Concept

Boundedness

The property of a set or sequence that it stays within some fixed distance of a point. Whether an orbit stays bounded or escapes to infinity is often the one question about it that can be settled, and it separates the inside of the Mandelbrot set from the outside.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The Mandelbrot set. Points of the complex plane shaded by how long the iteration takes to escape, with the set itself the innermost region.

The shape in every picture of itself

One line of arithmetic, repeated, with a single complex number as its only input. Sort the numbers by whether the result stays bounded and the boundary between the two answers is the most complicated object anyone draws from a rule this short.

dynamics · Complex numbers

Every orbit round a square closes. 5 outer-billiard orbits about a square, each drawn as its closed ring of points, with periods 4, 8, 12, 20, 24 growing outwards.

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 · Billiards

Named alongside it

The objects these essays reach for when they reach for this one.

Fractal Billiards Complex numbers Connectedness Escape-time Invariant Irrational rotation Iteration Lattice Mandelbrot set Orbit Outer billiards

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