World’s first aperiodic tiling with a single shape

3D Printing & Imaging Science
World’s first aperiodic tiling with a single shape

From MIT’s Technology Review:

The problem of tiling a plane has fascinated builders and mathematicians alike since time immemorial. At first glance, the task is straightforward: squares, triangles, hexagons all do the trick producing well known periodic structures. Ditto any number of irregular shapes and combinations of them.

A much trickier question is to ask which shapes can tile a plane in a pattern that does not repeat. In 1962, the mathematician Robert Berger discovered the first set of tiles that did the trick. This set consisted of 20,426 shapes: not an easy set to tile your bathroom with.

With a warm regard for home improvers, Berger later reduced the set to 104 shapes and others have since reduced the number further. Today, the most famous are the Penrose aperiodic tiles, discovered in the early 1970s, which can cover a plane using only two shapes: kites and darts.

The problem of finding a single tile that can do the job is called the einstein problem; nothing to do with the great man but from the German for one– “ein”–and for tile–“stein”. But the search for an einstein has proven fruitless. Until now.

The new tile is from Joshua Socolar and Joan Taylor at Duke University. The image above shows seven of the roughly-hexagonal tiles interlocked together, with each color corresponding to a single “tile.” As you can see, there are a couple of caveats: 1) Their 2D “tile” consists of several areas separated by empty space, and 2) (not so obvious) the tiling only works when you allow both mirror images of the shape. Socolar and Taylor point out, however, that both objections can be overcome if you allow the tile to have a third dimension, as shown below (the colors are only illustrative and not required for the tiling to “work”):


This, in other words, is a single 3D shape you could cover a surface with, forever, and never have the pattern repeat itself. Bragging rights to the first person to get a printable model up on Thingiverse.

You can read the full text of the Socolar-Taylor paper here.

12 thoughts on “World’s first aperiodic tiling with a single shape

  1. John Baichtal says:


  2. Gelada says:

    A couple of models and a blender file are now up on Thingiverse:

    The models were made for the images for this blog post:

  3. ernesto amezcua m. says:


  4. ernesto amezcua m. says:


  5. ernesto amezcua m. says:


  6. Riva says:

    This is a
    great article.The problem of finding a single tile that can do the job is
    called the einstein problem.This is absolutely right.I am agree to this
    content.Thanks to share this blog with us.Keep it up.I will keep share in

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I am descended from 5,000 generations of tool-using primates. Also, I went to college and stuff. I am a long-time contributor to MAKE magazine and My work has also appeared in ReadyMade, c't – Magazin für Computertechnik, and The Wall Street Journal.

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