Fractal Ir-rep-tiles of Order Two

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type
1
2
3
4
5
6
7
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9
iterations
10000
100000
200000
This Demonstration shows nine ir-rep-tiles of order two with fractal boundary using the iterated function system (IFS) chaos game method.

Details

A shape that can be tiled with smaller congruent copies of itself is called a rep-tile[1]. If the tiling uses
n
copies, the shape is said to be of order
n
. A shape that tiles itself using copies of different sizes is called an irregular rep-tile or ir-rep-tile. This Demonstration is limited to ir-rep-tiles of order two with compact disk-like shape and fractal boundary. Only a few such ir-rep-tiles are known. Hinsley[2] has listed seven such ir-rep-tiles of order two. Here, nine such ir-rep-tiles are shown.
All examples are related to roots of the polynomial
3
x
+qx+r
with
q,r∈±1
. Approximations of the fractal tilings are produced by an iterated function system (IFS) using the chaos game method. Example 2 is the well-known Hokkaido tiling, named by Shigeki Akiyama in reference to the Japanese island with the same name.

References

[1] Wikipedia. "Rep-tile." (Feb 14, 2018) en.wikipedia.org/wiki/Rep-tile.
[2] S. Hinsley. "2 Element Ir-Rep-tiles." (Feb 14, 2018) web.archive.org/web/20111212001755/http://www.meden.demon.co.uk/Fractals/dimerIRR.html.

External Links

Tiling Dragons and Rep-tiles of Order Two
Irreptiles
Rep-Tile (Wolfram MathWorld)

Permanent Citation

Dieter Steemann
​
​"Fractal Ir-rep-tiles of Order Two"​
​http://demonstrations.wolfram.com/FractalIrRepTilesOfOrderTwo/​
​Wolfram Demonstrations Project​
​Published: February 19, 2018