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Transcendental Spheres

Posted 2 years ago
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One of the most beautiful equations in the whole of mathematics is the identity (and its derivation): enter image description here

I recently came across another beautiful mathematical concept that likewise relates the two transcendental numbers e and Pi.

We begin by reviewing the concept of a unit sphere, which in 3-dimensional space is the region of points described by the equation: enter image description here

We can some generate random coordinates in Mathematica that satisfy the equation, to produce the expected result: enter image description here

The equation above represents a 3-D unit sphere using the standard Euclidean Norm. It can be generalized to produce a similar formula for an n-dimensional hyper-sphere:

enter image description here

Another way to generalize the concept is by extending the Euclidean distance measure with what are referred to as p-Norms, or L-p spaces:

enter image description here

The shape of a unit sphere in L-p space can take many different forms, including some that have “corners”. Here are some examples of 2-dimensional spheres for values of p varying in the range { 0.25, 4}: enter image description here

which can also be explored in the complex plane:

enter image description here

Reverting to the regular Euclidean metric, let’s focus on the n-dimensional unit hypersphere, whose volume is given by:

enter image description here

To see this, note that the volume of the unit sphere in 2-D space is just the surface area of a unit circle, which has area V(2) = ?. Furthermore:

enter image description here

This is the equation for the volume of the unit hypersphere in n dimensions. Hence we have the following recurrence relationship: enter image description here

This recursion allows us to prove the equation for the volume of the unit hypersphere, by induction.

The function V(n) take a maximal value of 5.26 for n = 5 dimensions, thereafter declining rapidly towards zero: enter image description here

enter image description here

In the limit, the volume of the n-dimensional unit hypersphere tends to zero:

enter image description here

Now, consider the sum of the volumes of unit hypersphere in even dimensions, i.e. for n = 0, 2, 4, 6,…. For example, the first few terms of the sum are:

enter image description here

These are the initial terms of a well-known McClaurin expansion, which in the limit produces the following remarkable result:

enter image description here

In other words, the infinite sum of the volumes of n-dimensional unit hyperspheres evaluates to a power relationship between the two most famous transcendental numbers. The result, known as Gelfond’s constant, is itself a transcendental number:

enter image description here

3 Replies

Dear Jonathan, thank you so much for sharing! Could you please attach the notebook with the code?

Two comments and a question arising from the visualizations of the unit p-sphere in the plane....

Comment 1: Curiously, the built-in function Norm in its two argument form Norm[vec, p] prohibits values of p that are less than 1. Of course one may define a generalization:

 norm[p_][x_List] := Total[Abs[x]^p]^(1/p)

Comment 2: It would be desirable to include all code!

Question: What exactly are you doing in the form of the visualization that uses complex numbers?

Hello Murray,

I was just using a MMA notebook as a "scratchpad" and didn't save it. I think most of the code is in the post though (at least that was my intention). As for the complex number graphic... I don't recall. I will experiment and see if I can recreate it.


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