Isogonal polyhedra have been studied since antiquity--the Platonic solids are examples. Described simplistically, they are polyhedra that "look the same" from every vertex. But the Platonic solids, such as the cube, are finitely bounded. Here we describe some that extend infinitely, yet still retain their isogonal quality, and which we refer to as sponges. In these examples, we restrict ourselves to those with only five squares adjacent to each vertex, as described below in more detail. To enable them to be better appreciated, we include files showing them in 3-D. These files use VRML97 (Virtual Reality Modeling Language, 1997) as the modeling language to display them. VRML97 is the current name for what used to be called VRML 2.0. The new standard for displaying 3-D objects on the Web is X3D, but it has been designed to also read VRML97 files so these images should still be viewable for some time. In any event, in order to view the images you will need a VRML97 or X3D-compatible standalone program or Web browser, or a plug-in to your regular browser. The Web3D organization maintains a Web page listing such software. To use the standalone programs you will first need to download the VRML files linked below (they are just plain text files) to your own computer.
We describe isogonal polyhedra in more detail, our notational description of them, how this complete list was constructed, and a historical review of the research into them in a paper entitled, "The {4, 5} isogonal sponges on the cubic lattice." It can be found in this archive as well, or published online in The Electronic Journal of Combinatorics, volume 16 (2009), issue number 1, article R22.
Steven Gillispie
Branko Grünbaum
As the {4, 5} name suggests, this vertex star (the basic building block set of polygons) has 5 squares, or 4-gons, arranged around a central vertex. All of the squares are aligned along the planes of a cubic lattice, so that all of the dihedral angles are either 90° or 180°. The 'a' edge is the one perpendicular to the three coplanar squares. As can be seen from its VRML model, this vertex star has a reflective symmetry across the plane through the 'a' vertex edge. Here we have drawn the 'a' edge using an apricot (orange) color and the 'b' edge using a blueberry (blue) color for easy identification. (The 'c', 'd', and 'e' edges continue similarly around the vertex star.) Altogether we find 35 different labeled sponges in 15 different shapes, as listed below. Using the vertex star's reflectively symmetric vertex symbol, 3 different adjacency symbols result in constructible sponges, and are listed as S1 to S3 below. On the other hand, when each of the edges are treated as unique, there are 32 constructible sponges, but 20 of them are just labeled versions of the symmetric ones so their shapes are the same as the three above. (There are 4 with the S1 shape, 8 with the S2 shape, and 8 with the S3 shape.) That leaves 12 non-symmetric sponges, listed as N1 to N12 below. Since their adjacency symbols cannot be made reflectively symmetric, these sponges are similarly also not reflectively symmetric, and they exist only in a single, asymmetrically labeled form. Therefore the 15 shapes separate into 3 groups with multiple labelings (having 5, 9, and 9 labelings, respectively) and 12 shapes with single labelings. The sponges are listed with their incidence symbols, whose first part (the vertex symbol) describes the symmetry of the vertex star and whose second part (the adjacency symbol) describes the connections between labeled edges. These two symbols are then enclosed in square brackets and separated by a semicolon to complete the incidence symbol.
Some of the sponges are infinite in only two dimensions, but most are infinite in three. All of the sponges are periodic, including the non-symmetric ones. The VRML models of the 15 different shapes listed below show only a section of them excised from their infinite extent. The sponges display a remarkable range of patterns, in spite of the restrictive rules imposed by isogonality.