No, the title isn't a list of random technical terms. I'm putting this here as a fast way to say "I found it first."
[And you'll really see why I need to post pictures and learn to type symbols!]
The smallest unit of the tessellation is a pentagon with side lengths and angles in the order: 1, 120 degrees, 2, 120 degrees, 1, 90 degrees, square root of 3, 120 degrees, square root of 3, 90 degrees.
This particular pentagon tiles in more than one way, one being a subset of the previously described Type 3. Another can be non-repetitive; this tiling consists of 2 parts. Part 1 is the "medallion." The medallion is formed by uniting three of the pentagons at one vertex with sides of length square root of 3 meeting, giving a hexagon of side length 2. Part 2 is the "hexagonal annulus" created by adding one pentagon to each side of the medallion (side of length 2 facing the medallion), creating a second hexagon of side length 2x square root of 3.
The annuli can be combined into a regular hexagonal tessellation and, if all the medallions are oriented in the same direction, the whole becomes a regular tessellation of pentagonal units. It is possible to rotate any of the medallions 60 degrees and still tile the plane, so it is possible to create tessellations that are of wallpaper groups not described in the 14 types of convex pentagons that tile the plane. In addition, it is possible to orient the medallions randomly, which creates a non-repeating tiling using one basic unit, something not previously described.
2 days ago