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HyperFlight
Circle
division by five draw the perfect star
Divide
a circle by .. .. design the perfect family of all-stars
Construct
the real and the virtual objects -- and take them to 3D
Geometric
construction of the pentagon, pentagram, and the five pointed star
How
to draw the pentagram using compass and straightedge;
How
to make the pentacle directly on a given circle; and
How
to create the five pointed star indirectly via pentagon tiling
Perfect
three-pointed
star can be had on a circle or semicircle |

"Give
me a ring and I will make it sing"
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How
many perfect stars
and star shapes can you make with geometry? With arithmetic?
Geometry
vs. Arithmetic is a very old topic, very unresolved, and very
relevant; and
Numbers
are not just symbols -- in a circle they make (1)
stars on paper and (2)
atoms in space
Pentagon
Pyramid. Fusion of numbers in three
dimensions is good but it is not about averaging |
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Take this page out
of frames
Historically,
the geometric creation of the pentagram star was considered a
secret. One
can appreciate keeping the "formula" and directions secret
because the five point star construction 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
pentagon, the segment must be exactly one fifth going around the circle. |
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Not
all numbers can divide a circle into exact and equal
segments, and this is the first differentiator between geometry and
arithmetic. When it comes to dividing the 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 finite number of steps and achieve exact
division, we would then claim executability and, therefore, the
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 exact division of a circle in
the atomic construction below.
If
the length of a 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 segment's length were
an irrational number then its 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. Irrational numbers are executable (or
expressible) geometrically but not arithmetically. You might skip on
the diff between length and distance right now -- later you may avail
to the explanation and construction of the (in)commensurable
numbers.
Instructions
on dividing a circle by five -- in five steps:
1.
Draw a horizontal line and erect a vertical line. The intersect
becomes the origin point O
2.
Make point A on the horizontal line at any unit distance 1
from O
3.
Draw a circle of radius 2 around O. This makes point V
4.
Draw an arch around point A through V and make point B
5.
Distance VB is the length of cord c that makes sides
of the pentagon
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There are two sources
-- and therefore more than one meaning -- of
the five pointed pentacle star: One meaning has its origin in the
exact division of a circle and is discussed here. It is a fairly
complex though rewarding topic that leads to the symbolism of a star
inside a single circle. The other root comes from orbits (hence two
circles/rings) of Venus and Earth, and is discussed there.
Venus, while most prominent through the five pointed star, is also
associated with the number eight and with the meaning of the diagonal. |
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Rational
numbers are
commensurable numbers -- that is, they all
have finite or repeating 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 finite
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 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? |
New
book you will thoroughly enjoy
QUANTUM
PYTHAGOREANS
More
.. |
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The
radius measure of 2
(diameter of 4) in our pentagonal construction 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.
Some
Pythagoreans view the number two as a problem
number because it "divides the unity." Pythagoreans
discourage division of the unit 1 because it could create
problems elsewhere -- but the number 2 is certainly 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 sub-unity will become applicable to atomic orbitals and
hence the unit 1 is the Great Divide between macro-cosmic and
the micro-atomic. [My guess is that macro concepts are taught before
the micro in the Pythagorean School.]
For the four sided pyramid,
particularly the Great Pyramid, you want to start with the irrational
and rational golden
numbers and their proportions. The pyramid template results from
the new way of the golden numbers construction that also reveals
geometries of the inside of the pyramid. The meaning and benefits of
proportion(ing) and rationing are also there. Again, we would not
pass up an opportunity to talk about the one-up of geometry over
arithmetic when it comes to irrationals.
When
you do 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 length of 4.
The base of the Great Pyramid is then 4 times of some unit of
measure [and this can get cosmic]. Pyramid's base periphery and base
area then also carry the same square number 16. The number 2
(half of the base) is the denominator of the unreduced
golden ratio.
If
you like Japanese calligraphy, you might
appreciate directions starting with the horizontal and then moving to
vertical. Then left, right. The lower closure happens last. The
horizon is also important to the ancient Egyptians.
If
you want to have more fun, consider that the
unit distance OA
could be an irrational number. Even Euclid did not think of the
number 1 as just a counting number. |
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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.
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 the circle, for it is about working the straight and curving
geometries. You will like the connection between the exact
construction and the (exact) conservation of energy. We did not
forget the ancient Egyptians either. |
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Construction
(continued):
If you want to construct the penta~ with the circle radius of your
choice, point A is one-half of the radius away from origin O.
[Every time you double something -- think octave. Every time you
halve something -- think node (or fit) for standing waves. Every time
you rotate by a right angle -- think..]
When
you divide a circle into exact fifths along the periphery you now
have the template and:
Make
the pentacle
by connecting every other (second) point. If you do it counter-clockwise,
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 mechanics are clockwise. [Pentacle is the easiest and the
simplest to make because it does not require erasing. Does it mean it
is the most fundamental?]
- Make
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
Make
the pentagon
by connecting every neighboring point and erase the circle. [Connect
it counterclockwise if you want to be disharmonious (ratio 9/5),
clockwise if harmonious (6/5).] The ~gon identifies the stars that do
not skip points -- polygon in general. When you say regular
polygon you are emphasizing that all segments have the same distance.
(Distance is more general than length.)
All
points of all perfect stars are on the circle and are equally 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. As to the
actual angles between points -- see below, for these angles could be
irrational numbers. (For now, ask yourself if a computer can give you
an irrational number.) |
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There
are other geometric ways of constructing a pentagon or pentagram. 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. Other constructions make
the side c
a rational
distance, which is better suited for the Great Pyramid's purposes.
(The Quantum Pythagoreans
book presents the pentagon construction with a rational segment
length.) In the pyramid, one half of the side of the base is (must
be) a rational unit of measure, for such measure is executable.
(Transcendentals may put a kink in that.) The connection between the
five sided 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.
Some
star constructions speak of fixed length sticks, which, technically,
can construct any size polygons. Here is where
the executability
of angles comes in. 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 case can be made that geometry takes precedence (has
priority) over arithmetic -- see below. [If you are a scientist, you
may think of Emmy Noether who ignored 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.]
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 (1
+ SQRT(5)) and one rational number (2)
and, because they 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.
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 physical entities if you know what they are. |
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Some basic geometry.
From a square angle to a square |
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Here is a simple yet powerful
construction that
(1) Divides
any distance exactly in half;
(2) Erects
the perfect right and square angle; and
(3) Makes a
true square using any circle centered at O (at the intercept
of horizontal and vertical axes). A square is also a four pointed star
Only straightedge
and compass are needed. (Straightedge is an unmarked ruler.)
Both semicircles
have the same radius.
Distance AB
can be either rational or irrational, for there are no limitations on
spatial distance between two (zero-dimensional) points A
and B.
If distance AB
is irrational, should it be dashed? If so, why? |
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Perfect
star family |
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Using
geometry's tools, a straightedge and compass, a circle can be
divided exactly into 2,
3, 5,
15, and 17
equal segments, technically called regular polygons. 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. 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 (below) you can make the exact hexagon and from there the
twelve point star of the Zodiac or do a layout for a twenty four
point Feng Shui star.
Starting
with a 2 point star -- the only direct even star -- you can
construct 4, 8, 16, etc. stars that 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 6, 12, 24, etc. stars. From the five point star you can make 10,
20, or 40 point stars. Every perfect star will have some symmetry
about a point and about an axis.
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 star is 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. {Mar
21, 2006}. The most
interesting (and important) aspect is that even if a start is
constructible exactly via geometry, the very same star cannot be
constructed exactly via arithmetic.
Carl Gauss
"recently" added the 17 sided polygon. The 15 sided polygon
is in Euclid's
Elements, Book 4, Proposition 16. |
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Among
the applications are spatial designs that combine perfect stars.
In watch design, for example, the circle being divided by twelve
looks fine but, in addition, it is harmonious to overlay the 12 point
layout with triangular 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 the
disharmony. Six gets "tricky" because it is disharmonious
with larger numbers but is harmonious with five, making a pentagon.
When using but a single star you choose one from the perfect star
family. 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.. |
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QUANTUM
PYTHAGOREANS

If
a is harmonious with b, and b is harmonious
with c, is c
harmonious
with a? Not always. The book explains harmony through
geometry and star construction. While much talk is about harmony, Quantum
Pythagoreans provides the formula for notes that actually are harmonious.
In
the beginning was the number .. and the power of numbers begins ..
Ready to construct
the atom? You will need knowledge and not just raw power.
More.. |
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One interesting property of the
perfect star family is that they do not intersect. A sequence growing
from each of the direct perfect star number does not cross (overlap)
with another sequence. That is,
from 2 we get 4, 8, 16, 32, 64,
128, 256, 512, etc
from 3 we get 6, 12, 24, 48, 96,
192, 384, 768, ..
from 5 we get 10, 20, 40, 80,
160, 320, 640, ..
from 15 we get 30, 60, 120, 240,
480, 960, ..
from 17 we get 34, 68, 136, 272,
544, ..
Each member of the perfect
all-star family has but one origin.
The perfect star family of
numbers introduces some changes to our perception of
universe building and how the everyday reality happens to come about.
Mathematicians can make all kinds of star constructions, in 2D and 3D
-- but only the perfect star family can begin
to bridge the straight line energies such as photonic energy with circular
orbits and orbitals. Because the vast majority of energy in the
universe is in the form of a spinning or an orbital energy -- that
is, energy that has angular momentum, the perfect star family of
numbers takes 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. |
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Construct the
perfect triangle on a circle and another triangle on a semicircle |
Instructions:
-
Draw horizontal
and vertical lines. The intersect is the origin O
-
Draw semicircle of
radius r around O. This makes point V
-
Draw circle around V
of radius r
Larger triangle
divides the circle into three exact cords of length c.
Using the
Pythagorean theorem, verify that the relation between cord c
and radius r
is: c2 = 3·r2 or (1/3)·c2
= r2
What physical
entity is proportional to r2?
If you know what that is, consider that the square of cord c
is three times that.
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The puzzle of a
bad souffle:
Given a square, construct a new square that is exactly one third of
the original square.
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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.
Could squares with
irrational sides be included in the general
division of a square into many other squares? If so, does it mean
that geometry does one up on arithmetic once again because arithmetic
cannot give you an exact irrational number? |
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The six-pointed
star could be of some interest regarding energy accumulation
and we included it in the numerology
section on the Pythagorean page. While the circle is divisible by
three exactly, we need six nodes for three wavelengths to wrap around
the center (or around the nucleus in the case of an atom). We are
then talking about the hexagon. |
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Pentagon
tiling does not
involve direct construction but only translation and/or rotation in
two dimensions. Translations are linear (straight) motions and are
always symmetrical about a line (edge) [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). |
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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. |
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If
you take five, five-pointed stars and arrange them around with their
points touching, do you get a pentagon in the center? (yes)
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.) |
The
pentagon template for the illustration was obtained with MS
PowerPoint by selecting AutoShapes .. Basic Shapes. Pick the
pentagon object.
Tiling
of five pentagons to make a five pointed star was (first?) published
by Kepler
in Astronomia Nova |
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The
tiling construction (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 the left-brain way at that). A good way is
to see this as the act of creation of the virtual object.
(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. Now, for
homework: (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?)
When
working the Great Pyramid, you may want to think of the chambers and
passageways as virtual objects or "spatial" objects. It
really helps.
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Geometry
versus Arithmetic
Ancient
Greek-speaking 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 the 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.
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? |
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What's
your angle? |
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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 the circle depends on 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.
Consequently, a 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, 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 nine-wavelength, or 18-node, 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 but it cannot be used to divide a circle exactly.
[Does this mean that 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
all-star 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. 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. So, if you use a protractor to construct the angle of 72
degrees, say, this angle (and the corresponding mark) cannot be
obtained exactly if this protractor was created or calibrated by a
computer. Arithmetically, no star
creation is perfect. Worse yet, arithmetic operations with real
numbers give no clue if a particular number is or is not divisible
into a circle exactly. A circle's periphery divided by any rational
number remains a transcendental number -- that is, arithmetic keeps
transcendental numbers unchanged. But what if you could geometrically
divide one incommensurable (transcendental) number by another
incommensurable (irrational) number and get an exact result? There
may be more to it than just being nifty, for you are dividing one
infinite (mantissa) by another infinite and get a rational
number with finite mantissa as a result. [Does it mean we could do
one up on the gods?] |
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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
24 (hours in a day) is a good example as it is divisible by 2, 3, 4,
6, 8, and 12. As convenient (abundant) as the number 24 is, 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. 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
geometrically-obtained angles do not necessarily have a whole numbers
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. [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.] |
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What
is the meaning of a circled star? The 'star inside a single circle'
is about the atomic creation symbolism. 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.
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 standing-wave electron and
with the standing-wave proton. Why, could you then do a precision
surgery on the atom?
The
eight
pointed star has diagonal (semicardinal)
directions and if the meaning of a diagonal is in some respects
special (it is) then the eight pointed star could be about something
else besides counting points. It is about Venus, too. |
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While many books
talk about Georg Riemann
accomplishments, they are written mostly by mathematicians.
It is likely Riemann did not even try to find applications for his
math and the science writers don't try very hard either. Mainstream
physicists bring up Riemann applications in a superficial way of
"curved space is possible." So far, nobody applied his work
to atomic closed topology orbitals and to atomic computability of
curved pathways.
So far, nobody linked his discovery of incomposite (prime) number
availability to the "forbidden" electron orbitals. This
task would be simpler if the incomposite numbers of the Pythagorean
origin were not renamed 'prime,' for incomposite numbers cannot compose
while prime has no
relevant applications association. The label prime
makes its descriptive value so weak nobody even thinks of it as a
hole in the wall. We have three book
reviews on Riemann. |
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This brings us to wider
applications of rational and irrational (incommensurable) numbers.
These numbers are important if you want to appreciate the interplay
between geometry and incommensurable numbers. A nice definition and
some apps are on their own page, with the following few lines serving
as summary:
For
commensurable (rational) numbers the applications are clean. Any
whole number divided by another whole number could also become a
fraction, yet such fraction always remains a rational number because
any fraction (such as the decimal fraction) sooner or later starts to
repeat, and such fraction can be converted back into a ratio of the
two whole numbers from which the fraction can again be made. The
ratio of any two whole numbers or its computed fraction can be reversibly
converted into each other without loss in fidelity. If you have
difficulty visualizing why a repeating fraction is really finite, you
will need to get into the ancient Egyptian way of representing
fractions [yes, this is loaded]. The computer, then, can store
rational numbers just fine (at least in principle), while the present
computer does not, and never will, manage the infinite mantissa of
the irrationals. Numbers with infinite non-repeating mantissa are
incommensurable numbers. Applications of incommensurables are rooted
in their infinite mantissa. In the space
of geometry, moreover, distances can actually be irrational.
Here is also where the 'number line' reaches the limit of its
usefulness, for the number line is about the magnitude while the
incommensurables are not. (Advanced -- need to differentiate between
'distance' and 'length/magnitude,' and the reason it is advanced is
not because it is complicated but because Euclid did not do it.
Euclid did not get primed very well in the Pythagorean tradition even
thought he understands that 1
is not just a counting number. Distance vs magnitude benefits are
treated in Quantum Pythagoreans
book and applied to energy.) |
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)
do 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.
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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 straight-moving energy to close upon
itself and make an atomic orbital? Think pyramid geometry. |
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Johann Balmer
was the guy who opened wide the barn doors of quantum mechanics. He
came up with the equation that produced a sequence of numbers
matching wavelengths of light coming from hydrogen. These were not
just any wavelengths -- they were particular jumps of particular
wavelengths and no other. Balmer did for quantum mechanics what Kepler
did for gravitation: He came up with the 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. Whole numbers and HIS theorem are
also the source of the quantum behavior of atoms. |
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Squaring
The Circle
This topic is
expanded and now has a page of its own. To reconcile
the straight and the curving, you will be dealing with the squaring
of a circle. Algebra works fine when things are straight or
polynomial but 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 the circle to this page's perfect division of a circle is in
the possibility of linearizing the curved segments (arches) 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. |
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Summary,
Cosmic
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 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
non-periodic system such as the free economy system, the time
parameter 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?"
[Think those pesky irrationals and glorious transcendentals.]
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Summary,
Atomic
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
energy-reconciled 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 'light-is-real-and-puts-pressure-on-mirror.' 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
harmonics-series energy components just might get the atomic
understanding going again -- perhaps in another country, perhaps by
another group of professionals. |
The
other Five
If
planetary orbits interlock in 8/5 or 6/5 ratios, a five
point star orbit
results when the two planets are computationally combined. Venus and
the pentacle get their say and, because the pentagram/pentacle also
originates from the orbits, there is yet another way of drawing the
pentagram/pentacle that deals with the "proper sequence"
and "proper orientation" of making the points. Different
ratios correspond to particular clockwise and counterclockwise stars.
A good question for orbit ratios 6/5 and 8/5 is:
Arithmetically,
we can calculate the angles of the pentagon and derive the equation
that allows the computer to draw the pentagon. You may have also
constructed stars on paper, but how does Nature draw the pentagon or
a pentagram on the canvas of space?
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Pentagon
Pyramid |
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Self-test:
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 double-ended 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. |
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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 pyra-mid? The Central fire? The hearth of Zeus?
Archimedes' fulcrum of the heaven and the earth? Just 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? Something even smaller than the infinitesimal of
Newton and Leibniz? A computational construct for all of the above
from "all of the above?"
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 four-sided 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 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 four-sided pyramid is the queen (moon, silver).
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.)
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QUANTUM
PYTHAGOREANS
Book by Mike Ivsin
Pythagoreans
use the knowledge of numbers with straight and curving geometries to
arrive at stable -- that is harmonious -- systems. Numbers'
properties under different symmetries impose specific solutions.
Numbers create the duality, and engagement of the two components of
the duality are responsible for organization. The Quantum Pythagoreans
book also treats the reversibility of interactions and
transformations as components of stability and creation.
In Quantum Pythagoreans
the force of gravitation is one of the variables that applies the
Tetractys template and results in all observable cosmic topologies.
Yes, the book proposes the gravitational mechanism as well as the
computational mechanism.
What it takes to
transform energies. Your body is the 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.
You will
appreciate the simplicity and the power of numbers. Pythagorean
management of numbers and operators then takes you on the road to
reality and invites you to drive it as well.
More
.. |
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