Monday, December 19, 2016

Qustions no one is asking part two

(This post continues a previous post.)

Moving on to hexagonal types, the way to form an all-glided type TTTTTT using as many asymmetric edges as possible is with isohedral class IH12, which is also type TG1G1TG2G2 with mirroring over the midpoint of the TT edges. The mirroring requires that the G1 and G2 edges be identical.
 The all-glided type TG1G1TG2G2 below is similar to the tiling above but uses a different shape for the G1 and G2 edges. It no longer fits TTTTTT.
 Below is an example of TG1G2TG2G1 with the TT edges glided with reflective symmetry.
 The type TCCTCC example below formed with glided edges also fits type TG1G1TG2G2. If the translated edges were straight, it would be an example of isohedral class IH15.
 An alternative way of forming a glided TCCTCC will fit type TG1G2TG2G1 and is an example of IH13. As such, it also fits CG1CG2G1G2.
 TCCTGG has a translation unit of 4x1. This example uses all three types of edges that can be glided: asymmetric edges, edges with reflection over the midpoint, and edges with center-point rotation.
 Because the tile used to form this all-glide version of TCCTGG has the CC part glided, it will also tile as TG1G1TG2G2.
 To form an all-glided version of type C3C3C3C3C3C3, the edges are formed with edges of mirror reflection arranged as G1G1G2G2G3G3.
 Notice how the two CC edges are mirrored in this all-glided version of CG1CG2G1G2.
We have finished looking at tiles with all edges formed with glides. We will continue in part three to look at another question no one is asking, how many of the Heesch types can be formed with tiles in which all edges are translated.

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