The infinite polyhedra of Wachman, Burt, and Kleinmann

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.

The infinite polyhedra of Wachman, Burt, and Kleinmann

The Coxeter Polyhedra
1. Page III: the {4,6} sponge
2. Page III: the {6,4} sponge
3. Page III: the {6,6} sponge
Cylindrical Infinite Polyhedra
1. Page VIII: {4,4} prismatic rods with various polygonal cross-sections
2. Page VIII: {3,6} antiprismatic rods with various polygonal cross-sections
  - Rods
  - Incidence symbols
    1. [a⁺ b a^- a⁺ b a^-; a⁺ b]
    2. [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺; a⁺ b⁺ c⁺]
    3. [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺; c^- b⁺ a^-]
    4. [a⁺ b⁺ c⁺ c^- b^- a^-; a⁺ b^- c⁺]
    5. [a⁺ b⁺ c⁺ c^- b^- a^-; c^- b^- a^-]
    6. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ e⁺ c⁺ d⁺ b⁺ f⁺]
    7. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ e⁺ d^- c^- b⁺ f⁺]
    8. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; c^- e⁺ a^- f^- b⁺ d^-]
    9. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d⁺ e⁺ f⁺ a⁺ b⁺ c⁺]
    10. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; f^- e⁺ d^- c^- b⁺ a^-]
    11. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b⁺ f⁺ d⁺ e⁺ c⁺] (only for even-edged cross-sections)
    12. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ e^- c⁺ f^- b^- d^-] (only for even-edged cross-sections)
    13. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; c^- e^- a^- d⁺ b^- f⁺] (only for even-edged cross-sections)
3. Page VIII: 3.3.3.4.4 rods made with alternating prisms and antiprisms
  - Rods
  - Incidence symbols
    1. [a b⁺ c⁺ c^- b^-; a b^- c⁺]
    2. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ e⁺ c⁺ d⁺ b⁺]
    3. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- e⁺ c⁺ d⁺ b⁺]
    4. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ e⁺ d^- c^- b⁺]
    5. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- e⁺ d^- c^- b⁺]
4. Page VIII: {3,6} rods with square, hexagonal, and octagonal cross-sections
5. Page VIII: 3.3.3.4.4 rods (longitudinal strips of squares and triangles)
  - Rods
    1. Square cross-section (3.3.3.4.4)
    2. Octagonal cross-section (3.3.3.4.4)
  - Incidence symbols
    1. [a⁺ b⁺ b^- a^- c; a^- b⁺ b^- a⁺ c]
    2. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ b⁺ c⁺ a⁺ e⁺]
    3. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ b⁺ c⁺ a⁺ e^-]
    4. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c^- b^- a⁺ e⁺]
    5. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c^- b^- a⁺ e^-]
6. Page VIII: {3,6} helicoid rods with various polygonal cross-sections
  - Rods
  - Incidence symbol
    1. [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺; a⁺ b⁺ c⁺]
    2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b⁺ f⁺ d⁺ e⁺ c⁺]
    3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d⁺ e⁺ f⁺ a⁺ b⁺ c⁺]
Corrugated Polyhedra
1. Page IX: {4,4} folded planes (two types)
2. Page IX: {3,6} folded planes (two types)
3. Page IX: 3.3.3.4.4 folded planes (three types)
  1. Strips of squares parallel to the plane (3.3.3.4.4)
  2. Strips of triangles parallel to the plane (3.3.3.4.4) (same as above)
  3. No strips parallel to the plane (3.3.3.4.4)
    1. [a⁺ b⁺ b^- a^- c; a^- b^^- b^⁺ a⁺ c^]
    2. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ b^^- c^^- a⁺ e^⁺]
    3. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ b^^- c^^- a⁺ e^^-]
    4. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c^⁺ b^⁺ a⁺ e^⁺]
    5. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c^⁺ b^⁺ a⁺ e^^-]
Mono-Layered Polyhedra
1. Page 2 (t,b): Two (S2 and N5) of the fifteen {4,5} cubic lattice polyhedra
2. Page 3 (t): A 4.4.8.8 slab polyhedron (4.4.8.8)
  1. [a b⁺ c b^-; a b^- c b⁺]
  2. [a⁺ b⁺ c⁺ d⁺; a⁺ b^- c⁺ d^-]
  3. [a⁺ b⁺ c⁺ d⁺; a⁺ b^- c^- d^-]
  4. [a⁺ b⁺ c⁺ d⁺; a^- b^- c⁺ d^-]
  5. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]
  6. [a⁺ b⁺ c⁺ d⁺; a⁺ d⁺ c⁺ b⁺]
  7. [a⁺ b⁺ c⁺ d⁺; a⁺ d⁺ c^- b⁺]
  8. [a⁺ b⁺ c⁺ d⁺; a^- d⁺ c⁺ b⁺]
  9. [a⁺ b⁺ c⁺ d⁺; a^- d⁺ c^- b⁺]
3. Page 3 (b): A 4.4.4.8 slab polyhedron (4.4.4.8)
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]
4. Page 4: A 3.3.4.3.4.4 slab polyhedron (3.3.4.3.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; e⁺ b⁺ d⁺ c⁺ a⁺ f^-]
5. Page 5 (t): A 3.3.3.3.3.4.4 slab polyhedron (3.3.3.3.3.4.4)
  1. [a⁺ b⁺ c⁺ c^- b^- a^- d; a^- c⁺ b⁺ b^- c^- a⁺ d]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; f⁺ c⁺ b⁺ e⁺ d⁺ a⁺ g⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; f⁺ c⁺ b⁺ e⁺ d⁺ a⁺ g^-]
6. Page 5 (b): Polyhedron (3⁸)P₁ of the Hughes Jones polyhedra
7. Page 6: A 4.4.6.6 slab polyhedron (4.4.6.6)
  1. [a⁺ b a^- c; a^- b a⁺ c]
  2. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d⁺]
  3. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]
  4. [a⁺ b⁺ c⁺ d⁺; c⁺ b⁺ a⁺ d⁺]
  5. [a⁺ b⁺ c⁺ d⁺; c⁺ b⁺ a⁺ d^-]
8. Page 7: A 4.4.12.12 slab polyhedron (4.4.12.12)
  1. [a b⁺ c b^-; a b^- c]
  2. [a⁺ b⁺ c⁺ d⁺; a⁺ d⁺ c⁺ b⁺]
  3. [a⁺ b⁺ c⁺ d⁺; a⁺ d⁺ c^- b⁺]
  4. [a⁺ b⁺ c⁺ d⁺; a^- d⁺ c⁺ b⁺]
  5. [a⁺ b⁺ c⁺ d⁺; a^- d⁺ c^- b⁺]
9. Page 8 (t): A 6.3.6.4.4 slab polyhedron (6.3.6.4.4)
  1. [a⁺ b⁺ b^- a^- c; a^- b^- b⁺ a⁺ c]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e^-]
10. Page 8 (b): A 6.3.6.3.3.3 slab polyhedron (6.3.6.3.3.3)
  1. [a⁺ b⁺ b^- a^- c⁺ c^-; a^- b^- b⁺ a⁺ c⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d⁺ c⁺ b⁺ a⁺ e⁺ f⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d⁺ c⁺ b⁺ a⁺ f^- e^-]
11. Page 9 (t): A 4.3.4.4.4 slab polyhedron (4.3.4.4.4)
  1. [a⁺ b⁺ b^- a^- c; a^- b^- b⁺ a⁺ c]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- c⁺ b⁺ d^- e⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- c⁺ b⁺ d^- e^-]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e^-]
12. Page 9 (b): A 4.6.4.4.4 slab polyhedron (4.6.4.4.4)
  1. [a⁺ b⁺ b^- a^- c; a^- b^- b⁺ a⁺ c]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ b^- c^- a⁺ e⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ b^- c^- a⁺ e^-]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e^-]
13. Page 10 (t): A 4.4.4.12 slab polyhedron (4.4.4.12)
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]
14. Page 10 (b): A 4.4.6.12 slab polyhedron (4.4.6.12)
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]
15. Page 11: A 4.4.4.6 slab polyhedron (4.4.4.6)
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]
16. Page 12 (t): A 3.3.6.3.4.4 slab polyhedron (3.3.6.3.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; e⁺ b⁺ d⁺ c⁺ a⁺ f^-]
17. Page 12 (b): A 3.3.3.3.4.4 slab polyhedron (3.3.3.3.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; e⁺ b⁺ d⁺ c⁺ a⁺ f^-]
18. Page 13: Not an isogonal polyhedron (as indicated in the book by the word 'nonuniform') (3.3.3.4.4.4)
Multi-Layered Polyhedra
1. Page 15: Polyhedron S3 of the {4,5} cubic lattice polyhedra
2. Page 16: Polyhedron N7 of the {4,5} cubic lattice polyhedra
3. Page 17: Polyhedron N11 of the {4,5} cubic lattice polyhedra
4. Page 18: A 4.4.8.8 multi-layered polyhedron (4.4.8.8) (same as page 3(t))
  1. [a b⁺ c b^-; a b^- c^]
  2. [a⁺ b⁺ c⁺ d⁺; a⁺ b^- c^⁺ d^-]
  3. [a⁺ b⁺ c⁺ d⁺; a⁺ b^- c^^- d^-]
  4. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^⁺ d^-]
  5. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^^- d^-]
  6. [a⁺ b⁺ c⁺ d⁺; a⁺ d⁺ c^⁺ b⁺]
  7. [a⁺ b⁺ c⁺ d⁺; a⁺ d⁺ c^^- b⁺]
  8. [a⁺ b⁺ c⁺ d⁺; a^- d⁺ c^⁺ b⁺]
  9. [a⁺ b⁺ c⁺ d⁺; a^- d⁺ c^^- b⁺]
5. Page 19: A 4.4.4.8 multi-layered polyhedron (4.4.4.8) (same as page 3(b))
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^⁺ d^-]
6. Page 20: A {4,5} (not cubic lattice) multi-layered polyhedron (same as Wells (4,5)-(4+2)tf)
7. Page 21: A 3.3.4.3.4.4 multi-layered polyhedron (3.3.4.3.4.4) (same as page 4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; e⁺ b^^- d^^- c^^- a⁺ f^-]
8. Page 22: A 4.3.3.4.4.3.4 multi-layered polyhedron (4.3.3.4.4.3.4)
  1. [a⁺ b⁺ c*⁺ b^⁺ a^⁺ d^⁺ d⁺; a^- d⁺ c*⁺ d^⁺ a^^- b^⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a^- f^⁺ c⁺ g^⁺ e^- b^⁺ d^⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a^- g⁺ c⁺ f⁺ e^- d⁺ b⁺]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; e^^- f^⁺ c⁺ g^⁺ a^^- b^⁺ d^⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; e^^- g⁺ c⁺ f⁺ a^^- d⁺ b⁺]
9. Page 23: A 4.3.4.4.3.3.4 multi-layered polyhedron (4.3.4.4.3.3.4)
  1. [a⁺ b⁺ b^- a^- c⁺ d c^-; a^- c^⁺ c^^- a⁺ b^⁺ d]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a^- e^⁺ g^⁺ d^- b^⁺ f⁺ c^⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a^- g^^- e^^- d^- c^^- f⁺ b^^-]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; d⁺ e^⁺ g^⁺ a⁺ b^⁺ f⁺ c^⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; d⁺ g^^- e^^- a⁺ c^^- f⁺ b^^-]
10. Page 24: A 3.3.3.3.4.3.3.4 multi-layered polyhedron (3.3.3.3.4.3.3.4)
  1. [a⁺ b⁺ c b^- a^- d⁺ e d^-; d^⁺ b⁺ e^ b^- d^^- a^⁺ c^]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; f^⁺ b⁺ g^⁺ d⁺ h^⁺ a^⁺ c^⁺ e^⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; f^⁺ d^- g^^- b^- h^⁺ a^⁺ c^^- e^⁺]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; h^^- b⁺ g^^- d⁺ f^^- e^^- c^^- a^^-]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; h^^- d^- g^⁺ b^- f^^- e^^- c^⁺ a^^-]
11. Page 25: Polyhedron (3⁹)P₁₁ of the Hughes Jones polyhedra
12. Page 26: Polyhedron (3⁹)P₂ of the Hughes Jones polyhedra
13. Page 27: A 4.3.4.4.3.3.3.4 multi-layered polyhedron (4.3.4.4.3.3.3.4)
  1. [a⁺ b⁺ b^⁺ a^⁺ c^⁺ d^⁺ d⁺ c⁺; a⁺ c⁺ c^⁺ a^⁺ b^⁺ d⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; a⁺ h⁺ e⁺ d⁺ c⁺ g⁺ f⁺ b⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d^⁺ h⁺ e⁺ a^⁺ c⁺ g⁺ f⁺ b⁺]
14. Page 28: A {3,10} multi-layered polyhedron (3.3.3.3.3.3.3.3.3.3)
  1. [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺ d⁺ e⁺ e^⁺ d^⁺; b^- a^- d^⁺ d⁺ a^^- b^^- c^⁺ e^⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺ i⁺ j⁺; b^- a^- j⁺ g⁺ f^- e^- d⁺ i⁺ h⁺ c⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺ i⁺ j⁺; e^^- f^^- j⁺ g⁺ a^^- b^^- d⁺ i⁺ h⁺ c⁺]
15. Page 29: Two similar but different (as in the book, and the other) {3,10} multi-layered polyhedra (3.3.3.3.3.3.3.3.3.3)
  1. (In the book) [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺ i⁺ j⁺; b⁺ a⁺ f^^- h^^- i⁺ c^^- j^^- d^^- e⁺ g^^-]
  2. (The other) [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺ i⁺ j⁺; b⁺ a⁺ h^⁺ f^⁺ i⁺ d^⁺ j^⁺ c^⁺ e⁺ g^⁺]
16. Page 30: A 3.3.3.12.12 multi-layered polyhedron (3.3.3.12.12)
  1. [a⁺ a^- b⁺ c b^-; a⁺ a^- b^- c^]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ b⁺ e⁺ d^⁺ c⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ b⁺ e⁺ d^^- c⁺]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺; b^- a^- e⁺ d^⁺ c⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺; b^- a^- e⁺ d^^- c⁺]
17. Page 31: A 3.3.3.6.3.3.3.6 multi-layered polyhedron (3.3.3.6.3.3.3.6)
  1. [a⁺ b⁺ b^- a^- a^^- b^^- b^⁺ a^⁺; a^- b⁺]
  2. [a⁺ b⁺ c⁺ d⁺ d^⁺ c^⁺ b^⁺ a^⁺; d⁺ b⁺ c⁺ a⁺]
  3. [a⁺ b⁺ c⁺ d⁺ d^⁺ c^⁺ b^⁺ a^⁺; d⁺ c^- b^- a⁺]
  4. [a⁺ b⁺ b^- a^- c⁺ d⁺ d^- c^-; a^- b⁺ b^- a⁺ c^- d⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d⁺ b⁺ c⁺ a⁺ h⁺ f⁺ g⁺ e⁺]
  6. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d⁺ b⁺ c⁺ a⁺ h⁺ g^- f^- e⁺]
  7. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d⁺ c^- b^- a⁺ h⁺ f⁺ g⁺ e⁺]
  8. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d⁺ c^- b^- a⁺ h⁺ g^- f^- e⁺]
  9. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d⁺ f^^- g^^- a⁺ h⁺ b^^- c^^- e⁺]
  10. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d⁺ g^⁺ f^⁺ a⁺ h⁺ c^⁺ b^⁺ e⁺]
18. Page 32: A 3.3.3.4.6.4 multi-layered polyhedron (3.3.3.4.6.4)
  1. [a⁺ a^- b⁺ c⁺ c^- b^-; a⁺ a^- b^- c^⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b⁺ f⁺ d^⁺ e^⁺ c⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b⁺ f⁺ e^^- d^^- c⁺]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; b^- a^- f⁺ d^⁺ e^⁺ c⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; b^- a^- f⁺ e^^- d^^- c⁺]
19. Page 33: A 3.3.3.3.3.3.6 multi-layered polyhedron (3.3.3.3.3.3.6)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; b^- a^- g⁺ e^^- d^^- f^^- c⁺]
20. Page 34: A second 3.3.3.3.3.3.6 multi-layered polyhedron polyhedron (3.3.3.3.3.3.6)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; b^^- a^^- f^⁺ d^^- g^- c^⁺ e^-]
21. Page 35: A second 4.3.4.4.3.3.3.4 multi-layered polyhedron (4.3.4.4.3.3.3.4) (same as page 27)
  1. [a⁺ b⁺ b^⁺ a^⁺ c^⁺ d^⁺ d⁺ c⁺; a^- c⁺ c^⁺ a^^- b^⁺ d⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; a^- h⁺ e⁺ d^- c⁺ g⁺ f⁺ b⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d^^- h⁺ e⁺ a^^- c⁺ g⁺ f⁺ b⁺]
22. Page 36: A second {3,10} multi-layered polyhedron (3.3.3.3.3.3.3.3.3.3) (same as page 28)
  1. [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺ d⁺ e⁺ e^⁺ d^⁺; a⁺ b⁺ d^⁺ d⁺ b^⁺ a^⁺ c^⁺ e^⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺ i⁺ j⁺; a⁺ b⁺ j⁺ g⁺ e⁺ f⁺ d⁺ i⁺ h⁺ c⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺ i⁺ j⁺; f^⁺ e^⁺ j⁺ g⁺ b^⁺ a^⁺ d⁺ i⁺ h⁺ c⁺]
23. Page 37: A 4.4.4.6.4 multi-layered polyhedron (4.4.4.6.4)
  1. [a⁺ b*⁺ a^⁺ c⁺ c^⁺; a^- b*^- a^^- c^-]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- c^- d^- e^-]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^^- c^- e^^- d^^-]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺; c^^- b^- a^^- d^- e^-]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺; c^^- b^^- a^^- e^^- d^^-]
24. Page 38: A 4.4.12.12 multi-layered polyhedron (4.4.12.12) (same as page 7)
  1. [a b⁺ c b^-; a b^- c^]
  2. [a⁺ b⁺ c⁺ d⁺; a⁺ d⁺ c^⁺ b⁺]
  3. [a⁺ b⁺ c⁺ d⁺; a⁺ d⁺ c^^- b⁺]
  4. [a⁺ b⁺ c⁺ d⁺; a^- d⁺ c^⁺ b⁺]
  5. [a⁺ b⁺ c⁺ d⁺; a^- d⁺ c^^- b⁺]
25. Page 39: A 4.4.6.4.4.6 multi-layered polyhedron (4.4.6.4.4.6)
  1. [a b⁺ b^⁺ a^ b^^- b^-; a b^-]
  2. [a⁺ b⁺ b^⁺ a^⁺ c^⁺ c⁺; a⁺ c⁺ c^⁺ a^⁺ b^⁺ b⁺]
  3. [a⁺ b⁺ b^⁺ a^⁺ c^⁺ c⁺; a^- c⁺ c^⁺ a^^- b^⁺ b⁺]
  4. [a b⁺ c⁺ d c^- b^-; a b^- c^- d c⁺ b⁺]
  5. [a b⁺ c⁺ d c^- b^-; d^ b^- c^- a^ c⁺ b⁺]
  6. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ f⁺ e⁺ d⁺ c⁺ b⁺]
  7. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ f⁺ e⁺ d^- c⁺ b⁺]
  8. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a^- f⁺ e⁺ d^- c⁺ b⁺]
  9. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d^⁺ f⁺ e⁺ a^⁺ c⁺ b⁺]
  10. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d^^- f⁺ e⁺ a^^- c⁺ b⁺]
26. Page 40: A 4.4.6.4.4 multi-layered polyhedron (4.4.6.4.4) (same as page 9(b), but with different edge labels)
  1. [a b⁺ c⁺ c^- b^-; a b^- c^⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ e⁺ c^⁺ d^⁺ b⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- e⁺ c^⁺ d^⁺ b⁺]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ e⁺ d^^- c^^- b⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- e⁺ d^^- c^^- b⁺]
27. Page 41: A 4.4.4.12 multi-layered polyhedron (4.4.4.12) (same as page 10(t))
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^⁺ d^-]
28. Page 42: A 4.3.3.6.3.4 multi-layered polyhedron polyhedron (4.3.3.6.3.4) (same as page 12(t))
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; e⁺ b^^- d^^- c^^- a⁺ f^-]
29. Page 43: A 4.4.4.4.6 multi-layered polyhedron (4.4.4.4.6)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; e^^- b^- c^^- d^- a^^-]
30. Page 44: A 4.3.3.3.4.4.3.4 multi-layered polyhedron (4.3.3.3.4.4.3.4)
  1. [a⁺ b⁺ c⁺ c^- b^- a^- d⁺ d^-; a^- d^⁺ c^- c⁺ d^^- a⁺ b^⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; f⁺ g^⁺ d⁺ c⁺ h^⁺ a⁺ b^⁺ e^⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; f⁺ h^^- d⁺ c⁺ g^^- a⁺ e^^- b^^-]
31. Page 45: A {4,6} (not cubic lattice) multi-layered polyhedron (4.4.4.4.4.4)
  1. [a b⁺ c⁺ d c^- b^-; a b^- c^- d]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b^- e⁺ d⁺ c⁺ f^-]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b^- e⁺ d^- c⁺ f^-]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a^- b^- e⁺ d⁺ c⁺ f^-]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a^- b^- e⁺ d^- c⁺ f^-]
  6. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ f⁺ e⁺ d⁺ c⁺ b⁺]
  7. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ f⁺ e⁺ d^- c⁺ b⁺]
  8. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a^- f⁺ e⁺ d⁺ c⁺ b⁺]
  9. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a^- f⁺ e⁺ d^- c⁺ b⁺]
32. Page 46: A second {4,5} (not cubic lattice) multi-layered polyhedron (4.4.4.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- c^- d^- e^-]
33. Page 47: A different 4.3.4.4.3.3.4 multi-layered polyhedron (4.3.4.4.3.3.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a^- g⁺ e⁺ d^- c⁺ f⁺ b⁺]
34. Page 48: A 3.3.3.4.4.4 multi-layered polyhedron (3.3.3.4.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a^- b^- f⁺ d⁺ e⁺ c⁺]
35. Page 49: A 4.4.4.4.8 multi-layered polyhedron (4.4.4.4.8)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- c⁺ d^- e^-]
36. Page 50: Polyhedron N6 of the {4,5} cubic lattice polyhedra
37. Page 51: Polyhedron N12 of the {4,5} cubic lattice polyhedra
38. Page 52: Polyhedron N5 of the {4,6} cubic lattice polyhedra
39. Page 53: Polyhedron N4 of the {4,6} cubic lattice polyhedra
40. Page 54: Polyhedron N3 of the {4,6} cubic lattice polyhedra
41. Page 55: A second 4.4.4.4.8 multi-layered polyhedron (4.4.4.4.8) (same as page 49)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- c^- d^- e^-]
42. Page 56: A third 4.4.4.4.8 multi-layered polyhedron (4.4.4.4.8) (same as page 49)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- c^- d⁺ e^-]
43. Page 57: A 4.6.4.4.4 multi-layered polyhedron (4.6.4.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- c^- d^- e^-]
44. Page 58: A second 4.6.4.4.4 multi-layered polyhedron (4.6.4.4.4) (same as page 57)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- c^- d^- e⁺]
45. Page 59: A 4.12.4.4.4 multi-layered polyhedron (4.12.4.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- c^- d^- e^-]
46. Page 60: A 3.3.3.4.4.4.4 multi-layered polyhedron (only when asymmetrically marked) (3.3.3.4.4.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a^- b^^- c^- g^^- e⁺ f^^- d^^-]
47. Page 61: Not an isogonal polyhedron (as indicated in the book by the word 'nonuniform') (3.3.3.4.4.4) (same as page 13)
48. Page 62: A different 3.3.3.4.4.4.4 multi-layered polyhedron (3.3.3.4.4.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a^- e⁺ c⁺ d⁺ b⁺ f^- g⁺]
49. Page 63: A 3.3.4.4.3.4.4 multi-layered polyhedron (3.3.4.4.3.4.4)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a^- f^- c⁺ g^- e^- b^- d^-]
50. Page 64: A 4.4.8.4.8 multi-layered polyhedron (4.4.8.4.8)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- e^- c^- d^- b^-]
Multi-Directional Polyhedra
1. Page 66: A 4.8.6.8 multi-directional polyhedron (4.8.6.8)
  1. [a⁺ a^⁺ b⁺ b^⁺; a^- a^^- b^-]
  2. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]
  3. [a⁺ b⁺ c⁺ d⁺; a^- b^- d^^- c^^-]
  4. [a⁺ b⁺ c⁺ d⁺; b^^- a^^- c^- d^-]
  5. [a⁺ b⁺ c⁺ d⁺; b^^- a^^- d^^- c^^-]
2. Page 67: The {4,6} Coxeter sponge
3. Page 68: The {6,4} Coxeter sponge
4. Page 69: A 4.4.6.6 multi-directional polyhedron (same as Wells 4.4.6.6)
5. Page 70: A 4.3.4.4.3.4 multi-directional polyhedron (4.3.4.4.3.4)
  1. [a b⁺ b^- a b⁺ b^-; a b^-]
  2. [a b⁺ c⁺ d c^- b^-; a c⁺ b⁺ d]
  3. [a b⁺ c⁺ d c^- b^-; d c⁺ b⁺ a]
  4. [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺; a⁺ c⁺ b⁺]
  5. [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺; a^- c⁺ b⁺]
  6. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ c⁺ b⁺ d⁺ f⁺ e⁺]
  7. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a^- c⁺ b⁺ d^- f⁺ e⁺]
  8. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d⁺ c⁺ b⁺ a⁺ f⁺ e⁺]
  9. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d^- c⁺ b⁺ a^- f⁺ e⁺]
6. Page 71: A 4.3.4.4.4 multi-directional polyhedron (4.3.4.4.4)
  1. [a⁺ b⁺ b^- a^- c; a^- b^- b⁺ a⁺ c]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- c⁺ b⁺ d^- e⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- c⁺ b⁺ d^- e^-]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e^-]
7. Page 72: A 4.4.4.6 multi-directional polyhedron (same as Wells 4.4.4.6)
8. Page 73: A 3.3.4.3.4.3 multi-directional polyhedron (3.3.4.3.4.3)
  1. [a⁺ a^- b⁺ c⁺ c^- b^-; a^- a⁺ c^^- b^^-]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; b⁺ a⁺ d^^- c^^- f^^- e^^-]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; b⁺ a⁺ e^⁺ f^⁺ c^⁺ d^⁺]
9. Page 74: A {3,8} multi-directional polyhedron (same as Wells (3,8)-I_6t)
10. Page 75: A 3.3.3.4.4.3 multi-directional polyhedron (only when asymmetrically marked) (3.3.3.4.4.3)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ c⁺ b⁺ f⁺ e^- d⁺]
11. Page 76: The {6,6} Coxeter sponge
12. Page 77: A 4.6.4.6 multi-directional polyhedron (4.6.4.6)
  1. [a⁺ a^⁺ a⁺ a^⁺; a^-]
  2. [a⁺ a^⁺ b⁺ b^⁺; a^- a^^- b^-]
  3. [a⁺ a^⁺ b⁺ b^⁺; b^- b^^- a^-]
  4. [a⁺ b⁺ a⁺ b⁺; a^- b^-]
  5. [a⁺ b⁺ a⁺ b⁺; b^^- a^^-]
  6. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]
  7. [a⁺ b⁺ c⁺ d⁺; b^^- a^^- d^^- c^^-]
  8. [a⁺ b⁺ c⁺ d⁺; c^- d^- a^- b^-]
  9. [a⁺ b⁺ c⁺ d⁺; d^^- c^^- b^^- a^^-]
13. Page 78: A 4.4.3.4.4 multi-directional polyhedron (4.4.3.4.4)
  1. [a b⁺ c⁺ c^- b^-; a b⁺ c^-]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ b⁺ d⁺ c⁺ e⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ e^- d⁺ c⁺ b^-]
14. Page 79: A 3.3.3.6.6 multi-directional polyhedron (3.3.3.6.6)
  1. [a⁺ a^- b⁺ c b^-; a⁺ a^- b^- c]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a⁺ b⁺ e⁺ d⁺ c⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; b^- a^- e⁺ d⁺ c⁺]
15. Page 80: A 4.3.4.6.6 multi-directional polyhedron (4.3.4.6.6)
  1. [a⁺ b⁺ b^- a^- c; a^- b^- b⁺ a⁺ c]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- c⁺ b⁺ d^- e^-]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c⁺ b⁺ a⁺ e⁺]
16. Page 81: A second 3.3.4.3.4.3 multi-directional polyhedron (3.3.4.3.4.3) (same as page 73)
  1. [a⁺ a^- b⁺ c⁺ c^- b^-; a⁺ a^- b^- c^-]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b⁺ f⁺ e⁺ d⁺ c⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; b^- a^- f⁺ e⁺ d⁺ c⁺]
17. Page 82: A 3.4.3.4.3.4.3.4 multi-directional polyhedron (3.4.3.4.3.4.3.4)
  1. [a⁺ a^- a^^- a^⁺ a⁺ a^- a^^- a^⁺; a^-]
  2. [a⁺ a^- b⁺ b^- a⁺ a^- b⁺ b^-; a^- a⁺ b^-]
  3. [a⁺ b⁺ c⁺ d⁺ a⁺ b⁺ c⁺ d⁺; b⁺ a⁺ d⁺ c⁺]
  4. [a⁺ a^- b⁺ c⁺ d⁺ d^- c^- b^-; a^- a⁺ b^- d^^- c^^-]
  5. [a⁺ a^- b⁺ c⁺ d⁺ d^- c^- b^-; a^- a⁺ d^⁺ c^- b^⁺]
  6. [a⁺ a^- b⁺ c⁺ d⁺ d^- c^- b^-; b^⁺ b^^- a^⁺ c^- d^-]
  7. [a⁺ a^- b⁺ c⁺ d⁺ d^- c^- b^-; c^^- c^⁺ b^- a^^- d^-]
  8. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; b⁺ a⁺ e^⁺ g⁺ c^⁺ h^⁺ d⁺ f^⁺]
  9. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; b⁺ a⁺ f^^- g⁺ h^^- c^^- d⁺ e^^-]
  10. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; b⁺ a⁺ h⁺ e^^- d^^- g^^- f^^- c⁺]
  11. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; b⁺ a⁺ h⁺ f^⁺ g^⁺ d^⁺ e^⁺ c⁺]
  12. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; c^⁺ e⁺ a^⁺ f^⁺ b⁺ d^⁺ h⁺ g⁺]
  13. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; d^^- e⁺ f^^- a^^- b⁺ c^^- h⁺ g⁺]
  14. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; f⁺ c^^- b^^- e^^- d^^- a⁺ h⁺ g⁺]
  15. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; f⁺ d^⁺ e^⁺ b^⁺ c^⁺ a⁺ h⁺ g⁺]
18. Page 83: Polyhedron (3⁹)P₁ of the Hughes Jones polyhedra
19. Page 84: A different {3,8} multi-directional polyhedron (same as Wells (3,8)-O_4t)
20. Page 85: A {3,7} multi-directional polyhedron (same as Wells (3,7)-I_4t)
21. Page 86: A second 4.4.4.6 multi-directional polyhedron (4.4.4.6)
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c⁺ d^-]
22. Page 87: Polyhedron S1 of the {4,5} cubic lattice polyhedra
23. Page 88: A {4,5} (not cubic lattice) multi-directional polyhedron (same as Wells (4,5)-4tb)
24. Page 89: A 4.4.4.8 multi-directional polyhedron (4.4.4.8)
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d⁺]
25. Page 90: A {3,12} multi-directional polyhedron (same as Wells (3,12)-O_8t)
26. Page 91: A {3,9} multi-directional polyhedron (same as Wells (3,9)-I_8t complement)
27. Page 92: A 3.3.3.4.3.3.3.4 multi-directional polyhedron (3.3.3.4.3.3.3.4)
  1. [a⁺ a^- b⁺ b^- a⁺ a^- b⁺ b^-; a⁺ a^- b^-]
  2. [a⁺ a^- b⁺ c⁺ d⁺ d^- c^- b^-; a⁺ a^- b^- c^- d⁺]
  3. [a⁺ b⁺ c⁺ d⁺ a⁺ b⁺ c⁺ d⁺; a⁺ b⁺ d⁺ c⁺]
  4. [a⁺ b⁺ c⁺ d⁺ a⁺ b⁺ c⁺ d⁺; b^- a^- d⁺ c⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; a⁺ b⁺ h⁺ g⁺ e⁺ f⁺ d⁺ c⁺]
  6. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; a⁺ b⁺ h⁺ g⁺ f^- e^- d⁺ c⁺]
  7. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; b^- a^- h⁺ g⁺ f^- e^- d⁺ c⁺]
  8. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; e⁺ f⁺ h⁺ g⁺ a⁺ b⁺ d⁺ c⁺]
  9. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺ h⁺; f^- e^- h⁺ g⁺ b^- a^- d⁺ c⁺]
28. Page 93: A 3.3.3.8.8 multi-directional polyhedron (3.3.3.8.8)
  1. [a⁺ b⁺ b^- a^- c; a^- b⁺ b^- a⁺ c]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ b⁺ c⁺ a⁺ e⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ b⁺ c⁺ a⁺ e^-]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c^- b^- a⁺ e⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺; d⁺ c^- b^- a⁺ e^-]
29. Page 94: A 3.3.3.4.4.4 multi-directional polyhedron (3.3.3.4.4.4)
  1. [a⁺ a^- b⁺ c⁺ c^- b^-; a⁺ a^- b^- c^-]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b⁺ f⁺ d^- e^- c⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; a⁺ b⁺ f⁺ e⁺ d⁺ c⁺]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; b^- a^- f⁺ d^- e^- c⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; b^- a^- f⁺ e⁺ d⁺ c⁺]
30. Page 95: A 3.3.3.3.3.4.3 multi-directional polyhedron (3.3.3.3.3.4.3)
  1. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺ g⁺; a⁺ e⁺ d^- c^- b⁺ g⁺ f⁺]
31. Page 96: A 4.4.8.8.4 multi-directional polyhedron (4.4.8.8.4)
  1. [a⁺ a^- b⁺ c b^-; a^- a⁺ b^- c]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- e⁺ d⁺ c⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺; a^- b^- e⁺ d^- c⁺]
  4. [a⁺ b⁺ c⁺ d⁺ e⁺; b⁺ a⁺ e⁺ d⁺ c⁺]
  5. [a⁺ b⁺ c⁺ d⁺ e⁺; b⁺ a⁺ e⁺ d^- c⁺]
32. Page 97: A 4.8.4.8 multi-directional polyhedron (same as Wells 4.8.4.8)
33. Page 98: A second 4.4.4.8 multi-directional polyhedron (4.4.4.8) (same as page 3(b))
  1. [a⁺ b⁺ c⁺ d⁺; a^- b⁺ c^- d^-]
34. Page 99: A 3.4.4.4.4.4 multi-directional polyhedron (3.4.4.4.4.4)
  1. [a⁺ b⁺ b^- a^- c⁺ c^-; a^- b^- b⁺ a⁺ c^- c⁺]
  2. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d⁺ b^- c^- a⁺ f⁺ e⁺]
  3. [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺; d⁺ c⁺ b⁺ a⁺ f⁺ e⁺]
35. Page 100: A 6.6.8.8 multi-directional polyhedron (6.6.8.8)
  1. [a⁺ b a^- c; a^- b a⁺ c]
  2. [a⁺ b⁺ c⁺ d⁺; c⁺ b⁺ a⁺ d⁺]
  3. [a⁺ b⁺ c⁺ d⁺; c⁺ b⁺ a⁺ d^-]
36. Page 101: A 4.4.6.8 multi-directional polyhedron (4.4.6.8)
  1. [a⁺ b⁺ c⁺ d⁺; a^- b^- c^- d^-]

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Last updated: July 10, 2020