Thursday, April 9, 2015

Exploring tessellating birds (Revised)

In my a previous post I mentioned that I had updated  A Tessellating Coloring Book with about 25 new designs and that I would do a follow-up post on bird tessellations.

The design below was used in the original edition of A Tessellating Coloring Book. It is a simple type TTTT tessellation of a quadrilateral. The top is repeated on the bottom and the right side is repeated on the left side. M. C. Escher had a tessellating bird that is used the same TTTT tessellation and is similar.
(People doing tessellations use a terminology named after German mathematician Herman Heersch. A T or translation moves one side to the opposite side. A G or glide moves a side to the opposite side with a flip or to an adjacent side with a flip and a rotation. A C or center point rotation has one half of a side rotated to form the other half. There are also corner rotations of 60, 90, and 120 degrees. One place that this terminology is explained with examples is here.)

Using TesselManiac! I found some other ways to tessellate a standing bird. Below is one based on a hexagon using a type TTTTTT tessellation. Each of the six sides of the hexagon is repeated on the opposite side: top to bottom, etc.
Below is a type TGTG tessellation. The left is repeated on the right, but the top is flipped and shifted to the bottom.
The tessellation below is formed from a hexagon and TesselManiac! tells me that this is a TG1G2TG1G2 transformation. The top of the image is repeated on the bottom, the T part. The upper right side is flipped vertically and horizontally to form the upper left side, and the lower right side is also flipped vertically and horizontally to form the lower right side.
The above patterns appear in A Tessellating Coloring Book. However, once started, I wondered how many other Heersch types could be made to work with a similar standing chicken/duck tessellation (bird standing on the back of a bird).

 Below is a type CGCG transformation of a quadrilateral. The top is flipped horizontally to form the bottom. Both tops of sides are rotated 180 degrees to form the bottoms of the sides.

Another pattern I found that worked used a G1G2G1G2 transformation. The top is flipped horizontally to form the bottom and the right side is flipped vertically to form the left side. (The sides look like they could be C sides, but if they were, the birds would be head-to-head and tail-to-tail.)

Here is a TCTC type on a quadrilateral.

I was not happy with what I got from a pentagon type TCCTC in TesselManiac!but I later got a much better result of the type in TesselManiac! with a TCTC starting point. The better result is below.

Type TCTGG, another pentagon type, is one of the patterns that lines up four different positions of the shape in a row rather than in a box.

A third pentagon type, a CG1G2G1G2, will work better for mazes than the two above. The C part is in the tail.

Below are four more based on hexagons. The first of type CG1CG2G1G2 with very short G2 pairs.

It took a some time and effort to get something decent from the TG1G1TG2G2 type.
A type TCCTGG has four different positions of the birds in a row.

A hexagonal TCCTCC standing bird turned out better than I expected. Two of the C sides are very small, the one that forms the top of the bill and the one that forms the back of the tail. (More interesting shapes are possible if one does not constrain the results to line up in a square grid as I did below.)
Finally, it is even possible to get standing birds with a triangle CGG tessellation.

A similar shape will tessellate for a quadrilateral CCGG

 and also for a quadrilateral G1G1G2G2.

There are 28 Heersh types and at least 17 will produce recognizable birds standing on the backs of other birds: seven of the quadrilaterals, three of the pentagons, and six of the hexagons, and one of the triangles. When I began this exploration, I did not expect to get that many. All eleven Heesch types not included above are composed solely of various types of rotations, either center point rotations or corner rotations of 60 degrees, 90 degrees, or 120 degrees.

Some of the Heesch types above can be yield additional ways of tessellating a standing bird and often when using TesselManiac! the end result is not the Heesch type that the template says it is. Those are topics for another post.

(Revised 04-20-1015)

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