“There’s too much space in this space” - Frank Farris [2].

The confounding nature of hyperbolic geometry can be witnessed throughout nature from the frills of lettuce leaves and the curves of coral. It was born in rebellion by those who defied Euclid’s parallel postulate. Despite seeming initially counterintuitive, hyperbolic space can be made visual and tactile. When we embed small sections of 2d hyperbolic space, which is infinite, into flat 3d space. When we do this, we recover the shape of a saddle. We cannot embed much of the hyperbolic plane into our world as it will fill space too quickly and intersect with itself [3]. It turns out this occurs for the same reasons that we cannot wrap a flat paper, or a map, onto a sphere without it overlapping, creasing, and buckling. Artist Daina Taimana crochets pieces of hyperbolic planes “by increasing the number of successive stitches in a row, though finding a workable rate of increase required trial and error” [2]. Another artistic capture of the hyperbolic plane is the Poincare disk. This disk is heavily distorted, for each tile is the same size in the hyperbolic universe. Moreover, the boundary is infinitely far from the center, so this projection can be misleading without careful provisions [3]. Interestingly, the artist M. C. Escher used the Poincare disk in a great deal of his work [2].