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

thingiverse

A more substantial rendering of this design can be viewed in figures 4.13 and 4.14 of Visualizing Mathematics with 3D Printing. This 3D printed depiction of the hyperbolic plane was a collaborative effort with Saul Schleimer. The hyperbolic plane exhibits a constant negative curvature, giving it the appearance of a saddle at every point. Unlike the terrestrial mapmakers who must navigate the distortions that result from mapping a curved surface to a flat one, this illustration provides an up-close look at the fundamental geometry of the hyperbolic plane. By employing advanced lighting techniques in three different ways, shadows are cast that reveal three additional models: the Poincaré disk model, the Klein model, and the upper half space model. Every triangle in the (7,3,2) tiling exhibits angles measuring pi/2, pi/3, and pi/7. This stands in stark contrast to triangles found in Euclidean geometry where the sum of their angles must be equal to pi. Notably, each triangle within this illustration is identical; those that appear smaller than others do so only as a result of distortion. A larger version of this model, along with additional detail, are also available on Shapeways for anyone seeking a more in-depth exploration of the subject.