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3D model of a small stellated dodecahedron. In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {5 ⁄ 2,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.
In the second chapter is the earliest mathematical understanding of two types of regular star polyhedra, the small and great stellated dodecahedron; they would later be called Kepler's solids or Kepler Polyhedra and, together with two regular polyhedra discovered by Louis Poinsot, as the Kepler–Poinsot polyhedra. [8]
The small and great stellated dodecahedron can be seen as a regular and a great dodecahedron with their edges and faces extended until they intersect. The pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces.
The Great Stellated Dodecahedron. Part 2. Section C. Shaping Space. Boston: Birkhauser, 1988. Messer, P., jt. author. Symmetry and Polyhedronal Stellation. II. Computers and Mathematics with Applications (Pergamon Press) 17:1-3 (1989). 1990-99 Polyhedrons and the Golden Number Symmetry 1:1 (1990).
If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron. The great stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, by attempting to stellate the n-dimensional pentagonal polytope which has pentagonal polytope faces and ...
This is an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes.
It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron. Its 24 vertices are all on the 12 axes with 5-fold symmetry (i.e. each corresponds to one of the 12 vertices of the icosahedron). This means that on each axis there is an inner and an outer vertex.
Compound of great icosahedron and stellated dodecahedron Type: stellation and compound: Coxeter diagram: ∪ : Convex hull: Dodecahedron: Polyhedra: 1 great icosahedron 1 great stellated dodecahedron: Faces: 20 triangles 12 pentagrams: Edges: 60 Vertices: 32 Symmetry group: icosahedral (I h)