Rank-3 Projections of a 4-Cube

TL;DR

This paper investigates the geometric properties of the projection of a 4-dimensional cube onto a random 3-dimensional subspace, analyzing volume, surface area, and mean width, and introduces new mathematical constants and functions.

Contribution

It provides a detailed study of the convex polyhedron resulting from projecting a 4-cube onto a random 3D subspace, including new constants and advanced mathematical functions.

Findings

Derived explicit formulas for volume, surface area, and mean width distributions.

Identified a new mathematical constant approximately 7.11856.

Connected geometric measures with hypergeometric functions, elliptic integrals, and Catalan's constant.

Abstract

The orthogonal projection of a 4-cube onto a uniform random 3-subspace in R^4 is a convex 3-polyhedron P with 14 vertices almost surely. Three numerical characteristics of P -- volume, surface area and mean width -- are studied. These quantities, along with the Euler characteristic, form a basis of the space of all additive continuous measures that are invariant under rigid motions in R^3. While computing statistics of {vl, ar, mw}, we encounter the generalized hypergeometric function, elliptic integrals and Catalan's constant. A new constant 7.1185587167... also arises and deserves further attention.