The crinkled {3,6} polyhedron

In 1995, R. Hughes Jones [Enumerating uniform polyhedral surfaces with triangular faces, R. Hughes Jones, Discrete Mathematics, Vol. 138 (1995), pp. 281-292.] listed the {3,6} crinkled plane as one of the polyhedra formed from triangles arranged according to paths on a cuboctahedron. However, this polyhedron was known to Grünbaum at least since 1992 and discussed in a doctoral report (unpublished) on a different topic by W. T. Webber, one of his students. It has the interesting property that it is collapsible, while retaining both its isogonal and acoptic characteristics.

To form the vertex star, start with six coplanar triangles arranged around a central vertex as in the {3,6} planar tiling. Consider the six triangles as forming three pairs of coplanar triangles, which distinguish three tri-divisional lines of the combined hexagon. Fold two of these pairs toward each other along one of the tri-divisional lines. Of course, this cannot be done without bending the third pair of triangles. When the fold extends along the first tri-divisional line, bending the third pair of triangles and without any other bends, the folded {3,6} vertex star and the polyhedra it forms are the result. However, if this additional bend is instead made in the opposite direction, and simultaneously the triangles along the other two tri-divisional lines are bent inward, the vertex star folds up into itself, eventually collapsing into a (non-acoptic) two-dimensional shape. (The situation is somewhat similar to what happens to the side of a square-bottomed paper bag when it is folded flat.) Not surprisingly, this vertex star can be connected isogonally in its original planar tiling form to other such vertex stars, but what is somewhat amazing is that as the star collapses inward, the whole plane can do so as well, all the while remaining isogonally connected and acoptic. Eventually, as the dihedral angle between the two coplanar triangle pairs approaches zero, the plane approaches an infinitely dense slab, extending infinitely in two dimensions, but still acoptic. This same thing happens with the folded plane, which folds up isogonally like a road map, but it is a bit surprising that the plane can be folded up isogonally in this alternate manner, as well.

Below we show VRML models of the polyhedron and the vertex star, starting from a dihedral angle of 170 degrees between the two coplanar triangle pairs and collapsing inward to a dihedral angle of 10 degrees. (The volume enclosed by the displayed VRML models has gradually been reduced as the dihedral angle decreases, in order to avoid displaying too many vertices. The density of vertices would otherwise increase quite dramatically.) The vertex star has a single reflective symmetry, and produces a single labeled symmetric symbol. But because it allows an asymmetric labeling and because the polyhedron is also reflectively symmetric, there are four possible asymmetric labelings.

The "crinkled" {3,6} polyhedron

The labeled incidence symbols:
  1. [a b+ c+ d c- b-;  d^ b^- c+ a^]
  2. [a+ b+ c+ d+ e+ f+;  d^- b^- c+ a^- e+ f^-]
  3. [a+ b+ c+ d+ e+ f+;  d^+ b^- e- a^+ c- f^-]
  4. [a+ b+ c+ d+ e+ f+;  d^+ f^+ c+ a^+ e+ b^+]
  5. [a+ b+ c+ d+ e+ f+;  d^- f^+ e- a^- c- b^+]

and the polyhedral shapes, by dihedral angles:

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Last updated: April 18, 2019