Table Of ContentTEXTS AND READINGS 21
IN MATHEMATICS
Convex Polyhedra with
Regularity Conditions and
Hilbert's Third Problem
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Convex Polyhedra with
Regularity Conditions and
Hilbert's Third Problem
A. R. Rajwade
Panjab University
Chandigarh
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HINDUSTA N
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right Law and is subject to legal action.
ISBN 978-81-85931-28-9 ISBN 978-93-86279-06-4 (eBook)
DOI 10.1007/978-93-86279-06-4
Contents
1 Introduction 1
2 Definitions and Notations 12
3 Theorems of Euler and Descartes 18
4 The Regularity Restrictions and the five bodies of Plato 27
5 Metrical Properties of the five Platonic Polyhedra 33
6 The fourteen Bodies of Archimedes 40
7 Another Method of Enumerating the Semi-regular
Polyhedra 48
8 The eight Deltahedra 54
9 Finiteness of the number of convex Regular Faced
Polyhedra (RFP) and the remaining cases of regularity
restrictions. 58
10 Star Polyhedra and Plane Tessellations 65
11 A Theorem of Johnson and Grunbaum 72
12 Description of the ninety two RFP and their derivation
from the simple ones 80
13 Hilbert's Third Problem 96
BIBLIOGRAPHY 114
INDEX 117
Preface
Amongst the very first set of convex polyhedra, introduced to a
novice, satisfying some conditions of regularity, are the five regular
solids: the tetrahedron, the octahedron, the cube, the icosahedron
and the dodecahedron and it is very intriguing for one to see the proof
of the fact that there are no more than just five of these. One next
meets the set of thirteen (or sometimes fourteen, including the pseudo
rhombicuboctahedron) semi-regular solids and their case by case
enumeration. One reads of the eight deltahedra and of the ninety
two regular faced polyhedra. One naturally wonders how many
such sets of convex polyhedra are there. Is there a systematically
classified set of conditions of regularity so that each of these sets of
polyhedra satisfies some or all of these conditions? Our main aim
is to write out four conditions of regularity: two related to faces
and two related to solid angles. It will be seen that imposing various
combinations of these conditions of regularity leads to very interest
ing sets of solids, each set satisfying the chosen subset of the set of
four regularity conditions. We take up each such set of conditions in
chapters four to nine. In chapter 1 we give a general introduction to
polyhedra, mentioning pyramids and prisms in particular and the
five Platonic (or regular) polyhedra with their history. Chapter 2
deals with definitions and notation, chapter 3 includes a few elemen
tary theorems in solid geometry which will be extensively used in the
book. In chapter 10 we briefly look at the four regular star polyhe
dra and make an application to tHings of the plane. In chapter 11,
a beautiful theorem of Johnson and Grunbaum, connected with regu
larity conditions, is proved and in chapter 12, we give a description of
the ninety two so called regular faced polyhedra and their deriva
tion from the so called simple ones. Finally in chapter 13 we give a
solution of Hilbert's third problem. In this, our text borrows heavily
from V.K. Boltianski's delightful little book [18].
Some very beautiful topics, intimately connected with this sub
ject have been included, as recreation, as for example, Pick's theorem,
star polyhedra, plane tessallations. Many others could find a place
here but then one has to stop somewhere! The book presupposes per
haps a little more than A-level mathematics; certainly no more than
undergraduate mathematics and of course that invaluable quality :
Mathematical maturity; in particular visualising in three dimensions.
I should like to thank the referee for making excellent sugges
tions which have gone a long way in improving the book. I also thank
Dr.Vipin Bhatnagar for converting the manuscript from MS Word to
Latex and to Ms.Monika Randhawa, and Mr.Saurabh Bhatia for incor
porating all the changes suggested by the referee, in the text. Finally
I thank Ms.Harleen Dahiya, Ms.Rajwant Kaur, Dr.Dinesh Khurana
and Mrs.Anjana Khurana for their help in a little bit of everything.
A.R. Rajwade,
Centre for Advanced Study in Mathematics,
Panjab University,
Chandigarh.
1. 8. 2000.
Chapter 1
Introduction
Polyhedra are three-dimensional analogues of polygons. Thus a
polyhedron is a solid figure (or the surface of such a solid figure) with
a finite number of plane polygonal faces, straight edges and vertices.
Commonest instances of polyhedra are the pyramids (figure 1.1 (a), (b))
and the prisms (figure 1.1 (c) ,( d)).
In this monograph we shall be considering only convex polyhedra
(whose faces, therefore, are necessarily convex polygons). Just .as in
general, little can be said about a general polygon, so in general little
can be said about a general polyhedron. In case of polygons one
demands regularity, i.e. all sides to be equal and all angles to be
equal; only then interesting results can be deduced about them. For
polyhedra, we have more variables and a larger degree of freedom viz.
the vertices, the edges and the faces. How does one define regularity
of polyhedra? Some restrictions like
1. regularity of faces,
2. equality of faces,
3. regularity of solid angles,
4. equality of solid angles,
