In 1977, A. F. Wells wrote a book summarizing what was known at that
time about the structure of crystals. [*Three-dimensional Nets and
Polyhedra*, A. F. Wells (1977); Wiley, New York.] In a crystal,
atoms or molecules bond together along directional lines in repeating
patterns to construct the crystalline material. The study of these
structures is called crystallography. If we refer to the atoms or
molecules as points and the bonds between them as links we can model the
crystal with points interconnected via links and these are called nets.
Most of the book deals with all of the known (and many as of yet
unknown) nets that appear in crystals. By "inflating" the points and
links, three-dimensional shapes appear, though with curved surfaces.
Surrounding the points with polygons on these surfaces, they can be
turned into polyhedra, though they are still fit onto curved surfaces.
But some of these curved surface polyhedra can also be constructed with
planar polygons and these then become relevant to our search here for
isogonal polyhedra. Wells deals with these planar polyhedra only at the
end of the book, in chapters 16-18, and primarily chapters 17-18.
Almost all of the nets produce Platonic polyhedra only (chapter 17) and
are shown on this page. We discuss the polyhedra created from mixed
polygons (chapters 16 and 18) on another page. Nevertheless, the
earlier book chapters on nets do help to suggest which {m,n} sponges
simply may not exist; that is, if no such net is known to exist then the
likelihood of an existing planar polygon polyhedron is considerably
reduced.

On the other hand, though this method of constructing polyhedra produces rather pretty and elegant-looking sponges, it does so because the polyhedra it generates are composed of assemblages of finite Platonic and/or Archimedean solids. This is natural, since the shapes were originally nets, and so they arise from interconnected points and links. But not all sponges will necessarily have such obvious constructions so the Wells list of sponges misses many that have been found through other means. Nevertheless, it is a substantial collection, and we definitely want to include them all here.

Another thing that arises from the approach used by Wells is that he focuses on the shape that is constructed by assembling the octahedra and tetrahedra, etc. These shapes spread throughout space, infinitely in three dimensions, but one can also consider the volume of space that they do not fill and Wells refers to this as the complementary polyhedron. Sometimes constructing the complementary polyhedron leads to a different-looking shape and so Wells shows examples of both polyhedra. (Many of the polyhedra are self-complementary, that is, their complementary polyhedron is the same as the original polyhedron.) Moreover, since Wells describes the polyhedra in terms of the finite building blocks that make up the shapes, his pictures usually show only partial vertex stars at the boundaries of his models. This reveals the underlying finite building blocks but tends to make identification of the vertex stars more difficult. Of course, in our approach here we focus on the vertex stars since this method allows a more general treatment of isogonal polyhedra rather than just those constructed from finite building blocks and so our pictures do not cut off the vertex stars at the peripheries. But, more importantly, we consider the polyhedron to be the surface itself rather than the enclosed volumes. In general, infinite isogonal polyhedra divide space into two regions and it is the surface of this dividing "line" that is what is of interest, not the enclosed volume. Thus, seen in this way, the polyhedron and its complement simply represent two ways of looking at the same polyhedron, either from the "outside" in or the "inside" out, and so there is no need to separately discuss the polyhedron and its complement. However, in order to accomodate those who wish to compare our models with the pictures in the book, in those cases where Wells has shown photographs of both the polyhedron and its complement we also show two VRML models. We have tried to distinguish them by switching the colors or perspectives, or selecting different volumes to enclose the two models, which may allow the assemblages of finite polyhedra to be more easily recognized. For models that appear in Figures in the book, we note below the corresponding Figure number. We also note, when Wells described them, whether the polyhedron is self-complementary or not. Of course, Wells did not include the possible labelings of the sponges which we have found and do show below. In describing the sponges Wells used (m,n) instead of our {m,n} and the letters 'T', 'O', 'I', and 't' which stand for Tetrahedral, Octahedral, Icosahedral, and tunnels, respectively, and describe the geometrical shapes of the "inflated" points and number of exiting tunnels (links).

As an interesting side note, Wells considered the regular {6,6} Coxeter-Petrie sponge to be inadmissible as one of his infinite polyhedra, even though he did include it in his book. It was considered inadmissible because it was constructed of two kinds of polyhedra--tetrahedra and truncated tetrahedra--at each vertex. Yet it is clearly a valid isogonal polyhedron with congruent and vertex-transitive vertex stars at each point. This helps explain how various researchers have missed finding some isogonal polyhedra in the past: they imposed (explicitly or implicitly) unnecessary criteria on their potential polyhedral candidates.

Another interesting aspect here is that the (3,8)-I6t polyhedron can be constructed in two different ways, with both incidence symbols shown below. Even though the two sponges are asymmetric and have different incidence symbols, they have the same shape. This can be seen not only by studying the VRML models but by noticing that the reflected edges in the second adjacency symbol are the same as in the first, unreflected one. Yet the two sponges are not alternate labelings of the same sponge because there is no common symmetric sponge between them. The reflective symmetry of the vertex star and the chirality of the snub cube combine to produce this result.

A number of other researchers have created Web pages that show off some of these sponges, either in pictures or also in VRML models. In doing so they have created them by assembling the models in the same way as Wells, using colored assemblages of finite Platonic or Archimedean solids, rather than colored vertex stars. Since viewing the sponges using this point of view can sometimes give a different (and possibly more understandable) perspective on their structure, you may want to search the Web for such pictures. Two people who were particularly involved in creating such models were Steven Dutch and Melinda Green.

Finally, we note that some of the Wells sponges are the same as those found by Hughes Jones in 1995. For these we also note below the name given them by Hughes Jones.

- (3,7)-I
_{4t}(3.3.3.3.3.3.3) Figure 17.10- [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}; b^{+}a^{+}f^{+}e^{-}d^{-}c^{+}g^{+}]

- [a
- (3,8)-O
_{3t}(3.3.3.3.3.3.3.3) Figure 17.2 (Hughes Jones (3^{8})P_{3})- [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}; a^{+}c^{+}b^{+}h^{+}f^{+}e^{+}g^{+}d^{+}]

- [a
- (3,8)-O
_{4t}(3.3.3.3.3.3.3.3) Figure 17.7- [a
^{+}b^{+}b^{-}a^{-}a^{+}b^{+}b^{-}a^{-}; a^{+}b^{-}] - [a
^{+}b^{+}c^{+}d^{+}a^{+}b^{+}c^{+}d^{+}; a^{+}c^{+}b^{+}d^{+}] - [a
^{+}b^{+}c^{+}d^{+}a^{+}b^{+}c^{+}d^{+}; d^{-}c^{+}b^{+}a^{-}]

- [a
- (3,8)-I
_{6t}(3.3.3.3.3.3.3.3) Figure 17.11 (complementary polyhedron Figure 17.12)- [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}; b^{+}a^{+}g^{+}e^{+}d^{+}f^{+}c^{+}h^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}; b^{+}a^{+}g^{-}e^{+}d^{+}h^{-}c^{-}f^{-}]

- [a
- (3,9)-T
_{4t}(3.3.3.3.3.3.3.3.3) Figure 17.4 (Hughes Jones (3^{9})P_{1})- [a b
^{+}b^{-}a b^{+}b^{-}a b^{+}b^{-}; a b^{+}b^{-}] - [a
^{+}b^{+}c^{+}a^{+}b^{+}c^{+}a^{+}b^{+}c^{+}; a^{+}b^{+}c^{+}] - [a
^{+}b^{+}c^{+}a^{+}b^{+}c^{+}a^{+}b^{+}c^{+}; a^{+}c^{-}b^{-}] - [a b
^{+}c^{+}d^{+}e^{+}e^{-}d^{-}c^{-}b^{-}; a e^{+}c^{+}d^{+}b^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}e^{+}c^{+}d^{+}b^{+}i^{+}g^{+}h^{+}f^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}e^{+}h^{-}g^{-}b^{+}i^{+}d^{-}c^{-}f^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}f^{-}c^{+}g^{-}i^{-}b^{-}d^{-}h^{+}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}f^{-}h^{-}d^{+}i^{-}b^{-}g^{+}c^{-}e^{-}]

- [a b
- (3,9)-T'
_{4t}(3.3.3.3.3.3.3.3.3) Figure 17.5 (Hughes Jones (3^{9})P_{3})- [a b
^{+}c^{+}d^{+}e^{+}e^{-}d^{-}c^{-}b^{-}; a e^{+}c^{+}d^{+}b^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}e^{+}c^{+}d^{+}b^{+}i^{+}g^{+}h^{+}f^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}e^{+}h^{-}g^{-}b^{+}i^{+}d^{-}c^{-}f^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}f^{-}c^{+}g^{-}i^{-}b^{-}d^{-}h^{+}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}f^{-}h^{-}d^{+}i^{-}b^{-}g^{+}c^{-}e^{-}]

- [a b
- (3,9)-I
_{8t}(3.3.3.3.3.3.3.3.3) Figure 17.13 (complementary polyhedron Figure 17.14)- [a b
^{+}c^{+}d^{+}e^{+}e^{-}d^{-}c^{-}b^{-}; a e^{+}d^{-}c^{-}b^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}e^{+}d^{-}c^{-}b^{+}i^{+}h^{-}g^{-}f^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}; a^{+}e^{+}g^{+}h^{+}b^{+}i^{+}c^{+}d^{+}f^{+}]

- [a b
- (3,10)-O
_{6t}(3.3.3.3.3.3.3.3.3.3) Figure 17.8- [a
^{+}b^{+}c^{+}d^{+}e^{+}e^{-}d^{-}c^{-}b^{-}a^{-}; d^{+}b^{+}c^{+}a^{+}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}; d^{+}b^{+}c^{+}a^{+}f^{+}e^{+}j^{+}h^{+}i^{+}g^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}; d^{+}i^{-}h^{-}a^{+}f^{+}e^{+}j^{+}c^{-}b^{-}g^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}; g^{-}b^{+}h^{-}j^{-}f^{+}e^{+}a^{-}c^{-}i^{+}d^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}; g^{-}i^{-}c^{+}j^{-}f^{+}e^{+}a^{-}h^{+}b^{-}d^{-}]

- [a
- (3,12)-O
_{8t}(3.3.3.3.3.3.3.3.3.3.3.3) Figure 17.9- [a
^{+}b a^{-}a^{+}b a^{-}a^{+}b a^{-}a^{+}b a^{-}; a^{+}b] - [a
^{+}b^{+}c^{+}a^{+}b^{+}c^{+}a^{+}b^{+}c^{+}a^{+}b^{+}c^{+}; a^{+}b^{+}c^{+}] - [a
^{+}b^{+}c^{+}a^{+}b^{+}c^{+}a^{+}b^{+}c^{+}a^{+}b^{+}c^{+}; c^{-}b^{+}a^{-}] - [a
^{+}b^{+}c^{+}c^{-}b^{-}a^{-}a^{+}b^{+}c^{+}c^{-}b^{-}a^{-}; a^{+}b^{-}c^{+}] - [a
^{+}b^{+}c^{+}c^{-}b^{-}a^{-}a^{+}b^{+}c^{+}c^{-}b^{-}a^{-}; c^{-}b^{-}a^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}a^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{+}e^{+}c^{+}d^{+}b^{+}f^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}a^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{+}e^{+}d^{-}c^{-}b^{+}f^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}a^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; c^{-}e^{+}a^{-}f^{-}b^{+}d^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}a^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; d^{+}e^{+}f^{+}a^{+}b^{+}c^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}a^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; f^{-}e^{+}d^{-}c^{-}b^{+}a^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}f^{-}e^{-}d^{-}c^{-}b^{-}a^{-}; a^{+}b^{-}f^{+}d^{+}e^{+}c^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}k^{+}l^{+}; a^{+}b^{+}l^{+}g^{+}e^{+}f^{+}d^{+}k^{+}i^{+}j^{+}h^{+}c^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}k^{+}l^{+}; a^{+}e^{+}c^{+}d^{+}b^{+}l^{+}i^{-}k^{-}g^{-}j^{+}h^{-}f^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}k^{+}l^{+}; a^{+}e^{-}g^{-}l^{-}b^{-}f^{+}c^{-}k^{+}i^{+}j^{+}h^{+}d^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}k^{+}l^{+}; d^{+}k^{-}j^{-}a^{+}h^{+}g^{-}f^{-}e^{+}l^{+}c^{-}b^{-}i^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}g^{+}h^{+}i^{+}j^{+}k^{+}l^{+}; f^{-}b^{+}g^{-}l^{-}e^{+}a^{-}c^{-}k^{+}j^{-}i^{-}h^{+}d^{-}]

- [a

- (4,5)-4ta
(4.4.4.4.4)
Figure 17.15
(complementary polyhedron is (4,5)-8th)

{4,5} S1 sponge

- (4,5)-4tb
(4.4.4.4.4)
Figure 17.16
(complementary polyhedron is (4,5)-8ti
Figure 17.20)
- [a b
^{+}c^{+}c^{-}b^{-}; a b^{+}c^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{+}c^{-}d^{-}e^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{+}b^{+}d^{+}c^{+}e^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}e^{-}c^{-}d^{-}b^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{+}e^{-}d^{+}c^{+}b^{-}]

- [a b
- (4,5)-(3+2)tc
(4.4.4.4.4)
Figure 17.18
(complementary polyhedron is (4,5)-(6+2)tg)
- [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}c^{-}d^{-}e^{-}]

- [a
- (4,5)-(4+2)te
(4.4.4.4.4)
Figure 17.19a (self-complementary)
- [a
^{+}a^^{+}b^{+}c*^{+}b^^{+}; a^{-}a^^{-}b^{-}c*^{+}b^^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}c^{-}d^{+}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}c^{-}d^^{+}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}e^^{-}d^{+}c^^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}e^^{-}d^^{+}c^^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; b^^{-}a^^{-}c^{-}d^{+}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; b^^{-}a^^{-}c^{-}d^^{+}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; b^^{-}a^^{-}e^^{-}d^{+}c^^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; b^^{-}a^^{-}e^^{-}d^^{+}c^^{-}]

- [a
- (4,5)-(4+2)tf
(4.4.4.4.4) (same as previous)
Figure 17.19b (self-complementary)
- [a
^{+}a^^{+}b^{+}c*^{+}b^^{+}; a^{-}a^^{-}b^{-}c*^{-}b^^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}c^{-}d^{-}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}c^{-}d^^{-}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}e^^{-}d^{-}c^^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; a^{-}b^{-}e^^{-}d^^{-}c^^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; b^^{-}a^^{-}c^{-}d^{-}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; b^^{-}a^^{-}c^{-}d^^{-}e^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; b^^{-}a^^{-}e^^{-}d^{-}c^^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}; b^^{-}a^^{-}e^^{-}d^^{-}c^^{-}]

- [a
- (4,6)-(3+2)t
(4.4.4.4.4.4)
Figure 17.22
(complementary polyhedron is (4,6)-(6+2)t, also Figure 17.22)
- [a b
^{+}c^{+}d c^{-}b^{-}; a b^{-}c^{-}d] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{+}b^{-}e^{+}d^{+}c^{+}f^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{+}b^{-}e^{+}d^{-}c^{+}f^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{-}b^{-}e^{+}d^{+}c^{+}f^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{-}b^{-}e^{+}d^{-}c^{+}f^{-}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{+}f^{+}e^{+}d^{+}c^{+}b^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{+}f^{+}e^{+}d^{-}c^{+}b^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{-}f^{+}e^{+}d^{+}c^{+}b^{+}] - [a
^{+}b^{+}c^{+}d^{+}e^{+}f^{+}; a^{-}f^{+}e^{+}d^{-}c^{+}b^{+}]

- [a b
- (4,6)-6t
(4.4.4.4.4.4)
Figure 17.21a (self-complementary)

Regular {4, 6} Coxeter-Petrie sponge

- (6,4)-6t
(6.6.6.6)
Figure 17.21b (self-complementary)

Regular {6, 4} Coxeter-Petrie sponge

- (6,6)
(6.6.6.6.6.6)
Figure 17.21c (self-complementary)

Regular {6, 6} Coxeter-Petrie sponge

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