Dual ModelsIn Dual Models, written in the same enthusiastic style as its predecessors Polyhedron Models and Spherical Models, Magnus J. Wenninger presents the complete set of uniform duals of uniform polyhedral, thus rounding out a significant body of knowledge with respect to polyhedral forms. He begins with the simplest convex solids but then goes on to show how all the more difficult, non convex, uniform polyhedral duals can be derived from a geometric theorem on duality that unifies and systematizes the entire set of such duals. Many of these complex shapes are published here for the first time. Models made by the author are shown in photographs, and these, along with line drawings, diagrams, and commentary, invite readers to undertake the task of making the models, using index cards or tag paper and glue as construction materials. The mathematics is deliberately kept at the high school or secondary level, and hence the book presumes at most some knowledge of geometry and ordinary trigonometry and the use of a scientific type small electronic calculator. The book will be useful as enrichment material for the mathematics classroom and can serve equally well as a source book of ideas for artists and designers of decorative devices or simply as a hobby book in recreational mathematics. |
Contents
The five regular convex polyhedra and their duals | 7 |
The thirteen semiregular convex polyhedra and their duals | 14 |
Stellated forms of convex duals | 36 |
The duals of nonconvex uniform polyhedra | 39 |
Duals of semiregular nonconvex uniform polyhedra | 40 |
Other nonconvex uniform polyhedral duals | 54 |
Duals derived from other Archimedean forms | 55 |
Duals derived from variations of Archimedean forms | 74 |
Duals of hemipolyhedra | 101 |
Duals of nonconvex snub polyhedra | 118 |
Some interesting polyhedral compounds | 143 |
Epilogue | 149 |
| 150 | |
References | 153 |
List of polyhedra and dual models | 154 |
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Common terms and phrases
Archimedean duals Archimedean forms calculation cells center of symmetry compound of five construction and stellation convex hull core cuboctahedron deltoidal hexecontahedron dodeca Dorman Luke construction drawings dual model dual of 119 dual of 67 dual of 70 duality Duals derived duals of nonconvex faces and vertices facial planes final stellation five regular solids geometry hedral hedron hemipolyhedra hence hidden vertices hull Dual icosa icosidodecahedron icositetrahedron intersecting mathematical medean Medial model shown nonconvex uniform duals nonconvex uniform polyhedra number of edges numerical data octagrammic octahedron pattern for duals pentagonal hexecontahedron pentagrammic vertices polar reciprocal formula poly Polyhedron models pyramids regular dodecahedra rhombic dodecahedron rhombic triacontahedron rhombitruncated shaded portions shown in Fig shown in Photo spikes stel stellated dodecahedron stellated forms stellation pat stellation pattern stellation process Stellation to infinity tern for dual thirteen semiregular tion triakisoctahedron triakistetrahedron triambic icosahedron trigonal vertices truncated tetrahedron variations of Archimedean vertex figure Wenninger
Popular passages
Page xi - Some of these are given in the list of references at the end of this book.
Page xii - FRS, for the interest he has taken in this work and for the valuable suggestions he has made.
Page xii - Only when you handle a model yourself will you see the wonders that lie hidden in this world of geometrical beauty and symmetry.