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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

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?

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

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.

Jonathan

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