Thursday, January 5, 2017

Questions no one is asking part four

(Part three is here. It begins the exercise of examining tilings with tiles that have all edges translated.)

Moving to the hexagonal types, a TTTTTT tiling by definition has translated edges.
Isohedral class IH8 fits both the TTTTTT type and the TCCTCC type. The tile has twofold rotational symmetry.
Isohedral class IH12 is simultaneously TG1G1TG2G2  and TTTTTT. IH12 reflects over the midpoint of its TT edges.
Isohedral class IH10 is both type C3C3C3C3C3C3 and TTTTTT. All edges are identical and each is rotated 120ยบ to form the adjacent edge. (Also fitting both of these types are isohedral classes IH11 and IH18. IH11 is formed with identical edges that reflect over their midpoints and IH18 with identical edges of central rotation.)
Isohedral class IH14 fits both types TG1G2TG2G1 and TTTTTT but it has a pair of straight, unshaped edges that give the tile reflective symmetry over a diagonal. If the straight edges are replaced with asymmetric edges, the reflective symmetry of the tile is lost but the tile can still fit types TTTTTT and TG1G2TG2G1, though no longer simultaneously. As a TG1G2TG2G1 type it has a translation block of two.
Below is a tiling that fits type CG1CG2G1G2 and also type TCCTGG.
For both types a C edge must be paired with a G edge for the edges to translate so these pairs must be formed with center-point rotation. In the case of CG1CG2G1G2, the G1 edges are the edges that reflect over their midpoints. Because of symmetry, the translation block for both types is reduced to two.

In the beginning of this exercise I stated that I would use an asymmetric edge when possible. If an asymmetric edge is used for the TCCTGG type, the tile has no symmetry and the result is a translation block of four rather than the two in the above figure.