Generator

A billiard path of slope 0.618, folded and unfolded

A generator in the dynamics library, called 45 times across 9 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

billiard is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

A billiard path of slope 0.618, folded and unfolded. A ball bouncing inside a square table, and the same trajectory drawn as one straight line through reflected copies of the table, so that the bounces disappear.

A billiard path that closes, and one that does not

A closed billiard path in a circle, and the disc it never enters

The one path an acute triangular table always has

A room a trajectory cannot get out of, and one it can

the L: 4 directions, and a surface of genus 2

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

golden slope: exactly n + 1 blocks of length 1 ×14
slope √2 − 1: exactly n + 1 blocks of length 1 ×14
every block of length 1 holds one of two neighbouring numbers of H ×12
in direction (1, √2, √3) there are n² + n + 1 blocks of length 1 ×9
with X erased, 2 blocks of length 1 ×8
with Y erased, 2 blocks of length 1 ×8
with Z erased, 2 blocks of length 1 ×8
4-cube (1, √2, √3, √7): Baryshnikov's count for blocks of length 1 ×6
cube (1, √2, √3): Baryshnikov's count for blocks of length 1 ×6
square (1, √2): Baryshnikov's count for blocks of length 1 ×6
a trajectory in this table takes exactly 4 directions and no more ×3
the path bounces correctly off side 1 ×3
2 to 40 jumps drawn ×1
4 to 9 steps ×1
40 to 160 cells a side ×1
a block length from 2 to 12 ×1
a closed path is one that returns to within a tolerance ×1
a direction with a whole-number relation among its components has fewer ×1
a jump round a circle keeps the distance from the centre ×1
a named direction ×1
a named slope ×1
a path closing on itself was found within 1e-9 — the search at this depth found none otherwise ×1
a square or a triangle ×1
a table the family knows ×1
and arrives and leaves at the same angle to the radius ×1
and does so after twice the sum of the two whole numbers ×1
and it is the word a line of slope 1/φ spells, started at the corner ×1
and its angles add to a straight angle ×1
and its invariant is the same at every bounce ×1
and never returns exactly to where it started ×1
and no nearby triangle inscribed in the same three sides is shorter ×1
and one H for every horizontal one ×1
and that genus is a whole number ×1
and the direction along the wall is untouched ×1
and the path does not close before it is supposed to ×1
and the two rare blocks never do ×1
and they grow more common the further it is nudged ×1
and turns it by twice the angle whose cosine is one over that distance ×1
and with the obstacle the gap grows by far more than the distance does ×1
at least one drawn orbit closes, so the two behaviours are both on the page ×1
at the value it was aimed with ×1
between 10 and 200 bounces ×1
between 20 and 200 bounces are followed ×1
between 3 and 24 bounces are drawn ×1
between 3 and 40 chords are drawn ×1
each altitude meets its side between the corners ×1
each block's frequency is one of the arcs, largest to largest ×1
each bounce lands on a wall of the kind its letter names ×1
each jump lands as far beyond the vertex as it started before it ×1
each word begins with the one before ×1
every chord passes the same distance from the centre ×1
every orbit about a table with whole-number corners closes ×1
every point outside the table has a vertex it jumps through ×1
in both coordinates ×1
in direction (√2, √3, √5) only forty-one blocks of six appear in a million letters ×1
in four dimensions the count is n³ + 2n + 1 ×1
no orbit started in the picture wanders far from the table ×1
nudge the third component by half a percent or more and all forty-three appear ×1
on the empty table the gap grows no faster than the distance travelled ×1
on the empty table the gap stays proportional to the distance travelled ×1
only the fastest letter ever comes twice running ×1
orbits further out take at least as long to close ×1
so no part of it is below the axis ×1
the angle of arrival equals the angle of departure at every wall ×1
the angles of the table add to what a polygon's angles must ×1
the arcs take at most three lengths ×1
the ball is moving and has a wall to reach ×1
the ball starts inside the table ×1
the bouncing path and the folded straight line agree everywhere along their length ×1
the drawn triangle has the angles it was asked for ×1
the escaping trajectory's invariant is below it ×1
the first hit is somewhere on the rim ×1
the irrational path enters nearly every cell of the grid ×1
the irrational path is followed for between 60 and 600 bounces ×1
the lengths are Fibonacci numbers ×1
the obstacle fits inside the table with room to pass ×1
the path is followed for a drawable number of bounces ×1
the rational slope is a ratio of small whole numbers ×1
the rational slope is in lowest terms ×1
the rational slope's path returns to its exact starting state ×1
the rational slope's word never has more than seven blocks of any length — it repeats every seven letters ×1
the reflected copies drawn fit inside the panel they are drawn in ×1
the search runs to between three and sixteen bounces ×1
the slope is drawable ×1
the square's word has n + 1 ×1
the star closes exactly ×1
the star polygon is drawn in a single closed loop ×1
the star wraps between 1 and 6 times ×1
the stem is between a fifth and a whole radius deep ×1
the stem is between a twelfth and half the cap wide ×1
the surface the table unfolds into has a whole genus ×1
the sweep is a grid of a workable size ×1
the sweep produced something to compare ×1
the table is a triangle ×1
the table is one the family knows ×1
the trajectory is followed for between 40 and 600 bounces ×1
the trajectory stayed inside the table ×1
the trapped trajectory never touches a stem wall ×1
the trapped trajectory's invariant is above the stem's half-width and inside the cap ×1
the triangle is acute, which is when this path exists ×1
the Tribonacci word has 2n + 1 ×1
the two invariants sit on opposite sides of the stem's half-width ×1
the two numbers bracket n times the share of H ×1
the two paths start very nearly together ×1
the view is one the family draws ×1
the word has one V for every vertical line crossed ×1
there are 7 blocks of length 6 ×1
thirteen of the twenty-seven blocks of three appear ×1
two angles of a triangle, in degrees, leaving a third ×1
two to five starting distances outside the unit circle ×1
while the trajectory aimed under the half-width does go down the stem ×1
with the obstacle the same two paths end up hundreds of times further apart ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Dynamics

A bounce is a fold of the table

Reflect the room instead of the ball and every bounce disappears — the trajectory becomes a straight line through a tiled plane, and questions about what a ball does forever become questions about the slope of that line.

Dynamics

A room that cannot be lit

Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.

Dynamics

A table folded into a surface

Unfolding a square billiard gives a straight line on a torus. Unfolding any table whose angles are whole fractions of half a turn gives a straight line on some surface — and which surface it is decides how hard the dynamics will be.

Dynamics

A twist that cannot avoid two points

Turn the two edges of a ring in opposite directions without changing any area, and something in between must stay exactly where it is — not one point, but at least two, and the reason is that two loops enclosing the same area have to cross.

Dynamics

The ball that stays outside the table

Turn billiards inside out. A point outside a convex table looks at the corner on its right, jumps straight through it, and lands as far beyond as it started before. Round a square every orbit closes; round a circle every orbit keeps to its own circle; round a regular pentagon the orbits form islands with a fractal between them. Whether some table lets a point wander off to infinity was Moser's question, and the answer — yes, for a kite — took until 2007.

Dynamics

The obstacle that makes a table chaotic

Put one round post in the middle of a square table and every trace of order goes. Two paths that start a hundred-thousandth of a degree apart end up on opposite sides of the table, and the reason is that a wall curving outwards multiplies a gap where a flat one only adds to it.

Dynamics

The triangle nobody can settle

Does every triangular billiard table have a path that closes on itself? Acute triangles do, right triangles do, triangles with rational angles do — and for the rest the question has been open since it was asked.

Dynamics

The word a line spells in a cube

A ball bouncing in a cube hits three kinds of wall, and the order spells a word in three letters. On a square table the word had n + 1 different blocks of length n; in the cube it has $n^2 + n + 1$ — three, seven, thirteen, twenty-one — for a direction in general position. What general position has to exclude turns out to be more than any rational relation among the three speeds.

Dynamics

The word a straight line spells

A ball rolling across a square table, forever, hits walls in some order: V for a side wall, H for the end wall. Unfold the table and the ball becomes a straight line across a grid, and the order of walls becomes a word — VHVHVVHVHVVHV… for the golden slope. That word has exactly n + 1 different blocks of every length n, the fewest any word that never repeats can have; every stretch of it holds its fair share of H's to within one; and its blocks occur with at most three different frequencies.

The whole library · What the figures prove