Hyperbolic

Tile a floor with squares and four meet at every corner. Try the same with regular pentagons and they will not close: flat space has no room. Step onto the hyperbolic plane and suddenly there is too much room, and tilings the flat world forbids snap together perfectly.

The concept

The Poincaré disk holds an entire infinite plane inside a finite circle. Distances stretch the closer you get to the rim, so the boundary is infinitely far away no matter how it looks. Straight lines, the shortest paths, are arcs that meet the rim at right angles. Drag the disk and you "walk" across a plane whose area grows exponentially, not quadratically, with radius.

The math

In flat space a regular $\{p, q\}$ tiling (q $p$-gons at each vertex) exists only when the angles fit exactly, $(p-2)(q-2) = 4$: hence the three Euclidean tilings $\{3,6\}, \{4,4\}, \{6,3\}$. The hyperbolic plane hosts a tiling for every pair with

$(p - 2)(q - 2) > 4$

an infinite family. It can afford them because its angles run small: a hyperbolic triangle's interior angles sum to less than $\pi$, and the deficit is its area,

$\text{area} = \pi - (\alpha + \beta + \gamma)$

so bigger triangles are angrier-cornered, and there is always enough angular slack to seat one more polygon around a vertex.

Why it stays strange

In this geometry there is no scaling: you cannot enlarge a shape and keep its angles, so "similar but bigger" does not exist. The parallel postulate fails, through a point run infinitely many lines that never meet a given one, and yet the place is perfectly consistent. It was the first proof that Euclid's fifth axiom was a choice, not a law.

Further reading

• Coxeter, Non-Euclidean Geometry.
• Thurston, Three-Dimensional Geometry and Topology.

@book{coxeter1998noneuclidean,
  author    = {Coxeter, H. S. M.},
  title     = {Non-Euclidean Geometry},
  edition   = {6th},
  publisher = {Mathematical Association of America},
  year      = {1998}
}