Width Distributions for Convex Regular Polyhedra

Details Width Distributions for Convex Regular Polyhedra Width Distributions for Convex Regular Polyhedra

2011-10-04

arxiv.org/abs/1110.0671v1

Mathematics

Metric Geometry

9 pages

Scientific paper

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.

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