The {5,5} polyhedra

One of the most unusual polyhedra, perhaps because it is so unexpected, is the {5,5} polyhedron first reported by J. R. Gott, III, in 1967. [Pseudopolyhedrons, J. R. Gott, III, The American Mathematical Monthly, Vol. 74, No. 5. (May, 1967), pp. 497-504.] The vertex star consists of three coplanar regular pentagons, with two additional upright regular pentagons fitting perfectly into the remaining planar gap. This vertex star creates only one sponge, infinite in three dimensions. But because the vertex star has a reflective symmetry, it can be asymmetrically labeled in four different ways.

The {5,5} polyhedron   (5.5.5.5.5)

1. [a b⁺ c⁺ c^- b^-;  a b⁺ c^^-]
2. [a⁺ b⁺ c⁺ d⁺ e⁺;  a⁺ b⁺ c^^- d^^- e⁺]
3. [a⁺ b⁺ c⁺ d⁺ e⁺;  a⁺ b⁺ d^⁺ c^⁺ e⁺]
4. [a⁺ b⁺ c⁺ d⁺ e⁺;  a⁺ e^- c^^- d^^- b^-]
5. [a⁺ b⁺ c⁺ d⁺ e⁺;  a⁺ e^- d^⁺ c^⁺ b^-]