Poincare Hyperbolic Triangulator
Symmetry:
This tool tiles triangles over a hyperbolic surface with a constant Gaussian curvature of -1.
The image is projected onto either the Poincaré disk or upper half-plane. Although the
triangles appear to shrink as they are tiled outward, they are all the same size inside the
hyperbolic geometry. Use the symmetry box to specify the degree of rotational symmetry at
each of the three vertices. This determines the three angles of the triangle, which are
π / N for each vertex where N is
the degree of symmetry. In hyperbolic geometry, the angles also determine the size of
the triangle, and their sum is less than π, because the
three edges are geodesic circular arcs. In the image, click and drag the vertices or
the whole triangle to change what part is painted onto the hyperbolic tiles. The portion
of the image inside the selected triangle is stretched to fit the tile triangle for the
current symmetry values.
If you enter a tile symmetry that cannot exist in hyperbolic geometry, it will draw a different
geometry. For example, a symmetry of 5,3,2 corresponds to a dodecahedron and must be drawn on a
sphere, which has a curvature of 1. (The sphere is projected onto the screen using stereographic projection.)
A symmetry of 3,3,3 corresponds to hexagons and must be
drawn on a Euclidean plane, which has a curvature of 0.