Greg Egan on Nostr: Every parallelepiped that you place around an ellipsoid whose faces are tangent to ...

Every parallelepiped that you place around an ellipsoid whose faces are tangent to the ellipsoid at their centres has the same volume for a given ellipsoid: 8 a b c, where a, b and c are the semi-axes of the ellipsoid.

It’s not hard to prove that this generalises to n dimensions.

Suppose we have an n-ellipsoid given by the equation:

∑ᵢ₌₁ⁿ𝑥ᵢ²/𝑎ᵢ² = 1

where the 𝑥ᵢ are coordinates (x,y,z,...) and the 𝑎ᵢ are the semi-axes in each direction.

A vector normal to the surface can be found from the gradient of the left-hand side of this equation; since that function is constant on the surface, the gradient is orthogonal to the surface.

norm(𝑥) = (𝑥ᵢ/𝑎ᵢ²)

Suppose we choose n points on the surface of the ellipsoid, 𝑥ˢ, s=1,...n, such that:

𝑥ᵗ·norm(𝑥ˢ) = 0 when t≠s

We will also have:

𝑥ˢ·norm(𝑥ˢ) = ∑ᵢ₌₁ⁿ𝑥ˢᵢ²/𝑎ᵢ² = 1

If we define the matrix 𝑋 to have the vectors 𝑥ˢ as its rows, and the matrix 𝐴 to have diagonal elements 1/𝑎ᵢ² and zeroes elsewhere, then the dot products above correspond to:

𝑋 𝐴 𝑋ᵀ = 𝐼

If we take the determinants of both sides of this equation, we have:

det(𝑋)² det(𝐴) = 1

So det(𝑋) will always be the same, the product of all the semi-axes, and the volume of the parallelotope that encloses the ellipsoid will be 2ⁿ det(𝑋), with each of its faces formed by taking ±𝑥ˢ as the centre and adding ±𝑥ᵗ, for all the choices of t≠s. Since those 𝑥ᵗ will be orthogonal to norm(𝑥ˢ), each face will be tangent to the ellipsoid at its centre.