Hyperbolic geometry

The picture is a map of the hyperbolic plane, which goes on for ever in every direction, like the flat plane. The map is the Poincaré disc: the whole plane fits inside the circle, and the circle, marked infinity, is not part of it. All the tiles are the same size in the plane; they look smaller towards the edge only because of the map. Map draws the same plane two other ways, with the same lengths, angles and areas.

When the walker takes equal steps out, the steps look shorter and shorter in the picture. A short piece $|dz|$ of the picture at the point $z$ stands for a length

of the plane, which grows without bound as $z$ nears the edge. Walking a distance $d$ out from the middle brings the walker only $\tanh(d/2)$ of the way to the edge. The edge is not a wall but the place where the picture runs out of room.

Any point can be carried to any other by a motion that keeps lengths and angles, and the plane has the same curvature, −1, everywhere. Follow uses such a motion to bring the walker back to the middle.

Straight lines are the shortest ways between points. The disc draws them as arcs meeting the edge at right angles, or as diameters. They bow towards the middle because there a length in the picture stands for less of the plane. In Klein every line is a straight chord, but away from the middle angles are generally distorted. In Half-plane lines are half-circles standing on the bottom edge, or upright half-lines, and angles are drawn right, as in the disc. No flat map can draw the lines straight and the angles right at once, or the picture would obey flat geometry.

A triangle’s angles add up to less than 180°, and the shortfall, in radians, is its area:

As the three corners approach three different points of the edge, all three angles tend to 0 and the area to π, the most any triangle can have. So a triangle cannot be enlarged keeping its angles. Two triangles with the same angles are congruent, and there are no similar triangles of different sizes.

In Parallels, a line through P is red while it meets ℓ and blue while it does not. When P is off ℓ, infinitely many lines through P miss ℓ. The boundaries are the two blue lines through ℓ’s ends, which miss it too, since the ends are at infinity. Each makes with the perpendicular from P the angle of parallelism

where $d$ is the distance from P to ℓ. As P comes close to ℓ, Π approaches the flat plane’s 90°; far away it approaches 0. None of the lines that miss ℓ keeps a constant distance from it. The two boundary lines close in on ℓ towards one end, and every other line that misses ℓ moves away from it on both sides of a common perpendicular.

The red dashed curve in Walk is the circle about the start through the walker. A circle of radius $r$ has circumference

growing exponentially, where the flat plane has $2\pi r$. The extra room shows in the tiles, each ring round the middle holding many more than the ring inside it.

The tiling $\{p, q\}$ has regular $p$-gons, $q$ at each corner. It is hyperbolic exactly when $1/p + 1/q < 1/2$.

Nikolai Lobachevsky and János Bolyai independently published this geometry around 1830.