The shortest line between two points is a straight line. The shortest lines on a sphere are arcs of great circles. These are the straight lines of spherical geometry. The equator and lines of longitude are examples of great circles. All great circles cross at two points, for example lines of longitude cross at the north and south poles, so there are no parallel lines in spherical geometry. One of the simplest results of Euclidean geometry is that the angles of a triangle sum to 180°. This result does not hold for spherical geometry. In fact, on a sphere the sum of the angles of a triangle is always greater than 180°.

Shown above is a spherical triangle. One edge follows a line of longitude from the north pole to the equator, the next edge goes 90° around the equator and the third edge goes from the equator back up to the north pole. All three edges are straight lines as they are arcs of great circles, but the angle between each pair of edges is 90°, so the sum of the angles of the triangle is 270°.

The non-Euclidean geometry developed independently by Nikolai Lobachevski and Farkas Bolyai in the 19th century is known as hyperbolic geometry. It differs from both Euclidean geometry and spherical geometry. In hyperbolic geometry the sum of the angles of a triangle is always less than 180°. Henri Poincare devised a way to represent infinite two-dimensional hyperbolic space as the interior of a disc.

The Poincare disc is a map of the entire hyperbolic plane. When we map the globe most of the great circles, which are straight lines on the sphere, are represented by curved lines on the map. Similarly, straight lines in hyperbolic space are represented by curved lines in the Poincare disc.

Shown above is a tessellation of triangles in hyperbolic space as represented on the Poincare disc. Each of the triangles is the same shape in hyperbolic space with angles of 90°, 45° and 30°, and each has the same area in hyperbolic space. The sum of the angles of these triangles is just 165°. (The size of the angles may be deduced by noting that the sum of all the angles meeting at a point must be 360°.)

Hyperbolic geometry opens up new possibilities for producing decorative tessellations, some of which were explored by the master of tessellation M. C. Escher.

In his woodcut Circle Limit I (1958) shown above, Escher produces a hyperbolic tessellation of birds based on a tessellation of isosceles triangles with one 60° angle and two 45° angles. An example of such a triangle is coloured blue in the illustration.

In Circle Limit III (1959), shown above, Escher's tessellation of fish is based on a hyperbolic tessellation of squares and triangles in the Poincare disc. Three squares and three equilateral triangles meet at each vertex in this tessellation, which is clearly impossible in Euclidean space.

Hyperbolic geometry is not restricted to two-dimensional space. In three-dimensional hyperbolic space it is possible to construct regular dodecahedra with dihedral angles, the angles between two faces, equal to 90°. An example of such a dodecahedron is shown in the animation above. These dodecahedra will stack up to fill space, just as cubes will in ordinary Euclidean space, with four dodecahedra arranged around each edge.

Text and computer-generated animations by Nick Mee.

Hyperbolic Space Tiled with Dodecahedra by Charlie Gunn.

Escher illustrations © 2006 The M.C.Escher Company - Baarn - Holland. All rights reserved. www.mcescher.com