Every acute triangle has a closed billiard path. Nobody knows whether every obtuse triangle does — computers have checked up to 100°
Does every triangular table have a path that closes up and repeats? It sounds like a puzzle for a rainy afternoon. It has been open for more than two hundred years, and it is still open for triangles with an angle over 100°.
Drag the corners. The demo fires balls at random, and whenever one repeats a pattern of sides it checks exactly whether that pattern closes. Each closed path comes with a band of parallel ones.
Drag a corner to reshape the triangle. The search fires 90 balls in random directions; whenever one starts to repeat a pattern of sides, the pattern is checked exactly by unfolding (next demo). The faint lines are other members of the same parallel family. Finding nothing is not a proof that nothing exists.
In 1775 Giovanni Fagnano asked for the triangle of smallest perimeter with one corner on each side of a given acute triangle. The answer is the orthic triangle, joining the feet of the three altitudes, and it is also a billiard path: at each foot, the two sides of the orthic triangle make equal angles with the wall. So every acute triangle has a closed billiard path with three bounces.
When the triangle becomes right-angled, two feet of the altitudes slide into the right angle and Fagnano’s orbit collapses. Right triangles still have periodic paths — that was settled in the 1990s — but different ones. For obtuse triangles the feet of two altitudes fall outside the sides, and Fagnano’s construction gives nothing at all.
Every periodic path in a polygon comes with a whole band of parallel neighbours that close after the same number of bounces (or twice as many, for odd paths like Fagnano’s). The faint lines in the demo are that band. It is why a random search can find them at all.
Reflect the triangle instead of the ball. The zig-zag path becomes one straight line through a chain of mirror images, and the sides it crosses spell out a word.
The word of sides decides the whole chain. A path is periodic exactly when, for some word, the last copy is the first one slid along without turning and a straight line in that sliding direction threads every shared side — the test the triangle search runs.
Unfolding turns geometry into combinatorics. A path is periodic when its word of sides, unfolded, produces a final triangle that is the first one slid along without any rotation, and when a straight line in that sliding direction can thread every shared side of the chain. The first condition is a statement about angles alone; the second asks for a nonempty corridor.
Some words are stable: the angle condition holds for every triangle, not just the one in front of you, so the path survives small changes of shape and works for a whole open region of triangles. Fagnano’s word, 0-1-2 run twice, is stable for all acute triangles. Proving that every triangle in some range has a periodic path means covering that range with regions like these.
Every triangle appears once on this map, by its two smaller angles. The demo searches each one in turn and colours it by the shortest closed path it finds. Click any cell to look closer.
One cell per whole-degree triangle. Green: acute, so Fagnano’s three-bounce orbit. Other colours: the length of the shortest periodic path the quick search found. Grey: nothing found within 26 random shots of 36 bounces — click one to search harder. Grey cells cluster near the thin triangles on the left, where the shortest periodic paths get long.
Richard Schwartz, with the McBilliards program he wrote with Pat Hooper, turned the atlas into a proof. He found a finite list of stable words whose regions together cover every triangle with largest angle up to 100°, and checked each region rigorously, partly by hand and partly by exact computer arithmetic. The result, published in 2009, is that every triangle with all angles at most 100° has a periodic billiard path.
The method gets harder the further you go. Near the thin triangles at the left of the map the shortest paths get very long, and the regions of stable words shrink, so covering the rest of the map this way would need ever more pieces.
For triangles whose angles are rational multiples of π the answer is yes: Howard Masur proved in 1986 that every rational polygon has periodic paths, in a dense set of directions. His proof goes through the surfaces of the next lesson. For general obtuse triangles past 100°, it is not known even whether a single periodic path always exists.
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