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11 - Four-Dimensional Polyhedra
from III - Polytopes of Nearly Full Rank
Peter McMullen, University College London
Book:

Geometric Regular Polytopes

Published online:

30 January 2020

Print publication:

20 February 2020, pp 383-430
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Summary

As in the case of the 3-dimensional regular apeirohedra described in the previous chapter, the mirror vector plays an important role in the classification of the 4-dimensional regular polyhedra. Thus the first task is to determine the possible mirror vectors of such polyhedra. The polyhedra with mirror vector (3,2,3) and their relatives under standard operations such as Petriality form a specially rich family. One particular family of these polyhedra is treated in detail, with a description of their realization domains. With the mirror vector (2,3,2), most of the standard operations lead to polyhedra in the same class. Though there is a close analogy between the infinite and finite cases, those with mirror vector (2,2,2) have symmetry groups that need not be related to reflexion groups; the treatment here employs quaternions. There are various connexions among these regular polyhedra, the most interesting being the way that the skewing operation takes certain polyhedra in class (3,2,3) into polyhedra of class (2,2,2).

7 - Physical causes: Kepler’s new astronomy
Todd Timberlake, Berry College, Georgia, Paul Wallace
Book:

Finding our Place in the Solar System

Published online:

18 March 2019

Print publication:

28 March 2019, pp 175-210
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Summary

Johannes Kepler was working as a mathematics teacher in Austria when he had a vision of how the universe must be constructed. Using the Copernican system as his model, Kepler thought that between each planetary orbital sphere was nested a regular polyhedron. There are only five regular polyhedra, so there could be only six planets. The relative sizes of the planetary orbits were set by the shapes that lay between them. Kepler’s idea caught the attention of Tycho Brahe and eventually he became Tycho’s assistant. When Tycho died, Kepler inherited Tycho’s accurate planetary data and he used these data to propose a new theory of planetary motion. Kepler found that the planets move in elliptical orbits with the Sun at one focus of the ellipse. Furthermore, Kepler believed the motion of the planets was powered by a force from the Sun that caused the planets to speed up when closer to the Sun and slow down when farther away. Kepler also discovered a curious mathematical relationship between the orbital periods of the planets and the size of the planetary orbits.

Kepler-Poinsot-Type Realizations of Regular Maps of Klein, Fricke, Gordan and Sherk
E. Schulte, J. M. Wills
Journal:

Canadian Mathematical Bulletin / Volume 30 / Issue 2 / 01 June 1987

Published online by Cambridge University Press:

20 November 2018, pp. 155-164

Print publication:

01 June 1987
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The paper describes polyhedral realizations for Felix Klein's map {3, 7}8 of genus 3, for Gordan's map {4, 5}6 of genus 4, and for two maps of genus 5, the Klein-Fricke map of type {3, 8} and Sherk's map of type {4, 6}. The polyhedra have self-intersections but high symmetry and thus are close analogues to the Kepler-Poinsot-polyhedra.