For the Pi Day

This post may be a bit technical for general audience (as if anybody is reading this!). Although, if you do and happen to have any popular interest in Pi, the mathematical constant π=3.141592…, since it’s the day of Pi, consider to scan it through and get to the end point, if I manage to make it!

From 20 to 18 years ago I attempted to build an algebraic axiomatic system to reformulate geometry. The goal was to generalize the theorems of Euclidean geometry to a version independent from the number of its dimensions.

I didn’t know how big the world is and that my work must be redundant. So I put the effort and called it “geometry beyond dimensions”. Soon after renamed to “multidimensional Euclidean geometry” (word by word translation from Persian).

My ambitions were beyond speculating about the “flatlanders” and generalizing their problem: Oh, poor flatlanders don’t know about us three dimensional beings, so we too must learn about four dimensions and higher.

No, the point was that there is a lot more to linking geometry and algebra. Still an unacomplished mission.

Anyhow, learning two and three dimensional geometry was mandatory at school and I extended it from N=2, 3 to any number. It was a mechanical and labor intensive work using the principles of induction and a minimal set of “bridging” axioms on top of the existing literature, our school books.

Not only the concept was beyond my intuitive perception, the formulation could also get weird quickly, but it was possible after all to get familiar and use tricks to grasp the concepts and proceed.

To see how it looked like before it escalates, here is an example axiom (a bulding block for more complicated structures and proofs that came later in the book):

There exists exactly one N-dimensional space passing through any N+1 points not lying on the same straight N-1-dimensional space.

A bit weird, huh? But you could put N=1 to get the following axiom in planar geometry, more intuitive:

There exists exactly one line passing through any two distinct points.

Or N=2:

There exists exactly one plane passing through any three points not lying on the same line.

It took some 80 theorems till it covered a satisfactory area and I wrapped it up. And although I was quite obsesssed with its mechanical accuracy, I remember it still had few holes and gaps.

Now let’s get closer to the Pi:

One of the wheels I reinvented in that work was calculating the volume of n-ball, or a multidimensional hypersphere. Of course I didn’t just write an integral to solve it; I proved dozens of theorems to justify that my integral is legit and comes only from the few axioms that were introduced at the start of the book, and assumes no more.

The final result was mysterious in terms of its connection with the Gamma function and Pi. And this is where it can take us beyond a dimension-agnostic theory of geometry: discovering the nature of Pi!

Now, I refer to the pages 45 to 56 in my book (Sorry it’s all in Persian!) But I will make a simpler point here. Let’s try to formulate:

0. Consider the volume of a 0-sphere: How many dots are in a dot? 1 (or 1.R0)

1. And the volume of a 1-sphere with radius R: What is the length of a line segment with radius R: 2 R1

2. The volume of a 2-sphere: What is the surface of a circle with radius R: R2

3. How about the volume of a 3-sphere? 4/3π R3

4. And it turns out that the volume of a 4 dimensional sphere (all the points on a 4D space that are as far from one point in a 4D space) is: π2/2 R4

N. In general the volume of N-ball, an N-dimensional hypersphere with radius R turns out to be: πN/2 / Γ(N/2+1) RN

You can find the full proof in the book in Persian (pages 45 to 56), and perhaps somewhere on the net in English. Now, ignoring the trivial part of the formula (RN) we end up with a magical co-efficient as a function of N:

πN/2 / Γ(N/2+1)

Where Γ  is the Gamma function. Now the value of this function for its integer arguments is straight ahead. It ends up equal to the famous factorial function, multiplication of all integers from 1 to that number [minus one]:

Γ(n) = (n-1)! = 1*2*3*…*(n-1)
Γ(n+1) = n* Γ(n) (n) = 1*2*3*…*n

For non-integers though it will take on funny values to interpolate the factorial results between two integers. For example for the half values right in the middle of two integers, it ends up a rational number (a number that can be written in a form of an integer devided by another one) multiplied by an irrational number which is Γ(½) and happens to be the square root of π, that is not only irrational but transendental:

Γ(n+½) = (n+½)*(n-½)*…*Γ(½)

Now the strange part is that the argument of the Gamma function in our formula is N/2+1. It gets one unit higher for every second added dimension! And that for odd dimensions it will not be an integer or a rational and will include the term Γ(½)=√π.

On the other hand the gamma function in our formula is multiplied by another term of πN/2 which also introduces a √π for every added dimension. Thus, for even number of dimensions none of the terms πN/2 and Γ(N/2+1) introduce a √π and we end up with a rational number multiplied by πN/2 where N/2 is an integer. For odd numbers both of these terms introduce a √π that divides and vanishes. So, there will not be a √π in any of the integer dimensions, even or odd.

It is not a √π introduced to the formula for every added dimension, instead is it an extra π coming to multiply, for every even number of dimensions. Odd dimensions (extending from a point to a line, or from a circle to sphere) do not introduce a new π to the co-efficient, only a rational number. The even numbers (going from a line to a circle) bring in a π to the play! A strange asymmetry between the odd and even dimensions, I would say.

Ignoring the rational part of our magical co-efficient, for every second added dimension there will be just one π introduced and the co-efficient for dimensions from 0, 1, 2, 3, … will be as the following:

0 -> 1
1 -> 2
2 -> 2π
3 -> 4/3.π
4 -> 1/2.π2
5 -> 8/15.π2
6 -> 1/6.π3
7 -> 16/105.π3
8 -> 1/24.π4
9 -> 32/945.π4

Where does π come from? One intuitive way is that it comes from the comparison of the space a hypersphere takes to that of a hypercube. But one π for every second dimension. Why every second? Well, this happens in Euclidean geometry where distances are Euclidean and the ball is defined as a set of points equally far from a center, using a “two” norm distance metric. You take another distance measure and the math will change. But I would argue that Euclidean distance is the only legit metric at least when it comes to defining a ball, as it is the only metric that maintains the shape of the ball when we rotate the axes. So the key is that when you go beyond one dimension something called “shortcut” comes to existance. And there’s a straight shortcut that for some reason follows the Pithagorean theorem and that defines the perfect curvature. I couldn’t reveal how these are connected, but if I ever want to speculate about the nature of π, here would be my starting point.

p.s. I read a bit more on the topic. I opened that back door in my head and it was two decades of silence and spiders ran off quickly. My friend Sajad gave me a torch, albeit a map: Quite surprisingly the Pi day coincided this news on some weird statistical behavior of the Prime numbers. I realized that I was brought up in a typical middle class (and 3-dimensional!) family. Dimension-deprievation is the evolutionary intution of 0, 1, 2, 3 only. That is too few to realize that all dimensions do not have to be symmetric because they are all numbers. The number of dimensions, even or add, prime or divisible, affects how N-space behaves and just like number theory it doesn’t have to inherit it all from N-1-space. Do all numbers exhibit the same properties cause they are all numbers? so why should they when they count dimensions. I think this is actually what numbers are made for: counting dimensions. And the historic fact that we count 1, 2, 3 and we “…” the rest is not pure coincidence. Sounds poetic, but read it logically:

3 doesn’t get every property of 2, neither does a ball from a circle. To my previous wonder, a ball (3-sphere) did not inherit an additional π from a circle in the calculation of the volume, but 4-sphere did. Is it weird? No, 3-sphere introduces singularity too, two poles in the hairly ball theorem, that are the two ends of a segment (1-sphere), but 4-sphere doesn’t: A circle (2-Sphere) can go round on another full circle around a point and you get a 3-torus or a 4-Sphere that you can comb (no singularity) and they both happens to have π2 in the volume and surface formula. Now you try to rotate the circle, not like you just did on both dimensions of a full circle and around one point, but instead around a segment on its own disk space. And you get a ball (3-sphere) with two inevitable South and North poles (singularity) and this time it does not give you that extra π. So, 3-sphere is just a product of a circle and a line segment (thus singularity, thus no extra π). The product of two circles (3-torus or 4-sphere) gets that extra π and you can also comb it (no singularity)!
This is a short summary of the common stories that two formal proves tell. The same thing happens in both: The multiplication of a new π in the volume of n-sphere on every second dimension in my [redundant] proof (A), and the generalized hairy ball theorem for 2n-spaces (B).

Is there an established field on the intersection of algebraic topology and number theory?

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