In 1974, three researchers at the Faculty of Architecture and Town
Planning of the Technion produced a most amazing collection of isogonal
polyhedra. Even more stunning was that the examples appear not to be
mere drawings but instead photographs of physically created models for
each polyhedron. They published their collection in the book
*Infinite Polyhedra*, later reprinted in 2005. While it might be
supposed the book was made in order to highlight what could become
interesting architectural design elements, it is clearly a work of
mathematics. But, very disappointingly, the book was overlooked by the
mathematical community, or at least those interested in isogonal
polyhedra. For example, this book was published three years before the
much more widely known and cited (in terms of isogonal polyhedra) one
by A.F. Wells. Branko discovered this book and showed it to me and I
then wrote to the Technion asking where I could purchase one. However,
Professor Wachman instead sent me a complimentary copy, for which I am
most appreciative. It is my hope that the extraordinary efforts made by
these three researchers, along with the students and staff at the
Technion who helped them, will now see their discoveries made available
to a much wider audience.

The book groups the polyhedra into sections according to their shaped appearances and these groupings are the same as used in the listings below on this page. The theory that was used for finding them was the same theory of nets that Wells used. The book begins with a preface that describes the overall structure of the book and includes some of the simpler polyhedra. The majority of the book then describes the more complex polyhedra. Thus the pages below that have Roman numerals refer to those polyhedra that appear in the preface section. Some pages have more than one polyhedron, and these are listed below as 't' or 'b', depending on whether they are on the top or bottom of the page. Polyhedra that have already appeared someplace else on some other web pages at this site are linked to those earlier appearances.

Because of the very large number of polyhedra in the book, in order to get them displayed on the Web as soon as possible they are shown here only as unlabeled ones. Eventually it is hoped they will be described along with all their labeled versions just like the other polyhedra on this site. Also, the list shown here at this time is incomplete because the remainder of the polyhedra are still being constructed. Because the book contains so many polyhedra this page is being posted while still unfinished with the missing VRML models to be filled in as they are completed.

- The Coxeter Polyhedra
- Page III: the {4,6} sponge
- Page III: the {6,4} sponge
- Page III: the {6,6} sponge

- Cylindrical Infinite Polyhedra
- Page VIII: {4,4} prismatic rods with various polygonal cross-sections
- Page VIII: {3,6} antiprismatic rods with various polygonal cross-sections
- Page VIII: 3.3.3.4.4 rods made with alternating prisms and antiprisms
- Page VIII: {3,6} rods with square, hexagonal, and octagonal cross-sections
- Page VIII: 3.3.3.4.4 rods (longitudinal strips of squares and triangles)
- Page VIII: {3,6} helicoid rods with various polygonal cross-sections

- Corrugated Polyhedra
- Page IX: {4,4} folded planes (two kinds)
- Page IX: {3,6} folded planes (two kinds)
- Page IX: 3.3.3.4.4 folded planes (three kinds)

- Mono-Layered Polyhedra
- Page 2 (t,b): Two (S2 and N5) of the fifteen {4,5} lattice grid polyhedra
- Page 3 (t): A 4.4.8.8 slab polyhedron
- Page 3 (b): A 4.4.4.8 slab polyhedron
- Page 4: A 3.3.4.3.4.4 slab polyhedron
- Page 5 (t): A 3.3.3.3.3.4.4 slab polyhedron
- Page 5 (b): A {3,8} slab polyhedron
- Page 6: A 4.4.6.6 slab polyhedron
- Page 7: A 4.4.12.12 slab polyhedron
- Page 8 (t): A 6.3.6.4.4 slab polyhedron
- Page 8 (b): A 6.3.6.3.3.3 slab polyhedron
- Page 9 (t): A 4.3.4.4.4 slab polyhedron
- Page 9 (b): A 4.6.4.4.4 slab polyhedron
- Page 10 (t): A 4.4.4.12 slab polyhedron
- Page 10 (b): A 4.4.6.12 slab polyhedron
- Page 11: A 4.4.4.6 slab polyhedron
- Page 12 (t): A 3.3.6.3.4.4 slab polyhedron
- Page 12 (b): A 3.3.3.3.4.4 slab polyhedron
- Page 13: Not an isogonal polyhedron (as indicated in the book by the word 'nonuniform')

- Multi-Layered Polyhedra
- Page 15: Polyhedron S3 of the {4,5} lattice grid polyhedra
- Page 16: Polyhedron N7 of the {4,5} lattice grid polyhedra
- Page 17: Polyhedron N11 of the {4,5} lattice grid polyhedra
- Page 18: A 4.4.8.8 multi-layered polyhedron
- Page 19: A 4.4.4.8 multi-layered polyhedron
- Page 20: A {4,5} (non-lattice grid) multi-layered polyhedron
- Page 21: A 3.3.4.3.4.4 multi-layered polyhedron
- Page 22: A 4.3.3.4.4.3.4 multi-layered polyhedron
- Page 23: A 4.3.4.4.3.3.4 multi-layered polyhedron
- Page 24: A 3.3.3.3.4.3.3.4 multi-layered polyhedron (only when asymmetrically marked)
- Page 25: Polyhedron (3
^{9})P_{11}of the Hughes Jones polyhedra - Page 26: Polyhedron (3
^{9})P_{2}of the Hughes Jones polyhedra - Page 27: A 4.3.4.4.3.3.3.4 multi-layered polyhedron
- Page 28: A {3,10} multi-layered polyhedron
- Page 29: A chiral (right, as pictured in the book, and left) pair of {3,10} multi-layered polyhedra
- Page 30: A 3.3.3.12.12 multi-layered polyhedron
- Page 31: A 3.3.3.6.3.3.3.6 multi-layered polyhedron
- Page 32: A 3.3.3.4.6.4 multi-layered polyhedron
- Page 33: A 3.3.3.3.3.3.6 multi-layered polyhedron
- Page 34: A second 3.3.3.3.3.3.6 multi-layered polyhedron
- Page 35: A second 4.3.4.4.3.3.3.4 multi-layered polyhedron
- Page 36: A duplicate of the polyhedron on page 28
- Page 37: A 4.4.4.6.4 multi-layered polyhedron
- Page 38: A 4.4.12.12 multi-layered polyhedron
- Page 39: A 4.4.6.4.4.6 multi-layered polyhedron
- Page 40: A 4.4.6.4.4 multi-layered polyhedron
- Page 41: A 4.4.4.12 multi-layered polyhedron
- Page 42: A 4.3.3.6.3.4 multi-layered polyhedron
- Page 43: A 4.4.4.4.6 multi-layered polyhedron
- Page 44: A 4.3.3.3.4.4.3.4 multi-layered polyhedron (only when asymmetrically marked)
- Page 45: A {4,6} multi-layered polyhedron
- Page 46: A second {4,5} (non-lattice grid) multi-layered polyhedron
- Page 47: A different 4.3.4.4.3.3.4 multi-layered polyhedron
- Page 48: A 3.3.3.4.4.4 multi-layered polyhedron
- Page 49: A 4.4.4.4.8 multi-layered polyhedron
- Page 50: Polyhedron N6 of the {4,5} lattice grid polyhedra
- Page 51: Polyhedron N12 of the {4,5} lattice grid polyhedra
- Page 55: A second 4.4.4.4.8 multi-layered polyhedron
- Page 56: A third 4.4.4.4.8 multi-layered polyhedron
- Page 60: A 3.3.3.4.4.4.4 multi-layered polyhedron (only when asymmetrically marked)
- Page 61: Not an isogonal polyhedron (as indicated in the book by the word 'nonuniform')
- Page 62: A different 3.3.3.4.4.4.4 multi-layered polyhedron
- Page 63: A 3.3.4.4.3.4.4 multi-layered polyhedron
- Page 64: A 4.4.8.4.8 multi-layered polyhedron

- Multi-Directional Polyhedra
- Page 67: The {4,6} Coxeter sponge
- Page 68: The {6,4} Coxeter sponge
- Page 69: A 4.4.6.6 multi-directional polyhedron (same as Wells 4.4.6.6)
- Page 70: A 4.3.4.4.3.4 multi-directional polyhedron
- Page 71: A 4.3.4.4.4 multi-directional polyhedron
- Page 72: A 4.4.4.6 multi-directional polyhedron
- Page 76: The {6,6} Coxeter sponge
- Page 83: Polyhedron (3
^{9})P_{1}of the Hughes Jones polyhedra - Page 86: A second 4.4.4.6 multi-directional polyhedron
- Page 87: Polyhedron S1 of the {4,5} lattice grid polyhedra
- Page 88: A {4,5} (non-lattice grid) multi-directional polyhedron
- Page 89: A 4.4.4.8 multi-directional polyhedron
- Page 90: A {3,12} multi-directional polyhedron
- Page 91: A {3,9} multi-directional polyhedron
- Page 92: A 3.3.3.4.3.3.3.4 multi-directional polyhedron
- Page 93: A 3.3.3.8.8 multi-directional polyhedron
- Page 94: A 3.3.3.4.4.4 multi-directional polyhedron
- Page 96: A 4.4.8.8.4 multi-directional polyhedron
- Page 97: A 4.8.4.8 multi-directional polyhedron
- Page 98: A second 4.4.4.8 multi-directional polyhedron
- Page 99: A 3.4.4.4.4.4 multi-directional polyhedron
- Page 100: A 6.6.8.8 multi-directional polyhedron
- Page 101: A 4.4.6.8 multi-directional polyhedron

Return to main page