cube.md

Pólya noted the solution for Problem 8 (Axes of a Cube) in How to Solve It:

... the "average width" of the cube, which is, in fact, 3/2 = 1.5

I guessed two definitions of the "average width":

Diameter equivalent to an isovolumic sphere

According to the volume of cube and sphere:

the solution is:

Integral over all directions

Here we use spherical coordinates:

Then the average radius can be defined as:

Here let's choose a cube with edge length 2, centered at the origin, then calculate the average radius (half of width) of one of the symmetric 48 parts of the cube:

Then this part of the cube is a tetrahedron with 4 planes:

• OAB: y = 0, or ϕ = 0
• OAC: x = y, or ϕ = π/4
• OBC: x = z, or cosϕ sinθ = cosθ
• ABC: z = 1, or r cosθ = 1

So we have:

Because of the symmetry, we can just integrate the average radius of the tetrahedron:

Let's calculate the integral:

(0 ≤ ϕ ≤ π/4)

Although the antiderivative of this function doesn't look analytic, we can get the numeric solution: ^[1]

This is close to the isovolumic-sphere definition but far different from Pólya's note.

Note

1. Here is the calculation:

from numpy import *
from scipy.integrate import dblquad

result, abserr = dblquad(lambda t, f : tan(t), 0, pi/4, 0, lambda f : pi/2 - arctan(cos(f)))
print(12/pi*result)