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.
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.