Danzer Face Adjacencies and Vertex Classes

Face Adjacencies

Faces are given the same number as the vertex opposite them. Each face can be matched with its mirror image, although A1-a1 and B3-b3 do not appear in a normal tiling. In addition, the following faces are equivalent under Danzer's equivalence relation (~), and so can be placed against each other:

Lower case denotes a left-handed tile. Note that the handedness of both tiles can always be reversed; so the fact that A1 and C1 match implies that a1 and c1 also match, and c3-K4 implies C3-k4. These mirrored pairings always occur together, so they are not listed separately. Of the inter-tile pairings in this list, all are found in an infinite tiling except B3-C3.

The three adjacencies which do not occur in an infinite tiling are absent because it is impossible to place tiles around them without violating the matching rules. This is most easily seen by constructing models of the tiles and trying to build around these pairings, but we give a brief explanation of why each of them is impossible here.

The A1-a1 pairing leaves an A3 face and its mirror image sharing an edge (the A24 edge). Since A3 can only match its inverse, two more A tiles must be placed here, but they would have to overlap. The only way in which the A1-a1 pairing can occur in a vertex configuration is if the A3 faces are on the outside. (See the Group 2 example, 48n, and 54n). The argument is similar for the B3-C3 pairing. If we take the B3-C3 pairing and place face 1 of a (left-handed) b tile against face 1 of the B tile (the only pairing allowed for that face), then face 3 of the b tile partially overlaps face 1 of the C tile, which is illegal. Again, the only way this can occur in a vertex configuration is if all B1 faces point outward. (See 36n, 38n, and 44n). Finally, B3-b3 does not admit even a vertex configuration. Both the B1 and B2 faces must match their inverses, and they cannot do so without overlapping.

Below is a table which gives for each face pairing a vertex configuration in which it can be found. This is not necessarily the "best" or "minimal" location, in terms of either inflation level or the number of tiles in the configuration.

Face pairing Location Face pairing Location Face pairing Location
A1-a1 Group 2 C1-c1 40a A1-C1 16
A2-a2 16 C2-c2 36 A4-B4 60a
A3-a3 16 C3-c3 16 B3-C3 36n
A4-a4 28 C4-c4 36 B3-K4 8b
B1-b1 8b K1-k1 8a c3-K4 36
B2-b2 8b K2-k2 8a

B3-b3 N/A K3-k3 8a

B4-b4 36 K4-k4 36

Vertex Classes

The sixteen vertices of the tiles fall into four classes such that only vertices in the same class can meet. The members of each class are given below, along with the vertex configurations which contain them.

Class 1 (Green): A3, B3, B4, C2, K3
Instances: 8b, 18, 28, 60a, 120a
Class 2 (Blue): A2, B1, C1, C3, K1
Instances: 36, 38, 40b, 42, 64, 90a, 120b
Class 3 (Red): A1, A4, B2, C4, K2
Instances: 16, 40a, 60b, 70, 80, 90b, 100, 110, 120c
Class 4 (Yellow): K4
Instance: 8a

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