Poincaré

A plane folded into a circle of radius one

In 1882 Henri Poincaré pressed an infinite plane into the inside of a unit circle. The trick is in the ruler: distances stretch as you approach the rim, so a step taken near the centre is a stride, and the same step taken near the edge is a hair’s width. Measured from inside, the boundary circle is infinitely far from every point of the disk. Nothing ever arrives there.

Every heptagon you can see — down to the ones a fraction of a pixel wide — is exactly the same size in the disk’s own metric. The shrinking is in your eye, not in the geometry.

Seven walls, three to a corner

On a flat table a regular heptagon insists on corners of 128.57°, and three of them can never share a vertex. Hyperbolic space absorbs the excess: here each heptagon closes at exactly 120°, three tiles meet cleanly at every corner, and the pattern — Schläfli symbol {7,3} — repeats without end. Each ring of tiles outnumbers the ring before it, all the way to a rim that is never reached.

Escher saw this figure in 1954, in a book of H. S. M. Coxeter’s, and cut the four Circle Limit woodblocks from it. This page keeps his ink: woodcut black, paper lines, one ember tile in seven.

One map moves the whole world

The motion above is a single Möbius transformation —

z  ↦  (z + a) / (1 + āz)

— the only rigid way to slide a hyperbolic world seen through this disk. Watch any tile near the rim: as the drift carries it inward it unfurls to full size, because it always was full size. The flow can run forever and the supply of heptagons never thins.

One tile wears a small ember ring: the tile this drift began on. The flow wanders, and keeps finding its way back.