Width Distributions for Convex Regular Polyhedra
Metric Geometry
2016-03-15 v2 Probability
Abstract
The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the regular tetrahedron and the cube. Higher-order moments of f_tetra and f_cube have not been examined until now. Assume that each polyhedron has edges of unit length. We deduce that the mean square width of the regular tetrahedron is 1/3+(3+sqrt(3))/(3*pi) and the mean square width of the cube is 1+4/pi.
Cite
@article{arxiv.1110.0671,
title = {Width Distributions for Convex Regular Polyhedra},
author = {Steven R. Finch},
journal= {arXiv preprint arXiv:1110.0671},
year = {2016}
}
Comments
9 pages