Extremum Problems for the Motions of a Billiard Ball. IV. A Higher-Dimensional Analogue of Kepler's Stella Octangula.
Abstract
Let us consider a square billiard table gamma 2 = ABCD, and let A'B'C'D' be a concentric square half the size of ABCD. Let alpha, beta, gamma, delta, be the midpoints of the sides of gamma 2. We observe that the path of a billiard ball moving along the sides of the square alpha beta gamma delta does not penetrate inside the square A'B'C'D'. However, it can be shown that the path of any other billiard ball must penetrate into the square A'B'C'D'. By 'any other' we mean (1) That the path is not parallel to any side of gamma 2. (2) That the path is different from alpha beta gamma delta. In the present paper this curious property of plane billiard ball motions is extended to a certain class of skew polytopes in the n-dimensional space R superscript n. These polytopes reduce to plane billiard ball motions if n = 2. If n = 3 the above property of the square alpha beta gamma delta is taken over by Kepler's Stella Octangula. This is an 8-pointed star which is explained in the paper.
Document Details
- Document Type
- Technical Report
- Publication Date
- May 01, 1980
- Accession Number
- ADA086550
Entities
People
- Isaac Jacob Schoenberg
Organizations
- University of Wisconsin–Madison