ISEGORIA / MATH ENCYCLOPEDIA
Hyperbolic geometry: where parallels multiply
Distance in the Poincaré disc, triangles whose angles fall short of π, and tilings no flat plane can hold.
Before you begin: Complex numbers and differential geometry
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Distance, lines, and many parallels
In the Poincaré disc, lines are arcs of circles that meet the rim at right angles, and the rim itself is infinitely far away. Drag P and Q: the dots are equally spaced in hyperbolic length and crowd towards the edge. Then drag R off the line. Through R pass two limiting parallels and infinitely many other lines that never meet PQ, so Euclid’s parallel postulate fails.
Worked example. From the centre, a point at Euclidean radius \(r\) lies at hyperbolic distance \(2\operatorname{artanh}r=\ln\frac{1+r}{1-r}\). At \(r=0.9\) that is \(\ln 19\approx2.94\), and the rim is infinitely far.
Watch out. The disc shows angles faithfully but not lengths: arcs near the rim look short but are long. Because the model is conformal, the angle \(\Pi\) can be read straight off the picture.
What happens to Π(d) when R is very close to the line?
It tends to 90°, and the two limiting parallels merge into the single Euclidean parallel. Small regions of the hyperbolic plane look Euclidean.
2. Triangles: the angle sum falls short
Drag the vertices of a geodesic triangle. Its angles always add to less than π, and the shortfall is exactly its area: the Gauss–Bonnet theorem for a surface of constant curvature −1. Press Play to grow the triangle from a point. Tiny triangles are almost Euclidean; the defect appears as the triangle gets large.
Worked example. A triangle whose three vertices sit on the rim (an ideal triangle) has three zero angles, so its area is exactly \(\pi\), the largest any hyperbolic triangle can have.
Watch out. Area means hyperbolic area, not the Euclidean area of the shaded region on the screen. With curvature \(-K\) the rule becomes \(\alpha+\beta+\gamma=\pi-K\cdot\operatorname{Area}\).
Can two hyperbolic triangles have the same angles but different sizes?
No. The angles fix the area, and in fact fix the triangle up to an isometry: there are no similar triangles of different sizes.
3. Regular tilings {p, q}
Choose p and q. The tiling {p, q} uses regular p-gons, q of them at every corner. On a flat plane only {3, 6}, {4, 4} and {6, 3} work; in the hyperbolic plane every pair with 1/p + 1/q < 1/2 does. The picture is built by reflecting every point into one triangle with angles π/p, π/2 and π/q. Drag the gold centre to move the whole tiling by an isometry of the disc.
Worked example. For \(\{7,3\}\): heptagons with 120° corners, three at each vertex. Each has area \(5\pi-\tfrac{14\pi}{3}=\tfrac{\pi}{3}\).
Watch out. The two tones mark mirror-image triangles, not separate tiles: each tile is made of \(2p\) of them. Near the rim the tiles blur together because they are smaller than a pixel in this picture, not because they change.
Why can’t {7, 3} tile the flat plane?
A flat regular heptagon has corners of \(900^\circ/7\approx128.6^\circ\), and three of them add up to more than 360°. Only negative curvature lets a polygon’s corners shrink to 120°.