Aperiodic Tiling in Three Dimensions

The Danzer Tiling

There are 22 vertex configurations which occur in an infinite (global) Danzer tiling produced by inflating an initial finite patch an infinite number of times. Danzer says in his paper that there are 27 vertex configurations total, but says nothing about the characteristics of the five configurations which do not appear in a global tiling. We have identified a total of 174 vertex configurations by exhaustive search. At present we are unsure whether Danzer's remark is an error or whether some 5 of these are special in some way.

Links to pictures of the 22 "naturally occurring" vertex clusters are given in the first table below. 'A' tiles are drawn in red, 'B' tiles are green, 'C' tiles are blue, and 'K' tiles are yellow. In this first table, there is no distinction between left- and right-handed versions. The remaining 152 "artificial" configurations fall into 3 groups. Two of the groups, one with 121 members and the other with 21, are variations on a theme, all the permutations of one or two local replacements. The ten members of the third group are the unique new configurations. In the second table below we give pictures of the ten unique artificial configurations and a representative of each of the other two groups. In these pictures, left-handed tiles are drawn darker.

These pictures were rendered on an SGI Indigo 2 using powerflip, a graphics demo program packaged with IRIX 5.3. We provide the powerflip source for each vertex configuration for people who have access to an SGI. We also give Persistence of Vision (POV) ray tracer source and our own (much more compact) file format. You can download them in any of these three forms.

The tiles in these pictures and in the flip and POV files have all been shrunk by 30% toward their centers of mass to allow us to see the internals of the vertex clusters. The single original tile (a right-handed A tile for all the clusters below) is called the level 0 tile. Thus A1 denotes the 11-tile patch obtained by inflating an A tile one time, without rescaling the subtiles. Vertex 1 of the level 0 tile is placed at the origin, vertex 2 is on the positive x-axis, and vertex 3 is in the x-y plane. The lengths of the sides of the level 0 tile and the vertex numbers are those given by Danzer.

The following table gives pictures of the ten unique artificial vertex configurations and a representative of each of the two groups of permutations. Since these were constructed rather than found in an actual tiling, there is no location given. In these pictures, left-handed tiles are colored darker than right-handed tiles, though lighting effects can make them hard to distinguish.

Things to notice about these configurations:

• 40na and 40nb are mirror images of each other. All of the 22 global configurations have at least one plane of symmetry, usually more. These two are completely asymmetric. In their pictures they are rotated in the same direction by 90 degree increments.
• 44n has five different types of vertex at its central point, which is the maximum possible (see vertex classes). It is the only configuration (besides the trivial K-octahedron) to use all the members of its class.
• 48n and 54n are variations on the globally occurring 60a. A group of 12 B tiles can be replaced by a divot of 6 A tiles. Starting with the ball of 120 B tiles, all the permutations of this replacement generate 121 new configurations.
• 60n is a natural extension of 90a, replacing both sides of the C-ball with medallions of 10 A tiles instead of only one side.
• 100n is similar to the 90b - 100 - 110 series. It is rather surprising that this benign-looking configuration does not occur in a global tiling.
• 36n, 38n, and 44n contain the face pairing B3-C3, which does not occur in a global tiling because it cannot be extended (see face adjacencies for the extensibility of face pairings). 48n, 54n, and all of the members of Group 2 contain the divot of 6 A tiles, which uses the A1-a1 pairing, so none of these can be extended either. It is quite possible that none but the 22 global configurations will admit a tiling, but we have no proof that the remaining 5 unique configurations or the 21 members of Group 1 cause a conflict.
• Group 2 has 121 members. Group 1 has 21 new members, but in fact six of the global configurations (60b, 70, 80, 90b, 100, and 110) and the new 100n fall in this group as well, so it has a total of 28 members. Since none of the members of Group 2 can be extended (see the previous comment), they are all new.

Other Reference Info

• Dihedral angles and side lengths for the four prototiles, taken from Danzer's paper, and the solid angle for each vertex.
• Inflations: a table giving the result of an inflation at each vertex and a graph showing the result of inflating each vertex configuration.
• Face adjacencies and vertex classes: a list of all legal face pairings and vertex configurations where they can be found, and the division of vertices into four classes such that only vertices in the same class can meet.
• Cutout patterns for the tiles. Also available in PostScript, which prints two nicely packed pages of them, one of each handedness.

Back home