Fractal Ir-rep-tiles of Order Two

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows nine ir-rep-tiles of order two with fractal boundary using the iterated function system (IFS) chaos game method.

Contributed by: Dieter Steemann (February 2018)
Open content licensed under CC BY-NC-SA



A shape that can be tiled with smaller congruent copies of itself is called a rep-tile [1]. If the tiling uses copies, the shape is said to be of order . 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 with . 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.


[1] Wikipedia. "Rep-tile." (Feb 14, 2018)

[2] S. Hinsley. "2 Element Ir-Rep-tiles." (Feb 14, 2018)

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.