Kepler-Poinsot Polyhedra
The Kepler-Poinsot polyhedra are a special class of convex polyhedra discovered by mathematicians Johannes Kepler and Louis Poinsot in the early 17th century. There are four Kepler-Poinsot polyhedra, each with its own unique set of properties:
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The Great Icosahedron: This polyhedron is composed of twenty equilateral triangular faces. Five triangles meet at each vertex. The Great Icosahedron complements the Great Dodecahedron. The Schläfli symbol for the Great Icosahedron is \(\{3,\ 5/2\}\).
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The Great Dodecahedron: This polyhedron consists of twelve regular pentagonal faces. The angles between adjacent faces are equal, and three faces meet at each vertex. The Great Dodecahedron is dual to the Great Icosahedron. The Schläfli symbol for the Great Dodecahedron is \(\{5,\ 5/2\}\).
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The Small Stellated Dodecahedron: This polyhedron has twelve regular pentagram-shaped faces. Three faces meet at each vertex, forming a corner. The Small Stellated Dodecahedron is dual to the Great Icosahedron. The Schläfli symbol for the Small Stellated Dodecahedron is \(\{5/2,\ 5\}\).
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The Great Stellated Dodecahedron: This polyhedron also has twelve regular pentagram-shaped faces. Five faces meet at each vertex. The Great Stellated Dodecahedron complements the Small Stellated Dodecahedron. The Schläfli symbol for the Great Stellated Dodecahedron is \(\{5/2,\ 3\}\).
These Kepler-Poinsot polyhedra are distinct as they are the only regular star polyhedra. A star polyhedron is characterized by self-intersecting or star-like extending faces. They possess symmetrical structures and exhibit captivating geometric properties.
The discovery of the Kepler-Poinsot polyhedra was a significant contribution to the field of mathematics, advancing geometric understanding. Even today, these polyhedra continue to captivate mathematicians and enthusiasts with their intricate designs and aesthetic appeal.
Kepler-Poinsot Polyhedra (following the list from left to right):