Loading AI tools
Abstract regular polyhedron with 10 triangular faces From Wikipedia, the free encyclopedia
In geometry, a hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 10 triangles), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.
Hemi-icosahedron | |
---|---|
Type | abstract regular polyhedron globally projective polyhedron |
Faces | 10 triangles |
Edges | 15 |
Vertices | 6 |
Euler char. | χ = 1 |
Vertex configuration | 3.3.3.3.3 |
Schläfli symbol | {3,5}/2 or {3,5}5 |
Symmetry group | A5, order 60 |
Dual polyhedron | hemi-dodecahedron |
Properties | non-orientable |
It has 10 triangular faces, 15 edges, and 6 vertices.
It is also related to the nonconvex uniform polyhedron, the tetrahemihexahedron, which could be topologically identical to the hemi-icosahedron if each of the 3 square faces were divided into two triangles.
It can be represented symmetrically on faces, and vertices as Schlegel diagrams:
It has the same vertices and edges as the 5-dimensional 5-simplex which has a complete graph of edges, but only contains half of the (20) faces.
From the point of view of graph theory this is an embedding of (the complete graph with 6 vertices) on a real projective plane. With this embedding, the dual graph is the Petersen graph --- see hemi-dodecahedron.
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.