Seeking name of $90-135-90-45$ deg angled kite which forms a non-periodic Fractal Tessellation

Question

In Barnsley and Vince's paper Self-Similar Polygonal Tiling https://people.clas.ufl.edu/avince/files/SSPfinal.pdf

fig $6$ shows a non-periodic tiling made from a kite in 6 sizes. The angles of the kite are $90, 135, 90, 45$. The sides are in ratio $2: 1: \sqrt2: \frac3{\sqrt2}$ (in no particular order). The reduction in size for each level is $\frac1{\sqrt2}$.

I'd show you the figure but I'm not allowed to post images yet but you can click on the link. There are other cool fractal tessellations there too and math that is over my head.)

This kite is the same as taking a square with sides of $2$ and cutting off a corner through the midpoints of two edges that are joined and cutting through the perpendicular diagonal. A number of patchwork designs are based on a $4\times4$ grid of squares with the corners cut to approximate an octagon but I've never seen the octagon partitioned into the non-symmetric kites in quilting circles.

The kite is also easy to knit due to its angles being $90$s, $135$ and $45$. I played around with the shape a bit and found some other tilings that it makes including two that are periodic. The set of kites has some interesting properties and I think it could be used for both quilting and knitting.

Questions:

1) Have you seen tilings with this shape before? (I'd like to see other examples)

2) Does the kite have a name? (would like to call it something other than Wonky Kite in the knitting pattern I'm working on)

3) Is there a web site where one could find all (or a large subset of) known tessellations like there is for number series?

Answer 1

I have not seen this tiling outside that paper before (I did come across that paper though in an investigation related to these questions: Can any number of squares sum to a square? and Can any number of squares sum to a square? ).

On your third question: There are a few tiling databases:

• http://www.tilingsearch.org/index1.htm
• https://tilings.math.uni-bielefeld.de/

(I tried to look for the titling of your question there, but could not find it, but as you will see the search facilities are not great.)

Neither projects are quite comparable to OEIS. Partly it is because it is so much harder to organize tilings than sequences; also, there is much less about and use for tilings in the literature.

Answer 2

Thank you. Those databases look like they will be fun to look through.

I received an email reply from Andrew Vince after I posted this question and he also does not know of any other examples of this tiling nor a name for it. He did give me the information that the kite can be dissected into 9 smaller kites of 4 sizes which was left out of the paper. I found that a kite of the largest size in the tessellation can be dissected using the 4 smallest sized ones. Also, the 4 largest or the 4 smallest can be assembled into a parallelogram with the large one twice the size of the small one. There is also a periodic tiling that assembles the kites into a hexagon.