## Friday, December 4, 2015

### FabTwo?

The recent post on the FabFours typefaces examined tiles based on a square template with identical edges formed with central rotation. These tiles could take only four distinct shapes and these four shapes would, when arranged vertex to vertex, produce voids that must have the same four shape as the tiles.

What happens if we replace the square template with the template of an equilateral triangle? For these triangular-based tiles only two distinct shapes are possible. One of the two shapes fits the criteria for isohedral class IH90. The other is formed by taking one edge of this shape and flipping it. Many of the visually-interesting edges from the tiles with square templates will fail when put on the triangle template because adjacent edges will overlap.

These two shapes form two sets of tiles, each with eight orientations. In each set of shapes, the IH90 shape has two orientations and the other has six. One set has its base on the bottom and the other is a 180º rotation of the first set. When one set is used, the other set forms the voids.  Below are examples of one set, with the IH90 shape in the first two positions.
Here is the second set.
In the first example below the red tiles leave voids that have shape of the IH90 tile. Notice that the red tiles are from the the first set and the voids are from the second.

The next example uses white tiles, so the tiles are not readily distinguishable from the voids.

If a rhombus (diamond) is fitted with identical sides of center-point rotation, there are exactly seven distinct shapes that result. The pattern above from a triangle template has all seven shapes based on the 60º-120º rhombus if we can erase some of the lines that separate a tile and an adjoining void. The seven edges that need to be erased to reveal the seven possible rhombus shapes are numbered in the picture above. (If we squeeze the rhombus template until it becomes a square, which is a special case of a rhombus, shapes 2 and 3 converge, as do shapes 4 and 6, and 5 and 7.)

Below are a couple more examples with different edges. They are taken from pages of Delightful Designs: A Coloring Book of Magical Patterns.

The visual interest from patterns formed with tiles of these sets may be more in seeing these rhombus-based shapes than in the patterns of the triangular shapes. Other interesting shapes result from combining blocks of six that share a common vertex and blocks of four centered on a middle tile.

Will what works for square and triangular templates work for hexagonal templates? The answer is,"No". Patterns based on the square and triangular templates have an even number of lines meeting at each vertex, four and six. The allows the tiles and the voids to alternate, a TVTV or TVTVTV sequence. Patterns based on a hexagonal template have three edges meeting at each vertex, so the alternating sequence of tiles and voids is not possible. In addition, all the shapes that can be formed with an identical side of center-point rotation using either a square or triangular template will tessellate. The same is not true when shapes are formed in the same way with a hexagonal template.