Ein Stein

I’ve been fascinated by aperiodic tilings of the plane since Martin Gardner first wrote about them in Scientific American. In the 1960s, Robert Berger discovered a set of 20 426 prototiles or tile-types that can tile the plane but only with no translational periodicity — a wonderful mix of the expected and the surprising, a kind of visual music.

Over the years, the number of required prototiles has been greatly reduced. In the 1970s, Roger Penrose discovered a set of just two concave aperiodic prototiles. Robert Ammann then dissected these to discover a set of three convex aperiodic prototiles. Can a single prototile, one tile or stone, literally ein stein in German, force a nonperiodic tiling? Despite several near misses and potential applications to quasicrystals, the existence of an ein Stein remains a fascinating unsolved problem.

Three convex Ammann tiles force a non periodic tiling of the plane.

Three convex Ammann tiles force a nonperiodic tiling of the plane.

Two concave Penrose tiles force a nonperiodic tiling of the plane.

Two concave Penrose tiles force a nonperiodic tiling of the plane.

About John F. Lindner

John F. Lindner was born in Sleepy Hollow, New York, and educated at the University of Vermont and Caltech. He is an emeritus professor of physics and astronomy at The College of Wooster and a visiting professor at North Carolina State University. He has enjoyed multiple yearlong sabbaticals at Georgia Tech, University of Portland, University of Hawai'i, and NCSU. His research interests include nonlinear dynamics, celestial mechanics, and neural networks.
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3 Responses to Ein Stein

  1. Dave Smith says:

    Excellent examples – thanks.

  2. Henning says:

    Still remains a fascinating but not unsolved problem anymore

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