(7,3,2) triangle tiling (small)

prusaprinters

A larger version of this model is featured in figures 4.13 and 4.14 of Visualizing Mathematics with 3D Printing.This 3D printed illustration of the hyperbolic plane is joint work with Saul Schleimer.The hyperbolic plane has constant negative curvature - it is shaped like a saddle everywhere. The hyperbolic plane cannot be "seen" directly. This is similar to the problem faced by terrestrial mapmakers: any flat map of the earth necessarily distorts either size or distance.This 3D print is a model, called the hemisphere model, of the hyperbolic plane. Lighting the piece, in three different ways, casts shadows of three further models: the Poincaré disk model (first image), the Klein model (second image) and the upper half space (third and fourth images) model.Each triangle in the (7,3,2) tiling has angles pi/2, pi/3 and pi/7. This is very unlike triangles in Euclidean geometry, where the angles must add to pi. In fact, all of the triangles in our illustration are identical! Some look smaller than others, but this is an artifact of the promised, and unavoidable, distortion. This model, and the larger, more detailed version are also available from Shapeways.Category: Math