbers, but the geometric way can be
exact and have much utility just because of that. We will apply the
division of a circle in the atomic construction below.
Yes, the exact construction is about the (exact) conservation of energy.
If
the length of a circle's straight segment (cord) is exact, then the
segment's length is unambiguous and can be expressed as a finite
number. A finite number is also a rational number. If the cord's
length were an irrational number then such number's subunity portion
(mantissa) is infinite and we could not agree on its exact length —
but we could agree on the exact distance between the two end
points because the two end points can be constructed exactly
geometrically. Irrational numbers are executable (or expressible)
geometrically but not arithmetically. This is the second
differentiator between geometry and arithmetic.
You might skip on the diff between length and distance right now —
later you may avail to the explanation and construction of the incommensurable
numbers (irrationals and transcendentals).
Instructions
on geometric division of a circle into five equal and exact parts —
in five steps:
Construction (continued):
If
you want to construct the penta~ with a circle radius of your choice,
point A is one unit distance (onehalf of the radius) away
from origin O.
When
you divide a circle into exact fifths along the periphery you now
have the template and:
Finish
the pentagon
by connecting every neighboring point and erase the circle. Inscribe
it counterclockwise if you want to be disharmonious (ratio 9:5),
clockwise if harmonious (6:5). (Cw and ccw makes a big difference —
think enhanced modulo
math. Ccw is acw for the Brits.)
or
Make
the pentacle
by connecting every other (second) point. If you do it
counterclockwise, you will be doing it in the 8:5 ratio, which is
harmonious, and you should not have a problem with that once you
understand that the underlying pattern mechanics are clockwise —
it's on the Venus page.
Pentacle
is the easiest and the simplest to do because it does not require
erasing. Does it mean it is the most fundamental?
 Complete
the pentagram
from pentacle by erasing the circle
 Make
the five pointed star from pentagram by erasing the inside
(smaller, upside down) pentagon. Also see pentagon tiling, further on
Unit
distance
The
unit distance u
is the shortest distance used in construction of a particular star.
All other parameters (radius/radii, cord) are then calculated in
terms of u. This is new {Sep 2012}.
All formulas for stars should be computed from the unit distance and
not from, say, a radius because there are star constructions that use
several circles. In our example, the equation for the length of the
cord c in the above construction comes from two Pythagorean
relations and is quite involved:
but
for the five point hyperstar
cord
c = 2u.
In
either case, you can now substitute any distance for u and
get the measures of the rest of star's parameters. There is more to
this. Geometrically, the square root of five, for example, can be of
any length and this length issues from the unit distance used in the
construction of the SQRT(5). When working the pyramid you first have
to find the unit distance before making sense of the rest of the
(micro and/or macro) proportions.
Polygons
and stars
The
~gon identifies the point connections that go straight from one
point to the nearest next point on
a single circle — polygon
in general. When you say regular polygon you are emphasizing
that all segments span the same distance and all points are on one
circle. Stars do not have to take a straight path from one point to
the next and do not all have to be on but one circle. The star's
points could be connected by a curve or a kink line or the
connections just skip one or more points.
As
you go around the center from one point to the next, you might keep
turning in but one direction and then the star is said to be convex.
Regular polygons are always convex. Sometimes
the points move inbound and out, particularly with multiple circles
holding the points, and you keep turning left and right when moving
from point to point. In such case the star is said to be concave.
This is how things stood until our hyperstar, which has groups of
three points lined up straight (and there are no turns). The
hyperstar is discussed in the right column. The hyperstar is not regular.
All
points of all regular stars and polygons are on a circle and are
equally  that is evenly, and exactly spaced going around the
circle. The best way of seeing it is that they have the exact
fraction of a circle between them. However, only some stars are
constructible exactly — see Perfect
Families. As to the actual angles between points —
see below, for these angles could be irrational numbers. (For now,
ask yourself if a protractor can give you any irrational angle.)
Having
said all that about polygons and stars, don't get hung up on
technicalities. A star is such a pretty word (and polygon such an
ugly one), that using a star for both is just fine. Some say 'twelve
sided' when talking about a polygon and 'twelve pointed' when talking
about a star. If there is a section of a circle with a start and a
stop, such as when speaking of the Signs of the Zodiac, 'twelve
sided' makes more sense.
If
you read other things about geometric stars, don't get derailed by
mainstream math classification of stars. Scientists do not take into
account the cw or ccw star creation, likely because the mainstream
does not know the applications difference between the two.
(Harmonydisharmony enters here but there is more to it besides cw or
ccw.) Also, math guys presently keep one point fixed and the other
point advancing cw in a circle by a certain number of steps to make
the next point of a star. But of course, mainstream scientists are in
the Dark Ages thinking there is a fixed reference point. The Quantum
Pythagoreans book gives all planets orbiting frequencies from
which the stars are made and, gee, the Earth is orbiting while the cw
or ccw pointtopoint trace arises mathematically from that.
Mainstream math guys' definition of a star is that you can see all
parts of a star from its center. This is a great example of a
definition of arm chair convenience because the VenusEarth interplay
makes a cool curlicue between points of a five pointed pentacle star
and this does not match the scientists' definition of a star —
so here is ours.
Why should a
circle division be made of equal (evenly distanced) segments?
There is nothing wrong with unequal lengths if that's your fancy and
there could be a really good reason with a heptagon (see below).
However, an electron's wavelength is proportional to the electron's
energy and if several wavelengths of one
electron were to fit (were to close) around the nucleus then they
have to do so in whole multiples of the same distance. It's about the
numbers (in a circle). 
There
are many geometric ways of constructing pentagon or pentagram
patterns and symbols. The construction presented here has the length
of the side of the pentagon c
the incommensurable
(irrational) distance —
that is, the measure of the side's distance is composed of an
infinite number of digits (that do not repeat individually or as a
group). Other constructions make the side c
a rational
distance, which is better suited for the Great Pyramid's purposes. In
the pyramid, one half of the side of the base is (must be) a rational
unit of measure, for such measure is executable
and can become. (For transcendentals you
may have to put a kink in that.) The connection between the five
sided pentagon/pentagram and the four sided Great Pyramid is through
the golden
proportion, a subject that lets you
understand how to draw and construct the Great Pyramid, and include
the pinch on its side.
Angles in a five pointed
star and how they relate to the golden proportion
Determination
of pentagram's angles is easy — on the
interior or the exterior. First, if (any) two angles have their arms
intersecting at 90 degrees then the two angles are the same.
The central
(Egyptian) star below has 360/5=72 degrees between its arms. Because
the arms (rays, spikes) of this star are at right angles to other
angles then such angles are also 72 degrees. You will get to 108º
angle inside the pentagon with the subtraction of the 72º angle
from 180º (from a straight line). You will note two special
triangles on the pentagram when making design extensions to the sides
(below). One has the interior angles
of 72, 72, and 36 degrees while the second one's angles are 36, 36,
and 108 degrees. Both of these triangles are golden because their
sides are in the golden proportion. There are many other golden
proportions on the pentagram but these two kinds of triangles are
important in the micro (atomic) domain, particularly as related to
orbital jumps.
In multiples of
one tenth of a circle
All angles
internal and external to the pentagram or the five pointed star are
in multiples of
1/10 of a circle: from 36º on to 72º, 108º, 144º,
180º ... 324º, 360º. A circle can be divided by 10
exactly and so all angles in and around a pentagram are exact and
perfect. A pentagram, aka pentalpha, is a prominent symbol of the
Pythagoreans. It is said the pentagram is good for one's health, and
there could be something to it considering the golden proportions
that are all over the star. I think it is also likely the pentagram's
prominence is due to having an abundance of the number 10 while
adding a circular aspect to the many number 10 aspects
of the Pythagorean Tetractys. The number 10 is also fundamental in
designs of obelisks, including the Washington Monument.
Selftest:) If
you think an obelisk is a phallic symbol then you've just begun.
The angle of
144º seems obscure (it's on the outside of the star) but it's
the only angle you'll need if you want to draw a pentagram with a
computer program the likes of 'Turtle,' 'Scratch,' or 'Logo.' The
program would run like this (in pseudocode):
If these computer
instructions do not mean much to you, you are doing well. First you
want to create the geometric foundation — yes, in your mind,
before resorting to a computer.
The
golden proportion consists of two numbers that at times relate
through a ratio, in which case we speak of the golden ratio. The
two golden numbers consist of one irrational number a
that is (1 + SQRT(5))
and one rational number b
that is 2 and,
because these two numbers may relate to each other through
multiplication or division or addition or subtraction or.., they
should not be reduced into a single number. Reduction into a single
number severely limits the application of the golden
proportion and that is one reason
scientists like to reduce it as the Phi [scientists have reductionist
tendencies – perhaps not a disease but it could be a handicap].
Reduction into one number hides other relationships the two golden
numbers might have. 
The golden spiral
Golden spiraling
happens naturally in 2D or 3D as you observe the golden proportions
and figure out how you could grow them. Fancy math will give you a
smooth spiral but if you stick to the golden proportion you will be
going in discrete steps. (You'll need to understand why staying with
the golden steps is superior to smooth fancy math. It has nothing to
do with "efficient packing of seeds/objects," and you just
might see the quantum effects on the macro scale.) Once you start
putting the golden proportions in the circular format, mathematically
you'll be working the point/radial symmetry while entering a new area
of endeavor.
Pentagonal tiling
Also known in
general as tessellations, doing it in 5fold rotational symmetry is
favorite of Dürer, Kepler, Penrose.. .. All
pieces that make up sets for the fivefold 2D tiling have their
angles in multiples of 1/10 of a circle {July
2011}. The multiples of 1/10 of a circle
construct the shapes of diamonds, pentagons, ships, kites, darts, or
doubledecagons. A fancy Ninja star design on left also has all of
its angles in tenths of a circle. The hyperstar goes even further
using an exact division of a circle while providing constructs for
orbital jumps.
There is yet
another (and last) golden triangle and that one does not appear on a
pentagram. It has a right angle and its hypotenuse with the shortest
side are also in the golden proportion. This triangle is onehalf of
the Great Pyramid going across the mid face and into the center of
the base. You can see it below on our Golden Eye design.
Pentagon
Dimension Priority: Diameter Dim or Pointtopoint Length Dim
There is a dual
approach to a geometric pentagon star construction. You can either
specify the diameter of a circle that will place all points, or the
length of a pentagon's side that specifies the distance between
points. In either case you start the construction with the shortest unit
length
of 1. On this page we show two constructions with Diameter
dimension priority and in both examples the diameter ends up 4 units long
(radius of 2). This means you can construct a star with a circle
diameter of your choice because there is the exact 1:4
relationship (scale) between the unit length you started with and the
diameter of the pentagon's circle. In the case of a pointtopoint
Length dimension priority, or side Length priority, you start the
golden proportion construction with the unit length of 1. When
finished, the pentagon will scale to the side length of 2.
Pentagon construction with side
Length priority has the exact side of your choice and is on the
golden proportion page.
A circle has many
positive connotations. What would be the idea of dividing it? Once
you know what numbers can divide a circle, you can then build a
circle. Not [yet] from real things such as wood or metal but from
waves. It turns out that the waves must have a particular wavelength
count (a particular multiple of particular energies) before these
waves are able to close in a circle  and thus be symmetrical about
a point. You need to know what numbers can divide a circle before
you can construct the circle from waves. You might think this is
something witches do, and you would be right, but an electron is a
wave that wraps around the nucleus, too.
There is (always)
a bit more to this. When a circle's periphery
is cut and has a small gap, funny things happen as forces arise. One
could call this a circle corruption and in a way it is. Yet the
forces that arise are not corrupting, for they attempt to close the
circle and .. (think free energy).
The making of a circle
is also about taking a step from 1D to 2D. There, you will find the
friendly transcendental number Pi.
To round it off, you may want to learn more about the squaring
of a circle, for it is about the straight and curving
geometries. We did not forget the ancient Egyptians and use the
example of the five pointed star as one of the steps in working the
circle and the square. The golden proportion and Pi get very close to
each other. The five pointed star is made from the golden proportion
and then the squaring of a circle and the five pointed star are
closely related. 
Ah, geometry
Not everybody likes geometry. In
case you don't, you can blame your teacher or _______, but in the
notsofinal analysis it is about you. Geometry is about movement and
placement in space, from an atom in your body to your ship as a
whole. Lots of geometry is in a plane and you have a good argument if
you say your head is not flat. So let me cut to the chase. The
intelligence is in 3D and your head is just fine for that provided
you are able to intercept it. Lots of free energy is in 2D and it can
be harnessed there once you figure out how to relate 3D to 2D. Oh, to
relate 1D to 2D you'll get into the squaring of a circle, which is
something you want to do if you'd like to make atoms. (Light is in 1D
and energy of an atomic electron cannot be in 1D —
it would leave the atom.) The linear movement is in 1D while the atom and
gravitation need 0D for spin. Your challenge, desire, need, or
necessity is to understand and work the Pythagorean tetra(ctys) of 0D
through 3D because that is how the universe is built and you want to
continue to be a nifty and smart participant in it.
Symmetry
in a five pointed star
Symmetry has appeal. It makes
things look nice but it could be difficult to move beyond that. There
are similarities with reflections in a mirror —
or refractions through a focus of a lens. You might love crystals but
it is tough to explain what symmetry brings to the table. Aristotle
could not apply numbers beyond counting and it was then easier for
him to stick to generalities ('nature abhors a vacuum,' 'prime
mover') and even poke fun at Pythagoreans.
There are two kinds of symmetries:
even and odd. The even symmetry duplicates things about the (usually
vertical) axis while the odd symmetry duplicates things by
halfcircle rotation about a point at the origin, which is the center
of the circle used to make the star. Symmetries issue from geometry
(and geometry issues from numbers). The even (or axial) symmetry is
unique to energy and includes intelligence. Esoterically it is the
head of the Sphinx while the empty space between the Sphinx' front
paws is the (virtual) line of the axis of the even symmetry. The odd
(or point) symmetry is unique to charge and matter. The even symmetry
is inclusive while the odd symmetry is exclusive. Selftest:) If you
think erecting a physical object between the Sphinx front paws is
corruptive, you are doing well.
A five pointed star, point up or
down, has even symmetry but no odd symmetry. You could also have
heard of 'rotational' symmetry. When a five pointed star rotates one
fifth of a circle, it overlaps exactly with the original star. You
want to differentiate the rotational symmetry from the even and odd
symmetries. Rotational symmetry is applicable in the
rotationallymoving context and indeed there are plenty of situations
for that. I like to call the even and odd symmetries the placement
symmetries while the rotational symmetry is one of the movement
symmetries. Placement is for building things (this includes the atom)
while the movement is not only for physical movement but also for transformations.
Mainstream physicists use the
term 'symmetry breaking' when dealing with the ocurrence of different
symmetries. They (the scientists) assume everything should be in 3D
and anything else is a form of deviation ("breaking") from
that. This is very stupid. The fundamental thing to look for is
computability. If the system is computable, it will happen. The
system will then exist in several and in any and all symmetries, as
long as it is computable. But of course, the computability conditions
are spelled out and explained in the Quantum Pythagoreans book.
This talk about symmetries
becomes introductory once you begin to appreciate that the (law of
the) conservation of energy is based on the conservation of
symmetries. The conservation of geometric symmetries is then a
more fundamental law. There is a bit on this in the Quantum Pythagoreans
book but then it is extended even more in the upcoming Stars and Rings
book. What? Who needs cables? 

Five pointed star
in a circle
There are three sources – and
therefore more than one meaning –
of the five pointed star. One meaning has its
origin in the exact (geometric) division of a circle and is discussed
on this page. It is a fairly complex though rewarding topic that
leads to the symbolism of a star issuing from a single circle.
Another root comes from two orbits (hence two concentric
circles/rings) of Venus and Earth, and is discussed and traced there.
(Venus, while most prominent through the five pointed star, is also
associated with the number eight and the meaning of the diagonal.)
The third source of the five pointed star calls on nonconcentric yet
interlocking rings {Dec 19, 2009}.
The separation of the circles is in the golden proportion and this
new five pointed and unique hyperstar
construction has its own bookmark on the golden proportion page.
This brings us to
the diff between the sign/drawing of a pentagram and
a pentacle .
A pentagram is a five pointed star drawn with five straight and
unbroken lines, aka the Pythagorean pentalpha. A pentacle
has a circle(s) around the star.
Yet, these are but technical differences. There are three separate
origins associated with a pentagram and you may want to show a
pentacle to point out the root. In other words, a pentagram always
issues from orbits/orbitals/circles/rings/rotation and there are
three separate ways to do so, as follows:
1)
A single circle around the star makes it the classical or atomic
pentacle that comes from atomic construction, and a single
circle shows the standing wave around the nucleus  the orbitals.
Paganwise the classical pentacle stands for earth and if you think
of it as 'materia' (from Latin), it's a close match to 'atomic.'
2)
Two concentric circles around the star make it the cosmic or planetary
pentacle and the two circles are the orbits of Venus and Earth
(can be computed via modulo math from the clockwise 8:5 orbit ratio
and in the illustration on left the orbits are to scale). Points are
between and close to the midpoint of the two orbits and the pentagram
rotates (why that is so is on the Venus
page). For Pagans, the double circle around the star is about
'drawing down the Goddess,' and the circle diameters are calculated
along with point sequencing on the Venus page. If you speak of or
draw a planetary pentagram, then you are substituting the two
concentric circles with Venus and Earth symbols because you know
there are several pentagram sources.
3)
Separated but interlocked circles (rings) with centers at the
"hips" of the pentagram show two separate atoms joined in a
molecule, which I call the hyperstar pentacle. This is new and
I don't presently know of anybody applying the hyperstar pentacle.
Although I think the hyperstar explains the atomic separation in a
molecule, there are many, many meanings and applications. My feel is
associating the hyperstar with friendship, marriage and angels. I
like using it in Tai Chi and here is an
example. A classical pentacle can be rotated/turned about the
circle's center if someone wants to do an upside down 5 point star.
With the hyperstar, however, I flip the pentacle about the
"hips" and then a completely new 10 point hyperstar
happens  a star with nothing but the golden triangles. See below.
[Flipping something about an axis is a feminine operation.]
The abundance of
the golden proportions associated with all penta~ constructions
deserves an analysis of its own. Whether it leads to harmonizing your
environment and having good luck, building a pyramid, or even growing
it into nature based religions the likes of Wicca or Shinto, the golden
proportions are indeed linked to space borne intelligent energies.
Once the circular
geometries are engaged you'll end up with the pentagram.
It is then okay to draw just the pentagram but you want to draw the
various pentacles if you want to show where the pentagram is coming
from. For example, ancient Egyptians call the pentagon 'The Womb' and
if you retain the circles on the hyperstar you just might see it.
Okay guys, another story has Isis looking for Osiris' parts. She
found all except the phallus, which was eaten by a fish. Strange
story. Well, if you retain the circles you just might see a fish
doing its thing. [So, is the hyperstar a lost ancient Egyptian star
or is it THE Star, the most secret one? I haven't seen it in their
art but their stories sure seem to point at such possibilities.
Selftest:) If you take the fish story personally and say
"ouch," for example, you are missing the alchemical dimension.]
Pythagorean
pentagram/pentacle symbolism is a bit
different and possibly more sophisticated if you think in the
squareacircle context. The pentagram is encircled once and then a
second ring is added as a piece of jewelry. The second ring is then
at the right angle to the first. (The illustration is from Secret Teaching
by Manly Hall.)
Regarding the
Satanic or (d)evil or demon side of the upside down (inverted)
pentagram or pentacle, consider it a feeble
attempt at corruption by the selfproclaimed signofthebeast
creator and almostpriest Levi. You want to know where corruption
comes from and then you are in position to overcome it. In
this case the upside down pentagram could issue from natural
rotation of the EarthVenus cosmic/planetary
pentacle in the solar plane  with the sun in the center  and
then the satanic notion loses its meaning once you appreciate that
the cosmic pentacle's rotation does not stop and there is no 'up' and
'down' of the solar plane to begin with, just as you cannot tell if
the whole coin on the right having odd (point, "sun")
symmetry is up or down.
(Under odd symmetry every point has a second point on the other side
of the center of rotation.) If you want to unambiguously show the
star's pointup or pointdown placement on a coin, you have to place
the star next to a nonsymmetrical reference such as an animal (or
introduce the even/mirror symmetry, which is the case in nature). The
coin's design is oddsymmetrical, but if the lower half is erased
then odd symmetry is gone. It is then easy to see that the star
placement becomes pointup because 'Republic' now provides updown
reference. So you can appreciate that "up side down" is not
a given under odd symmetry alone because when your reference is but a
point (such as the Sun in the center), you could see the star one way
looking from the center and the opposite way looking into
the center.
Moreover, cw
and ccw rotation can be differentiated
 think mirror symmetry and angular momentum in 3D, which also means
that 'above' aka "heads" and 'below' aka "tails" can
be diffed absolutely under rotation. Geometric stars' origin is
from orbits (macro) or orbitals (micro). Micro includes both the
atomic and/or molecular (valence) orbitals. This means that the stars
issue from odd [masculine] symmetry. Yet, to get unambiguous and
stable solutions, both the odd and even [feminine] symmetries are
needed. Ancient Egyptians include a horizontal bar with their oval
cartouche just for that reason, and the bar  which is the axis of
the even symmetry [it is not "the horizon"]  is very
explicitly tied to the oval. Now, I am extending it past the
Pharaohs' cartouche, and there are more practical and new examples in
the 5 and 10 point hyperstars below and, of course, the Golden Eye.
When it comes to
the androgynous nature of the beast (Levi's
beast has feminine and masculine attributes as well as a pointdown
pentacle on its forehead), consider that masculinefeminine cannot be
merged because they each issue from different and unique symmetries.
Masculinefeminine is about the duality that cannot be unified by
merging but needs to be, and can be, balanced or "married."
(Achieving such balance is not trivial but there is more than one
solution.) In essence, the "evil" side arises from
conflicts that ignore nature's duality, and one of the gateways to
its understanding is to ask, 'Why and how is the human brain separated
and joined
at the corpus callosum?' Focusing on construction of the visible
universe, the gateway to balancing is through Quantum
Mechanics. Oh, if you don't like the status quo get into vortex
and free energy.
Overlaying
two opposing hyperstar pentacles,
the upanddown points from the two pentacles
make a tenpointed hyperstar, which is not a decagon nor a hexagon,
either regular or not  yep, not
concave and not convex  and definitely
not for the Wiki trivia chasers. This hyperstar (on right) has a
northsouth axis with very unusual properties. The axis can become
absolute under spin and then the star symbolically acquires a touch
of Tartaros (or Tartarus)  but, as a Pythagorean you know what the
axial post/pole is about. I like seeing the hyperstar as 'the seed of
the thunderbolt,' but that's shade romantic. Did you notice all
triangles on the hyperstar are golden?
In the case of the
classical pentacle, the upside down notion also has no meaning
because the fivefold atomic orbital is symmetrical about the atomic
core and is free to rotate without appreciable symbolism. (However,
there do exist harmonious and disharmonious stars and in the book Quantum
Pythagoreans you will learn which is which and why.)
A
human body has a close resemblance to a five pointed star. Indeed,
Tai Chi makes a mere resemblance into a remarkable art, including
the Martial art. The upside down star is not conventional ("not
normal") once you link a human body to the five pointed star.
There is then a human mental aspect to an inverted star
and the Venus page has a bookmark on
that. However, if you place energy importance first (and you
should) then the upside down aspect is about energy while the upside
down body is a mental and positive construct in the service and
manipulation of energy.
How do you make your
own pentacle or pentagram? Buy or make is okay
but to have your own you have to have it in your mind. And add a cw
pentagon, too. [Feeling better?] Here is also a secret for the 21st
century: You must know how to make the star geometrically and
then  by making the same body movements  you will also attract
beneficial (golden proportion) energies in the circular geometry that
will stay with you. Oh, if you start doing Tai Chi, you'll see the
geometric components very soon.
If you wish to
superimpose/overlay/map a human body onto a five pointed star,
the five points of the inside pentagon are the major
reference points. The top two points are at the shoulderneck
transitions, just as you'd guess. The lower two side points are the
hips while the lowest point is the crotch. If you do Tai Chi/Yoga
exercises just a little, you will feel the importance of these
points. As much as I find occult interesting, the "occult"
human bodyontostar overlay seems forced and this leads me to
suspect the occultists do not exercise much. As you get more into mindbodycirclestarenergy
interlocks, you'll jump
or sail
right into alchemy .
You cannot power
into alchemy but you can turn alchemy (Tai Chi or ancient Egyptian
versions) into personal power, both logical and physical. There is
the Buddhist view, too. Ever seen an angel? 

The
book you will thoroughly enjoy
QUANTUM
PYTHAGOREANS
Harmony is talked
about ever since antiquity but it is this book that actually allows
you to predict whether any two tones will be harmonious. Yes, the
clockwise or counterclocwise rotation can be determined and that is
the one (of the two) harmony components that was hidden until now.
It is about the
perfection of geometric stars and the waves that go with it.
More
.. 

The
radius measure of 2
(diameter of 4) in our pentagonal construction on the left is the
outcome of using the shortest applied distance as the unit 1.
This is not because you couldn't divide by two (you can  and work
with ½ as the distance OA, for example), but if you
construct other structures such as the the Great Pyramid with the
shortest distance as the unit 1, you will always be in sync
with your numbers from one structure to the next. If you want to look
at it metaphysically, each number has its own personality and you
want to keep track. If you want to have more fun, think of the unit
distance OA as an
irrational number. Even Euclid
did not think of the number 1 as just a counting number.
On the political
side, particularly in the association with Communist power,
you want to be cognizant that the five pointed star issues from
orbits and the star is always a 2D entity made from 1D constructs.
Making the star into a 3D star (the likes of the Kremlin) points to
the lack of understanding on the root of the star's creation. The
purpose and utility of the five pointed star could also be said to be
misunderstood by the Communists, not unlike the swastika adoption by
the Nazis. [If the Nazi technology was as advanced in the flying
saucer technology as is rumored then the swastika would not be far
off.] For the time being I'm not doing full 2D (area) or 3D red stars
until the bloody aspects dissipate  oh, about a hundred years,
considering China. These stars are fancies anyway.
Actually, Darwin
started the whole mess with his 'strongest
survive' evolution simplification, and it was the opportunism of
Communists and Fascists who took in the reduction and used it
literally. (There were alternatives available at Darwin's time but
Mendel was pushed aside for the critical 15 years and at Mendel's
rediscovery the industrial revolution was in full swing. Book review: The
Monk in The Garden.)
What to look for.
People bent on power will promote and support the following:
1. Nature
is on our side
Best represented
by Dawkins' The Selfish Gene, the idea is that being loaded
with money is natural (you guessed it, it's Gates' favorite). A virus
is selfish, a gene is not. In the UK the Oxford U nicely masks its
ignorance with arrogance. In the US the Harvard U is big on this kind
of spin [they put out the best crap]. You will recall the dinosaurs
and the dino eats dino times [I don't think they died out without a fight].
2. Running
out of energy
Usually about
("peak") oil, this mentality replaced "Running out of
food" from the last century. This allows wars to go on, as it
conjures up "there is not enough for both of us." This is
but one example of reductionism. [If you don't see God, there are
plenty of bosses to work under – here and on the other side.]
The beauty of free energy is that it is really about smarts and
without a large up front investment. Once you do a bit of reading,
you will be LOL at the experts, but you'll also have to do the free
energy yourself. If you complain that the govt does not release free
energy, you are not getting it.
3. Socialized
medicine
The mind job here
is to convince you that you need the institutions of state and
selected private hospitals and clinics for you to stay healthy but
there is an obligation to pay extra premiums. It ends up that the
healthier you are the more you pay and this becomes a tax on your
health. Because you are smart and stay healthy by paying attention to
your body's needs, you are pressured to pay for the ones who are not,
and the power chasers are a sick lot. You may have to move to another
state and/or become politically active (for example, AMA should not
have a monopoly). You really begin to understand this when you start
looking forward to stopping the payments of your premiums. You may
find out it is the best thing for actually being healthy. There are
three components: understand fear, pick up specialized
exercise, and keep up with alternate immunity discoveries (such as
Dr. Jan Raa's) as well as the revival of Raymond
Rife technology. Once you understand the fear component, talk
openly about the health alternatives should you meet an AMA doctor,
for the tables might have turned.
4. Culture
or religionembedded "values" of guilt, fear, and intolerance
Once embedded they
can be invoked without explanation (prejudice will not be seen for
what it is). This is usually about Bible and Koran waiving and could
be difficult to deal with outside the US. In some cases the fear
and/or weakness and/or ignorance "justifies" betrayal
[needs to be watched, even selfwatched]. Betrayal is also a
component of power, and power has its own category (power is reversed
in the virtual domain and there it is not based on specific might but
on infinity). In the Pythagorean and Buddhist traditions the
friendship works well. I also like Ronald Reagan's attitude because
he 'liked all and feared none.' You might get hurt but you will have
the last word. 

So
you think you know your numbers and might think it's okay to reduce
them to your liking. But if you construct the Great
Pyramid with the golden numbers and use the shortest
distance as the unit 1, you will arrive at the pyramid's base
as having the side length of 4.
The base of the Great Pyramid is then 4 times of some unit of
measure. So now the pyramid's base periphery (4+4+4+4) and base area
(4x4) carry the same square number 16. You see, if you
reduce the numbers and think of the pyramid's base as having the unit
length of, say, two, the base periphery would have eight units of
length but the base area would be but four (square) units. If you do
not reduce the numbers you can think of the number 16 in the context
of acceleration (unit of measure per time squared) and derive the
unit of length that is most appropriate for this planet [yeah, it's a foot].
Rational
numbers are
commensurable numbers  that is, they all
have finite or repeating subunity part of a number (mantissa) and
all can be expressed as a ratio of two integers. Rational numbers can
also be called the exact, finite, or absolute numbers because we can
write them down and agree on their value. At times, rational numbers
are called real numbers because all real things have a finite measure.
Rational
numbers happen when we ratio two integers. All
mainstream mathematicians define the rational number as the ratio of
any two integers. So, a mainstream math guy would say, "Of
course the rational number is a ratio of two integers  it is
defined that way." Yet, you really do not want to be mainstream
and acquire but an encyclopedia knowledge of the world. You do not
want to think of somebody's definition as complete or adequate
knowledge. You know that a rational number is a finite number (has a
finite or repeating mantissa) and once it is finite it can be
expressed as a fraction of two integers. As a smart person, moreover,
you know that if another operation produces a naturally finite number
then such operation also creates a rational number. The circumference
of a circle is a transcendental number. Many of circle's round
segments (arches) are transcendental numbers and their straight cord
could be an irrational number  and both of these numbers have an
infinite mantissa (infinite precision). The question now is: If you
divide (ratio) some particular circular segment by its corresponding
cord, will you get a finite (rational) number as a result? 
Some
Pythagoreans view the number two as a problem
number because it "divides the unity." Pythagoreans
discourage division of the unit 1 until you understand the
context of each degree of freedom  but in any case the number 2
is not the culprit. (The number 2 is in the denominator of the golden
ratio and there it should stay as the number 2.) As you get familiar
with this site the subunity will become applicable to atomic
orbitals and hence the number 1 is the Great Divide between
the macrocosmic and microatomic. [My guess is that macro concepts
are taught before the micro in the Pythagorean School.]
Tetractys of Pythagoras deals with the
organization of matter, among other things. What makes you a
Pythagorean? Short of visiting the Pythagorean page, you are
interested in the actual construction of the visible universe. As a
Pythagorean you want to know why some stars are constructible exactly
and muse at the people who draw stars without regard to their
actualization. Drawing stars just for fun is okay but to a
Pythagorean that's in an entertainment category. The ~Wiki~
compilations about stars are trivia, which quickly become distracting
as well. Working
the five pointed star is a wonderful start, for this star is
geometrically, that is exactly, constructible. Constructing the real
universe is, moreover, the greatest show there is. Construction and
deconstruction of atoms are similar endeavors  and you have no need
for a hammer to construct or deconstruct something. The forces
therein can add up in controlled fashion, too. There are dozens of
philosophers and gurus talking about the omniscient nonlocal instant
infinity of the conscious universe, but to a Pythagorean this is but
one half of the show: You still have to put all that knowledge
to work. Yes, we have pictures. The upside down star has no negative
meaning in the micro domain. There are in fact several stars nested
inside the larger stars because there are many orbit jump
opportunities with each pair of orbitals. Yes again, ALL triangles
are golden.
Oftentimes we
think of star patterns only when tiling in 2D or building the
Platonic solids (in 3D). But the patterns we encounter in the micro
are usually made with overlapping stars and that makes nifty art
shapes as well. The overlap comes from the inclusiveness of waves.
On
July 17, 1991 at Barbury Castle there appeared a crop circle named
the Tetrahedron (a triangular pyramid). For its 20th anniversary we
have a summaryupdate. 
Here
is a poster for a Graphics show I had in Prague (where I teach now)
in February 2013.
Largeprint format
is becoming affordable. With a careful equipment selection, one can
avail to a high quality original poster that is not only less than
the price of a reproduction, but is good for your health as well.
Just about ready
(second picture).

Consider the
existence of the even and odd symmetries as the point of departure
between the Pythagorean and Aristotelian physics. Pythagoreans
continue basing new concepts on numbers and then the even aka twofold
symmetry issues from and relates to the number two (and the
feminine). Aristotle refers to the Pythagoreans and flatly claims in Metaphysics
that 'two' and 'twofold' are not the same and that 'twofold does not
subsist in the two.' (Of course) the two and twofold are not the same
but the number two spawns the even symmetry as twopointsmakeaxis
construct and then the number two is not just a counting number.
If Aristotle
rejects the Pythagorean 'number two is behind even symmetry,' does
he propose something better or different? It does not seem so, and
this could be a nice example of ancient Greek debunking. Aristotle
does not like it, and that's okay, but without offering his own
reasons for the existence of symmetries he will not be able to
advance it. Sure enough, we don't hear from Aristotle on symmetries.
Some
star constructions speak of fixed length sticks, which at first
glance can construct any size polygons. Here is
where the executability
of angles comes up. In space, the
irrational angle is constructible only approximately and only
some angles will be actualized –
think snowflake formation. Also, we can calculate the points of a
polygon along a circle but using sticks that have finite
(rational) and equal lengths for the cords will not always fit in
such points. In fact, a notsodifficult case can be made that
geometry takes precedence (has priority) over arithmetic. [If you are
a scientist, you may think of Emmy Noether who ignored the nature's
beauty of snowflakes and made simplifying assumptions about space
that proved the 'ignorance is bliss' postulate – for in her
world everything is reduced and snowflakes and crystals don't exist.]
You
can calculate the area of any polygon by taking the area of the
triangle and multiplying by the number of sides. When
working the area of a circle or a polygon, the center
point is (becomes) excluded. (If you are
metaphysically inclined, think Isis looking for all parts.) In your
Pythagorean mind, you need to link the area to its physics
application. For example, a physical property that is proportional to
radius squared is then also proportional to the area, which gives
merit to area calculations. This is bigger than it seems. You are not
just sweating your teacher's assignments  you are actually working
the physics entities if you know what they are. 

One other
Penta~ construction
This one comes
from Yosifusa Hirano of 19th Century
Japan. It is elegant and also constructs the pentagon or pentacle on
radius 2.
The angle
of 72 degrees
If you think the
construction of a 72 degree angle above is special, consider it the
exact one fifth of a circle rather than a number with some magical
properties. The angle of 72º is 360/5 but the number 360 is arbitrary
to begin with. Geometric one fifth of a circle gives you the exact
division of 2Pi and you now have the exact division of a transcendental
number — something your computer cannot do. Yes, the arithmetic
division of a circle by a whole number (or any rational number) must
use the circumference or an area of a circle, which is based on the
number Pi having an infinite mantissa — and the arithmetic
result is never exact because the result has an infinite number of
nonrepeating subunity digits. The modified Hirano construction below
begins to work the transcendentals in the squareacircle context.
Visit the proofs page.
"All
pentagram
angles are
arithmetically
divisible by nine."
"What?"
"Or by four."
So, what's the big
deal if the number 360 (degrees in a circle) is no big deal?
If you line up all angles from the pentagram: 36, 72, 108,
144, etc. and sum their individual digits you will always get 9. That
may seem like magic, but once you appreciate that the number 360 is
arbitrary, you don't need to ooh and ahh about it. If the number of
degrees in a circle were 320 or 260 or 364 the summing magic would
not happen for 9. The circumference of a circle is 2Pi and putting in
a number to stand for degrees is purely a practical consideration. In
the case of 360, this number is 4x9x10 and now, because the angles of
a pentagram are in tenths of a circle, the number that is left is a
multiple of 9 (and 4) and so it will be always divisible by 9
(or 4). All numbers divisible by 9 have their digits sum up to 9
(modulo 9 — thank you, Gauss).
So, always work with fractions of a circle (or fractions of 2Pi) even
if the numerical subunity pushes your right brain into infinities.
There is plenty of real magic left in this subject, particularly if
you get into the squaring of a circle. 
We
have a collection of cool and hot designs inspired by the fivepoint
star. View select
designs
 or visit our
store at Zazzle (.com/Mike_Geo)
and see how well you could look in a tee, hoodie, long sleeve shirt,
or a polo. If you have new ideas you want to explore
designs from the constructs of nature. All our designs come from
nature and center on the golden proportion.
Design on right is
called the Adventure of The Red Sun
The Golden Eye.
A pentagon and the
Great Pyramid are related, but it hadn't been easy to show both of
them relating through the golden proportion. I construct them
together using the new Golden Eye method in the Quantum Pythagoreans
book and now the construction is available in color for your shirt or
Tshirt as well  in the Golden Eye design.
The golden
proportion is between the pyramid side and the halfbase. The Great
Pyramid is shown in a vertical cut through the midface. Much detail
is on the golden
numbers page.
Energies
The five pointed
stars attract energies on account of the golden proportions. Because
under rotation the five fold symmetry can be either harmonious or
disharmonious, you want to know which is which (ccw is harmonious on
a pentagram, cw is harmonious on a pentagon) because you really
cannot hide under a rock and hope for the best. Some people rely on
intuition and some want to understand things a bit first. Either way,
I wish this site and the book will do it for you. 




Two pentagons
make a decagon, a regular ten pointed star
A
regular pentagon made with the Hirano method can be used to easily
make a regular decagon — an exact tenpoint star. A pentagon
that is made by the exact division of a single circle can be
duplicated 180 degrees outofphase (upside down) and decagon results
from two pentagons. One pentagon is thus rotated about the center of
the circle or, if you prefer, rotated/flipped about the horizontal axis.
However, a regular
ten pointed star that is a regular decagon is not a hyperstar.
(Hyperstar is discussed in the right column.) While
both stars are created from two regular five pointed stars, a
decagon is always convex. A hyperstar has some straight segments
spanning three points. A decagon has its points on a single circle. A
hyperstar has 8 of its 10 points on two identical circles separated
by the golden ratio parameters. 
[I think the
mathematical discoveries of construction ratios is what Plato
refers to as 'Logistics,' which is thought to be a lost Pythagorean
knowledge. In the case of a pentagon Diameter
priority the ratio is with square numbers (1:4) because we go from 1D
(unit length) to 2D (circle). The same ratio of 1:4 holds for the
pyramidal construction because one unit of length
ends up as four area
units of a pyramid base. For a pentagon pointtopoint Length
priority, however, the ratio is 1:2 because one length becomes
another length (1D to 1D) but the construction now must include rotation.
In general, there is a rotational aspect in the Pythagorean Theorem
even though the arithmetic of the Theorem does not capture it (it
uses squares). Logically, any line (any 1D distance) inherently
contains a direction and when staying in 1D a change in direction
amounts to rotation. Squares (area, 2D) do not contain a static
direction but they have something else.] 
Every
time you double something –
think octave. Every
time you halve something – think node (or fit) for standing
waves. Every time you rotate by 45 degrees – think
transformation. Every time you rotate by a right angle – think..
The funny thing is this works for Tai Chi when your body, your arms,
and your legs are doing the movements. Geometry and movement is about
your health too. 


Harmonious ten
pointed star
The star below, if
drawn counterclockwise, is classified as a (10+7)/10
star using my method that is (x+y)/x in general {while y
is between 1 and x — that is, x>y>=1}.
The
decagram star looks nice — it is unicursal, is regular during
construction (advances by the same angle), has parallel sides and is
harmonious. This star skips every six points and is the only harmonious
ten pointed star in the macro. (Decagon and all other tenpoint stars
are not harmonious, cw or ccw.) The Pythagorean style rationing
(really proportioning) method (x+y)/x as well as the
corresponding musical harmony or disharmony is explained in the book,
and you'll know why the proportioning approach presented here is much
more useful than what the mainstream puts out. You noted x and y
are integers but if x and y are the golden
numbers then (x+y):x
is the golden proportion — and enter both the micro and the
macro. (Rationing is not commutative because A/B is not B/A.
Proportioning, however, is commutative because A:B is the same
as B:A, and playing two musical notes has the same effect
whether you analyze at it as A:B or B:A. Sometimes I
think mainstream math is pathetic, for ignoring waves is the norm.)
Not done yet
There exists
symmetry about one point called the point symmetry (or odd or
radial or rotational symmetry) [masculine]. There also exists
symmetry about two points called the even symmetry (or axial
or twofold or mirror line symmetry) [feminine] — the two points
making an axis by which the original image rotates. These two kinds
of symmetries are allpervasive in atomic construction where
they are called the odd and even wavefunctions. Yes, everything is
coming up numbers. Now, how would you marry these two symmetries?
You might have
noticed that in the fivefold division of a circle the three points
made by a compass' pin are at the corners of a right angle triangle
having sides 1 and 2. (A pin of a compass centers the
radial symmetry [masculine].) It is no coincidence that the Great
Pyramid's Grand Gallery has the vertical height (rise) of 1
and the horizontal length of 2
while the Trough is the hypotenuse spanning the distance of SQRT(5).
(This also establishes the unit length 1 of this pyramid.)
Is it a
coincidence that to define
Pi we need distances 1 and 2?
Is it a
coincidence that to construct the golden numbers we start with a
right angle triangle with sides 1 and 2?
There is more
to 5
It is very easy to
get excited about the number 5 and begin to associate each of the
five points with other things. This is a Pythagorean site and we love
the number five but Pythagoreans also know that using numbers for
counting is introductory to the power of numbers while
correspondences are about the elemental — that is
differentiating, power of numbers. The five discrete elements in
Wicca and Tao are fine, yet the visualgeometric imagery based on the
Penta~ is about the infinity (irrationality) of the golden
proportions stemming from the square root of five. With the
golden numbers you construct many shapes that extend the Penta~
beyond counting. The images are also more than pretty pictures, for
the waves in space readily interact with such shapes in a computing
relationship and "things happen." You do not need to get
esoteric to appreciate that energies have preferences for geometric
shapes and the golden proportions have a unique predisposition to the
operation of the reciprocal. (Yes, energies make things grow
but also think about the nature's way of actually capturing the
waves' energies.) So
it is nice to recognize the golden shapes around you even though
neither the star or the number five jump out at you. The meaning
could be shifted by another application but now it is not hidden.
[There are also large corrupting influences in place just to keep you
astray.] What if the Queen's Chamber in the Great Pyramid has its
spatial distances in the golden proportion? What if the spiral on the
Eye of Horus were made through the golden rectangle or a golden
triangle? How can you visualize the golden
ratio in financial terms? 
Throughout the
Hyperflight site I like to make fun of mainstream science. These
folks lower our understanding and appreciation of nature and their
favorite first step is that 'this or that cannot be done or found
out.' The best modern example is Einstein and I don't know of any
of his concepts that are correct: This includes the special and
general relativity theories (stemming from his inability to determine absolute
motion/rest); photonic work function (a photon puts no pressure
on a mirror); Brownian motion (molecules vibrate in place); and
energymatter mechanics (not reversible, i.e., nova or supernova).
Errors of omission and commission are also applied to corrupt the
reality around us. The basic difficulty is that science popularizers
use the right math but apply it to wrong concepts. It may take a
while to "get to the bottom of this" and what really helps
is that equations cannot make wrong concepts right, and therefore are suspect.
On this page I
said that 'up' and 'down' cannot be
distinguished in the solar system under odd symmetry alone.
Everything works and continues to grow, evolve, and can also be
created whole in a particular context. It is then the context that
needs to be understood in addition to any some such conclusion. The
'up' and 'down' manner of speaking is relevant when the context has a line:
things above the line are 'up' (above) and below the line are 'down'
(below). However, orbits and orbitals and spin in general have a point
as its construct of rotational/spinning existence and it is then
appropriate to say that when dealing with point symmetry there is no
fixed 'up' or 'down' because there intrinsically is no (fixed) line.
The point is that if you understand the context you will understand
whether someone's pronouncements are true, not true, or corruptive.
For example, ignoring 0D (a point) as the fourth dimension of freedom
 which provides a movement construct just as 1D, 2D, and 3D do 
is corruptive because it does not recognize the spin/orbit that
accounts for 99% of the moving energy found in the universe. There
are several ways of getting to the truth, too. Geometry is one. Tarot
works nicely and in this case it's The Wheel of Fortune card. Tarot
has four suits, which stand for .. (your brainwork). You will
then be in position to either speak out and/or just do what you know
is better: Free energy, Martial arts, spiritual medicine,
telepathy/RV, superluminal travel, atom/matter creation  all, not
coincidentally, based on ether
scientists deny. Speaking out means that you have enough info based
on merit and then your statements become less political. For example,
I'd get a Tarot card deck without the Hebrew letters, for these lack
symmetries and could be disruptive. (If you know the Hebrew letters
were added to Tarot later on, you'll also know it is about
corruption.) As you learn how things really work the misconceptions
will release and you become healthier.
After reading a
book on the history of the Chinese philosophy
that weaves through the Confucian, Taoist, and Buddhist pursuits over
2300 years there, it seems the Chinese never got to the
understanding, and then the differentiation, of the symmetries. Yet I
would not relax too much after this. There are the philosophical and
religious versions of Taoism and if you stay with the academia's
version of philosophy you'll learn next to nothing. Once you get to
the religious side you will read a lot about circles and spirits and
demons, about keeping centered (weighed) in a point and having light
crystallized  but all this is pure alchemy and it takes some
dedication for the ideas to work their way through (just as you would
do with the ancient Egyptians). While the Chinese did not push the
three philosophies/religions into the objective realm we call
science, the Chinese pursuit in the improvement of the individual's
mindbody yielded Tai Chi with most significant implications  and
leaving a formidable gap for "the West." Wang
Fuzhi (b. 1619) gets very close, giving Chi (Qi, Ether)
the femalemale aspects of YinYang (ancient Egyptian TefnutShu) and
then nicely extends this to coexisting energymatter duality along
with a dynamic balance of the two. Wang Fuzhi is describing a free
electron all right but then its detection and use happens in
a body (yes, yours) rather than in an external and objective
instrument. (Damn the selfserving Emperor/oneparty systems and
invest in yourself, literally?) Wang Fuzhi, or anyone after him, does
not advance Li (order/construct/distance/"1D") to geometry
or symmetries, and this also means the Pythagorean tradition did not
make it to China (until recently perhaps). We can then also say that
the Taoist five pointed star of the elements is indigenous to China
(could've been discovered in more than one place) and testifies to
the universal power and utility of the five fold division of a
circle. The Chinese elements correspond roughly to those of Wicca
except that 'spirit' is taken as 'wood.' This seems strange but think
of wood as the circles of tree rings, too. This also gives you an
idea how "hidden" Chinese alchemy is. (Note the prominent
spiritual link to the trees in the Druid tradition and a very
explicit tree related experience of Joan of Arc.) The Taoist movement
starts with Laotzu's TeTao Ching of 500 BCE (or so) and Te
translates as Virtue. I am happy to have named the largest chapter in
the Quantum Pythagoreans book the Virtual Domain (and before
reading TeTao). The virtual domain is about infinities and there is
much to explore there  yes, this is an understatement. (One can
also destabilize in John Nash fashion. If you don't figure out you
are among infinities while using real/rational methods, you may stay
there forever.) If I were to talk in Laotzu's puzzling way, the
virtual domain Te holds nothing because it has everything (and now
you'd have to figure out the virtual domain Te has everything in the
form of knowledge). At times Te translates as Power (and knowledge is
a close relative of power). Laotzu described Tao in the most general
way. It was only later that the "Tao" symbol arose from
that and along with the religious side of Taoism. But of course, the
Tao symbol is but the Chinese version of what the infinite Tao is or
could be used for, including the label of 'Tao.'
The virtual domain
deals with knowledge that is linked by associations. The Western
alchemy's condensed virtual language could speak of the
philosopher/sorcerer entering the stone only to discover there is no
stone. Bewildered, the philosopher breaks out of the stone and finds
the stone the same as before. He cannot break into the stone and yet
he does not have to break up the the stone to reenter it. The
philosopher's stone can be had  but not physically. Yikes!
Here is s'more on
stability via organization. 
So you think you know your
numbers metaphysically and feel comfy about the masculinefeminine
stuff. You might be dividing by 2 and think it
feminine. Not so. Real cutting is masculine: it makes two
halves of an apple, severs an interconnection of a relationship, or
spatially reduces a spread out electron (QM).
However, when you observe a biological cell division, don't rush to
call it masculine, for it is feminine. You'll have to get into
symmetries to understand this. Meanwhile, don't make the silly
mistake of equating masculine with a man and feminine with a woman 
unless you want to give up on one half of your brain. Oh, and think
about The One. 
Analytically attacking all three
major pyramids at Giza as one layout can earn you a
label or two, but on this
site Jiri starts with a square and then looks for the golden
proportion –
and gets very, very close to the actual measurements. Ready to bury
the Pharaohs someplace else? 


Some
basic geometry. From a square angle to a square
There
does not appear much we can do with a square. A square is pretty,
has a lot of symmetries, but that's about it. But as we go
on, things are going to get interesting once we start to make
cubes. Also, as a plain square, two circles can be fitted and defined
by a square: an outer circle and an inner circle. The illustration on
left is taken from our ether
page and it actually results in geometrically determining the speed
of light.
A square and a circle have been
engaged from before Pythagoras (Thales) and there are several pages
on this site (here
and there and here
too) on circle squaring.
Pythagoreans love square numbers.
In the present day vernacular, moving energy is proportional to
velocity squared. Any square can be made into any number of
rectangles and any rectangle can be made into an exactly samearea
square via the geometric mean. Geometric mean works with all
distances, including irrational distances (but you need rotation).
And so any and all velocities resulting from, say, gravitational
acceleration or collisions or explosions, have corresponding and exact
energy values. Any square (any amount of energy) can be divided into
as many squares as you want — and so the
energy of one moving object is conserved exactly even if the object
is broken up into many other objects. 
Here is a simple yet powerful
construction that
1) Divides any distance exactly
in half;
2) Erects the perfect right and
square angle (making the Cartesian Coordinates); and
3) Makes a true square using any
circle centered at O (at the intercept of horizontal and vertical
axes that are the coordinates). A square is also a four pointed star
Only straightedge and compass are
needed. (Straightedge is an unmarked ruler.)
Both arcs (arches) have the same radius.
Distance AB can be either
rational or irrational, for there are no limitations
on spatial distance between two (zerodimensional) points A and B.
Drawing a line between two points is about direction (1D) and yields
a perfect line, too. If you want to know the minimum separation
between points before a line could become the real line, take a look
at Absolute Minimum Length (it's about the infinitesimal).
If distance AB is irrational,
should it be dashed? If so, why?
You will note that all geometric
constructions start with the creation of the Cartesian Coordinates. Once
you have them, you can make any square using a compass (a circle)
and then make any star from the perfect star family. A square,
then, is inherent in all star constructions.
A square is
feminine and a circle is masculine. Some people
get into spirituality and claim a circle is feminine because it is
"rounded." That is how you can tell a beginner. Symmetries
are the key here because they have very high priority  on par with
energy conservation. Ancient Egyptians, Native Americans, and Chinese
have no problem here. Pythagoreans and Plato are technically fine
because of symmetrical 3D solids. Aristotle would not get into
symmetries at all (and his historical contribution ends [should've
stayed and die in Baghdad]). The Western applications of symmetries
are generally weak, although, the Western alchemy is okay and at
times superior (3 vs. 4). 

Three pointed star
Construct
the perfect triangle on a circle and another triangle on a
semicircle –
in three steps
Triangle as a
logical and mathematical construct
Because
threepointed and sixpointed stars are geometrically perfect they
can be used, circled, as a symbol for 3 or 6 wavelengths wrapping
around the nucleus. However, a hexagon and hexagram reduce into a
triangle under modulo math for harmonious ratios and do not manifest
in orbits (macro) — that is, showing a sixpointed star with two
circles does not reflect nature.
A hexagon is also
prominent in free energy work. Hexagon is full of different
symmetries and, for better or for worse, enables rapid
transformations or projections. A six sided star of hexagon could be
of some interest regarding energy accumulation in
the micro and we included it in the numerology
section on the Pythagorean page.
Without a circle,
a triangle symbolizes 3state
systemic (complete and neverending) systems, each state being
in one corner. Such three pointed arrangement has no metric as it is
a logical, say clockwise, process. (Some systemic processes call for
quaternaries — think ancient Egyptians.) For Pythagoreans
a triangle provides bounds for the ten dots of Tetractys (a
triangular numeral 10), which also becomes one facet of a tetrahedron
(projection from the apex [or from your eye]). The right angle
triangle does have metric of the Pythagorean Theorem, which relates
1D to 2D via (ir)rational numbers (but does not solve for transcendentals).
There is a very
special place for the golden triangles. Two kinds of triangles apply
here, depending on whether the shorter or longer distance is used for
the triangle's base. Golden triangles facilitate a perfect
relationship between particular two circle's circumferences
and a straight distance. Yes, this is the fundamental reason for the
existence of particular atomic orbitals, which is driven by the
exchange in the electron's 2D and photon's 1D energy {Feb
13, 2011}. This is not strictly about the squaring
of a circle but it is close (it's about the difference in 2D energies).
Differentiate
by 3
Finally, there is
an aspect to the number 3 as the fundamental qualitative
differentiator of nature. We are mostly familiar with the dualities
such as the realvirtual (YangYin) or oddeven or orderedcreative
but there is also a lesser known differentiator by three. Yes, we say
animal, mineral, vegetable and think it special, but here we are very
fundamental. Preatomic, I'd say. So much so the crop circles are
worth studying just for that. [I would not go inside a crop circle
for more than a few seconds when the circle is less than a day old
— it is about the 3D energies.] Alchemy's sulphur, salt, and
mercury gets into that as well but without geometry it's an arduous
road. The Quantum Pythagoreans book explains what is at each
corner of the Tetractys and that's what it's about.
When you see a
triangle with some symbol in the center (a dot, an eye, dragon), take
such symbol into 3D of the apex of a tetrahedron to see if it means
something to you. 
Instructions:
 Draw
horizontal and vertical lines. The intersect is the origin O
 Draw a
semicircle of radius r around O. This makes point V
 Draw a circle
around V of radius r
Now that we
divided a circle into exact thirds, you can make a three pointed
("Mercedes") star or a three sided star of a perfect
triangle. In the illustration the larger triangle divides the circle
with three exact cords of length c
for a perfect three pointed star.
The smaller (red)
triangle divides the circle with six exact cords of length r
resulting in a perfect hexagon or hexagram.
You can verify
(using the Pythagorean Theorem) that the relation between the cord c and
cord (radius) r is:
c^{2}
= 3·r^{2}
What physical
entity is proportional to r^{2}?
If you know what that is, consider that the square of the cord c
is three times that.
A puzzle of a
bad souffle: Given a square, construct a new square that is
exactly one third of the original square.
You
can be fairly certain that the person will try to partition the
square in some way but the solution is to erect
a triangle on the square's side and then obtain the radius for the
circle that covers the triangle. Radius r
is the side of the new square. [In our case the bad souffle does not
cave in but runs over the rim.]
It is not possible
to reverse engineer a souffle without stepping
back and understanding the relationships between the ingredients and
their proportions –
as well as the irreversible nature of the baking process.
Can you apply the
construction of the geometric
mean in the solution of this puzzle? Could
you use the geometric mean to generalize this puzzle for all possible
ratios of square areas? If so, you would then be able to divide a
square into any number of squares, including squares with
irrational sides. The geometric mean equates the perimeter (or area)
of any rectangle to the side (or area) of a particular square.
Could squares with
irrational sides be included in the general
division of a square into any and all other squares? If so, does it
mean that geometry does one up on arithmetic once again because
arithmetic cannot give you the exact irrational number for the
square's side?
Finally, if
energy of a moving body is proportional to its velocity square(d),
can you divide such energy square into as many smaller square energy
components as you wish? (Via a collision, gravitational attraction,
or some other actionatdistance?) 

A six sided
star, a hexagon, is prominent in virus structures. Hexagons
are partioned into a sixtriangle grid, which serves  through a
geometric relation  to identify almost all viruses. This is a
complex topic. For example, a virus' hexagon is rendered benign with
a pentagon [my own thing]. The overall structure in 90+% of viruses
is icosahedron. Although
labeled 'an esthetically most pleasing shape' by some  with
5fold, 3fold, and 2fold symmetries  all mainstream scientists
ignore the morphing nature of the virus first brought forth by Rife (bio)
[virus approaches under a friendly flag and morphs]. Because the
current work on viruses discloses but the stained (i.e. dead)
viruses, the mainstream is still way behind Rife. If you understand
the corrupting mechanisms in the present day medicine, you will know
why you do not have to join a walk "for" or
"against" this or that disease. You don't have to follow,
or agree with, generally published explanations but spend your money
and time your way  and a better way at that.
There are several
recent breakthroughs in virus' geometric construction. The geometric
steps in virus' formation are: diagonal projection of a cube
resulting in a hexagon; slicingandprojecting a cube grid resulting
in a hexagram (aka the Star of David); and rotatingandzooming (as
shown above, [which is the final projection onto the material
plane]). I see some of these elements in the crop circles and, short
of taking sides, I'd like the virus construction viewpoint
represented in crop circle analysis. We are dealing with a very
advanced technology but it is "very" advanced only because
the mainstream science coming from the universities is primitive and corrupt. 


Twelve pointed
star
In practical terms the twelve
pointed construction is about the design of a clock's face. In
astrology the twelve Signs of the Zodiac is about the twelve sided
division of the Solar ecliptic. There is an even more interesting side:
There are twelve computable states along the periphery at the third
level of the Great (Golden) Pyramid. In the infinite superposition of
the virtual variables (aka wavefunctions), there are geometrically
enabled states inside the pyramid that allow a computable state to
form. Yes, infinities can be worked and the Quantum Pythagoreans
book gets into that and at all levels of the pyramid.
Eight pointed
star
Our construction for the twelve
fold division of a circle also divides a circle into eight exact
segments, or angles, because the 45 degree diagonals
are available for free from our twelve point construction if you
make full circles (and don't stop at the intersects with the central
circle). This is also true the other way: When constructing the eight
point division that is a regular octagon, the twelve fold division
happens as well if the central circle is drawn in full (and you don't
just mark the intersects with the cardinals). So here you have the
combined 8 & 12 point star construction. After a short Internet
search I did not find an 8 & 12 point star common construction {Apr
30, 2011}. I found one logical assembly having the Chinese
Zodiac on the outside circle and eight binary Paqua states on the
inside circle. (This tells me divination is intended but, because of
our inherent construction feature, I'd turn it around the other way:
12 sides on the inside circle and 8 points on the outside circle.
This is counterintuitive but there could be [is] a form of Tai Chi
in this.)
Music of Venus
and other
planets
The common construction of the
eight and 12 pointed stars starts with the combined two and three
pointed stars. As you double the count the ratio stays the same:
3 to 2. (As you double the points you go up by an octave.) This is a
musical ratio and a harmonious one at that. What planets subscribe to
this ratio? It is not VenusEarth but it is on the Venus
page.
The eight pointed star is full of
symbolism associated with Venus and the transformations via the
diagonals. The ccw octagon is harmonious and made by EarthMars
interplay, too. So enjoy the eightpoint star construction here via
the cardinal and semicardinal directions but the Venus page has a
lot more on the eight fold division of a circle, including the Hunab
Ku symbol.
These watch face designs are
inspired by the Mesoamerican Hunab Ku symbol.

Instructions
for drawing
a 12pointed/12sided star:
1.
Draw a horizontal line and erect the vertical line. The intersect is
the origin O
2.
Draw a full circle of radius R
around O. This will be the clock face
3.
At each of the horizontal and vertical intercepts draw additional
circles of the same radius R
4.
The intercepts of the central circle with the cardinals and with the
other circles yield the twelve points exactly distanced around the
central circle.
Can you see how you could make a
24 point star using the diagonals? Almost every month we offer quick
topics of general interest. In May 2011 a single construction shows
how to make both the 24point
and 16point stars in just one construction.



Perfect
star families
It
is easy to draw stars using geometry's tools, a straightedge and
compass. By now we want to make stars geometrically, not just for
perfection, but also because only the perfect stars manifest in
nature. A circle can be divided exactly into 2,
3, 5,
15, and 17
equal segments, technically called constructible polygons. (Some
exclude the 2segment division because it yields but a virtual line
of a circle's diameter — but I include
it.) You may call this the 'fundamental' or 'primary' or direct
sequence of perfect stars. Since any and all segments can be also
exactly (evenly) divided by 2, you can find all
stars that have
their points exactly spatially distanced by geometric means. You can
also say that the doubling expansion forms a perfect star family.
For
example, you can make an eight point star or a 64 point star from a
two point star through simple halving of distances. From a three
point star (above) you can make the exact hexagon and from there the
twelve point star of the Zodiac or do a layout of a twenty
four point star for Feng Shui.
Starting
with a 2 point star — the only direct
even star — you can construct 4, 8,
16, etc. stars you could also label the 'evenly even' sequence of
stars. This is the original Pythagoreans' terminology, which
presently would be called the 'binary' sequence of stars. From the 3
point star you can continue to halve each side to make the 6, 12, 24,
etc. point stars. From the fivepoint star you can make the 10, 20,
or 40 point stars.
Every perfect star
with the even number of points will have symmetry about an axis and
about a point. If you think there is no such thing as a two
point star, it is on the Venus page and it is formed by the
combined NeptunePluto 3:2 orbit.
The
stars that are left out from direct and doubling constructions
cannot be constructed exactly. For example, you cannot make a nine
point star directly — or indirectly
from a three point star. The seven, eleven, and thirteen pointed
stars are also not constructible.
Numbers
that divide a circle exactly could have a name of their own. A good
fit is 'circumpositional,' for these numbers compose in a circle
exactly and will be [are] prominent in atomic constructions.
Carl Gauss
"recently" added the 17 sided polygon as the perfect star.
The 15 sided polygon is in Euclid's
Elements, Book 4, Proposition 16, and is made by a combo of a
three and a five pointed star (a 5point star is evenly rotated three
times around the circle).
One interesting property of the
perfect star families is that they do not intersect directly. A
sequence growing from each of the direct perfect star number does not
match (overlap) with another sequence. That is,
from 2 we get 4, 8, 16,
32, 64, 128, 256, 512, 1024, 2048,
4096, 8192, 16384, 32768, etc
from 3 we get 6, 12,
24, 48, 96, 192, 384, 768, 1536,
3072, 6144, 12288, 24576..
from 5 we get 10,
20, 40, 80, 160, 320, 640,
1280, 2560, 5120, 10240, 20480..
from 15 we get 30, 60,
120, 240, 480, 960, 1920, 3840,
7680, 15360, 30720..
from 17 we get 34, 68,
136, 272, 544, 1088, 2176, 4352, 8704, 17408, 34816..
Each member of the perfect
allstar family has but one origin. Once a number becomes even it
stays even. 3, 5, 15, and 17 make the only exact odd stars.
A circle is a zero pointed star
having an infinite number of points. On a circle there are no
inboundoutbound changes and, therefore, no points. Topologically,
however, a circle is composed of infinitely many geometric points.
This is a technical play on words. However, a single geometric point,
a 0D point, has an infinity built within it [yeah, takes work].
Note that
360 does not divide a
circle exactly
260 does not divide a
circle exactly (Mayan Tzoltek calendar). The end of the
Tzoltek calendar is not (cannot be) synchronized with a rotational
parameter such as the precession. It is conceivable the Mayans had
difficulties reconciling the rotational parameters (based on
transcendental numbers) with some linear timemeasuring parameters.
The numbers' unexact nature would lead to "disaster"
conclusions. Well, yes, it could have been their own squareacircle problem.
320
does divide a circle exactly and the 1/320
(reciprocal) is called ro
by the ancient Egyptians
While a circle can be divided
exactly by hours (24) or minutes (60), a circle cannot be
divided by hours and minutes (24x60=1440).
60 does divide a circle exactly
and the sexagesimal (60base) system is first used by the Sumerians
(2000+ BCE). Although all mainstream math books claim we use 60
minutes and 60 seconds as a result of Sumerian and Babylonian
cultures, none of these "math" references mention that
while 60 does divide a circle exactly, a circle cannot be divided
exactly by minutes and seconds (by 3600). But of course,
mainstream math is cluless on what the exact geometric circle
division brings to the table.
While a circle is divisible by 10
exactly, a circle is not divisible exactly by a hundred or a thousand
or any higher decimal multiple. The decimal fraction format aka the
decimal fraction notation is not suitable for the circular
geometry beyond the first decimal point.
Is it strange the Mayans —
while having the positional notation of base 20 —
had no subunity base20 fractions? Perhaps not. Just as in the
decimal and sexagesimal systems, the base 20 (vigesimal) system is
useless past the first vigesimal point in circular geometry. 
One
star inside another star
Among
the applications are spatial designs that combine perfect stars.
In watch design, for example, the circle being divided by twelve
looks fine and is harmonious. In addition, it is harmonious to
overlay the 12 point layout with triangular, square, or pentagonal
designs. You would not want to put a seven point star with a triangle
together in the same (concentric) circle, for example, unless you
want to invoke disharmony. Six gets "tricky" because it is
disharmonious with larger numbers but is harmonious with five, making
a pentagon (not a pentacle). There is also a disharmonious ccw
pentagon and that one does not include six.
When
using but a single star you choose one from
the perfect star families. When combining stars, however, you also
must deal with harmony. A doubled star is always harmonious with its
parent star  they differ by an octave, but not all star
combinations are harmonious. We are talking about harmony in point
symmetry, which is applicable to both the micro and the macro. See
the book 
QUANTUM
PYTHAGOREANS
The
Book
If
a musical tone x
is harmonious with tone y
and y is
harmonious with z,
is z
harmonious
with x?
Not always. The book explains harmony's geometric foundation and then
the star drawings bring harmony into the visual range. Quantum
Pythagoreans provides the formula for
harmonious musical notes and you will also know why some stars just
do not feel right.
Harmony
is not just extended to the macro. Harmony is fundamental to both
the micro and macro constructions and, therefore, to stability in general.
Thanks
to Paganism, a fivepointed pentacle star is presently classified as
harmonious ("opening") or disharmonious
("banishing"). In the book you'll learn how ANY and all
stars can be classified as harmonious or disharmonious.
In
the beginning was the number –
and the power of numbers begins ..
More
.. 

The perfect star families of
numbers introduce some changes to our perception of
universe building and how everyday reality happens to come about.
Mathematicians can make all kinds of star constructions, in 2D and 3D –
but only the perfect star families can begin
to bridge the straight line energies, such as photonic energy, with
circular orbits and orbital energies. Because the vast majority of
the real energy in the universe is in the form of spinning or orbital
energy –
that is, energy having angular momentum, the perfect star families of
numbers take the front seat. Scientists can draw all kinds of curves
but these are usually fancies. Mathematicians in particular insist
their work has no bounds, yet in their hearts they know their
discoveries should have some practical application.
New
Timekeeping
Starting with 24
hours per day and having 80
minutes in an hour, every minute of every day (24x80=1920)
would divide a circle exactly. Such is not the case today. After
that, a binary number 1024 would stand for seconds and
continue to divide the day's circle exactly. One of the new seconds
would then be the exact 1,966,080th part of a day. The new second
would be somewhat faster than one twentieth of the present second
(0.05). While adequate for almost all sports without further
division, additional circleexact divisions are always available –
something that is not possible to do today. (Once you lose the exact
circle division it cannot be recovered.) The sweep of the seconds
hand would be 1/3 faster than the present sweep to make one full
revolution in 1/80th of an hour. At all numerical markings in the
illustration on right –
the hours, minutes, and seconds –
would not be fractions. For example, the sweep of the seconds hand
spanning one half of the quadrant adds up to 128 seconds {May
11, 2011}.
One half quadrant,
then, is 128 seconds or 10 minutes or 3 hours. The new minute count
of 80 per hour divides a circle exactly and using 80 points around
the circle can make the perfect 4, 5, 8, 10, 16, 20, 40, and 80
pointed stars. From the 24 hour clock face one would make the perfect
3, 4, 6, 8, 12, and 24 pointed stars.
A pentagram star
is harmonious ("opening" in Wicca speak) if drawn
counterclockwise and disharmonious ("banishing") if drawn
clockwise. Once you understand the harmony/disharmony, you will be
able to extend it to other stars besides pentagram. Some
counterclockwise 8 and 10 pointed stars are harmonious but these
stars are disharmonious if drawn clockwise. This does not mean the
clock's hands rotation should reverse. See the Venus page.
Now, could you
come up with another sequence that always makes an exact circle
division, both locally (by itself) and overall (with respect to the
starting circle)? Yes. Start with 20 hours per day and divide each
hour by 192 minutes. Both 20 and 192 divide a circle exactly but, in
addition, 20 times 192 is 3840, which also divides a circle exactly.
Is there more to this than making new and different watches? You bet.

Tiling of
pentagons and stars 

Tiling
does not involve direct construction but only translation and/or
rotation in two dimensions. Translations are linear (straight)
motions. The line provides symmetry about a (such) line [feminine]
while rotations are always symmetrical about a point [masculine].
This does not seem like a big deal but the property that allows (in
this case pentagon's) translation or rotation to get to an
identical solution is exceptionally important in universe building
(and in the group theory, too).
Tiling
of five pentagons to make a cool five pointed star was (first?)
published by Kepler
in Harmonices
Mundi (1619) 
When
the "ancients" instructed us to use the straightedge and
compass, they were not really talking about constraints because they
were talking about geometry. Rotation about a
point is about the use of the compass. Straight movement
(translation) is about a symmetry about a line and perhaps you could
see now that the line of symmetry is a virtual line – that is,
the line of symmetry is an empty slit. (Would you go as far as to
have Justice brandishing her sword with a slit down the middle of the
blade?) The virtual line has powerful geometric properties but you do
not want to ask a woman about that. Not that you couldn't, it's just
that the explanation is nonverbal. 
The
pentagon template for the illustration on the
left was obtained with MS PowerPoint by selecting AutoShapes ..
Basic Shapes. Pick the pentagon object. On the newer versions it is
Insert .. Shapes .. Basic Shapes.

If
you tile five pentagons you get the fivepoint star in the center.
Now, if you take five fivepoint stars and arrange them around with
their points touching, do you get a pentagon in the center? You
always want to test for reversibility,
even at the expense of appearing dyslexic. Relations are reversible
only under certain conditions and you want to know what they are. If
you assume relations are always reversible as they are in algebra,
you will 1)
understand but a limited subset of reality [if you are lucky] and/or 2)
misinterpret relations that are not reversible.
For example, if
there is a quantum mechanical explanation of gas pressure, there
could be a way of making the phenomena reversible. Now, how would you
reverse the rotation of a light mill? (Give it
a thought and get the
answer.)
We readily apply
force to get things moving. So, how would you reverse 'something' and
have the force arise?
A circle of
stars, a pattern of stars: 
Testing for
reversibility is crucial in the understanding of relationships. Dyslexia
is a condition that is constantly reversing relationships in all
modalities: verbal, tonal, geometric, written  to see if the
reversal possibly acquires another meaning, or if the reversal
carries no meaning. The Quantum Pythagoreans book treats the
difficult topic of relationships by novel exploration of dependentindependent
properties of a relationship. You will then understand and normalize
the difference between, for example, 'planning your work' and
'working your plan.' 



The
tiling construction —
that is, movement about a point and/or translation along a line, of
some objects may result in the appearance of another object. This is
at times referred to as "negative space." While it is true
that the original object is real and in some respects positive, the
'negative space' label is but an introductory way of looking at it
(and a leftbrain way at that). A good way is to see this as the act
of creation of the virtual object.
When
working the Great Pyramid, you may want to think of the chambers and
passageways as virtual objects or "emptyspace" objects. It
really helps. 
Self
test:) Straighten up two adjacent fingers. Do
you see a difference if you think of these fingers as two closely
spaced pencils – or as an empty slit or space that is between
the pencils? Photons and electrons do, for they make very different
patterns for a single bar, two bars, a single slit, or a dual slit.
For
brainwork: 1) How
is it possible, and 2)
What is the utility of the result that one pattern ends up in the
left side of the brain while the other in the right side? 

Geometry
versus arithmetic
Ancient
Greekspeaking scholars debated geometry and arithmetic, and
understood the complexities even without a PC.
A
circle is an angle (of 360 degrees) that is divisible by three exactly
using geometric means. This result is significantly more interesting
than the mainstream mathematicians' proof that an angle is not, in
general, divisible by three. If you think of a circle with the orbit
(cosmic) and/or orbital (atomic) applications in mind, you will see
there is lots of fun in figuring out what works[, rather than beefing
up your resume with things that don't]. So,
the ancient riddle about dividing the angle into thirds has more
than one answer and no answer is the wrong answer. It is, however, a
parting, or the "Tau" riddle that to some makes all the
difference. But again, no angle can be divided by three exactly
arithmetically. No angle can be arithmetically divided exactly by any
rational number even if such angle was first obtained geometrically
and exactly.
A
circle cannot be divided by 7 or 9 equally and exactly. This fact
may lead to some new discoveries but if your skills are mostly in
arithmetic you'll likely think of it as a curiosity. That is the
basis of reductionism, for a reductionist first makes a claim that
arithmetic is just as good as geometry (brain "grouping"),
and then happily ignores the advantages of geometry. Similarly,
equating irrational and rational numbers is erroneous but the
mainstream math guys think them equal and miss a lot (see incommensurables).
Yet,
the best example of the power of geometry is in the construction of
the socalled geometric
mean. Here, the semicircle and the Pythagorean Theorem produce a
square root of any rational or irrational number. Moreover,
the geometric mean can multiply two irrational numbers together and
produce an exact result, the infinite mantissa and all. No computer
can do that.
The
bottom line: Geometry always leads, arithmetic always follows.
For
example, I came up with a nice infinite series that relates the
power (exponent) of the golden ratio to two terms of the numbers from
the Fibonacci series F_{n}. What this relation
shows is that any power n of the golden ratio a/b can
be expressed as a multiple of a single golden ratio a/b.
IG_{n}
= (a/b)^{n1} = F_{n1}(a/b)
+ F_{n2}
In
case n=5, IG_{5} = (a/b)^{4} = F_{4}(a/b) + F_{3}
= 3(a/b) + 2
Initially
I called this the IG series for Instant Gold
series. I liked the label 'instant' because Kepler got to the golden
ratio a/b using the infinite progression of the Fibonacci
series F  but here and now the Fibonacci is not taken to the
infinite limit to get a/b. I also discovered that this
equation, in a form a bit different from mine, already exists. After
a while and once I figured out what is happening geometrically, I
could not help but calling it the Ivsin Gold series
[yes, thank you], simply because the geometric understanding opens up
a whole new world of applications {5/5/2011}.
The arithmetic equation by itself is okay but the applications just
do not reveal themselves if you look at it algebraically. Try it
yourself ..
In
another example, you know that 360 degrees in a circle is arbitrary.
Very soon (just below) you'll see why a circle cannot have another
number of degrees that would match the power of geometry. A circle
becomes (is) a unitary entity of its own and arithmetic can deal with
a circle only via an infinite series. Having said that, a circle
would not be possible to turn into a square —
but, by using the virtual numbers and understanding the infinite
superposition .. .. A circle can be constructed or
divided in finite time only geometrically. Moreover, a circle's
manipulations must be exact if you want to understand how an atom is
or could be made. 


It
is now time to visit the angles of a circle. Can we map the angles
in such a way as to obtain correspondence between geometry and arithmetic?
What's
your angle? 

Arithmetic
makes it strange
In
a calculator, the angle of 360 degrees is divisible by 9 without a
remainder, but this is but an arithmetic computation. In geometry,
the circumference of a circle issues from Pi,
which is a transcendental number and so you cannot be arbitrary
about the length of the circle or the exactness of an angle inside a
circle. Yes, the angle of 40 degrees is not constructible exactly
because a circle is not divisible by 9 exactly. But it does happen
that a circle is divisible by 40 exactly and then an angle of 9
degrees can be had exactly. Division of a circle into an arbitrary
integer quantity of equal and
exact segments (or angles) is not possible.
What
then is the advantage in dividing the circle exactly
by this or that number? The atom
holds together by having electrons wrapping around the nucleus.
Because the electron's momentum is also a wave (de Broglie),
the electron's wave must evenly, that is exactly, close upon itself
to form a standing and a round wave that is symmetrical about a
point. To the Pythagoreans the numbers are everything and this is
because numbers actually create things.
The
mainstream scientists' argument that "computer's representation
of an irrational number is close enough" is, unfortunately, not
relevant to atomic construction. Scientists just do not know how to
interpret 'precise' and 'exact' in an applications setting. The
scientist can divide the circle by nine to a very large number of
decimal places, but there will never
be a wavelength
that would fit
nine times around the circle of the orbital. Incidentally, 'fit' is
the original (superior?) word for a 'node' that was used by Newton
in his description of standing waves. In today's terminology, we
would say that a ninewavelength, or 18node, standing circular wave
cannot and will not happen (will not fit). Numbers 7, 9, 11, 13, 19,
21, 22, 23, 25 and others cannot divide a circle exactly. Most of
these numbers are incomposite (prime) numbers. Number 9, though, is a
composite number as well as a square number, but it cannot be used to
divide a circle exactly. [Does this mean the Chinese Emperors could
not sing? Having said that, they might have been good golf players.]
Number 5 is incomposite but can be used to divide a circle exactly.
What is needed, then, is a class of numbers that compose
in a circle,
instead of just being composite numbers (composed of products of
other numbers). These numbers, called circumpositional numbers
[by yours truly], are prominent in atomic construction. Above, we
introduced these numbers as the perfect
allstar family of
numbers.
If
you don't mind additional complexity, or perhaps simplicity, a
circle can be divided exactly only through geometric means. Another
way of saying 'geometric means' is 'spatial distance means.' Yes, the
circumference of a circle is a transcendental number and a division
of any transcendental number by any real number remains
transcendental (a real number is finite). The computer can use only
real numbers and the length of the circumference is then rounded off
if it is to be stored in a computer. What this also means is that a
computer cannot give you a perfect star. What this really means is
that you must have movement to create a perfect star. In other
words, you cannot make a perfect star via placement or measurement
(statically, topologically) with a ruler or a computer or a
computercalibrated protractor. As
a Pythagorean you might realize that you cannot construct a perfect
star without a compass –
that is, you need rotation to complete exact constructions. A compass
is the only tool that allows you to enter 2D from 1D. In other words,
a compass allows you to turn. A larger implication is that there must
be movement even at the atomic core level —
and under radial symmetry the movement is also about frequency. (You
might guess here is one of the gateways to gravitation and yes, the
philosopher's stone opens up to the alchemist.) The necessity of
movement permeates everything. Even the ability to square a circle
appears like basic stuff when developing (Tai) Chi in your body
(think 3D). 

Arithmetic
makes it practical
Pythagoreans
had a category of numbers they called 'abundant.' Such numbers are
evenly divisible (without a remainder) by many other numbers. Number
60 (minutes in an hour) is a good example as it is divisible by 2, 3,
4, 5, 6, 10, 12, 15, 20, 30 and 60  which, not incidentally, are
the number of points of stars one can make with minute watch face
markings. Another convenient (abundant) number is 24 (hours in a
day), but we presently divide the day only by 2 (am and pm) and by 3
(work part), and possibly by 12 (entertainment). Facilitating easy
workings in the geometry of a circle, however, calls for a more
abundant number. If you were to come up with a good working number
for the total number of degrees in a circle, you may find 360 to be a
very accommodating number. 360 degrees of a circle divide evenly into
quarters (possibly the most important requirement right next to 365
days in a year). It divides evenly by 5, 10, and 20, too. You've
already noted that the division by 5 (and consequently by 10) is very
practical because a circle division by five brings in the golden
proportion. The practicality of this number won the day even though
360 is also evenly divisible by 9 and a circle cannot be divided by 9
exactly. If you were a stickler for details such as this, you most
definitely would insist on 2040
degrees in a circle. 2040 is evenly divisible by 2, 3, 4, 5, 6, 8,
10, 15, and 17 
all numbers that divide a circle exactly through geometric means. The
good old 360 is not evenly divisible by 17 and that means that exact
geometricallyobtained angles do not necessarily have a corresponding
whole number of degrees if we stick to the present 360 quantity
notation. Both 360 and 2040 are not evenly divisible by 7, 11, and 13
 as it should be. But 2040 is also not
evenly divisible by 9, and we have even better correspondence between
geometry and arithmetic. The year 2040 could be the most harmonious
year coming up. [But don't tell IRS. They'll put this number on a
form and spoil it.]
Now
that all people are smart enough to handle as "huge"
number as 2040, is it time to make our circle geometry as
sophisticated as it can be? Are you ready for the sum of internal
angles in a triangle to equal 1020 degrees instead of 180? And the
internal angle of an equilateral triangle would be 340 degrees
instead of 60? Even if you could legislate the change – and
during the French Revolution they "legislated" 100 degrees
in a circle and 100 minutes in an hour  the bottom line is that
there is no perfect number for a quantity of degrees in a circle that
would be a whole number or even a rational number. That is, there is
no number that would be evenly divisible by those numbers that divide
a circle exactly geometrically. In the example of the 2040, this
number is not evenly divisible by 16. [Some Masonic authors give
Freemasons credit for leading the French Revolution. If so, they
would certainly be quite ignorant on what to do in the aftermath
– all their Gs notwithstanding.]
Whole
numbers and rational numbers are called real numbers – a good
name. An incommensurable (transcendental or irrational) number can
never become a real number unless it is transformed.
The transformation is irreversible because we cannot save an
irrational number such as SQRT(2)
in a computer and retrieve it as (convert it back into) the original
irrational number without first taking a nip
off the number – think Ouroboros and visit Circle
and Pi. Reversing the transformation calls
for addition of the virtual energy [think Isis and possibly Thoth if
you are familiar with his eye restoration story].
If
you want to get deeper into transformations of rational (real) and
irrational numbers – think ancient Egyptian fractions [here, you
will need to appreciate three things: 1)
Ancient Egyptian fractions are quite sophisticated; 2)
Present day scientists are clueless as to the ancient Egyptian
fraction applications or origin; and 3)
Our present civilization is not necessarily advancing.]
When
dividing a circle with the straightedge and compass, the goal is to
make the number to become,
for the number's geometric construction creates something specific
to that number. There is, then, more to numbers than philosophy, and
you may want to visit the original number apps guys, Pythagoras and
his fellow Pythagoreans.
There is a treatment there of real, virtual, and irrational numbers.
Irrationals and transcendentals are in the family of incommensurables
but transcendentals are not constructible through the Pythagorean
Theorem (from 2D of the curve to 1D of the hypotenuse), while
additional differences between irrationals is based on applications.
[There are good and bad numbers and some of them have an infinite mantissa.] 
What
does a star in a circle represent? What is the meaning and symbolism
of a star inside a single circle? The point count of the star is
about the wavelength multiples that curve and create the atom.
Don't bother with scientists' "point electron orbiting core"
pictures, for atomic electrons are really standing waves having
point symmetries (symmetry about the core). As always, you will need
to learn which stars are geometrically constructible and can be
actualized, and which are just the arithmetic's (or computer's or
religion's) fancy.
Is
the atomic core composed of standing waves? You
bet. Scientists have way too much invested in the solid and static
core hypothesis and so it is safe to talk about the pulsing and
standing waves of the core. Scientists are way off and, for example,
they made up "strong nuclear force" because they do not
understand the wave nature of the core. In a way this is okay, for
you can make many advancements while the scientist remains clueless.
For example, the core's shape is not necessarily spherical.
With
all their equations, scientists think highly
of whatever it is they describe with them. Saying that the scientist
will remain clueless is no idle talk, however. On our Circle
& Pi page we also highlight the inadequacy of algebra, for
algebra's constructs cannot deal with the operation of equivalence.
Algebra completely misses irreversibility, too.
As
a Pythagorean you want to figure out what entity will prevail in the
interaction with a standingwave electron and
with a standingwave proton. Why, could you then do a precision
surgery on the atom? 


Seven
pointed star
A circle
cannot be divided by seven exactly. Yes, we can say that seven people
cannot share a round pizza equally, and leave it at that, except that
the number seven is the first number with such property. When things
are happening inside a circle —
that is, when things are spinning and evolving, the wave folding
encounters the number seven and consequent inability to fit around
the circle (and make a perfect star). When you examine the
Mesoamerican Hunab
Ku symbol, there are two sevensided areas (heptagons) and now
the challenge is to explain that. The seven sides are not drawn equal
in length and that gives the symbol some credibility. The heptagons
pivot into 3D and now it gets really intriguing. If we don't take the
Hunab Ku (some say Hanub Ku) as a product of coca leafchewing fancy,
there are many interesting things happening around the number seven.
I'd be
careful in using the seven pointed star. Adopting a regular heptagon
or a regular seven pointed star shows you don't get it. So I'd draw
the star free hand or use an irregular heptagon. Mayans have a gap in
their sevensegment Ouroboros too.
The
Goddess Seshat's symbol from ancient Egypt is a contextualized
sevenleafed plant (she is a scribe, which means she is a magician).
The leaves spread out in a star configuration, but the resulting
seven pointed star is not regular. [If you think the symbol is a
cannabis plant then that's fine by me. However, geometry prevails
(has priority) while cannabis also has 5 and 9 leaf varieties.]
The Statue of
Liberty has a hair dress with seven rays. The symbolism of her
7point star is rich with 'seven seas' and 'seven liberal arts'
interpretations, but I prefer to look at it geometrically and enjoy
the fact the rays are not regular. 

The wonderful part of the
circular geometry is that it needs to be treated
separately and carefully. Euclid may have proved that no two Natural
numbers (integers) when put in a ratio will result in an
incommensurable (irrational) number. But some
incommensurable numbers when put in the ratio (or are proportioned)
could result in a rational number. You may want to reflect on what it
means. As far as Euclid goes, not much. After all, Euclid talks about
what does not happen. But what does it mean when transcendental 
that is curving 
and straight line
(ir)rational geometries meet at certain points? Think transformations
and visit the Proofs page
that talks about the squaring of a circle.
Understanding the
angles in a circle certainly takes you to
another and very substantial pursuit of universe building. Looking
back, is there some work that was done just in this area? But of
course, you'll have to step over the reductionists, yawn at the
popularizers, and laugh at people who allow to be called experts.
[I'd bet 10:1 they will talk about running out of energy.] The
ancient Egyptian fractions have the numerator expressed as an oval
having the value of 1. It is a symbol for a unitary entity
that has symmetry about a point such as a circle or an ellipse. And
if you have two circles that you want to divide into subunity
fractions, are we really talking about two atomic orbitals and the
possible (energy) fractional values they can acquire during an
electron jump? So now you have yet another quest to make. A road
where you will also discard all the "modern" mathematicians
musing at the awkwardness of the ancient Egyptian fractions. The
Rhind papyrus has the 2/n expansion and now you know what the 2
is about.
Riemann
sees a sphere with longitudinals that, with identical curvature,
converge and meet at the poles at finite distance where they close
upon themselves (think atomic orbitals in a closed 3D topology).
Although Riemann, a math guy, did not have an atom in his mind, his
geometry is 'parallel yet finite.' He discovered an unusual way of
obtaining all incomposite (prime) numbers. We have three book
reviews on this remarkable individual.
Then there are the
scales of the fish [yeah, the dumb fish]. The
sweeps of the scales subtend a particular angle. Do you think a fish
could fly or extract energy from the swirling water around it by
using the geometry of its scales? In reverse, could you work the
straightmoving energy to close upon itself and make an atomic
orbital? Think pyramid geometry and Schauberger. 
Illuminati,
Masons and New Order Guys. Pythagoreans, too
Symbolism
is about associating something with something similar. Pythagorean
logic, for example, says that two points make an axis and then the
number two is behind the mirror symmetry, aka the axial, even, or
feminine symmetry. As a Pythagorean you have no
problem with that, and wouldn't use the term "axis of evil"
because both the constructive and destructive energies exist about an
axis (about 1D). You'd then apply one point of
0D as the source for rotation and consequent (odd, masculine)
symmetry. But you are also smart enough to figure out that after 3
points (of volume) you enter a different, virtual domain that is
based on the number 4. Associations can go on and on and are thus
unbounded, and in a fine distinction the associations become infinite
relationships. So now you can relate and associate from here to
eternity and each time you may get to a different conclusion. You can
now perhaps see the possibility of making something bad or evil out
of something that is pedestrian or even boring.
So you want
to ask if the result you reached is beneficial or not. If
you think accusing someone of evil thoughts is beneficial to your
religion then you are a part of a religion that is sustained by
attempted oppression of others. In short, your religion is not based
on truth. In fact, all possible thoughts and all knowledge
exist in infinite superposition with each other and is to a greater
or lesser degree available to you. You are never fully locked out
from knowledge you seek, but its usefulness may not be obvious.
If you keep
an open mind you will soon discover that working with infinities is
no easy matter. Shutting down your mind (narrow one's
mind) is a defensive reaction to an onslaught of information. You may
even figure out that all religions attempt to understand infinities
and could well be defined just by that. So relax. Infinities can be
worked but they need different methods. Things that don't add up
simply won't happen.
[So what is
my Pythagorean take on the 7point irregular star? It is about the
creation or release (liberation) of energy. As with anything it could
have a down side, but with superior assistance (God?) it can be
managed and be useful. Oh, the thing about God is that you have to
ask. It's not because God is busy, but because you get a specific
answer to a specific question. Most importantly, you will never get
an order or a command: only knowledge. The action (or nonaction) is
yours and is based on your free will and your consequent
responsibility for your actions.] 
Historical
Note
Johann Balmer
was the guy who opened wide the barn doors of quantum
mechanics. Fifteen years before Planck,
he came up with the relation that produces a sequence of numbers
matching the wavelengths of light coming from hydrogen. These are not
just any numbers –
they are wavelengths corresponding to particular electron jumps and
no other. Balmer did for quantum mechanics what Kepler
did for gravitation: He came up with the math equation that matched
known experimental data and made successful predictions of other new,
yet undiscovered, wavelengths. But there is a bit more to it. Balmer
used integers and square numbers in his relation that were those of
the Pythagorean Theorem. Well, good ol' Pythagoras was not only right
all along but the breadth of his –
some say HIS –
teaching was also the foundation of quantum mechanics. Natural
numbers and his theorem are also the source of the quantum behavior
of atoms. 




This topic is
expanded and has a page of its own. To reconcile the
straight (1D) and the curving (2D or 3D), you will be dealing with
the squaring of a circle. Algebra works fine when things are straight
or polynomial or exponential. Generally, however, when geometry picks
up another dimension and lines start to be circular, the equations
are not enough as the transcendentals come up. The
relationship of the squaring
of a circle to this page's perfect division of a circle is in
the possibility of linearizing the curved segments (arcs) of a circle
and then making a tractable exchange between curving and
straight topologies. So there is a continuation to 'how to draw a
star' and it deals with energy. Can we say that geometry is about
energy? Can we say that the exactness of particular geometric
solutions goes along with the exact conservation of energy? Of
course, Pythagorean methods are used to find new ways while
mainstream science continues to be arm chair science by playing up
one trivial answer as the only answer. At times politicians pick up
the 'square a circle' analogy and then you should know they are
trying to find excuses and "explain" failures to their
supposedly dumb constituents. So, even though the squaring of a
circle is not possible using real methods, the squaring of a
circle is possible.
Alchemy
The
complexity of our environment is oftentimes worked through alchemy.
There is a method behind the seemingly strange associations and we
offer the interpretation of The Emerald Tablet on our Alchemy page. 



Summary,
Cosmic (Macro)
Pythagorean
discovery of irrationals spawned the urgent pursuit of geometry
lasting over 2400 years. Kepler brought arithmetic to the forefront
by establishing the mathematical and arithmetic relations of heavenly
orbits. In effect and in fact, Kepler introduced the parameter of
time in the mathematical context, which made it possible to make
planetary position forecasts —
forecasting being Kepler's life long passion. Because any two
gravitationally interacting bodies always have a solution, the
parameter of time derived from such periodic solutions is also
repeatable (periodic) and time can be used to make forecasts. Even
though time is always a derived (dependent) variable, the
mathematical solution establishes reversibility and allows the time
parameter derived from this
system to be used.
(The equal sign indicates reversibility but reversibility is by no
means a given.) Another way of seeing the mathematical solution and
consequent time reversibility is that the spatial distance (space)
and time form an overlay. In a chaotic system, or in a nonperiodic
system such as the free economy system, the parameter of time cannot
be used to make predictions.
Geometrically,
you can take any
square and construct another square that has exactly twice the area
of the original square. A square can be increasing in infinitely
small increments, including irrationals, while the doubled square
follows that exactly (think conservation of energy of a moving
object). This is something your computer cannot do. If you think this
is no big deal and it is something for the ancients to contemplate
– that's fine. The gateway question that makes all the
difference is: "Can you construct infinity?" Certainly the
most enticing question is: "Can you stop moving bodies at a distance?" 
Summary,
Atomic
(Micro)
Light
is understood as moving or standing linear  that is,
straight, waves following Newton's analysis of fringes (first
observed as fringe rings). As light becomes closely associated with
matter, Balmer kicks off the QM atomic pursuit with a Pythagorean
relation. A wavefunction is understood as a probability distribution
of an atomic particle — a great step
forward by Heisenberg and von Neumann. A moving particle has
momentum but momentum can also be worked as a wave —
a second great step, this time by de Broglie. A moving electron now
gets to become a wave as well, but this wave must curve –
that is, become circular, and close upon itself in a symmetry
about a point (about a core). The circular (or rounded) electron
orbital and the straight path of light need to be energyreconciled
through the squaring of a circle — the
first difficult hurdle.
Ether
is taken out of science's purview, which is the Great Reduction
making the scientist that much poorer in the end. Scientists cannot
make headway and talk about impossibilities. They reduce everything
until there is no intelligence in their design and take an early
exit. (In their last hurrah the angry mob bashes and trashes cold
fusion.) Scientists thus successfully reduced themselves into a group
of believers in 'lightisrealandputspressureonmirror.' While
much of today's physics rests on it, the scientist has no guts and no
brains to perform the actual experiment measuring the presumed
pressure light puts on a mirror. Scientists are not able to face up
to the truth that a light beam does not and cannot put pressure on a
mirror and so they are stuck perpetuating, defending, and
proselytizing their dogma.
Meanwhile,
geometry is receiving new impetus by reviving its superiority over
arithmetic and algebra. The golden proportion, the infinite and
instant wavefunction superposition, the understanding of irrational
and transcendental numbers, linearizing particular segments of a
circle, and the possibility of creating electron waves with
harmonicsseries energy components just might get the atomic
understanding going again – perhaps in another country, perhaps
by another group of professionals. 
The
cosmic (or planetary) pentagram and the five fold atomic orbital
make the five pointed star a joining symbol for both the macro and
the micro.
While
the atom's orbitals are symmetrical about a point (have radial
symmetry), the valence orbitals in a molecule need to close
around two points of symmetry (the cores of the atoms are some
distance apart). Yes, the Hyperstar
has an answer to that on our golden
proportions page. {Dec,
18, 2009}


Pentagon
Pyramid 
Selftest:)
If you are not happy about the pentagon base
and/or the pyramid being "split up" or "broken up"
or "separated," you are not getting the picture. You may
want to think about the red part as being the tangible component
while the blue part is the intangible (knowledge, virtual) component.
If that does not help, stay in 2D [earthbound?] where the pentagon is continuous.
If
you think Two rather than Four is feminine,
you are very close. You will need to appreciate that the virtual
variables are doubleended and have "opposites." Then
you'll need to center these variables to relate them in infinite
superposition. (Quantum Pythagoreans book helps in this area too.)
The
engagement of
Three and Four is just that: It can be
supportive in some contexts and in others it could be conflicting, in
which case rebalancing work is needed. 
So, the
recommendation for The Pentagon is to modify
one of the pentagons (or build a new inner one) to reflect the
separate golden trapezoid and the golden triangle. The new
construction slants upward toward a point. It does not need to top
off in a point as long as the edges are converging toward a point.
(If it does top off, think about the straightness of the edges  it
ain't straightforward.) 
This
pentagonal pyramid is on Mars. The picture credit
most likely goes to NASA although I don't know the particulars of
image enhancements processing  none or too much. Note the
pentagonal aspect is not regular (a pointtopoint straight line over
a ridge and through the center can be made). 
There
is one symbol that uses a circle framed by two vertical lines. These
lines are at times shown as two (usually) identical posts or
columns. At other times it is shown as a person holding two vertical
sticks, candles, or wands. This symbolism, however, is not about the
3 vs. 4. Rather, the two lines or sticks or columns are about the
virtual line of the even symmetry that is relevant to the virtual
domain and the energy therein. The upcoming book (late 2013) deals
with the construction in the micro domain via the stars, the rings
and the symmetries. Yes, knowledge is organized energy. 

Three and Four
must remain separate even though they are also joined. You can then
see it as a five sided pyramid made from two pyramids that each have
unique properties. The overall 3D structure has almost all numbers in
it but the point is that the numbers must play together to their best
advantage (rather than just being represented).
Do you see the
number zero? Or is it the infinity? Could it be both? Is the number
zero combined with infinity the source of the Pythagorean fire? The
root of the pyramid? The Central fire? The hearth of Zeus?
Archimedes' fulcrum of the heaven and the earth? Just a plain (free)
electron? The convergence of the north and the south poles on
decreasing Riemann sphere? A point of the zero dimensional (0D)
geometric construct? 'The One' of alchemy? Something even smaller
than the infinitesimal of Newton and Leibniz? The dot in the
semicircle on the AUM symbol? A computational construct for all of
the above from "all of the above?" The top dot of
Tetractys? My favorite: The eye Thoth sends to look for Tefnut
when she runs away to Nubia (yes, the eye finds her).
If you think this
pyramid is about marriage, you are on the right track. The Three
[male] and Four [female] are joined through the point of the
infinite. This joining is applicable to the actual marriage where the
joining is through God –
all there is. The alchemical marriage would have the female part
becoming more abstract although the foursided pyramid geometry
continues to be needed for dealing with the infinite. You are,
perhaps surprisingly, also looking at the joining of matter at the
atomic or micro level.
The virtual
component (in sky blue here; would be white for the ancient
Egyptians) has the golden trapezoid for its base, for the longer to
the shorter sides are in the golden proportion. The real component
has one of the golden triangles for its base.
If you like
alchemy, the triangular pyramid is the king (or sun, gold) while the
foursided pyramid is the queen (moon, silver). Yes, this is the 3
vs. 4.
In the ancient
Egyptian context, the crown of Egypt has two separate components: The
white (upper, virtual) and the red (lower, real). The gap between the
two is the ancient Egyptian blue crown and is invoked at war. (The
gap is white in our pentagonal pyramid.)

QUANTUM
PYTHAGOREANS
Book by
Mike Ivsin
Pythagoreans
use the knowledge of numbers to arrive at harmonious and stable
systems. Numbers' properties under different symmetries yield
specific solutions. Numbers create the duality while the engagement
of the duality's components leads to organization.
Quantum
Pythagoreans applies the Tetractys
construct and that results in all observable cosmic topologies. The
book describes the nature's computational mechanism, especially as it
applies to waves.
What
it takes to transform energies. Your body is a component and it is
not the only one. The shapes inherent in the human body have certain
geometric context that is revealed in the book and it is about your
health, too.
You
will like and appreciate the simplicity and the power of numbers.
The Pythagorean management of numbers takes you on the road to
reality and invites you to drive it as well.
More
.. 

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 2013 Backbone Consultants Inc. Copyrights Information
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12, 2013 