Name: Math Club: Prof. Andrew Knightly
Start: 2025-03-10T16:00:00-04:00
End: 2025-03-10T17:00:00-04:00
Location: 204 Neville

Math Club: Prof. Andrew Knightly

March 10 @ 4:00 pm - 5:00 pm

Title: High Dimensional Spheres
Location: 204 Neville Hall

In the spirit of pi day week, I will describe some counterintuitive features of spheres in higher dimensions. For example, if $C_n$ denotes the volume of the unit sphere in $\mathbb{R}^n$ , i.e., the sphere of radius 1, then:

$C_1 = 2$

$C_2=\pi\approx$ $3.14159265358979323846264338327950288419716939937510$

$C_3=\frac 43 \pi \approx 4.18879\ldots$

$C_4 = \frac{\pi^2}2 \approx 4.93480\ldots$

$C_5 = \frac{8\pi^2}{15}\approx 5.26379\ldots$

These seem to be growing with n, but this is misleading. In fact the unit 5-sphere has the largest volume among all of the unit spheres, and further, when $n$ is very large, the unit sphere has volume close to $0$ :