Small Stellated Dodecahedron
The small stellated dodecahedron is a captivating structure in the realm of geometry. It is classified as one of the four Kepler–Poinsot polyhedra, which are known for their nonconvex regular forms. Named by Arthur Cayley, this polyhedron is denoted by the Schläfli symbol {5/2, 5}.
Structure and Geometry
The small stellated dodecahedron can be visualized by attaching twelve pentagonal pyramids onto a regular dodecahedron, with each face replaced by a pentagram. The resulting structure boasts 12 pentagrams as its faces, meeting at 30 edges and 12 vertices. The surface topology of this fascinating form is akin to the pentakis dodecahedron, albeit with taller isosceles triangles serving as faces.
Mathematical Significance
The intricate design of the small stellated dodecahedron can be understood using Euler's formula, which relates the number of vertices, edges, and faces of a polyhedron. The small stellated dodecahedron can further be considered a branched covering of the Riemann sphere by a Riemann surface of genus 4. This concept was illuminated by Felix Klein in 1877, following initial observations by Louis Poinsot.
Dual and Related Polyhedra
The great dodecahedron is the dual of the small stellated dodecahedron. The dodecadodecahedron is noted as a rectification of the great dodecahedron, and is related to the small stellated dodecahedron through its truncation down to points. While the truncated small stellated dodecahedron may appear as a degenerate uniform polyhedron due to coinciding edges and vertices, it remains an integral part of understanding the complexities of geometric stellations.