The {4,4} polyhedra

The patterns found for the {4,4} polyhedra are very similar to those for the {3,6} ones. The {4,4} polyhedra are built from vertex stars consisting of four squares arranged around a central vertex. When the squares are all coplanar, the polyhedron formed is a plane. Treating this polyhedron as a two-dimensional plane made up of square tilings, Grünbaum and Shephard found 36 possible incidence symbols. Since we discuss these on another page, we do not repeat them here but instead consider what happens when the vertex star is bent along the central edges that separate the four squares into two pairs of coplanar squares.

Clearly, the dihedral angle under consideration can vary between 0 and 180 degrees. An obvious polyhedron that results is what we call the "folded" one, where each bend is in the opposite direction from the previous one. This results in a folded shape like a collapsible fan or the air chamber of a bellows or accordion. This shape can be labeled isogonally, and clearly it can be constructed for any angle. Considering its adjacency symbol, it must always be the case that red faces must be adjacent to red faces or black faces to black faces along the folded direction, so that the adjacency symbols must always contain the '^' marking for both edges. (We choose the 'a' edge to be along the bent, center line.)

Another shape that also exists as an isogonal polyhedron occurs when the folds change direction every second bend instead of every bend. This produces a "corrugated" appearance. For example, when the bend is 90° the polyhedron cross-section looks like a square wave. Clearly the isogonality is not affected by the bend angle so this polyhedron can also be constructed for any angle. Except that when the angle is 60° or less, the edges touch and the polyhedron is no longer acoptic. Since one of the bends alternates but the other does not, the adjacency symbol will have one edge with a '^' mark and one not. (Which one to choose is arbitrary, so each unique incidence symbol will always have a duplicate one.)

Only one other class of acoptic polyhedron exists for this vertex star, and it occurs when the angle is chosen to be one of the angles of a regular polygon. In this case the bends wrap around and form a rod with polygonal cross-section. These polyhedra are infinite in only one dimension, and their incidence symbols have no '^' markings. Consequently, we have chosen to consider six different bend angles below: π/3, π/2, 3π/5, 2π/3, 5π/7, and 3π/4 (60°, 90°, 108°, 120°, 128-4/7°, and 135°). For simplicity, since they form the 3- to 8-sided polygons in cross-section, we refer to them as the 3- to 8-angle rod polyhedra.

The folded polyhedra exist for all six angles, and their labeled versions are all identical. The corrugated ones exist for all angles except the 60° bend, because its faces bend back onto the adjacent ones and the vertices coincide; thus it is not acoptic. Otherwise, their labeled versions are also all identical. Note that because the corrugated ones are asymmetrical in their use of the '^' edges, they can only exist for vertex symbols that label one coplanar edge asymmetrically different from the other edge. The labeled rod polyhedra are more varied in relation to the bend angle: some of them exist for all angles, some only when the cross-sectional polygon has an even number of edges, and two of them only when the number of polygon edges is a multiple of four.

The ( vertex stars

The "folded" polyhedra

and their isogonal labelings (for all angles):
  1. [a b a b;  a b^]
  2. [a+ b+ a+ b+;  a+ b^+]
  3. [a+ b+ a+ b+;  a+ b^-]
  4. [a+ b+ a+ b+;  a- b^+]
  5. [a+ b+ a+ b+;  a- b^-]
  6. [a b+ c b-;  a b^+ c]
  7. [a b+ c b-;  a b^- c]
  8. [a b+ c b-;  c b^+ a]
  9. [a b+ c b-;  c b^- a]
  10. [a+ b a- c;  a+ b^ a- c^]
  11. [a+ b a- c;  a- b^ a+ c^]
  12. [a+ b+ c+ d+;  a+ b^+ c+ d^+]
  13. [a+ b+ c+ d+;  a+ b^+ c- d^+]
  14. [a+ b+ c+ d+;  a+ b^- c+ d^-]
  15. [a+ b+ c+ d+;  a- b^+ c- d^+]
  16. [a+ b+ c+ d+;  a- b^+ c- d^-]
  17. [a+ b+ c+ d+;  a- b^- c- d^-]
  18. [a+ b+ c+ d+;  a+ d^+ c+ b^+]
  19. [a+ b+ c+ d+;  a+ d^- c+ b^-]
  20. [a+ b+ c+ d+;  a+ d^- c- b^-]
  21. [a+ b+ c+ d+;  a- d^+ c- b^+]
  22. [a+ b+ c+ d+;  a- d^- c- b^-]
  23. [a+ b+ c+ d+;  c+ b^+ a+ d^+]
  24. [a+ b+ c+ d+;  c+ b^+ a+ d^-]
  25. [a+ b+ c+ d+;  c+ b^- a+ d^-]
  26. [a+ b+ c+ d+;  c- b^+ a- d^+]
  27. [a+ b+ c+ d+;  c- b^- a- d^-]
  28. [a+ b+ c+ d+;  c+ d^+ a+ b^+]
  29. [a+ b+ c+ d+;  c+ d^- a+ b^-]
  30. [a+ b+ c+ d+;  c- d^+ a- b^+]
  31. [a+ b+ c+ d+;  c- d^- a- b^-]

The "corrugated" polyhedra

and their isogonal labelings (for all angles):
  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+;  a- b- c- d^+]
  5. [a+ b+ c+ d+;  a- b- c- d^-]
  6. [a+ b+ c+ d+;  c+ b+ a+ d^+]
  7. [a+ b+ c+ d+;  c+ b+ a+ d^-]
  8. [a+ b+ c+ d+;  c+ b- a+ d^+]
  9. [a+ b+ c+ d+;  c+ b- a+ d^-]

The "rod" polyhedra

and their isogonal labelings (those for all angles):
  1. [a b a b;  a b]
  2. [a+ b+ a+ b+;  a+ b+]
  3. [a+ b+ a+ b+;  a+ b-]
  4. [a+ b+ a+ b+;  a- b+]
  5. [a+ b+ a+ b+;  a- b-]
  6. [a b+ c b-;  a b- c]
  7. [a b+ c b-;  c b- a]
  8. [a+ b+ c+ d+;  a+ d+ c+ b+]
  9. [a+ b+ c+ d+;  a+ d+ c- b+]
  10. [a+ b+ c+ d+;  a- d+ c- b+]
  11. [a+ b+ c+ d+;  c+ d+ a+ b+]
  12. [a+ b+ c+ d+;  c- d+ a- b+]
additional ones for the (even) 4-, 6-, and 8-angles:
  1. [a b+ c b-;  a b+ c]
  2. [a b+ c b-;  c b+ a]
  3. [a+ b a- c;  a+ b a- c]
  4. [a+ b a- c;  a- b a+ c]
  5. [a+ b+ c+ d+;  a+ b+ c+ d+]
  6. [a+ b+ c+ d+;  a+ b- c+ d-]
  7. [a+ b+ c+ d+;  a+ b- c- d-]
  8. [a+ b+ c+ d+;  a- b+ c- d+]
  9. [a+ b+ c+ d+;  a- b- c- d-]
  10. [a+ b+ c+ d+;  a+ d- c+ b-]
  11. [a+ b+ c+ d+;  a- d- c- b-]
  12. [a+ b+ c+ d+;  c+ b+ a+ d+]
  13. [a+ b+ c+ d+;  c+ b- a+ d-]
  14. [a+ b+ c+ d+;  c- b+ a- d+]
  15. [a+ b+ c+ d+;  c- b- a- d-]
  16. [a+ b+ c+ d+;  c+ d- a+ b-]
  17. [a+ b+ c+ d+;  c- d- a- b-]
and additional ones for the (4-multiple) 4- and 8-angles:
  1. [a+ b+ c+ d+;  a- b+ c- d-]
  2. [a+ b+ c+ d+;  c+ b+ a+ d-]

Return to main page

Last updated: April 18, 2019