The Mystery of π Solved



pi \pi\ n. pl pis \'piz\ [MGr, fr. Gk pei, of Sem origin; akin to Heb pe pe]  1 : the 16th letter of the Greek alphabet   2 a : the symbol π denoting the ratio of the circumference of a circle to its diameter   b : the ratio itself : a transcendental number having a value to fifteen decimal places of 3.14159 26535 89793



A transcendental number is one that is not the solution of a polynomial equation with rational coefficients.  It is a fact of mathematics that every transcendental number is also irrational since every rational number is the solution of a linear polynomial equation.  The converse is not true.  As an example, √2 is a solution of x2 − 2 = 0 and hence is not transcendental, but √2 is irrational.

For centuries people have attempted to prove that π is rational.  However, it is in fact not rational but is irrational.  Anyone who develops a proof otherwise only demonstrates that it is the person that is irrational.

For centuries people have attempted to square the circle; that is, prove that using a compass and unmarked straightedge a square with an area equal to a given circle can be constructed.  Since the operations with a compass and unmarked straightedge are operations that are equivalent to polynomial functions, squaring the circle requires that π be a solution to a polynomial equation.  However, it has been proven that π is a transcendental number and cannot so be constructed.  Anyone who develops a proof otherwise only demonstrates that it is the person that transcends reality.

The cosmic mystery of π is more subtle than the average psycho ceramic can perceive; yet the answer is contained in the definition of π for anyone to discern!  All other explanations are superfluous at best.


Pi and pie

If we append an e to pi we get pie which is still pronounced the same as pi.  Is this a coincidence?  Definitely not!

Pie is one of the tastiest constructs that can produced in a kitchen.  We get all sorts of fantastic edibles:

  • meat pies
  • potpies
  • fruit pies
  • cream pies
  • pizza pies
If we count other treats as filling the crusty definition of a pie, we can include:
  • cobblers
  • turn overs
  • pasty
  • beef Wellington
  • tarts
  • pop tarts
  • cheesecake
There are several fundamental pie truths.  The first is the mathematical relationship:  π  =  180°.  And 180° C is 356° F, a most perfect temperature to bake many kinds of pie!

If we compute the volume of a right circular cylinder of mozzarella cheese with radius z and height a, we obtain πz2a.  Or, rewriting the equivalent:  pi z z a!  A coincidence?  I think not!

An approximation to π is 3.14.  Even this crude approximation has fundamental tasty truth.  If we reflect it about a vertical line (π/2), we note:

All of these tasty treats involve enclosing (at least partially) something in dough and baking (endothermic chemical reaction).  Is it a coincidence that the term dough is used as a term for money?  Of course, not!  This is a basic truth of the universe.

We also have the fundamental note of our musical scale do.  This is pronounced the same as "dough."  Is this a coincidence?  Absolutely not.  The "music of the spheres" must be based upon π at a very harmonious level since π appears in many of the formulae for spheres!

Now if we take the three letters of pie and substitute π for the p, we get: π i e.  Reversing these letters we get: eiπ.  This last can be rewritten as (obviously using higher mathematics):

eiπ

and this is a familiar and famous (and beautiful and simple yet complex) relationship in mathematics:

eiπ + 1 = 0

which combines so many fundamental constants in mathematics in one equation built upon pie!

As long as humans enjoy pie they will enjoy π.   To consume pie is sublime.  To contemplate π is divine!


Pi and the Universe

The familiar value of π that we are considering comes from Euclidean geometry, first described by Euclid (ca 300 BC).  This is also know as flat geometry with straight lines, planes, and spaces.  Mathematicians also consider positively and negatively curved geometries where the value that π (as defined as the ratio of the circumference of a circle to its diameter) is different from the familiar value.

Physicists have discovered that these non-Euclidean geometries are very useful in describing the space around massive objects.  Gravity is most easily explained by a warping of the four-dimensional space-time continuum from a flat Euclidean geometry to a curved geometry.  It is the geodesic motion of an object in the curved space-time surrounding a massive object that causes it to orbit and fall.

The basic question of the universe is whether it is negatively curved, flat, or positively curved.  Measurements of the parameters of our universe suggest strongly that it is actually flat; that is, π indeed takes on its familiar value.  It this a coincidence?  Definitely not!  This π in the sky value is inherent in the very fabric of space-time of the universe.  It suggests a rational mind to circle the universe with an irrational value—one whose decimal expansion will encompass the cosmos by reaching all the way around the universe without repeating!  No rational value can do this.  Every rational value will come up short or will generate a boringly predictable repetition.


Pi and Squares

One of the most remembered formulas from geometry is:

π r2

This is the formula for the area of a circle.

Reading this, we get "pi are square."  Now, we will not quibble with the obvious grammar difficulty—it should be "pi is square," but nevermind.  And forget about the corny joke involving "pi are round".  (Indeed, this pun involves baking a sort of dough as a type of bread—both bread and dough terms are used for currency!)

The fundamental fact that is the important gem of truth is contained in the definition above:  π is the 16th letter of the Greek alphabet.  Sixteen, of course, is simply 42; that is, 16 is a perfect square!!

p is also the 16th letter of the English alphabet!  Not coincidently, i is the 9th letter of the English alphabet—another perfect square!

Thus the English pi is a double square and further the sum, 16 + 9 = 25, another perfect square!

Even though we cannot square the circle, with the multiple connections that pi has with perfect squares; we see that indeed "π are square!"

But wait, there is more!  The square roots of the numbers above, 4, 3, and 5, form a right triangle with sides 3 and 4 and a hypotenuse of 5.  This is one of the Pythagorean Triangles whose sides are all integers.  These right triangles obey the familiar:

a2 + b2 = c2

The angle opposite the hypotenuse is a right angle—the same angle that is found four-fold in a square!

Thus, it is in π that the two perfect shapes of a circle and a square come together tied together with a Pythagorean Triangle— a right triangle and not a wrong triangle!

This is not simply a fantastic coincidence.  The equation of a circle in analytic geometry is bound together by this very same Pythgorean relationship:

(x−h)2 + (y−k)2 = r2

where (h,k) is the center and the radius is r.

Of course, the distance around this circle is simply 2πr.

By the way, the letter e is the 5th letter of the English alphabet; that is, √25.  Is it any wonder that the combination pie is such a wonderous entity?


Calculating π

The number 16, which is 24, is a useful and important value in computers which explains why there is such a fascination among computer programmers, as well as mathematicians and bakers, in computing a decimal value for π.

A computer word of half this size consisting of 8 bits can contain exactly 256 different binary values.  This is termed a byte.  (A 16-bit quantity is a double byte or a mouthful.)  Half of a byte, or a 4-bit quantity can contain 16 different values—sufficient to represent all the ten decimal digits one at a time.  This is called a nibble.  Thus a decimal expansion, or slice, of π with 1,000 digits can be enjoyed as 1,000 nibbles, 500 bytes, or 250 mouthfuls!

Is it a coincidence that we describe eating pie with the same terminology?  Of course not!!  Once again we see how π is baked into the very nature of a computer!

As an aside, half a nibble is a 2-bit quantity.  This is, of course, a quarter—a metallic disk (in the shape of a circle) that is an integral part of "dough."  And do not forget, a quarter of a turn is found at each corner of a square as well

A bit diabolical is the fact that 16 is the base of the hexadecimal notation for expressing values in computer programming.  It is no wonder that this value of π has placed such a hex on programmers around the world?

Proof of this dark side of π is that the sum of the first 144 digits to the right of the decimal point of π is 666!

However, in spite of this, good can come from an obsession with π.  Setting a computer calculating a large number of digits of π can be a good test of the computer, and keeping a computer busy is at the heart of goodness since "idle computers are the devil's workshop."  In reality, computers are happier when doing something.  Letting one munch on π is the humane thing to do.

Also, the calculation of π does not involve a radical—we avoid even the appearance of the "root of any evil!"

And, again on the good side, requesting a malevolent entity residing in the ship's computer to compute the last digit of π permitted the crew to regain control of the Enterprise.  (Wolf in the Fold)

It is also built into our human psyche to use π as a measure of attractiveness—how many times have you heard or used the phrase:  "She's as cute as π!"


Conclusion

Since π is a non-repeating, non-terminating decimal value (because it is irrational) we will never be able exhaust all the quintessential truth contained within.  For millennia mathematicians and non-mathematicians, rational and irrational humans, men and women have attempted to discern the truth of the mystery of π.  It is only now that the cosmic significance can be revealed once and for all for the edification of sentient beings on this planet. All of the facts and proofs outlined above are irrefutable.  There are no coincidences!  Everything is significant.  Indeed, π has now been computed to more significant digits than any other irrational value.

It is within the historical collective consciousness that ties together all humans past, present, and future that the symbol π was chosen to represent this important value.  On the surface, there is no reason that ρ (standing for ratio) could have been chosen centuries ago; however, this would have led us all down the wrong stream of unconsciousness away from the truth that is.  We would have been up the proverbial "creek with the wrong paddle" to try to row or float aimlessly along.  It might have been a "merrily" experience, but it would been in blissful ignorance with nothing firm to stand on.  Indeed, we might have found ourselves neck deep in hot water.  How long would it have taken for humans to get back to the correct path?  It was the prescience of the entire human conscious experience that without any question the cosmically correct letter π was the chosen symbol.

And finally we come to the pièce de résistance!  (Note, how this term begins with pi as well as an e.)  There is a certain entity found on cattle ranches and dairies that is referred to as a "pie."  When the male bovine is the source, the specific name applied is also a synonym for the truth value of a statement.  Indeed, the designation of the highest degree of learning is taken from the initial letters of "Piled High and Deep."

Is this another coincidence?  By now, we all know the answer is a resounding "No!"  It is all part of the cosmic intertwining of the most fascinating number in the universe.  I am ecstatic—I am overjoyed—and I am humbled to be able to apply this last "pi" term to the present exposition!


© 2004, 2008, 2011, and 2012 by L. Turner — may be reproduced with suitable attribution by all citizens of the Republics of Texas and California without charge