Play billiards outside the table. Around a pentagon the orbits draw a fractal; around the Penrose kite, Schwartz proved in 2007 that one escapes to infinity
Turn billiards inside out. Stand outside a convex table, sight along a line that just touches it, and jump to the point the same distance beyond the touching corner. Can repeated jumps carry you arbitrarily far away? That question was open for about fifty years.
Click outside the table to start an orbit. Each jump is a half-turn about one corner of the table, with the table on the right.
Click outside the table to start a new orbit. The yellow segments are the latest jumps, each bisected by the highlighted corner. Around the square every orbit returns to its start; around the pentagon every orbit stays bounded and most come back, some only after a long time. The Penrose kite gets the last demo to itself.
Bernhard Neumann described outer billiards in popular lectures in the 1950s and asked whether an orbit can be unbounded. Jürgen Moser publicised the question in the 1970s as a toy model for the stability of the solar system: does a planet stay in a bounded region for ever?
For a smooth, strictly convex table the answer is no escape: Moser’s twist theorem, a relative of the KAM theorem from celestial mechanics, puts invariant curves around the table arbitrarily far out, and an orbit trapped between two of them is trapped for ever. For polygons there are no such curves, but for polygons with rational vertices every orbit is periodic, and for regular polygons every orbit is bounded. The hope that every orbit is bounded for every table lasted until 2007.
Colour every point outside a regular pentagon by how many jumps it takes to come home. Click to zoom in.
Each pixel is a starting point; its colour is the number of jumps until the orbit first returns. Black: not back within the limit — near the fractal boundary between tiles, periods grow without bound, and on the fractal itself orbits never return yet stay bounded. Click to zoom in by 2.5×; the same pentagonal patterns reappear at smaller and smaller scales.
Each jump is a half-turn, so a sequence of jumps that returns to the same corners in the same order is a translation or the identity. Points that come home therefore come home together, in polygonal tiles where every point has the same period. For the square the tiles cover everything.
For the regular pentagon they do not. Sergei Tabachnikov showed in the 1990s that between the tiles lies a fractal set of points whose orbits never come home yet stay bounded, and that the picture is self-similar: zoom in near a corner of the pattern and the same arrangement of pentagonal tiles reappears, scaled by a power of the golden ratio.
The orbit Richard Schwartz proved unbounded, around the Penrose kite, computed exactly for tens of millions of jumps.
The kite is drawn in the middle with vertices (−1, 0), (0, −1), (√5 − 2, 0), (0, 1); the orbit stays on the horizontal lines at odd heights. Coordinates are stored as (m + n√5)/2 with whole numbers m and n, and every left-or-right decision is made exactly, so the orbit shown is the true one. The computer can only ever show records growing for a finite time; that they grow forever is Schwartz’s theorem.
The Penrose kite is the kite-shaped tile from the Penrose tiling. After an affine change of coordinates it has vertices (−1, 0), (0, −1), (A, 0) and (0, 1) with A = √5 − 2 = φ⁻³, where φ is the golden ratio. In 2007 Richard Schwartz proved that the orbit of the point (φ⁻², −1) is unbounded in both directions, answering Neumann’s question. The proof is computer-assisted, and it rests on a self-similarity: a picture of the orbit’s structure, scaled by φ³, closely shadows itself, so the orbit keeps being pushed further out.
In a 2009 monograph Schwartz extended the result to every kite with an irrational A, and in 2011 he showed that for the Penrose kite the unbounded orbits together form a set of Hausdorff dimension 1. Dmitry Dolgopyat and Bassam Fayad then found unbounded orbits around a half-disc, a table with corners and a curved side. The escape is slow: the orbit keeps coming back near the kite, and its record distance creeps up, as the chart shows.
Every coordinate in the demo has the form (m + n√5)/2 with whole numbers m and n, and each jump only doubles and subtracts such numbers. Deciding which corner to jump about needs only the sign of numbers like m + n√5, which can be found exactly by comparing m² with 5n². So the orbit on screen is the real one, not an approximation.
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