A hemi-icosahedron is an
abstract regular polyhedron, containing half the faces of a
regular icosahedron
In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
It has five equilateral triangular faces meeting at each vertex. It ...
. 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.
Projec ...
(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 ...
of the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
by 10 triangles), which can be visualized by constructing the projective plane as a
hemisphere
Hemisphere refers to:
* A half of a sphere
As half of the Earth
* A hemisphere of Earth
** Northern Hemisphere
** Southern Hemisphere
** Eastern Hemisphere
** Western Hemisphere
** Land and water hemispheres
* A half of the (geocentric) celes ...
where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.
Geometry
It has 10 triangular faces, 15 edges, and 6 vertices.
It is also related to the nonconvex
uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruence (geometry), congruent.
Unifor ...
, the
tetrahemihexahedron
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...
, which could be topologically identical to the hemi-icosahedron if each of the 3 square faces were divided into two triangles.
Graphs
It can be represented symmetrically on faces, and vertices as
Schlegel diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the orig ...
s:
The complete graph K6
It has the same vertices and edges as the 5-dimensional
5-simplex
In five-dimensional geometry, a 5- simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°.
The 5 ...
which has a complete graph of edges, but only contains half of the (20) faces.
From the point of view of
graph theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
this is an embedding of
(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 is ...
with 6 vertices) on a
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
. With this
embedding, the
dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-lo ...
is 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 is n ...
--- see
hemi-dodecahedron.
See also
*
11-cell
In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...
- an abstract regular
4-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: vertices, edges, faces (polygons), an ...
constructed from 11 hemi-icosahedra.
*
hemi-dodecahedron
*
hemi-cube
*
hemi-octahedron
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 , pages
162–165
External links
The hemi-icosahedron
Projective polyhedra