Henri Poincaré1854–1912
Disk model
French mathematician whose disk model draws the whole infinite hyperbolic plane inside one circle, with angles kept true and distances shrinking toward the rim.
Poincaré Disk
Every tile here is the same size. The whole infinite hyperbolic plane is drawn inside one circle, and the rim is infinitely far away, so the tiles appear to shrink as they approach it.
About this pattern
Henri Poincaré · the hyperbolic plane in a circle
In the disk model a straight line is an arc meeting the boundary circle at right angles, and distance stretches without limit toward the rim. That leaves room for tilings the flat plane cannot hold: at {p, q}, q polygons of p sides meet at every corner, which is only possible when 1/p + 1/q is less than a half.
Each ring is produced by reflecting the one before it in its own edges. Coxeter's diagrams of exactly this sent Escher off to draw the Circle Limit prints.
Behind the mathematics
H. S. M. Coxeter1907–2003
Regular hyperbolic tilings
British-Canadian geometer who worked out the regular hyperbolic tilings and whose diagrams sent Escher off to draw the Circle Limit prints.