The {4,6} sponges

When 6 squares, or 4-gons, are arranged around a central vertex with all squares aligned along the planes of the cubic lattice, two non-isometric vertex stars can be produced. The first one is the regular Coxeter-Petrie {4,6} sponge. It has dihedral angles of 90° between every square. At each edge of the star there is a bend in the opposite direction from the previous bend, until the squares wrap around to the starting point. This makes the star extremely symmetric, so that its most symmetric vertex symbol is a a^ a a^ a a^. With such a high degree of symmetry, there are really no choices for edges to be joined to create different sponges, so it is no surprise that there is only a single geometric shape that can be created. On the other hand, so much symmetry allows a very large number of labeled versions. There are 16 different vertex symbols that can be assigned to this vertex star, but 6 of them are duplicates, leaving only 10 to analyze. Altogether, these 10 vertex symbols combine to produce 49 different labeled versions.

The second, non-regular, vertex star differs from the first in that one edge has no bend, but instead the next square continues on in the same plane. Then the same bending pattern as the regular star resumes until, on the opposite side of the star, the same coplanar dihedral angle repeats. This allows the squares to match back up on returning to the starting edge. In its most symmetric form, this vertex star has vertex symbol a⁺ b*⁺ a^⁺ a⁺ b*⁺ a^⁺. Note the use of the '*' notation within the vertex symbol, showing that the edges are the same when both flipped and unflipped, and which can only occur when the dihedral angle is 180°. Having a reduced symmetry, this vertex star is able to produce 5 different sponges. Only one of them is so asymmetric that it cannot support labeled versions.

The regular {4,6} sponge   (4.4.4.4.4.4)

1.    [a a^ a a^ a a^;   a]
2.    [a⁺ a^⁺ a⁺ a^⁺ a⁺ a^⁺;   a⁺]
3.    [a⁺ a^⁺ a⁺ a^⁺ a⁺ a^⁺;   a^-]
4.    [a⁺ a^^- a⁺ a^^- a⁺ a^^-;   a⁺]
5.    [a⁺ a^^- a⁺ a^^- a⁺ a^^-;   a^-]
6.    [a b a b a b;   a b]
7.    [a b a b a b;   b^ a^]
8.    [a⁺ b⁺ a⁺ b⁺ a⁺ b⁺;   a⁺ b⁺]
9.    [a⁺ b⁺ a⁺ b⁺ a⁺ b⁺;   a⁺ b^-]
10.    [a⁺ b⁺ a⁺ b⁺ a⁺ b⁺;   a^- b^-]
11.    [a⁺ b⁺ a⁺ b⁺ a⁺ b⁺;   b^⁺ a^⁺]
12.    [a⁺ b⁺ a⁺ b⁺ a⁺ b⁺;   b^^- a^^-]
13.    [a⁺ b a^- a^^- b^ a^⁺;   a^- b]
14.    [a⁺ b⁺ c⁺ a^^- b^^- c^^-;   a⁺ b^- c^-]
15.    [a⁺ b⁺ c⁺ a^^- b^^- c^^-;   a^- b^- c^-]
16.    [a⁺ b⁺ c⁺ a^^- b^^- c^^-;   a⁺ c^^- b^^-]
17.    [a⁺ b⁺ c⁺ a^^- b^^- c^^-;   a^- c^^- b^^-]
18.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   a^- b⁺ c^-]
19.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   a^- b^- c^-]
20.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   c⁺ b⁺ a⁺]
21.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   c⁺ b^- a⁺]
22.    [a b⁺ c⁺ d c^- b^-;   a b^- c^- d]
23.    [a b⁺ c⁺ d c^- b^-;   a c^^- b^^- d]
24.    [a b⁺ c⁺ d c^- b^-;   d^ b^- c^- a^]
25.    [a b⁺ c⁺ d c^- b^-;   d^ c^^- b^^- a^]
26.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ b^- c^- d⁺ e^- f^-]
27.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ b^- c^- d^- e^- f^-]
28.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- d^- e^- f^-]
29.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- d^- f^^- e^^-]
30.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ b^- e⁺ d⁺ c⁺ f^-]
31.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ b^- e⁺ d^- c⁺ f^-]
32.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- e⁺ d⁺ c⁺ f^-]
33.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- e⁺ d^- c⁺ f^-]
34.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ c^^- b^^- d⁺ f^^- e^^-]
35.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- c^^- b^^- d^- f^^- e^^-]
36.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ e^⁺ f^⁺ d⁺ b^⁺ c^⁺]
37.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- e^⁺ f^⁺ d^- b^⁺ c^⁺]
38.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ f⁺ e⁺ d⁺ c⁺ b⁺]
39.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ f⁺ e⁺ d^- c⁺ b⁺]
40.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- f⁺ e⁺ d^- c⁺ b⁺]
41.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^⁺ b^- c^- a^⁺ e^- f^-]
42.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^^- b^- c^- a^^- e^- f^-]
43.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^^- b^- c^- a^^- f^^- e^^-]
44.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^⁺ c^^- b^^- a^⁺ f^^- e^^-]
45.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^^- c^^- b^^- a^^- f^^- e^^-]
46.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^⁺ e^⁺ f^⁺ a^⁺ b^⁺ c^⁺]
47.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^^- e^⁺ f^⁺ a^^- b^⁺ c^⁺]
48.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^⁺ f⁺ e⁺ a^⁺ c⁺ b⁺]
49.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^^- f⁺ e⁺ a^^- c⁺ b⁺]

Non-regular {4,6} sponges   (4.4.4.4.4.4)

• N1
  1.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ b^- c^- d^- e^- f^-]
  
  
• N2
  1.    [a⁺ b*⁺ a^⁺ c⁺ d*⁺ c^⁺;   a⁺ b*^- a^⁺ c^- d*^- c^^-]
  2.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ b^- c⁺ d^- e^- f^-]
  3.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a⁺ b^^- c⁺ f^^- e^^- d^^-]
  4.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^⁺ b^- a^⁺ d^- e^- f^-]
  5.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^⁺ b^^- a^⁺ f^^- e^^- d^^-]
  
  
• N3
  1.    [a⁺ b*⁺ a^⁺ c⁺ d*⁺ c^⁺;   a^- b*⁺ a^^- c^- d*^- c^^-]
  2.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b⁺ c^- d^- e^- f^-]
  3.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b⁺ c^- d^- e^^- f^-]
  4.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- d^- e^⁺ f^-]
  5.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^⁺ c^- d^- e^^- f^-]
  6.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b⁺ c^- f^^- e^- d^^-]
  7.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b⁺ c^- f^^- e^^- d^^-]
  8.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- f^^- e⁺ d^^-]
  9.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- f^^- e^⁺ d^^-]
  10.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^⁺ c^- f^^- e^- d^^-]
  11.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^⁺ c^- f^^- e^^- d^^-]
  12.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^^- c^- f^^- e⁺ d^^-]
  13.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^^- c^- f^^- e^⁺ d^^-]
  14.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b⁺ a^^- f^^- e^- d^^-]
  15.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b⁺ a^^- f^^- e^^- d^^-]
  16.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b^- a^^- f^^- e^⁺ d^^-]
  17.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b^⁺ a^^- f^^- e^^- d^^-]
  
  
• N4
  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⁺;   a^- b^⁺ c^-]
  4.    [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺;   c^^- b⁺ a^^-]
  5.    [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺;   c^^- b^⁺ a^^-]
  6.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   a^- b⁺ c^-]
  7.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   a^- b^⁺ c^-]
  8.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   c^^- b⁺ a^^-]
  9.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   c^^- b^⁺ a^^-]
  10.    [a⁺ b*⁺ a^⁺ c⁺ d*⁺ c^⁺;   a^- b*⁺ a^^- c^- d*⁺ c^^-]
  11.    [a⁺ b*⁺ a^⁺ c⁺ d*⁺ c^⁺;   c^- b*⁺ c^^- a^- d*⁺ a^^-]
  12.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b⁺ c^- d^- e⁺ f^-]
  13.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b⁺ c^- d^- e^⁺ f^-]
  14.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^⁺ c^- d^- e^⁺ f^-]
  15.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b⁺ c^- f^^- e⁺ d^^-]
  16.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b⁺ c^- f^^- e^⁺ d^^-]
  17.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^⁺ c^- f^^- e⁺ d^^-]
  18.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^⁺ c^- f^^- e^⁺ d^^-]
  19.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- e⁺ c^- d^- b⁺ f^-]
  20.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- e^⁺ c^- d^- b^⁺ f^-]
  21.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b⁺ a^^- f^^- e⁺ d^^-]
  22.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b⁺ a^^- f^^- e^⁺ d^^-]
  23.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b^⁺ a^^- f^^- e^⁺ d^^-]
  24.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- e⁺ a^^- f^^- b⁺ d^^-]
  25.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- e^⁺ a^^- f^^- b^⁺ d^^-]
  26.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^- b⁺ f^- a^- e⁺ c^-]
  27.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^- b^⁺ f^- a^- e^⁺ c^-]
  28.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^- e⁺ f^- a^- b⁺ c^-]
  29.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^- e^⁺ f^- a^- b^⁺ c^-]
  30.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   f^^- b⁺ d^^- c^^- e⁺ a^^-]
  31.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   f^^- b^⁺ d^^- c^^- e^⁺ a^^-]
  32.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   f^^- e⁺ d^^- c^^- b⁺ a^^-]
  33.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   f^^- e^⁺ d^^- c^^- b^⁺ a^^-]
  
  
• N5
  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⁺;   a^- b^^- c^-]
  4.    [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺;   c^^- b^- a^^-]
  5.    [a⁺ b⁺ c⁺ a⁺ b⁺ c⁺;   c^^- b^^- a^^-]
  6.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   a^- b^- c^-]
  7.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   a^- b^^- c^-]
  8.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   c^^- b^- a^^-]
  9.    [a⁺ b⁺ c⁺ c^⁺ b^⁺ a^⁺;   c^^- b^^- a^^-]
  10.    [a⁺ b*⁺ a^⁺ c⁺ d*⁺ c^⁺;   a^- b*^- a^^- c^- d*^- c^^-]
  11.    [a⁺ b*⁺ a^⁺ c⁺ d*⁺ c^⁺;   c^- b*^- c^^- a^- d*^- a^^-]
  12.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- d^- e^- f^-]
  13.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- d^- e^^- f^-]
  14.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^^- c^- d^- e^^- f^-]
  15.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- f^^- e^- d^^-]
  16.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- c^- f^^- e^^- d^^-]
  17.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^^- c^- f^^- e^- d^^-]
  18.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^^- c^- f^^- e^^- d^^-]
  19.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- d^^- c^^- e^- f^-]
  20.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- b^- f^- d^- e^- c^-]
  21.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- e^- c^- d^- b^- f^-]
  22.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- e^^- c^- d^- b^^- f^-]
  23.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- e^^- d^^- c^^- b^^- f^-]
  24.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   a^- e^- f^- d^- b^- c^-]
  25.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b^- a^^- f^^- e^- d^^-]
  26.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b^- a^^- f^^- e^^- d^^-]
  27.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- b^^- a^^- f^^- e^^- d^^-]
  28.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- e^- a^^- f^^- b^- d^^-]
  29.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   c^^- e^^- a^^- f^^- b^^- d^^-]
  30.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^- b^- f^- a^- e^- c^-]
  31.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^- b^^- f^- a^- e^^- c^-]
  32.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^- e^- f^- a^- b^- c^-]
  33.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   d^- e^^- f^- a^- b^^- c^-]
  34.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   f^^- b^- d^^- c^^- e^- a^^-]
  35.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   f^^- b^^- d^^- c^^- e^^- a^^-]
  36.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   f^^- e^- d^^- c^^- b^- a^^-]
  37.    [a⁺ b⁺ c⁺ d⁺ e⁺ f⁺;   f^^- e^^- d^^- c^^- b^^- a^^-]