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Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
A R Rajwade, Panjab University, Chandigarh, India
A publication of Hindustan Book Agency.
Hindustan Book Agency
2001; 120 pp; hardcover
ISBN-10: 81-85931-28-3
ISBN-13: 978-81-85931-28-9
List Price: US$35
Member Price: US$28
Order Code: HIN/8
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Since antiquity, people knew that there are only five regular solids, i.e. polyhedra whose all faces are regular polygons and all solid angles are also regular. These solids are, of course, the tetrahedron, the octahedron, the cube, the icosahedron, and the dodecahedron. Later, much attention was drawn to the question of how to describe polyhedra with other types of regularity conditions. The author puts together many facts known in this direction. He formulates four regularity conditions (two for faces and two for solid angles) and for any combination of their conditions lists all the corresponding polyhedra. In this way, he obtains such very interesting classes of solids as 13 semiregular solids, or 8 deltahedra, or 92 regularly faces polyhedra, etc. In later chapters the author presents some related topics of geometry of solids, like star polyhedra and plane tessellations. In the concluding chapter, a complete solution of the Hilbert 3rd problem is given.

Supplied with many figures, the book can be easily read by anyone interested in this beautiful classical geometry.

A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels.


Advanced undergraduates, graduate students, and research mathematicians interested in geometry.

Table of Contents

  • Introduction
  • Definitions and notations
  • Theorems of Euler and Descartes
  • The regularity restrictions and the five bodies of Plato
  • Metrical properties of the five Platonic polyhedra
  • The fourteen bodies of Archimedes
  • Another method of enumerating the semi-regular polyhedra
  • The eight Deltahedra
  • Finiteness of the number of convex regular faced polyhedra (RFP) and the remaining cases of regularity restrictions
  • Star polyhedra and plane tessellations
  • A theorem of Johnson and Grunbaum
  • Description of the ninety-two RFP and their derivation from the simple ones
  • Hilbert's third problem
  • Bibliography
  • Index
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