To make this argument work, the group had to deal with many finicky mathematical details. For example, in most cases you have an infinite number of spots available to move to, not just four corners.

Let P be a locally finite disk pattern on the complex plane C whose combinatorics are described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are ...

“A Mirror Maze: Numbers in Nature,” at the Arizona Science Center in Phoenix through Sept. 4, features 1,800 square feet of a seemingly infinite pattern of mirrors, filled with surprises.

It is a shape that can tile an infinite two-dimensional flat surface but only in a non-repeating pattern. David Smith, a self-described shape hobbyist from England, was behind the discovery.

With infinite particle arrangements there would be infinite pattern repetitions. So there must also be countless duplications. Earths, solar systems, and yous and mes.

A simple mathematical model can account for lizard’s green-and-black pattern Tried-and-true Ising model can describe how scales color-switch as the lizard ages.