How
to Square a circle
Geometry
and its magic
Squaring
(quadrature) of a circle unwinds a circle in an attempt of making
the turns the same as straightaways
Making
a circle out of a square is about reversible transformations of the
circular geometry (2D, circle, compass) and the straight line
geometry (1D, square, straightedge). Yes, we can figure out the
symbols and the magic of Thoth
Pythagorean
pursuit in squaring of a circle must be relevant and useful.
Pythagoreans, then and now, apply math that describes the operations
of nature. Squaring of a circle is about energy and its meaning is
energy transformation
We
put Euclid on the Pythagorean platform, too, for his fifth
proposition is weak
So
you take the circumference of a circle and make a square having the
periphery that's exactly the same. Is there more to this than an
intellectual exercise? Keeping your brain in gear? Proving something
others said is impossible?
While
it is commonly accepted that squaring of a circle is not possible,
there are ongoing attempts to work this problem. Indeed, every person
should understand what is behind the squaring of a circle because the implications
of the process are richer than a philosophical discussion, a
belief, trivia, or the plethora of various approximate constructions.
To put it with a flavor of the East  It's not about the
destination, it's about the journey. Even though we are told that the
answer at the destination is 'no,' we want to make the trip. As it
happens, we will come to a fork in the road – a Pythagorean Y or
a Greek Tau – which you will never find if you jump to conclusions.
A
moving object has energy (technically called momentum) that is
determined by its speed, which also depends on the distance the
object is traversing (because speed is distance per unit of time).
Going on a straight path the moving energy of an object has some
particular value. If the curved path were to have the same distance
as the straight path, the moving object's energy would be conserved
at all times.
But
what if the distance of a straight path cannot be the same as the
distance of a curving path? The true question then comes up: Because
the moving energy of an object is conserved, what happens to
the object's energy if the path changes from being straight to
curved? Similarly, what happens to the object's energy if its curving
path is straightened?
The 'how to'
square a circle is about the exact match in length because the energy
is conserved exactly and if circle squaring is possible then the
speed on the straightaway would be the same as on the curve. One way
of seeing the 'straight' vs. 'curving' assignment is by taking a
square and then turn the sides into flexible lines such that the
circumference of the original square is the same as the circumference
of the resulting circle  or vice versa. This appears a simple thing
to do and possibly in the recreational category, too. Applying the
benefits of geometry  and if its exactness brings something special
to the table  then geometry may well help us decide this question.
In essence, if we can make the exact match
between a circular path and a straight line path, then the moving
energy is conserved and we could not use the parameter of energy to
tell us the difference on what is straight and what is curved. But
the straight line of a square and the curved line of a whole
circle or an ellipse cannot be matched in length. Every time
the geometry of a moving object changes between the straight and the
curving, energy is released. The energy is released as force in the
case of a gyro (really force times distance)  or the energy
radiates away or into the object.
We can also
say that a tire will always leave rubber
on the road even if we go perfectly straight because the 2Pi of the
circumference is a transcendental number  and no straight distance
has the value of Pi. You
will find the treatment of transcendentals on the Circle
and Pi page while all incommensurable numbers are detailed on
the golden proportion
page. All is number! 
After a short
search on the Internet, the Ankh symbolis
nowhere to be found as being associated with the squaring of a
circle. But it is. Ancient Egyptians practiced and likely understood
magic as it is embedded in geometry and in natural life forms. (A
feather, for example, is both.) We are going to give the ancient
Egyptians a lot of credit for understanding their environment and for
dealing with complex topics such as the eternal life and the soul.
Many interpretations of the Ankh
center on sex and reproduction, particularly in
connection with keywords such as life, the womb, and the phallus.
Well, the reproductive mechanics are simple enough and it certainly
would not explain why the ancient Egyptian gods run around clutching
the Ankh. Reduced to the basic mechanics of sex, the horizontal bar
on the Ankh could well stand in the way of doing it, too.
The square
(straight, 1D) aspect and the curving (2/3D) aspect also deal with
the intercept of cosmic energies and the Ankh symbol is about the
human's fundamental geometric duality: the circle and the
square  the curving and the straight geometries. The rounded top,
moreover, is not really a representative of a head or a body, but it
is the round rib cage that expands during breathing. The air moving
in and out (through your diagonalpassage nose, no less) is also
undergoing a transformation between linear and circular/spherical
geometries. It is nice to point out that Buddhists do practice and
possibly understand this aspect of the "breath of life."
Would
you believe there is an alphabet based on the squaring of a circle? Yep,
it's that important. If the squarish dot in the center were to stand
for a zerodimensional point then that would make sense. There may be
a bit more to it if you think the squarish dot has its sides in the
golden proportion. 
Mainstream
math guys cannot advance the squareacircle topic. To begin with,
all math guys struggle with the reputation that they do not pay
attention to applications. That is certainly the case. The easiest
example is their treatment of geometric stars. Although they
understand that some stars are constructible exactly and others are
not, they do not know what to make of it, applicationswise. In the
case of the squaring of a circle, many of them claim that the circle
squaring puzzle is artificial, contrived, or useless. But of course,
this is a super example of mainstream science logic: "If I
should care about the applications even though I really don't, then I
am safe if I say there exist no applications." On this page the
squaring of a circle is tied squarely to energy and movement and
atomic stability. But, really, don't expect the college professors to
wake up one day just because you might have suggested the energy
connection. It is within the individual person's ability and
character to be fair and to challenge himself or herself. By not
being able to deal with applications, mainstream science and math is
for the most part irrelevant. By proving to themselves that the
squaring of a circle is not possible, the mainstream science and math
guys have completely foreclosed dealing with atomic construction and
free energy  and that's fine by me. 


"Making
a turn is not straightforward"
Making of a circle is
also about taking a step from one to two dimensions. There, you will
find the friendly transcendental number Pi.
You can also stump
your teacher with easy questions about Pi.
A circle can be
divided into exact and equal segments that
allows you to construct various perfect stars and regular polygons.
Because not all numbers can divide a circle exactly, there is a page
devoted to perfect
stars construction  yes, it is also
about geometry vs. arithmetic.
Squaring of a
circle is about addition or subtraction of energy.
Because this energy is not mechanical (kinetic), we are dealing with
the virtual energy. Free
energy page gets into that while the summary
as well the roadmap
offer additional classification.
If you like alchemy
 and the ancient Egyptian paintings and texts are nothing but 
stop by our alchemy
page for an interpretation of The Emerald Tablet.
Alchemy also works
on Leonardo's
alwaysfamous Vitruvian
man in the context of circlesquare and the mysterious 3 vs. 4 
The
book you will thoroughly enjoy
QUANTUM
PYTHAGOREANS
More
.. 

From ancient Egypt comes a story
of magic. The guy, or shall we say a god, in charge of
magic was then Thoth, the Ibisheaded. He was also into records
keeping because he was there when it was time to weigh your soul and
that is when the truth and how much you did for truth is important. A
point can be made that 'truth' and 'thought' are the two components
of 'Thoth.'
When Osiris Jr.
was fighting to get his kingdom back he fought
Seth and both inflicted heavy damage on each other. Osiris Jr. lost
an eye. His compatriots were able to come up with pieces of it but
could not get all of it. Thoth took what was left of the eye and
restored Osiris Jr. eye through magic. No additional details are available.
A
case can be made that the eye restoration story
is about the making of a circle from a finite number of components
which, moreover, requires some magic. Thoth knows a lot about
geometry and you can tell by the way he holds his arms. The beak of
the bird Ibis has a nice curve to it, too, and so you will not find a
compass sticking out of Thoth's pocket.
You can have a
field day finding geometric constructs on Thoth.
Just a few: Hyperbolic cut on his skirt, diagonals across his chest
and skirt, papyrus scroll as a cylinder. Are the markings on his arm
one Royal cubit apart? Is the folded arm dividing the straight arm in
the golden proportion? Is the folded arm making 36 and 2x72 degrees
of the golden triangle? Why isn't the left hand thumb opposing the
rest of the fingers? Is he making ellipses with his right hand fingers?
Putting some of it together,
consider Archytas using a cylinder
to arrive at the doubling of a cube, which is necessary for the
elliptical orbits (having the variable of radial distance the power
of 3 and the variable of time the power of 2). Yes, Archytas needs
transcendental numbers of the cylinder and that's what this page is
about, too. [Let's do another one: I think the unclosed belt with the
vertical axis of symmetry ("tail") is about the "tree
hugging" posture in Tai Chi (has links to intelligent etheric energies).] 
Why did the Christian religion
drop the ball and just kept the straight cross? After
all, Coptic Christians in Egypt adopted the Ankh as their own symbol
and a circle is also integral to the cross of the early Celtic
Christians. A circle
is quite difficult to understand and might be elusive  and possibly
painful  even to the most ardent students of the Bible. The Ankh is
also considered 'the key' to the eternal life and two keys are on the
Pope's coat of arms. For the rest of the people (read followers),
however, the Beast of Reduction has made his first mark:
"Off with their heads. We will tell them what to think!"
One key would be in the hand of a priest who positions himself in the
gateway to your salvation.
It is quite amazing
how actively the Catholic Church wants their followers to stay away
from a circle. So much so a circle is to the Catholics as a cross is
to the devil. Perhaps a circle is the escapement from the dogma of
the Catholic Church, a dogma resulting from a nice piece of marketing.
It is really not
that difficult to figure it out. Christ was
given the crown of thorns, shown as a circle, and when you get into
the meaning of a circle you will have to get over the thorns. Then
there are the five wounds of Christ  could hurt as you work the
number five.
And so it comes
back from antiquity to the basic two questions:
Can people
govern themselves? and
Is an
individual responsible for his or her own salvation?
If so, is your
salvation a matter of learning or belief  or perhaps learning and belief?

The reproduction
of species is an exceptionally complex affair.
There is quite a functional difference between a womb and a vagina,
for example, and all of the components need to be looked at with
their geometric shape in mind. The function of the Couplex point is
not understood at all but reproductively it is critical. (The Couplex
is our English name for Chinese 'dantien' and Japanese 'hara' that is
coined, explained, and applied in the Quantum Pythagoreans book.) 
Applying
Pythagorean methods to square a circle, the
goal is to take a onedimensional straight line and convert it
unambiguously  that is commensurably  into a multitude of zerodimensional
points because a circle is composed of points. Since a point is
dimensionless and has no length, the conversion is not
commensurable and not possible. In other
words, you cannot take a point and measure distance with it. If you
think this proof is simple, then perhaps that is how it should be.
This proof is good for any curve such as an ellipse. However, this
proof is a general proof. We cannot square a circle in general
 that is, a circle cannot be squared directly as a whole under all
(arbitrary) conditions and under all circumstances, but that does not
mean that a particular circular segment cannot be linearized.
This
is more fundamental than you might think. When
something can be unambiguously mapped back and forth from one state
to another we can use the equal sign when we figure out the
relationship. A complete oneforone mapping between but two
states such as 'potential' and 'kinetic' energy however, is not what
usually happens in nature. We can force the environment to yield an
equation but at the expense of doing it in a closed system. So we
impose a lots of environmental constraints to get a twoway
(equationbased) relation to hold but in the universe there are no
such impositions taking place. Yet, the universe organizes
nonetheless. If you wish to understand how things really happen, you
will need to accept that a twoway backandforth mapping is an
exception. The relations are normally over a multistate (triangular
or greater) processes, particularly if a transformation between the
real and the virtual domains takes place, which means that the equal
sign of algebra will not do. In our case, when we force to equate
(use equal sign) between 'straight' and 'curving' to always hold, we
will do so at the expense of tractability  that is, the process
requires an infinite number of steps and becomes intractable.
If
you come up with a question, "Why would
the infinite number of steps be intractable and bad while the
infinite number of additions (superpositions) would not?," you
are doing well. The "steps" happen in the real domain and
each step takes finite time  and therefore the infinite number of
steps takes infinite time. Superposition is instantaneous in
the virtual domain (and much of QM happens in the virtual domain).
At
times the results of an analysis are easier  and possibly more
effective  to show in an art form. Yet, you'll have to know which
aspects are harmonious and which are disharmonious. If you don't feel
good (at the moment) you may get to like disharmonious designs. This
particular design on left deals with atomic construction and is
harmonious through and through. 
When
squaring a circle, every pebble on the road is a component of the solution
In two
dimensions the exact squaring of the circle through geometry or
arithmetic is not exact and so the squaring of the circle in 2D is
not possible. Even the Pythagorean Theorem does not do it. The
equivalence impossibility between 1D (square, straight line) and 2D
(circle, curve) shows nicely there is hard
separation between straight and curving geometries. In fact, the
impossibility of the squaring of a circle proves that the spatial
'straight' vs. 'curved' definition is not
by convention. Said another way, the straight and the curved are
individually enforceable. But you also know that the separation is
not insurmountable, for atoms with their curving orbitals and the
straightline photons manage to interact in some exact way because
the conservation of energy holds. There then must exist a
transformative agent or agency. (For the ancient Egyptians it could
be god Shu, usually depicted with his arms at the diagonal.)
A spinning
gyro can move in a straight line just fine, but how does the mindless
gyro know the path is curving? A gyro has an angular momentum and an
introduction of another curving  that is angular  momentum by the
curving path is readily differentiated from the linear momentum. So
it might be fun to work this anyway, for the squaring of a circle
could be possible with, say, 3D constructs (pyramid?, angular
momentum is always in 3D) or the transformative agent is an
additional step we will have to go through.
The ancient
Greek geometric constructions always call for a solution with a
finite number of steps. The idea is that solutions that have some
useful applications are those that are tractable. The point is
that you don't want to deal with intractable methods, for they
deliver solutions in some far away time in the future and only
approximately at that.
You also do
not buy the argument that it just might be okay to use intractable
methods, for intractable methods would at some point yield an error
that is very small. You do not succumb to the "small enough 
good enough" invitation, simply because you are thinking and
working with energy. When applying intractable methods the
result could be close all right, but because the energy is conserved
you will not be able to explain what is happening with the energy's
deficit or excess/surplus [and you may want to do better than Thoth,
who chalks it up to magic]. When making a wooden round table you will
finish in finite time and of course you are the practical guy who
gets the job done. But now we are working with energy and that is an
important context in its own right. [Yes, it is in your right brain.]
You also feel that for the paths of a circle or a sphere the energies
might be closely related to the atom because atomic orbitals have a
curved and closed topology.
Here is where
it gets a bit Zen (or metaphysical or alchemical).
The Zen part is that now we got ourselves into quite a mess. We
think we need to use tractable methods, for we want to be done by the
end of the day  but tractable methods are not available.
Intractable methods do not help us with energy  but working with
energy is very useful. The only option open to us is to work with
infinities while finishing in finite time! And for that we will need
all components to be available to us right at the beginning  the
full infinity of them. And then we will need instant action. Well
yes, we are entering the virtual domain of quantum mechanics where
the waves do it instantly. (Okay, Bunky, what this means you'll hafta
use your right brain.)
In the illustration above the
approximate construction of Pi is that which is found in the Great
Pyramid. (The angle alpha is on the golden
proportion page with comments on the insides of the Great
Pyramid.) While the Pi is approximate, you will need to figure out
why the circle could be closed with it (think SQRT(5)).
If you prefer the esoteric road, think "foam of Venus" to
collect the infinities you need. Instant action will happen with QM
superposition and that's doable, too.
When working the Great Pyramid
you may begin to appreciate how the Pi and the proportions of golden
numbers are tied together. You could start thinking it is one or the
other but after a while you just might see their relationship.
Squaring of a circle is not
strictly about the length equivalence of a circle and a square, for
it is about reversible transformations between 1Dstraight and
2D/3Dcurved geometries. So, when you are working this you may also
want to think about the exact division of a circle. You see, a
section of a periphery of a circle (an arch or an arc) and its
corresponding straight segment (cord) start and end at identical
points. A particular curved section and the corresponding straight
distance then also subtend the same angle. Such angle, moreover,
could be an integer multiple of the whole circle. Multiplication or
division by an integer or by a rational number is an operation that
finishes in finite time, and that is the reason why the circle
and the square could differ by a rational number while still
satisfying the squaring of a circle goal.
If the proportion between the
arch and the cord resulted in a transcendental number we could not
use multiplication or division and finish in finite time because we
would be dealing with an infinite mantissa and infinite computing
time. However, if the proportion resulted in an irrational
number, we could use the geometric
mean to multiply out the irrational number in finite time
even though irrationals also have an infinite mantissa [this goes
beyond cool]. Bringing the ancient Egyptian fractions into this
helps, too, for the components of the harmonics series represent
energy components.
As a Pythagorean you want to
think about the operation of multiplication and what physically
arises from multiplication  that is, how multiplication manifests
in nature. Along the way you may encounter some universal constants
as well.
Now, the curving portion and a
straight portion are in some rational or irrational
relationship via the whole circle. Still, the paths (segment
and cord) are not equal and the conservation of energy question comes
up again. It's back to Zen but this time it is not that difficult.
(Keep in mind a photon cannot be cut in two  a minor complication.)
The application here continues to be the atom  its stability,
energy extraction, and possibly gravitation. If you are a math guy,
think Dirichlet. If you learned
something about the hyperstar on this site, you can now also make a
bet that all orbitals (energy clouds) must have a point (odd) or
axial (even) symmetry.
Could we say
that only some circles can be linearized  and then in but a
particular way?
Is this enough?
Atoms do
it every day.
If you want to have more fun,
think about placing two circles (read orbitals) that would have their
straight segments in a mutual rational relationship. Rationing is
very Pythagorean. Yeah, you might need circumpositional
numbers (that divide a circle exactly). Bring along the ancient
Egyptian fractions, and you already know it's about energy. Happy
thinking! Happy workings!

QUANTUM
PYTHAGOREANS
Book by
Mike Ivsin
Pythagoreans
use the knowledge of numbers with straight and curving geometries to
construct stable  that is organized  systems. Atom building is
the first real construction and the means of construction are also
the means of maintenance and healing. Quantum
Pythagoreans book applies the Great
Pyramid geometry  along with the ancient Egyptians fractions  in
the atomic creation context.
Can
we build the technology that deals with the infinite and apply it in
the creation of real things? The Pythagorean way is the only road to reality.
Continue
.. 


Also
consider Euclid's definition of a point as
'that which has no parts.' Another way is through questions. a)
"If you remove a point from a line, will its length
change?"; and b) "If you remove a point from the inside of
a line, will you cut the line?" [Just a couple of Western Koans
from MI. Feb 2007]
Note
that the tasks of squaring the circle's periphery or the
circle's area are equivalent. Proving or
disproving one task is good for both topologies, for either topology is commensurable
(proportional) to the other through a constant that is the radius.
The picture showing the squared periphery looks a bit different than
the one showing the squared areas, though (radius associated with
periphery squaring is a bit larger). This issue is very similar to
the difference between the momentum and the energy of a moving
object. First formulated by Leibniz,
momentum is proportional to v
(velocity) while energy is proportional to v^{2}
(a square number). Momentum and the moving energy of an object are
commensurable with each other through v.
The implication here is that the conservation of energy holds for
the conservation of the object's energy as well as its momentum. (Quantum
Pythagoreans book also talks about gravitational and energy
vectoring and why momentum contains direction  it's a vector 
while energy does not.)
Just
to make it a bit more complicated (but far more interesting),
consider that the circle's circumference is, because of the Pi, a
transcendental number. The straight distance of
the square's sides can be either a rational or an irrational number
because no straight distance has the value of a transcendental
number. If a straight distance were transcendental we could solve for
transcendentals with the Pythagorean Theorem. We cannot take the
circumference of a circle that would result in a square (or a
straight line distance) with a rational side (or a rational length),
because the rational number has a finite mantissa while the
transcendental number has an infinite mantissa. Now the question
becomes: Can a transcendental number equal an irrational number? We
answer this as 'no,' because irrationals issue from the Pythagorean
Theorem (or an algebraic relation from among polynomials of any
degree) while the transcendentals do not. Are we there yet? Almost.
What happens if we ratio
transcendentals and irrationals? This just might be it. That guy
Pythagoras and his ratios are at it again!
Rationing is not only about rational numbers, you know. [Would you
rather be a Mayan, for Mayans had no fractions?!] 
Pythagoreans
established rationing in the framework of musical heavenly spheres
and in the harmony of tones. Using lengths
of various strings, their ratios
(proportion) resulted in particular harmonious and disharmonious
sounds. But what if the length (1D) of a string keeps shrinking and
becomes a point (0D)? Think infinity. Better still, think
nonlocality. Best yet, think electron. It then also appears that
rationing is an ongoing operation in nature and this could also point
to the Pythagoreans' admiration for rationing. (Scientists in general
think that Pythagoreans admired rational numbers but that just comes
from their reductionist thinking.)
Commensurable
and incommensurable numbers take you on a
tour from the finite to the infinite. 
The way of seeing
the squarethecircle issue is in the context of transforming the
linear geometry into curving geometry and vice versa, which
is about the separation and coexistence of the Euclidean
and Riemann
geometries. The separation of the two is in the Fifth Proposition of
Euclid's Elements
where two parallel lines take the straightonly path. Riemann,
however, opens the extra dimension by allowing a curvature. While
Euclid's Fifth works fine for 'parallel' as two lines in a plane
(think light beams), 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).
From the Fifth
Proposition separates yet another, a third, possibility and
that is of a line winding ("snaking") around a straight
line, for both of these lines are equidistant
 just as parallel lines are  but do not meet each other and do
not close on themselves (coaxial or helix topology  that is, open
3D topology). (All three
aspects are incorporated in the Quantum
Pythagoreans
book from the Pythagorean perspective  that is, each aspect has its
representative in nature that meshes with its unique geometry.)
The Fifth Proposition of Euclid,
it is said, was difficult to formulate for Euclid himself. This
proposition defines parallel lines and their properties. Euclid's
difficulties point to his (and others') weak understanding of the
Pythagorean concepts, which explicitly differentiate one dot (point,
0D), two dots (line, 1D), three dots (plane, 2D), and four dots
(volume, 3D) geometries. The increasing number of independent
dimensions provide different contexts
within geometry. On top of that, each of the zero, one, two, and
three dimensional constructs are individually enforceable. If Euclid
were to say his 5th proposition is valid for parallel lines in 2D 
that is, in any plane, he would have been okay [this is what leads me
to suspect that Euclid was a bit removed from the Pythagorean
teaching]. Sometimes you can but oftentimes you cannot increase the
dimensionality of the interacting geometry while holding your
previous proposition valid  and this is the basis of the group
theory that deals with variant and invariant attributes. Variances
are quite easy to see going from 1D to 2D, for example, but many
people expect the 3D environment to be just a bunch of 2Ds stacked up
like pancakes and essentially "the same." But there are
many things around us that are naturally flat, including our solar
system, and this is because tractable computing is easier in 2D. Carl
"The Prince" Gauss
knew there were deficiencies with the Fifth and encouraged Riemann
to work this area. (It is a worthwhile exercise to see if the 4D
environment is in some respects tractable and this is treated in the Quantum
Pythagoreans book.) Another way of seeing
the dominating nature of individually increasing 0, 1, 2, and 3
dimensions is that the numbers rule. In any beginning there is a number!
Even today, challenges to the
Fifth are summed up as 'nonEuclidean.' These
challenges are trying to "unify" geometry by claiming that
'if another [my] geometry holds then Euclid is wrong [and I am
right].' Not so. Simply said, the only diff between 'Euclid's
parallel' and 'nonEuclid parallel' is that Euclid is valid in 2D
while the rest belongs to 3D. It turns out that the addition of the
third dimension allows two new constructs for 'parallel:'
one for the open 3D
topology and one for the closed 3D
topology. In other words, the squaring of a circle is also an issue
for the cubing of a cylinder
and the cubing of a sphere.
In 3D we encounter the problems of 1)
Straightandround topology of a cylinder, which is in open 3D; and 2)
Alwayscurving topology of a sphere, which is in closed 3D.{1 Jan
2007} The spherical, closed 3D topology is successfully addressed by Riemann
while the cylindrical, open 3D topology is qualified but on this site
(also see Touching
in a Point).
There is a way to resolve the
squaring of a sphere as well as the squaring of a cylinder  all in
one swoop. It is closely linked to the interplay between the odd and
the even symmetries.
We
have a collection of designs inspired by the five pointed star. View select
designs  or visit our
store at Zazzle (.com/Mike_Geo) and
see how well you could look in a tee, a hoodie, or a long sleeve
shirt. This particular design deals with atomic orbital paths and
their energy exchange. Design on left is called All My Children while
to the right is The Abyss. 

Some people may run into
difficulties and instead of admitting they do not have all the
answers, they work hard to prove there exist no answers. Henri
Poincaré is the case in point. He
did not defer to geometry and claimed that straight and curving is,
well, subjective. If we were to take Henri seriously, a gyro would
have never been invented and perhaps not even attempted.
Mathematicians and scientists make their proofs from assumptions
and.. ..garbage in..
Okay, Henri's
copout happened a hundred years ago. In 2007, though, a new vaccine
had gone through clinical trials and is presently being marketed.
This vaccine emulates a known cancerpromoting virus (or a cancer
modalitymorphing virus if you are closer to Raymond Rife)
of the cervix  yet the virus emulation is based in the geometry of
the virus. No weakened virus or chemical similarities are present 
it is strictly the geometric form of the virus that supposedly
triggers the human antibody response. If you know something about Raymond
Rife's work, you would know that the geometric representation of
the virus' benign form keeps the virus from morphing. The virus'
geometric likeness is in the form of a virtual pentagon  that is,
the pentagon is hollowed out into a sphere rather than sticking out
as a five sided object. This means the virtual pentagon will attract
waves (energies) in the golden proportion, for a pentagonal real
object (that sticks out) cannot form standing waves.
Here
might be a fertile ground for psychology. We
could use a term for people who cannot figure things out and the way
they compensate for it is by trying to dumb down the audience. The
science writer who is clueless about the ancient Egyptian fractions
then describes ancient Egyptians as simple farmers, for example.
Einstein first dumbed down the audience by trying to show that the absolute
motion is not possible, and then produced his feeble relativity
theories. He denied ether
only to work equations to his liking with miserable and miserly results.
Could
you go on a limb and say that brain
functioning imbalance is due to poorly brainformed geometric factors
rather than the catchall "chemical imbalance?" If our
thought patterns have a geometric representation in the brain then
this would make sense. 
Very common among
scientists is the presumption that there is but one way of doing things.
Actually, Euclidean
and Riemann
topologies exist side by side and both can be individually enforced.
Transcendental numbers are the gatekeepers between the two [and
shedragons would make fine ones at that]. 
Archimedes
was the first to apply a method in the calculation of the
transcendental number Pi, which allows the computation
of Pi to result in a large quantity of decimal places. Pi is the
circumference of any semicircle (halfcircle). Archimedes' method is
intractable because we need an infinite number of steps (moves) to
reach Pi. Methods or formulas by Leibniz and Newton
that calculate Pi are intractable as well, and so is any other method
that obtains Pi through the infinite multiplication/division of
succeeding components. (Infinite addition can be accomplished in
finite time through superposition but you'll need to be in the
virtual domain.) But there is always but .. (think the golden proportion).
The number of accurate decimal
digits of Pi is not arbitrary because we will always
run out of paper when trying to express Pi. Can we say that Pi is in
some aspects inexpressible? Is such inexpressibility limited to real methods? 
While many books
talk about Georg Riemann accomplishments,
they are written mostly by mathematicians. (We've reviewed three books
on Riemann.) Mainstream physicists bring up Riemann's magnum opus
only to make a point about "curved space." So far, nobody
differentiated curved space from curved pathways. So far,
nobody applied Riemann work to atomic closed topology orbitals and to
internal atomic tractability. It is for this reason the 0, 1, 2, and
3D geometries are not firmly anchored as discrete
constructs for spatial behavior, which
issue from the unique spatial tractability associated with each
increasing degree of freedom.
Once you get into circle squaring
you will encounter a plethora of derisive comments written
about circle squarers. While you may go on knowing full well you are
onto something, you will also encounter a new aspect of the human
psyche nicely depicted as The Hanged Man of the Tarot. Some people
get stuck (hung up) and full of frustration strike out at others
around them. But you smile, for you just might find a way  the Tau
 the others in the [this] life of them cannot. (Some Tarot decks
have the Hanged Man suspended on a tree branch looking like a Tau.) 
About general proofs
Proving
something that is not possible to do in general
might be easy but such proof is usually
irrelevant while its negative connotation can be misleading too.

Proving that an angle is not, in
general, divisible by three is pathetic because there
are many angles that are. Because a whole circle (an angle of 360) is
divisible by three exactly, we can now analyze and work the atomic
orbitals with thirds, as well as with other circumpositional
numbers. Scientists' proof is irrelevant when we talk about all
possible standing waves forming integerwavelength orbitals that can
wrap around the nucleus.

Similarly, threebody
gravitational equations do not have a general solution and
that means that three or more bodies in general behave chaotically
(intractably). But we know there is a very specific solution the
likes of our very own solar system that is a nonchaotic multibody
system. All is quiet on the scientific front on this because  while
it is easy to prove that in general three or more mass bodies are not
tractable and are thus chaotic  the proof would also require an
explanation on why our solar system does not subscribe to the
scientists' proof. This topic is referred to as the threebody
problem. On top of that the scientists' explanation is empty
because they just don't know  likely they do not possess the
intelligence  about creating tractable Nbody systems. The
scientist is clueless about why and how the specific tractable
solution prevails over the chaotic state, which is the general and
provable state.

Some
mathematicians can prove that quintic equations have no solution
(polynomial equations with the power of five). However, the golden
numbers a and b do present solutions to pentic and even
higher order equations provided the golden ratio a/b is not
reduced into one number and a/b as well as b/a are
allowed as solutions. (Obtaining the reciprocal of the golden ratio
is easy.)
Finally, a
personal story. This is bit technical but the
idea is that there exists a much engraved and authoritative proof
regarding sorting that claims there exists the absolute minimum of
NLogN operations to complete a sort of N elements. This proof is not
wrong but it makes an assumption. In my (by now patented)
implementation of the concurrent sorting process, the advantage is in
realizing that even in a perfect random string some substrings
(groups of elements) are sorted already  that is, a perfectly
random string does not have its elements always alternating high and
low  and sorted substrings can be captured right in the beginning.
Suddenly the iron clad proof is not a proof, for there exists a
several percent chance of a sort completing earlier than (the
provable minimum of) NLogN while the new lower bound on sort becomes
N. Moreover, everyday sorting deals with updates, which contain a
previously sorted string and a new random string. Topics such as this
are way too specialized and they are not in the Quantum Pythagoreans
book. The idea, though, is that claims of a proof always rest
on assumptions and you should feel free to question their relevance
or applicability to what you want to do.
There are no
proofs without assumptions. The squaring of a circle is not possible
in realtime using real methods. The part 'using real methods' is an
important assumption and I've never seen it stated. There are plenty
of infinite series (from Newton, Leibniz, Euler ..) that converge
toward Pi. Now, the kicker is that when the components are waves 
that is, wavefunctions, the superposition (addition) is instantaneous
and even an infinity of additions can happen in finite time.
Assumptions issue
from a context, which in turn is created in your right brain from an
infinite number of variables. Context may be narrow or wide depending
on your ability to construct larger or smaller timeinvariant
entities. Think selforganization and yes, the book gets into that. 
