Final stellation of the icosahedron | |
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Two symmetric orthographic projections | |
Symmetry group | icosahedral (I_{h}) |
Type | Stellated icosahedron, 8th of 59 |
Symbols | Du Val H Wenninger: W_{42} |
Elements (As a star polyhedron) |
F = 20, E = 90 V = 60 (? = −10) |
Elements (As a simple polyhedron) |
F = 180, E = 270, V = 92 (? = 2) |
Properties (As a star polyhedron) |
Vertex-transitive, face-transitive |
In geometry, the complete or final stellation of the icosahedron^{[1]}^{[2]} is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron, or inside of it.
This polyhedron is the seventeenth stellation of the icosahedron, and given as Wenninger model index 42.
As a geometrical figure, it has two interpretations, described below:
Johannes Kepler researched stellations that create regular star polyhedra (the Kepler-Poinsot polyhedra) in 1619, but the complete icosahedron, with irregular faces, was first studied in 1900 by Max Brückner.
The stellation of a polyhedron extends the faces of a polyhedron into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as edges. The Fifty Nine Icosahedra enumerates the stellations of the regular icosahedron, according to a set of rules put forward by J. C. P. Miller, including the complete stellation. The Du Val symbol of the complete stellation is H, because it includes all cells in the stellation diagram up to and including the outermost "h" layer.^{[6]}
A polyhedral model can be constructed by 12 sets of faces, each folded into a group of five pyramids. |
As a simple, visible surface polyhedron, the outward form of the final stellation is composed of 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an Euler characteristic of 2.^{[9]}
The 92 vertices lie on the surfaces of three concentric spheres. The innermost group of 20 vertices form the vertices of a regular dodecahedron; the next layer of 12 form the vertices of a regular icosahedron; and the outer layer of 60 form the vertices of a nonuniform truncated icosahedron. The radii of these spheres are in the ratio^{[10]}
Inner | Middle | Outer | All three |
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20 vertices | 12 vertices | 60 vertices | 92 vertices |
Dodecahedron |
Icosahedron |
Nonuniform truncated icosahedron |
Complete icosahedron |
When regarded as a three-dimensional solid object with edge lengths a, ?a, ?^{2}a and ?^{2}a (where ? is the golden ratio) the complete icosahedron has surface area^{[10]}
and volume^{[10]}
Twenty 9 polygon faces (one face is drawn yellow with 9 vertices labeled.) |
2-isogonal 9 faces |
The complete stellation can also be seen as a self-intersecting star polyhedron having 20 faces corresponding to the 20 faces of the underlying icosahedron. Each face is an irregular 9/4 star polygon, or enneagram.^{[6]} Since three faces meet at each vertex it has 20 × 9 / 3 = 60 vertices (these are the outermost layer of visible vertices and form the tips of the "spines") and 20 × 9 / 2 = 90 edges (each edge of the star polyhedron includes and connects two of the 180 visible edges).
When regarded as a star icosahedron, the complete stellation is a noble polyhedron, because it is both isohedral (face-transitive) and isogonal (vertex-transitive).
Notable stellations of the icosahedron | |||||||||
Regular | Uniform duals | Regular compounds | Regular star | Others | |||||
(Convex) icosahedron | Small triambic icosahedron | Medial triambic icosahedron | Great triambic icosahedron | Compound of five octahedra | Compound of five tetrahedra | Compound of ten tetrahedra | Great icosahedron | Excavated dodecahedron | Final stellation |
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The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry. |