Hemi-icosahedron
Encyclopedia
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.
It has 10 triangular faces, 15 edges, and 6 vertices. It has the same vertices and edges as the 5-dimensional polytope, the 5-simplex, but only contains half of the (20) faces.
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.
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
.
Abstract polytope
In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...
, containing half the faces of a regular icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
. It can be realized as a projective polyhedron
Projective polyhedron
In geometry, a projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids....
(a tessellation
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
of the real projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
by 10 triangles), which can be visualized by constructing the projective plane as a hemisphere
Hemisphere
Hemisphere may refer to:*Half of a sphereAs half of the Earth:*Any half of the Earth, see Hemispheres of the Earth, see:...
where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.
It has 10 triangular faces, 15 edges, and 6 vertices. It has the same vertices and edges as the 5-dimensional polytope, the 5-simplex, but only contains half of the (20) faces.
It is also related to the nonconvex uniform polyhedron
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
, the tetrahemihexahedron
Tetrahemihexahedron
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....
, which could be topologically identical to the hemi-icosahedron if each of the 3 square faces were divided into two triangles.
The complete graph K6
From the point of view of graph theoryGraph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
this is an embedding of
(the complete graphComplete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
with 6 vertices) on a real projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
. With this
embedding, the dual graph
Dual graph
In mathematics, the dual graph of a given planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge in G joining two neighboring regions, for a certain embedding of G. The term "dual" is used because this property is symmetric, meaning that if H is a dual...
is the Petersen graph
Petersen graph
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named for Julius Petersen, who in 1898 constructed it...
--- see hemi-dodecahedron
Hemi-dodecahedron
A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and...
.
 The complete graph Complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:... K6 represents the 6 vertices and 15 edges of the hemi-icosahedron |
See also
- 11-cell11-cellIn mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
- an abstract regular polychoronPolychoronIn geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
constructed from 11 hemi-icosahedra. - hemi-dodecahedronHemi-dodecahedronA hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and...
- hemi-cubeHemi-cube (geometry)In abstract geometry, a hemi-cube is an abstract regular polyhedron, containing half the faces of a cube.It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
- hemi-octahedronHemi-octahedronA hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron.It has 4 triangular faces, 6 edges, and 3 vertices...