The Pinnacle of Mathematical Beauty — Volume in Higher Dimensions
For this article, I will try my best to convince you that mathematics possesses not only unbelievable elegance but also supreme beauty. And in order to achieve this, I’m going to take you to the world of higher dimensions. I know, I know, higher dimensions are difficult to perceive. Bear with me, though. I will try my best to make it as intuitive as possible.
What about higher dimensions are we going to consider? Well, imagine you have a ball. That ball clearly has 3 dimensions. If we were to project it onto 2 dimensions, we’d have a circle. And projecting that onto 1 dimension would give us a line segment. But what if, instead of going to lower dimensions, we go further up? What happens to the volume of a ball with 4 dimensions? What about 1,000,000 dimensions? What about infinite dimensions?
This article will answer these questions by presenting an exploration of the volume of the unit n-ball (a ball with unit radius in the n-th dimension). A “ball”, mathematically, is any region in space that comprises of all points a fixed distance from a given point and its volume is simiply the volume of its interior. Now, our consideration involves finding the volume of each n-dimensional ball and analyzing its behavior.
Deriving a formula for volume
In order to represent the volume in a way that’s more concrete, we turn to the spherical coordinate system. It is well-known that the volume of a 3-dimensional sphere in spherical coordinates is given by the equation:
If we adopt the reasoning involved in constructing the above triple integral, we can extrapolate this definition to higher dimensions. For a more rigorous consideration, I direct the interested reader here.
Following this line of reasoning results in the following multiple integral for the volume of an n-ball of radius R:
Intuitively, this represents the volume because this multiple integral is integrating over all the possible angles and radii. In a certain sense, it “covers” the entire volume of the hypersphere — almost in the exact same way as you would integrate a circle or a sphere. Also note that since our work only deals with the unit ball, we can later substitute for R=1.
Now, let’s begin chipping away at this nested integral. Sure it’s daunting but it’s nothing a little bit of systematic thinking cannot take care of. First, we notice that each factor in the integrand is only of one variable. So, the nested integral can be represented as a product of all the sub-integrals:
We then notice that the sine is symmetric about π/2. With deliberate foresight, we can now change the bounds of integration to [0, π/2] and introduce a factor outside:
Now we see the foresight at play. We’re going to make use of a very clever theorem (a proof of which can be found here):
Then, very conveniently, we see that our volume can be represented as a product of Euler’s Beta functions:
Now, we can use the following Beta-Gamma relationship to make things substantially easier:
Substituting this relationship into the equation for the volume yields a beautiful telescoping product:
Upon some elegant cancellations, our work becomes very straightforward:
Evaluating Γ(1/2) is well-known and computes to the square root of π. Combining this with the fact that nΓ(n) = Γ(n+1) yields the incredibly simple:
With this, we have arrived at a beautifully elegant formula for the volume of the unit n-ball. Some important points to note are the existence of the Gamma function and the square root of π. The way all of mathematics ties together is beyond explanation. It is truly something special. This perpetual state of inter-connectedness is something one can only find in mathematics and is the cornerstone of mathematical elegance. At least to me.
Analyzing the V
Now that we’ve derived the formula, we can begin analyzing it. The most obvious way to proceed is to graph the formula. Immediately, we see something very interesting:
The volume of the unit n-ball seems to peak in the 5th dimension and asymptotically approaches 0 very quickly after that. Upon some more analysis, one sees that in the limit, as n → ∞, volume → 0. This is incredibly counter-intuitive but truly something magnificent.
To prove the asymptotics, one can employ Stirling’s approximation for the Gamma function in the limit and upon some elementary manipulations, the result becomes highly apparent.
Well, with this, our consideration of higher dimensions reaches a conclusion. I hope I have managed to convince you that mathematics just exudes elegance. Cold and austere beauty. Comparable to that of art and poetry.
Thank you for reading and I hope you have a great day!
