Small stellated dodecahedron


In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {,5

Topology

If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar. The critical angle is atan above the dodecahedron face.
If we regard it as having 12 pentagrams as faces, with these pentagrams meeting at 30 edges and 12 vertices, we can compute its genus using Euler's formula
and conclude that the small stellated dodecahedron has genus 4. This observation, made by Louis Poinsot, was initially confusing, but Felix Klein showed in 1877 that the small stellated dodecahedron could be seen as a branched covering of the Riemann sphere by a Riemann surface of genus 4, with branch points at the center of each pentagram. In fact this Riemann surface, called Bring's curve, has the greatest number of symmetries of any Riemann surface of genus 4: the symmetric group acts as automorphisms

Images

In art

A small stellated dodecahedron can be seen in a floor mosaic in St Mark's Basilica, Venice by Paolo Uccello circa 1430. The same shape is central to two lithographs by M. C. Escher: Contrast and Gravitation.

Related polyhedra