Really,
now, what is Pi? Scientists
with algebra do not have a handle on it
Circle
and Pi, the definition
You
don't want to and you really cannot use the circular
reference like the scientists;
Easy
to define Pi the Pythagorean way and it
starts by drawing an arch of a semicircle. Yes, integers are in the
definition; and
The
operators are created as integers begin to move
Pi
is more than was said by all, and it has
to do with the even and odd symmetry
Pi
figures in the conservation of energy. Ouroboros
sums it up but not on the silver platter
"A
circle is the delimiter of context."
"What?
Never heard of it."
"That's
the idea.."
Everybody
has something nice to say about the circle, as if the circle were perfect
Carl Jung
put the circle in the archetype category -- you have to go really
deep to connect with it and the circle is something very basic and
innate. On the other hand, many writers go straight up and call the
circle a solar this or a cosmic that, possibly with the help of the
shaman's drum.
Geometers hang on
to the equidistance from a point and use a compass when dealing with
the circle. Arithmetic works the point equidistance through
mathematical relations by applying angles and the Cartesian
coordinates of Rene Descartes.
Each point is computed and then displayed, and if you use random
numbers the circle nicely fills in one dot at a time. The actual
circle has an infinite number of points and we (have to) stop the
computer when the circle is nicely visible. Or you can step through
the angles and keep drawing the circle over and over. Math guys also
like the stuff that has some periodicity in it because Fourier
picks on repeating things very fast.
For Pagans, a
circle is the symbol of closure that has repeating periods, and from
there it goes on to yet another harvest of nature's bounty and
certainly a good cup of wine. You do not need to be a magician, here
or in Tibet, to see the circle as the delimiter of context. A judge
will make a circle around the case to make the universe of evidence
admissible or inadmissible.
When all
descriptions of a circle are exhausted, some scientists give the
circle another name such as infinity. Scientists tend to think of
infinity as being stuck round and round in a rotary, and do not know
the difference between unbounded and infinite. With recent Eastern
imports, a circle is zero and
infinity, and maybe you should think which part is the good part
before making a corporate logo -- Lucent and Vodafone coming to mind.
All in all, we
understand Pi from school as something to use when going from the
straight line that is diameter to the curved line that is the circle,
or vice versa.
While the real energy is well
known as kinetic energy, the virtual energy takes some
work. Free energy
is about transforming the virtual energy into real energy.
When squaring
the circle we must reconcile the infinite
transcendental numbers (incommensurables) with naturally finite
numbers (rationals). The golden
numbers make certain proportions that just might help.
Incommensurable numbers are composed of points to help you with the
infinite addition (superposition) of the virtual energy.
New
book you will thoroughly enjoy QUANTUM
PYTHAGOREANS More
..
Careful
on how you define Pi.It
may seem easy to do that but you want to be alert. Pi is routinely
defined as the circumference C of a circle divided by its diameter D,
and there appears nothing wrong with such definition in the
scientist's mind. It is easy to paste the C/D definition of Pi all
over the Internet and print it in books (even in books about Pi), but
this kind of arithmetic is of limited use. Scientists say that Pi
stems from an algebraic relation between two numbers but the
numerator is no ordinary number.
It may not be
obvious, but even if you measured the diameter exactly and now you
want to put the circumference in the numerator, the circumference of
a circle will not be exact because the circumference is a
transcendental number. So, you cannot take a transcendental number
with the infinite mantissa to start your computation to get Pi
because this number does not fit in the computer to begin with, and
there is not enough paper in the universe to write down one
transcendental (or irrational) number. Pythagoreans also say that
irrational numbers are unspeakable -- that is, nonverbal, as you
would need an infinite number of words to express one irrational
number. You will find that you can obtain Pi -- or the
circumference of a circle -- only by superposition (subtraction
and/or addition) of an infinite quantity of components.
Unlike the
scientist, you know you cannot define a word by using the word itself
in the definition. Surprise, but Pi and the half-circle (semicircle)
are really equivalent. They are the one and the same. Using a
shorthand notation, the Pi and the circumference C would look like this:
DPi
equ C
and then also Pi equ C/D,
which means that you can get the circumference C of a circle
by using Pi and diameter D, but both Pi and C
are qualitatively equivalent
and the constant D is but a multiplier to get one from the other.
It is not possible
to explain, 'How to catch an elephant?' by 'Catch two and let one
go!' So, when the scientist says that Pi = C/D,
he or she is saying that Pi is DPi
divided by D.
For a circle of radius one the scientist says that Pi is 2Pi
divided by 2. Scientists are, well, not very smart.
Pi
and C
differ from each other by a constant D but because Pi
and C
are equivalent then you cannot use one in the definition of the
other. Whenever you speak of Pi you also speak of a circle. Whenever
you speak of a circle, you also speak of a sphere.
Pi is not possible
to define algebraically because Pi is a transcendental number, which
cannot be obtained from an algebraic relation. The best definition of
Pi is through integers
(coming up).
Leibniz(bio)
first came up with an infinite series that adds up to a half of a
circle's quadrant: Pi/4. Leibniz series issues
from a trigonometric function and is the first real improvement over Archimedes'
methods from 300 BC that approximated Pi from geometric polygon construction.
Let's say you ask
100 math teachers: "If Pi is irrational,
how could you teach that Pi is a ratio
(of C/D)?" How many of these teachers will answer with something
similar to, " .. perhaps the format of the circumference-Pi
relation is not clear."? Possibly one out of a hundred. How many
will defer to a committee to fix it? All of them! Yes, algebra cannot
do it but scientists do not mind reducing it until they are just
plain wrong while teachers just go along.
This topic is
complex because division (or rationing) -- in and of itself -- does
not always result in a rational number. Golden
ratio is a ratio but it is not a rational number. You may want to
take a look at the explanation
of commensurables and incommensurables (both transcendentals and irrationals).
Circle diameter D
could also be an irrational number but when
you-the-builder pick up the compass and declare the distance between
points to be of some length, D must now have a finite mantissa. This
is crucial to universe (atom) building. One may argue that even God's
tools become limiting to God, but a better way of looking at it is
that reality -- if it is to remain organized -- is finite and
God may wish to make reality that is organized. This logic has larger
implications and even contains a gateway to gravitation. Think of
gravitation as an extension to atomic construction.
The addition
mechanism to get Pi is qualitatively different from the procedure
that generates SQRT(2),
for example, and this may lead you to suspect
that transcendentals such as Pi and the irrationals -- really the
straight-line incommensurables -- belong to two different classes of
numbers. For starters, SQRT(2)
is constructible with the Pythagorean Theorem in finite
number of steps, but transcendentals are not constructible with the
Pythagorean Theorem. This starts another chapter on Pi that deals
with the squaring of a circle. The basic question is: Can the area or
the circumference of a circle be equated exactly with the area of
a square? You need to know the physical parameter represented by
area to appreciate this issue. The squaring
of the circle topic now has its own page and deals with straight
vs. curving geometries.
We have two book
reviews on Pi, both books authored by scientists.
It is a sad story on what today's scientists can say about Pi.
Archimedes
All
topologies of a circle and sphere -- the periphery, area, and volume
-- are equivalent
to Pi through
rational (exact) constants. This is
really a summary of Archimedes work from 300 BC but [as you might
agree] it has not survived in clean relations because of algebra. Archimedes
also added the cone
to the equivalence of Pi because he found a rational number that
relates the volume of a cone to the volume of the cylinder (1/3). It
is said he was very happy about that.
The
story has it that Archimedes had the cone volume formula inscribed
on his tombstone.
[While
this story could be difficult to verify, a much better story would
be that he threw a great party and gave everybody a hat. The
tradition of the cone shaped party hat would then have been started
by Archimedes and it would also have become the longest surviving
tradition in the history of the world. There. Then one guy built an arch
in the honor of Archimedes and the world was never the same
since. As you may well imagine, the guy called himself the architect.]
There
is more to equivalence
When
Newton (bio)
speaks of "equal and opposite force" (action-reaction), he
is speaking of equivalence
of forces that arise simultaneously and one (action) cannot exist
without the other (reaction). Because forces are equivalent, one
cannot arise before the other. Another way of getting to the same
result is by understanding that all forces arise from the even
(wave)functions. This is also the foundation of the
action-at-distance because mass bodies separated by a distance act
with the gravitational force that is equivalent. Yes, the earth is
exerting force on the apple that is equivalent to the force apple is
exerting on the earth. (Because the masses of the earth and the apple
are in a huge ratio, the resulting acceleration will also be hugely
lopsided -- but the forces are the same and pointing in the opposite
direction. This is true of any and all forces -- such as when a
photon is absorbed.)
Action-at-distance
is no big deal for quantum mechanics and
this happens when the even
wavefunction reduces: one big one for Newton. The
gravitational wavefunction is an even wavefunction.
Reversibility
of a math relation
The
equal sign of algebra signifies reversibility rather than
equivalence, but this can be applied only in situations when there
are but two
and mutually exclusive outcomes happening over a single
bidirectional path such as potential-kinetic energy (spring, weight
lifting) or pressure-volume (piston). It is easy to construct such
bidirectional paths -- and the equations with equal signs to go with
it -- in a closed system. But not all paths are bidirectional
because at times reversibility can be had only over a triangular path
or multi-state path having more than two states. For example, an absorbed
photon can create several forms of energy. Some of the resultant
energy quanta are not readily transformable back into photons and
need to be worked over several stages before a photon could be recreated.
In another example
of algebra inadequacy, the equation E=mc2
allows you to "calculate mass"
of the photon by placing its energy on the other side of the equal
sign. But a photon has no mass and then the algebra weakness is
"explained" by calling the mass the 'effective mass,' while
some algebra holdouts continue to hang on to photon's mass and coming
up with their own conclusions such as "tired" or
slowing-down light. This equation simply and nicely shows that
algebra is incomplete. The equal sign works fine in the real domain
where the commutative property holds (ab=ba). As soon as the virtual
entities are encountered -- and light is a virtual entity -- algebra
falls apart. This equation is incorrect in another way but the error
discussed now is that irreversible transformations are paraded as
reversible via the equal sign. Algebra is not representative of
reality when transformations (reversible or irreversible) take place.
Reversibility or
irreversibility of a mathematical relation,
then, is an exceptionally important determinant and algebra needs to
be replaced with another methodology. Presently, scientists muse
about some of their weird or unusual results they get from their
inadequate math while unable or unwilling to question math itself
[think the left and the right brain].
Algebraic
relation needs a bidirectional path for
the equation to hold. While a bidirectional path can be created
through a specific (closed system) context, a triangular path is the
more general case found in nature. The kicker is that when dealing
with virtual variables the system cannot be closed -- you cannot use
real things and real methods to confine virtual variables. This also
gets into intelligence and organization.
Easy definition of Pi,
the Pythagorean way
Now that we
criticized the scientists' definition of Pi,
we should offer our own, unencumbered, strict, literal, and
unambiguous definition of Pi:
Semicircle
is a curve that has the distance of 1 from a point that that
lies on the line. The semicircle has the starting and ending points
on the line. At the starting point it is possible to traverse either
a straight distance 2 or a semicircle distance Pi
to get to the same ending point.
The
Pythagorean definition of Pi makes it easy to see that Pi and the
semicircle are equivalent because the semicircular distance is
Pi. The semicircular distance is the value of Pi. [It is also
a nice definition because it gives you a choice. You can go straight
in 1D to get to the ending point and you know exactly how far it is.
Or you can take a more circuitous route that is not exact but you
will get to meet the Pi.]
It is said
Pythagoras started his studies with the semicircle. Not surprisingly,
we need integers 1 and 2 to define Pi. Perhaps more
accurately, we need the unit 1 and a doubling operator
to arrive at Pi. You can get metaphysical
here because 1 is odd and masculine while the even (2, doubling) is
feminine. If you start with the length of 2, which is feminine, you
will need the operation of halving to determine the point O at
which to draw the semicircle. Halving is masculine. (Yes, cutting is
masculine and you will find it in the left brain.)
What then is the actual value for
Pi? There is an infinite series by Leibniz that is quite simple and
it converges toward Pi/4 -- that is, this series converges to the
length of one-half of a circle's quadrant. This formula is also the
first qualitative improvement over Archimedes' method of
"exhaustion," which uses inscribed and outside polygon
construction to determine the ever-improving value for the circle's
circumference of 2Pi.
Pi/4 = 1/1 - 1/3 + 1/5 - 1/7 +
1/9 - ..
This formula happens to converge
to Pi very slowly. With an increasing number of new terms we get more
accurate (closer to Pi) but -- compared to other methods -- the
computer must do a lot of computing to show progress. (Self
test:¬) If this paragraph does not make much sense to you, you
are doing well.
So, as a Pythagorean, you are not
as much into expressing Pi on paper as you are into the actual
construction and actualization of Pi. You start by seeing that the
pluses and minuses can be implemented through superposition and in QM
the superposition is instantaneous. Nice!
Operators are well
understood by the present day mathematicians
(with a likely exception of i
and -1).
The treatment of operators is generally adequate and includes the
null operator, which can be applied as having a body at rest
[Aristotle would scratch his head on this one]. The doubling operator
is for some reason not included explicitly but there is, of course,
the operation of 'multiplication by 2.' You will note that
'multiplication by 2' and 'addition of the same' arrive at the
identical result but both of these are different operators because
they represent different domains. (There is a tangent here on the
golden proportion.)
When you use words
such as 'traverse' or 'measure distance,' you are describing a movement.
In order to unambiguously define Pi, you need to get from here to
there. It is the movement itself that uncovers the operators.
Metaphysically, men at times need to make a pilgrimage or a quest in
search of something -- for that is the only way to meet or
"slay" a dragon. (In the book Quantum Pythagoreans
the growth of the universe is accomplished through orbits, for the
continuous accumulation of matter in a single body unbecomes
tractable. The book explains why operators are forces.)
In our
definition of Pi we use 'distance' instead of 'length.' The straight
distance is in 1D while the semicircle exists in 2D. Both exist in
space (and all geometry exists is space). There are no restrictions
on the diameter, and the unit distance can be rational or
irrational. That is, the unit distance is not limited to finite
numbers such as integers and this opens yet another metaphysical
dimension. (Perhaps you understand now that the unit distance cannot
be transcendental because it is always a straight distance.) If the
unit distance were an irrational number the radius (and diameter)
should be dashed because we change the context from unit length to
unit distance but Pi will still be Pi (this is more important than
you might think). Even Euclid
thinks 1
is more than a counting number.
Semicircle
in the Buddhist tradition
There is a
nice dot and a semicircle at the top of the AUM (or OM) symbol found
in Buddhism. There are sparse interpretations of the AUM symbol, most
of them deferring to the infinite, the unconscious, or to taking your
own and individual path to its understanding. Because this site is
about numbers and geometry, and because the numbers are at the heart
of it all, we are not shy about delving into the aspects of the AUM,
no matter how infinite and all-encompassing it may be.
The dot is
indeed the nothing (indivisible, has no parts) as well as the
everything (infinity). Because this dot is not real -- it is a
geometric construct -- we prefer to draw it as a small circle. The
dot is then infinitely small inside the small circle. It is through
this dot the real and the virtual components are interconnected. This
is indeed a big leap and so you may need to take it as a given for now.
Intelligence
in general is virtual energy and can be thought of as organized
energy. One property of the virtual energy is that it is always
symmetrical about 1D axis (the axis is not shown on the AUM). The
total intelligence of the entire universe is symbolized by the
component that looks like the number three on the AUM symbol. This is
not the counting number 3 and neither it is a trinity but the two
humps deal with the even, or axial, symmetry that is inherent in the
formation of the virtual knowledge -- that is, intelligence. As you
look through (or move through or think through) the dot you will
extract, or copy, the kind of intelligence that is relevant to your
needs. This, by now subjective, intelligence issues from the total
intelligence of the universe and moves into the circle that could be
you, for the circle is the delimiter of context. [I'd put a little
curlicue between the total intelligence and the circle to indicate
spin. I'd also put horizon(tal) lines on each side of the point.]
Oh yes, the
semicircle. That is about reality, for all real entities have odd
symmetry and that means that all real entities are always symmetrical
about a point. You guessed it, it is the same point. You could
interpret the semicircle in that you want to make something real from
the intelligence you have received.
You are still
left with some unanswered questions such as "why is it a
semicircle and not a circle," and "why is the horizon
important and what qualitatively differentiates horizontal from
vertical," and "where does Pi fit in," but now the
answers are not all that difficult -- consider it your homework.
There
could be a tie-in between the Pythagorean idea and the AUM.
Pythagoras is reported to have said something similar to: "The
eternal world is revealed to the intellect but not to the
senses." It does not have the poetic qualities of "Don't
poke fire with iron," but it is conceivable Pythagoras was
teaching from behind a curtain just to make that point.
How do you make the sound
of AUM? (Words are mostly about objects.)
You start with A
as a broad sound and imagine it as opening a book. This is the energy
part. You are a smart person and you realize that the spine of the
book is the vertical axis of symmetry. Pages of the book are now free
to rotate about the axis.
You transition to U
as, of course, the semicircle. This tone then changes a bit as you go
sounding ("curving") the semicircle. This is the charge
part. Yes, the charge is always symmetrical about a point. (The
charge and matter are closely linked. There is no matter without charge.)
You move to
vibrating M, where
the spikes are the points that nail the two of the above components
together in zero-dimensional points.
The Latin letters
fit in nicely considering the geometry and associated symbolism.
A
circle and sphere are made from a semicircle
The
starting and the ending points of both the straight and the
semicircular distances are identical.
A circle exists as the doubling of a semicircle and so the
definition of a circle is also about the exact
division of a circle by 2. Are there
other numbers that divide a circle exactly? Or is it that any number
can divide a circle exactly? You will need geometry to answer that.
Metaphysically,
the semicircle is feminine because the entire
curve is symmetrical about the (vertical) axis. When making a full
circle by, again, doubling, the 1st and 3rd quadrants become
symmetrical about a point and, therefore, masculine. Metaphysically,
then, "masculine arises from the feminine via the reflection
about the horizon." If you get huffy about this consider that
masculine (feminine) is not the same as man (woman). If you still
cannot get over it, don't go on a date with a witch.
Perhaps
the best application of a semicircle is the geometric
mean. Because the geometric mean works for
both the rational and irrational numbers, there are significant
implications here to such (for some people esoteric) topics such as
stopping moving bodies at a distance. Incidentally, since the
geometric mean also constructs the square root, your attempt at
achieving the squaring of the circle now includes Pi2
because you could then readily obtain the square root via the
geometric mean. Pi2 has an infinite convergence series by Euler
(if you want to give chase via QM superposition).
Even and odd symmetry
There are two
basic functions applicable in the virtual domain of the atom: Even
and odd mathematical functions. The property of the even function is
that it is symmetrical about the vertical axis: f(x) = f(-x),
while the odd function is symmetrical about a point: -f(x) = f(-x)
(the equal sign
means the author of the equation thinks that the operation is
reversible, and it is). The even function has infinitely inclusive
properties such as those of a photon and it is an agreeable extension
to call even functions 'feminine.' If the operation of translation or
rotation have unique symmetries, then these operations can acquire
gender as well. Inclusiveness is at times called superposition and it
is truly infinite, not just unbounded. Generically, all even
functions are forces [and may she be with you]. The odd function, you
guessed it, has exclusive properties such as those of a proton and
calling the odd functions 'masculine' then also makes sense because
protons "butt heads" and displace each other -- at least in
the real domain. Generically, all odd functions are real things, and
bumpers with automobiles are just such things. Every atomic entity
has properties that fit either the even or the odd function.
The circle is the
one and only geometry that has even and
odd symmetry. This does not make a circle the gender neutralizer.
Instead, because the circle is acceptable as both the even and the
odd function in particular quadrants, the circle is the originator
and the go-between among the two. The circle is the vehicle for the
initial gender differentiation and the circle also facilitates the
transformation between the two. Moreover, whenever something is
symmetrical about a point it is easy to see we are dealing with
rotation or orbits (macro), or orbitals (atom). Whenever something is
symmetrical about the vertical axis we are dealing with a translating
motion and the even symmetry holds for straight linear motion at any
point on its straight trajectory. You now take your compass and an
unmarked straightedge, and attempt to be William Blake's
master geometer that is God.
With the odd
function you could begin to see how you could really explain the
two-body collision mechanism. You will need to
take the moving energy as a wave (easy) but then you will need to do
a transformation.
How can a circle become or stand
for zero? You will need to visit India and learn how
the zero came to be. Since a circle is the delimiter of context, the
circle you draw can be evacuated of every real thing -- and you have
zero. There are some who think this is not just an intellectual
exercise because the circle can be emptied of real and
virtual things.
How can a circle become (or hold)
infinity? This is a bit more complex than a simple
reversal of the above. Go for it. Yes, you will need geometry.
Transformation, as used by
mathematicians, is not very transformative. The math
guys see transformation as something that translates (moves, shifts)
some function (a graph) along a line either horizontally or
vertically to another spot. This just moves the reference and is not
really transforming anything. Another form of transformation math
guys claim is a transformation is to change the scale and this
amounts to zooming in or out. Again, not very transformative.
Finally, math guys speak of transformation when they flip a function
about the axis and make a mirror image. You can easily see that none
of these operations are transformational. For example, just because
you shift from meters to feet as the new scale the math guy would say
you are transforming but you know that the word 'transformation'
should be reserved for something bigger than that.
In our definition of
transformation we speak of a change in the entity --
or the function describing such entity -- from the even to the odd function
or vice versa. So, when a photon (even function) reduces by being
absorbed, the energy is now converted to momentum that moves a couple
of bodies such as molecules away from each other and the moving
molecules now acquired an odd function as the two are moving away
from a point. Similarly, when a real electron becomes the virtual
electron it transforms from an odd function entity that is a real
thing into an even (wave)function entity. (There is a bit more to
this, for an electron has two components.) Dirac,
a physicist and a mathematician, was pretty close and actually spoke
of electron transformations. Incidentally, math guys are quick to
point out that most functions are neither even or odd. What they do
not say is that in the atomic environment there exist only the
even and the odd functions.
We have a couple
of Pi stumpers
you can try on your math teacher. Because math teachers subscribe to
lame committee rules they are easy game.
QUANTUM
PYTHAGOREANS
Is i
a number or an operator?
Is -1
a number or an operator?
i is
a square root of minus 1.
For example, take
the equation i = 1/ i
If you square both
sides the equation is correct. If you multiply both sides by i
the equation does not hold. The book explains. More
..
If you try the
above equation & non-equation on a real or self-proclaimed
mathematician, chances are 99.9% he or she will not know where the
problem is. You can tell him that squaring and/or multiplying both
sides of an equation are both legal algebraic operations -- and
chances are he knows it. Don't push it if you don't want to confuse
the poor guy.
Ouroboros
In a simple and
practical explanation, Ouroboros shows the difference between the
straight and the curving geometry. If you have a string of some
length then the expression of such length is always finite --
that is, the number that is the length of the string always has
finite mantissa (it is a rational number). But the circle's
circumference made from such string cannot and will not be of the
same length because the circumference of any circle is not a finite
number. The circle's circumference will always be shorter or
longer than the length of any string. At times, then,
Ouroboros is shown with excess or with a deficiency in the length of
its tail.
What this also
means is that a circle cannot be made with a specific (exact)
circumference because a specific number is a finite number and does
not have an infinite mantissa. A more general case can also be made
because a circle's circumference is a transcendental number and even
the irrational number with an infinite mantissa cannot be made into a circle.
Circles, ellipses,
and other curving things are made from many zero-dimensional points.
Because a geometric point is dimensionless, the number of points
making a circle is infinite. Then there is also the question,
"what entity can exist in the circumference of a circle?"
By now you know it is not a real thing because no real thing such as
a string can be made into a circle. If such curving entity is not
real -- can it be virtual? If it is virtual -- does it exist? Can we
see virtual things or entities? Does the electron become the virtual
electron when it forms an orbital? Are dragons made of electrons?
Some comment on
Ouroboros as "eating its own tail." Perhaps you see now
that such is not the case. Some could argue that the material can at
times be transformed into virtual and that is the "eating"
part. While technically true, it says nothing about the creation of
the real world and why the creation of the real world prevails over
its destruction.
There are also
more esoteric interpretations of the Ouroboros. Skip the following if
you don't fancy esoterica.
Pi has an infinite
amount of components calling for an infinite number of decimal
places. You are pretty good if you asked a question:
"What happens
when its tail is chopped off?"
Why, you could
even discover there are different ways of drawing Ouroboros. As you
look at all of them the Ouroboros of the ancient Egyptians looks the
best, for it has a white belly on the inside. And then you might say:
"Pi is a
serpent? With scales or feathers, no less. Come on, get real!"
My point,
precisely. But this may all be too obvious and you do not see the
woman. There must be a woman.
"She is
holding the sword .. vertically. There is a pivot, too."
Yeah, the guy
should be in charge of that pivot. But you can do better.
"Engage the
Couplex on the diagonal."
There is not much
to say.
In summary,
Pi must facilitate energy conservation when entity's motion changes
between the curving and the linear motion. Because Pi is
transcendental then Pi's value is expressed with infinite quantity of
components [here she is again]. The real domain, however, consists of
object's values that have finite length (finite precision) that's
just right for the real world. The curving geometry expressed by Pi
is reconciled with the linear (straight) geometry of Euclid through
some form of transformation. That's the importance of Pi and the
basic differentiator between straight line irrationals and
transcendentals. Talking about the squaring
of a circle, eh?
Pythagoras would
certainly agree that classifying Pi as the transcendental number is a
good call.
If you wish to
give a test indicating the level of understanding of physics by
physicists, here is one question where the answer is not very
encouraging. Ask any physicist: "Why would the Planck constant h
be routinely divided by 2Pi?" and he or she will say:
"Because it is convenient."
Yikes!
We
are almost there. The classification of some numbers as
'transcendental' presents a very old challenge. All
non-transcendental irrational numbers the likes of the square root of
two are readily constructed with the Pythagorean Theorem. The
transcendentals, however, do not seem to care. The point is that
transcendentals take us from the Euclidean geometry into Riemann
geometry. Ready for another, and altogether different ride?
A
circle can take you places. Possibly the longest unanswered cry is
that of Archimedes: "Give me a point and I will move the
earth!" Is there such a point? Such fulcrum? [think golden proportion.]
Now
that we created the circle, it may be fun to explore the exact division
of a circle. Some numbers do it, some don't.
The Pythagorean Theorem does not
solve for transcendental numbers such as Pi but it does work with
addition or subtraction (superposition) of curved areas.
This particular rendering of the
Pythagorean Theorem may reveal powerful properties (read energy) when
dealing with spherical geometry. The illustration is from the back
cover of the Quantum Pythagoreans book.
QUANTUM
PYTHAGOREANS
Book by Mike Ivsin
If the atomic
component changes from linear to radial geometry, energy imbalance
arises because linear and curving paths cannot be exactly the same.
The atom maintains its stability by radiating or absorbing energy to
or from the environment because the geometric transformation must be energy-neutral.
Quantum Pythagoreans
describes linear and curving topologies, and the sustenance role of
light for the two topologies. The pyramid then also provides the
geometric constructs for the actual creation of new matter.
The duality of the
real and virtual energies is not only introduced but is fully
developed as two interacting domains.