## Monday, January 2, 2017

### Questions no one is asking part three

Two previous posts (here and here) explored tessellations formed using only glided edges. This post looks at tessellations formed with only translated edges. Like glided edges, translated edges must be paired and the pairs must be equally long, which limits them to quadrilateral and hexagonal types. Glided edges, however, can be aligned in several ways while translated edges must be opposite and parallel to each other. Hence, the possibilities when using tiles with only translated edges are even more limited than those when using only glided edges.

In what follows tilings that have a translation block of one are presented when possible. If asymmetric edges are possible, they are used, then edges with reflective symmetry over their midpoints, with edges formed using center-point rotation only as a last resort.

First, a TTTT type.
Isohedral class IH68 is both a TTTT type and a G1G1G2G2 type. All edges are identical and the tile mirrors over a diagonal.
In isohedral class IH57 edges are formed with center-point rotation and opposite edges translate. It fits three types: TTTT, CCCC, and TCTC.
IH62 is similar to IH57 but it requires all four edges to be identical. Each edge is rotated 90ยบ to form the adjacent edge. It fits type C4C4C4C4 in addition to the three types that IH57 fits.
Isohedral class IH74 is another tile that uses identical edges formed with central rotation. The tile mirrors over both diagonals. In addition to types TTTT and CCGG, it fits types CCCC, TCTC, and G1G1G2G2.
Although the topic of this exercise is translated edges, the last three examples highlight the flexibility of edges with center-point rotation. If we want an all-translated version of C4C4C4C4 formed with asymmetric edges, it will have a translation block of four. (In this and in some other cases below, it should be obvious how the tile can arranged as a TTTT pattern with a translation block of one.)
Similarly, if we want to use an asymmetric edge for the TT pair of the TCTC type, the tiling will have a translation block of two.
A TGTG tile with translated edges also has a translation block of two. Edges can be both glided and translated only when they mirror over their midpoints.
A CGCG type with all edges translated results in another tiling with a translation block of two.
An all-translated G1G2G1G2 type has a translation block of four.
Below is a C3C3C6C6 type in which the tiles have translated edges. For the edges to both translate to opposite edges and rotate to form adjacent edges, they must be formed with reflection over their midpoints. The tile is symmetrical over its long diagonal and the tiling fits isohedral class IH68.
The figure below shows the tile from the above figure arranged as a TTTT type. In this arrangement it fits isohedral class IH68.
Type C3C3C6C6 cannot be formed with translated edges because the template cannot be a parallelogram.

Part four will continue with hexagonal types.