Great Stellated Dodecahedron

The great stellated dodecahedron is a fascinating structure in the field of geometry, recognized as one of the four Kepler–Poinsot polyhedra. It holds the Schläfli symbol {(5/2,3)}, indicating its intricate structure composed of intersecting faces.

Structure

The great stellated dodecahedron shares its vertex arrangement with the regular dodecahedron, although it does not share the same vertex figure or vertex configuration. It can be considered a stellation of a smaller dodecahedron, and is the only dodecahedral stellation with this distinct property, apart from the dodecahedron itself.

Dual and Related Polyhedra

The dual of the great stellated dodecahedron is the great icosahedron, which is similarly related to the icosahedron. This relationship is part of the broader concept where polyhedra in the Kepler–Poinsot category are often dual to each other, a phenomenon also seen with the small stellated dodecahedron and the great dodecahedron.

Construction and Stellation

The construction of the great stellated dodecahedron can be likened to the formation of a pentagram, its two-dimensional analogue. By attempting to stellate a pentagonal polytope, the structure is extended until it reaches its final stellation. This process highlights the intricate design and complexity of forming such a polyhedron.

Truncation Process

The great stellated dodecahedron can undergo a truncation process, leading to a series of uniform polyhedra. Truncating the edges down to points produces the great icosidodecahedron, which can be seen as a rectified version of the great stellated dodecahedron.

Historical Context

The concept of non-convex regular polyhedra, such as the great stellated dodecahedron, was explored by Johannes Kepler and later formalized by Louis Poinsot. These polyhedra are named after them, recognizing their pioneering work in this domain of mathematics.

Related Topics

• Kepler–Poinsot polyhedra
• Regular polyhedra
• Polyhedron
• Stellation
• Dodecahedron

Explore these links to dive deeper into the fascinating world of geometric shapes and their profound implications in both theoretical and applied mathematics.