Generator

billiard

A generator in the dynamics library, called 7 times across 2 essays. Below: what it draws at its defaults 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.

At its defaults

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.

`show: "rational"`

A billiard path that closes, and one that does not. Two paths in a square table: one of rational slope, which returns to its starting state and repeats, and one of irrational slope, which fills the table without ever closing.

`show: "circle"`

`show: "fagnano"`

The one path an acute triangular table always has. An acute triangle with the feet of its three altitudes joined into a closed path, which obeys the reflection law at all three sides and therefore repeats forever.

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.

the path bounces correctly off side 1 ×3
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 its angles add to a straight angle ×1
and never returns exactly to where it started ×1
and no nearby triangle inscribed in the same three sides is shorter ×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
every chord passes the same distance from the centre ×1
the angle of arrival equals the angle of departure at every wall ×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 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 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 reflected copies drawn fit inside the panel they are drawn in ×1
the slope is drawable ×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 table is a triangle ×1
the triangle is acute, which is when this path exists ×1
the view is one the family draws ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

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 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.

The whole library · What the figures prove