Uniform Wells crystallography sponges

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 then 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. Almost all of the nets produce Platonic polyhedra, which we discuss on another page. As for the polyhedra created from mixed polygons, two of them are from chapter 16 and the rest from chapter 18. Wells pointed out that he had not made an exhaustive study of the possible infinite polyhedra and even less of those that used more than one type of polygon. As a consequence he listed very few of these. Nevertheless, we include them all here.

The method Wells used for generating polyhedra was to assemble them from finite Platonic and/or Archimedean solids. Following this technique he focused on the shape constructed by assembling the various finite solids which spreads throughout space infinitely in three dimensions. But one can also consider the volume of space that is left empty and Wells refers to this as the complementary polyhedron. For non-Platonic sponges the complementary polyhedron usually produces a different shape so Wells discussed both polyhedra. Moreover, since Wells described 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, 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. Therefore there is no need to separately discuss the polyhedron and its complement. 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 polyhedra are 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 letter 't', which stands for tunnels.

The first two polyhedra, even though they are made only from triangles, are here because they are made from two types of triangles, one equilateral and the other isosceles. The (3,10)-6t sponge below should really more properly be called the (3,4+6)-6t sponge since, even though its vertex star is made from ten triangles, 4 are equilateral and 6 of them are isosceles. In describing its shape Wells mentions the cubic icosahedron, which is only a finite polyhedron. Yet it is also a fully vertex-transitive isogonal polyhedron so we include it here as well.

The second set of sponges is constructed from space-filling arrangements of finite polyhedra. The first one is from an arrangement of cubes, truncated octahedra, and truncated cuboctahedra. Selecting just the cubes and truncated octahedra produces the shape below. The truncated cuboctahedra constitute the complementary polyhedron. The last two are constructed starting from a space-filling arrangment using truncated cuboctahedra and octagonal prisms. The first one uses one truncated cuboctahedron and one octagonal prism, while the second one uses just the octagonal prisms.

Sponges from space-filling arrangements

• Cubes and truncated octahedra   (4.4.6.6)     Figure 18.4    (complementary polyhedron:  side-by-side truncated cuboctahedra)
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+ d+ c+ b+]
5. [a+ b+ c+ d+;  a- d+ c+ b+]

• Truncated cuboctahedra and octahedral prisms   (4.4.4.6)     Figure 18.5    (self-complementary)
1. [a+ b+ b^+ a^+;  a- b-]
2. [a+ b+ c+ d+;  a- b- c- d-]
3. [a+ b+ c+ d+;  a- c^- b^- d-]
4. [a+ b+ c+ d+;  d^- b- c- a^-]
5. [a+ b+ c+ d+;  d^- c^- b^- a^-]

• Octahedral prisms   (4.8.4.8)    (complementary polyhedron:  corner-to-corner truncated cuboctahedra)
1. [a+ b+ a+ b+;  a- b-]
2. [a+ b+ c+ d+;  a- b- c- d-]
3. [a+ b+ c+ d+;  a- d- c- b-]
4. [a+ b+ c+ d+;  c- b- a- d-]
5. [a+ b+ c+ d+;  c- d- a- b-]