The {4,5} sponges

This vertex star has 5 squares, or 4-gons, arranged around its 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. Using the symmetric vertex symbol,  a b+c+c-b-, three different adjacency symbols result in constructible sponges. They 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 labeled versions of the symmetric ones so their shapes are the same as the three above. That leaves 12 non-symmetric sponges, listed as N1 to N12 below. Since their adjacency symbols cannot be made reflectively symmetrical, these sponges are similarly also not reflectively symmetrical, and they have only a single, asymmetric labeling.

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 shown below display 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. This vertex star has produced the largest number of different geometrical shapes of any of the stars we have examined so far.

The results of this {4,5} vertex search were published online in the paper, "The {4, 5} isogonal sponges on the cubic lattice," located at The Electronic Journal of Combinatorics, volume 16 (2009), issue number 1, article R22.

The 15 {4, 5} sponge shapes   (

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Last updated: June 10, 2019