https://www.johndcook.com/blog/2025/09/25/conways-pinwheel-tiling/

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Conway's pinwheel tiling

Posted on 25 September 2025 by John

John Conway discovered a right triangle that can be partitioned into
five similar triangles. The sides are in proportion 1 : 2 : [?]5.

[conway_pinwheel1]

You can make a larger similar triangle by making the entire triangle
the central (green) triangle of a new triangle.

[conway_pinwheel2]

Here's the same image with the small triangles filled in as in the
original.

[conway_pinwheel3]

Repeating this process creates an aperiodic tiling of the plane.

The tiling was discovered by Conway, but Charles Radin was the first
to describe it in a publication [1]. Radin attributes the tiling to
Conway.

Alternate visualization

It would be easiest to illustrate the tiling if we were standing
together in a room and placing new triangles on the floor, watching
the tiling expand. Given the limitations of a screen, it may be
easier to visualize subdividing the triangle rather than tiling the
plane.

Imagine the smallest triangles are a constant size and at each step
we're viewing the process from further away. We see a constant size
outer triangle at each step, but the triangle is growing and covering
the plane.

Here's an animated GIF of the process.

 

[conway_pinwheel_animated]

Related posts

  * Twin dragon fractile
  * Conway's mental exercise rituals
  * Conway's mental factoring methods

[1] Charles Radin. "The Pinwheel Tilings of the Plane." Annals of
Mathematics, vol. 139, no. 3, 1994, pp. 661-702.

Categories : Math
Tags : Geometry
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One thought on "Conway's pinwheel tiling"

 1.
    John Tromp
    27 September 2025 at 08:38

    The bottom of my home page has this as a PostScript signature:

    %!PS % -John Tromp http://tromp.github.io/
    /t{dup 1 sub gsave dup 0 gt{[.4 .2 -.2 .4 .4 .2]concat t
    currentgray
    .8 mul .2 add setgray -1 1 scale t -1 2 translate t 1 -1 scale t
    [0 1
    1 0 0 2]concat t pop}{0 moveto 1 0 lineto 0 2 lineto closepath
    clip
    fill}ifelse grestore}def 10 10 translate 600 600 scale 5 t
    showpage

    which outputs this pinwheel tiling https://tromp.github.io/img/
    pinwheel.pdf

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John D. Cook

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