### The Hughes Jones {3,•} cuboctahedron polyhedra

In 1995, R. Hughes Jones [Enumerating uniform polyhedral surfaces with triangular faces, R. Hughes Jones, Discrete Mathematics, Vol. 138 (1995), pp. 281-292.] noted that vertex stars made up entirely of regular triangles could be created using a cuboctahedron as a template. Consider a cuboctahedron centered at the origin where the distance from the origin to any vertex of the cuboctahedron is the same as the cuboctahedron's edge length. Then any two adjacent vertices of the cuboctahedron along with the center form a regular triangle. A Hamiltonian cycle is a path along the edges that visits a subset of the vertices, but each vertex only once (except for the final point), and that ends with the same vertex as its start. By then forming the triangles created using the edges of the path, each edge joined with the cuboctahedron center to make a regular triangle, a vertex star can be created. For example, one of the faces of the cuboctahedron is a triangle. Traveling around this triangle forms a Hamiltonian cycle with three vertices. Each of these three edges can be joined with the center to make three triangles attached adjacently around a central vertex (the origin), and which combined together form a vertex star. In fact, this vertex star is the one used to construct a tetrahedron. Many other cyclic paths exist on the cuboctahedron. As another example, a path around the diameter forms a planar hexagon and this vertex star creates the {3,6} planar tiling vertex star. In fact, Hughes Jones found 52 such paths, each forming a vertex star. Of these 52 vertex stars, 22 of them form polyhedra. There are 32 of them, though 2 are essentially the same; of these remaining 31, 2 are finite and 3 are infinite in two dimensions only and are described elsewhere here on other pages. That leaves 26 three-dimensionally infinite sponges. Of these 26, there is 1 with 7 triangles in its vertex star, 3 with 8, 13 with 9, and 9 with 12. (While the cuboctahedron generates vertex stars having 10 or 11 triangles, none of them form sponges.) Without a doubt, the sponges having 12 triangles in their vertex stars are very difficult to visualize!

The smallest vertex star created in this way is the one described above that forms the tetrahedron. The next largest one has four triangles and forms the octahedron. These are the two finite polyhedra. No polyhedron can be created from the five-triangle vertex star. Then there are four {3,6} vertex stars, each creating two-dimensional infinite polyhedra:  the first is the coplanar one that creates the {3,6} planar tiling (as mentioned above), the second is the crinkled {3,6} polyhedron, and the final two form folded {3,6} polyhedra. These last two differ only in the dihedral angle of their fold and since they form essentially the same polyhedra they can be considered to be identical for the purposes considered here. Together these form the 6 polyhedra separated from the possible 32 above, leaving 26 remaining infinite sponges formed using 16 different vertex stars. The sponges and their vertex stars are listed and named below using the order and notation Hughes Jones assigned them. In these cases, 3n corresponds to {3,n}, with Pi referring to polyhedra and Vi referring to vertex stars. Hughes Jones also used an adjacency labeling related to the dihedral angles rather than the edges of the vertex star. We do not report these below, but instead only our own assigned incidence symbols. Because of the large number of polygons some of them contain and the unusual symmetries they are thus able to possess, many of them use some of the most exotic vertex and adjacency symbol notations we have encountered so far.

Hughes Jones only reported the different geometric shapes but not the number of different labelings that the symmetric sponges could support. Therefore these have been added here. When only a single labeling appears it is because there are no symmetries that can support multiple labelings. Finally, for the first six polyhedra their multiple labelings are reported on the other pages where these polyhedra have already appeared.

#### The 32 Hughes Jones {3,•} cuboctahedron sponges

1. (33)V1   tetrahedron
2. (34)V1   octahedron
3. (36)V1   folded {3,6}
4. (36)V2   crinkled {3,6}
5. (36)V3   folded {3,6}
6. (36)V4   {3,6} planar tiling

7. (37)P1   (37)V3
1. [a+ b+ c+ d+ e+ f+ g+;  a+ f^+ d+ c+ g^+ b^+ e^+]

8. (38)P1   (38)V5
1. [a+ b+ c+ d+ d- c- b- a-;  b+ a+ c- d+]
2. [a+ b+ c+ d+ e+ f+ g+ h+;  b+ a+ f+ d+ e+ c+ h+ g+]
3. [a+ b+ c+ d+ e+ f+ g+ h+;  b+ a+ f+ e- d- c+ h+ g+]

9. (38)P2   (38)V6
1. [a+ b+ c+ d+ a+ b+ c+ d+;  a+ c+ b+ d+]
2. [a+ b+ c+ d+ e+ f+ g+ h+;  a+ c+ b+ h+ e+ g+ f+ d+]
3. [a+ b+ c+ d+ e+ f+ g+ h+;  e+ c+ b+ d+ a+ g+ f+ h+]

10. (38)P3   (38)V3
1. [a+ b+ c+ d+ e+ f+ g+ h+;  a+ c+ b+ h+ f+ e+ g+ d+]

11. (39)P1   (39)V6
1. [a b+ b- a b+ b- a b+ b-;  a b+ b-]
2. [a+ b+ c+ a+ b+ c+ a+ b+ c+;  a+ b+ c+]
3. [a+ b+ c+ a+ b+ c+ a+ b+ c+;  a+ c- b-]
4. [a b+ c+ d+ e+ e- d- c- b-;  a e+ c+ d+ b+]
5. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e+ c+ d+ b+ i+ g+ h+ f+]
6. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e+ h- g- b+ i+ d- c- f+]
7. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ f- c+ g- i- b- d- h+ e-]
8. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ f- h- d+ i- b- g+ c- e-]

12. (39)P2   (39)V4
1. [a b+ c+ d+ e+ e- d- c- b-;  a e^- c+ d^- b^-]
2. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e^- c+ d^- b^- i^- g^- h+ f^-]
3. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e^- h- g^+ b^- i^- d^+ c- f^-]
4. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ f^+ c+ g^+ i^+ b^+ d^+ h+ e^+]
5. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ f^+ h- d^- i^+ b^+ g^- c- e^+]

13. (39)P3   (39)V8
1. [a b+ c+ d+ e+ e- d- c- b-;  a e+ c+ d+ b+]
2. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e+ c+ d+ b+ i+ g+ h+ f+]
3. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e+ h- g- b+ i+ d- c- f+]
4. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ f- c+ g- i- b- d- h+ e-]
5. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ f- h- d+ i- b- g+ c- e-]

14. (39)P4   (39)V8 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  c^+ f^- a^+ g- i- b^- d- h+ e-]

15. (39)P5   (39)V2
1. [a b+ c+ d+ e+ e- d- c- b-;  a e^- c^- d+ b^-]
2. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e^- c^- d+ b^- i^- g+ h^- f^-]
3. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e^- h^+ g- b^- i^- d- c^+ f^-]
4. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ f^+ c^- g- i^+ b^+ d- h^- e^+]
5. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ f^+ h^+ d+ i^+ b^+ g+ c^+ e^+]

16. (39)P6   (39)V3
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e^- c+ d^- b^- i^- g+ h^- f^-]

17. (39)P7   (39)V3 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e^- g+ h^- b^- i^- c+ d^- f^-]

18. (39)P8   (39)V7
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e+ c+ d+ b+ i+ g+ h+ f+]

19. (39)P9   (39)V7 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ e+ g^- h^- b+ i+ c^- d^- f+]

20. (39)P10   (39)V9
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ g^+ c^- h- e^- i^+ b^+ d- f^+]

21. (39)P11   (39)V9 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ g^+ e^+ h+ c^+ i^+ b^+ d+ f^+]

22. (39)P12   (39)V1
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ g^+ c+ h^+ e+ i^+ b^+ d^+ f^+]

23. (39)P13   (39)V1 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+;  a+ g^+ e- h^- c- i^+ b^+ d^- f^+]

24. (312)P1   (312)V5
1. [a*+ b+ c+ d*+ c^+ b^+ a*+ b+ c+ d*+ c^+ b^+;  d*- b+ c^- a*-]
2. [a+ b+ c+ d+ e+ f+ a+ b+ c+ d+ e+ f+;  d^- b+ c^- a^- e^- f+]
3. [a+ b+ c+ d+ e+ f+ a+ b+ c+ d+ e+ f+;  d- b+ e- a- c- f+]
4. [a+ b+ c+ d+ e+ f+ a+ b+ c+ d+ e+ f+;  d- f^+ c^- a- e^- b^+]
5. [a+ b+ c+ d+ e+ f+ a+ b+ c+ d+ e+ f+;  d^- f^+ e- a^- c- b^+]
6. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  d^- b+ c^- a^- e^- l+ j^- h+ i^- g^- k^- f+]
7. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  d^- b+ c^- a^- k^- f+ j^- h+ i^- g^- e^- l+]
8. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  d^- h+ i^- a^- e^- l+ j^- b+ c^- g^- k^- f+]
9. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  d^- h+ i^- a^- k^- f+ j^- b+ c^- g^- e^- l+]

25. (312)P2   (312)V4
1. [a+ b+ c+ a^- b^- c^- a+ b+ c+ a^- b^- c^-;  a+ b^- c+]
2. [a+ b+ c+ d+ e+ f+ a+ b+ c+ d+ e+ f+;  a+ b^- f^- d+ e^- c^-]
3. [a+ b+ c+ d+ e+ f+ a+ b+ c+ d+ e+ f+;  a+ e+ c+ d+ b+- f+]
4. [a+ b+ c+ d+ e+ f+ a+ b+ c+ d+ e+ f+;  d^- b^- c+ a^- e^- f+]
5. [a+ b+ c+ d+ e+ f+ a+ b+ c+ d+ e+ f+;  d^- e+ f^- a^- b+ c^-]
6. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  a+ b^- l^- j+ e^- i^- g+ h^- f^- d+ k^- c^-]
7. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  a+ e+ i+ j+ b+ l+ g+ k+ c+ d+ h+ f+]
8. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  a+ h^- f^- j+ k^- c^- g+ b^- l^- d+ e^- i^-]
9. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  a+ k+ c+ j+ h+ f+ g+ e+ i+ d+ b+ l+]
10. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  d^- b^- c+ a^- e^- l+ j^- h^- i+ g^- k^- f+]
11. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  d^- b^- c+ a^- k^- f+ j^- h^- i+ g^- e^- l+]
12. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  d^- h^- i+ a^- e^- l+ j^- b^- c+ g^- k^- f+]
13. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  d^- h^- i+ a^- k^- f+ j^- b^- c+ g^- e^- l+]

26. (312)P3   (312)V2
1. [a*+ b+ c+ d+ e+ f+ g*+ f^+ e^+ d^+ c^+ b^+;  g*+ e+ c+ d+ b+ f+ a*+]
2. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g+ e+ c+ d+ b+ f+ a+ h+ l+ j+ k+ i+]
3. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g^+ e+ k^+ j^+ b+ h^+ a^+ f^+ l+ d^+ c^+ i+]
4. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g+ i^+ c+ j^+ l^+ h^+ a+ f^+ b^+ d^+ k+ e^+]
5. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g^+ i^+ k^+ d+ l^+ f+ a^+ h+ b^+ j+ c^+ e^+]

27. (312)P4   (312)V2 (same as previous)
1. [a*+ b+ c+ d+ e+ f+ g*+ f^+ e^+ d^+ c^+ b^+;  g*+ e- c+ f- b- d-]
2. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g+ e- c+ f- b- d- a+ j- l- h- k+ i-]
3. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g^+ e- k^+ h^- b- j^- a^+ d^- l- f^- c^+ i-]
4. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g+ i^- c+ h^- l^- j^- a+ d^- b^- f^- k+ e^-]
5. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g^+ i^- k^+ f- l^- d- a^+ j- b^- h- c^+ e^-]

28. (312)P5   (312)V2 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  g^+ i^- k^+ d+ l^+ f+ a^+ j- b^- h- c^+ e^+]

29. (312)P6   (312)V3
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  a+ b^- l^- d+ k^- f+ j^- h+ i^- g^- e^- c^-]

30. (312)P7   (312)V3 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  a+ b^- l^- f- k^+ d- j^- h+ i^- g^- e^+ c^-]

31. (312)P8   (312)V3 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  h+ i^- l^- d+ k^- f+ j^- a+ b^- g^- e^- c^-]

32. (312)P9   (312)V3 (same as previous)
1. [a+ b+ c+ d+ e+ f+ g+ h+ i+ j+ k+ l+;  h+ i^- l^- f- k^+ d- j^- a+ b^- g^- e^+ c^-]