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The hemidodecahedron
Projective polyhedra
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a hemi-dodecahedron is an abstract, regular polyhedron
A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In ...
, containing half the faces of a regular dodecahedron
A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the ...
. It can be realized as a projective polyhedron
In geometry, a (globally) 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.
Proje ...
(a tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
of the real projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
by 6 pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
s), which can be visualized by constructing the projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...
as a hemisphere
Hemisphere may refer to:
In geometry
* Hemisphere (geometry), a half of a sphere
As half of Earth or any spherical astronomical object
* A hemisphere of Earth
** Northern Hemisphere
** Southern Hemisphere
** Eastern Hemisphere
** Western Hemi ...
where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.
It has 6 pentagonal faces, 15 edges, and 10 vertices.
Projections
It can be projected symmetrically inside of a 10-sided or 12-sided perimeter: :
Petersen graph
From the point of view ofgraph theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
this is an embedding of the 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 i ...
on a real projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
.
With this embedding, the dual graph
In the mathematics, mathematical discipline of graph theory, the dual graph of a planar graph is a graph that has a vertex (graph theory), vertex for each face (graph theory), face of . The dual graph has an edge (graph theory), edge for each p ...
is
''K''6 (the 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. A complete digraph is a directed graph in which every pair of distinct vertices i ...
with 6 vertices) --- see hemi-icosahedron
In geometry, a hemi-icosahedron is an abstract polytope, 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), ...
.
See also
* 57-cell – an abstract regular4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: Vertex (geometry), vertices, Edge (geo ...
constructed from 57 hemi-dodecahedra.
*hemi-icosahedron
In geometry, a hemi-icosahedron is an abstract polytope, 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), ...
* hemi-cube
*hemi-octahedron
In geometry, a hemi-octahedron is an abstract polytope, abstract regular polyhedron, containing half the faces of a regular octahedron.
It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the Hemicube (geometry), hemicube ...
References
* {{citation , last1 = McMullen , first1 = Peter, author1-link=Peter McMullen , first2 = Egon , last2 = Schulte , chapter = 6C. Projective Regular Polytopes , title = Abstract Regular Polytopes , edition = 1st , publisher = Cambridge University Press , isbn = 0-521-81496-0 , date=December 2002 , pages162–165
External links
The hemidodecahedron
Projective polyhedra