| Hemicube | |
|---|---|
 | |
| Type | abstract regular polyhedron globally projective polyhedron |
| Faces | 3 squares |
| Edges | 6 |
| Vertices | 4 |
| Vertex configuration | 4.4.4 |
| Schläfli symbol | {4,3}/2 or {4,3}3 |
| Symmetry group | S4 , order 24 |
| Dual polyhedron | hemi-octahedron |
| Properties | non-orientable Euler characteristic 1 |
In abstract geometry, a hemicube is an abstract, regular polyhedron, containing half the faces of a cube.
It can be realized as a projective polyhedron (a tessellation of the real projective plane by three quadrilaterals), 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 three square faces, six edges, and four vertices. It has an unexpected property that every face is in contact with every other face on two edges, and every face contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets.
From the point of view of graph theory the skeleton is a tetrahedral graph, an embedding of K4 (the complete graph with four vertices) on a projective plane.
The hemicube should not be confused with the demicube – the hemicube is a projective polyhedron, while the demicube is an ordinary polyhedron (in Euclidean space). While they both have half the vertices of a cube, the hemicube is a quotient of the cube, while the vertices of the demicube are a subset of the vertices of the cube.
The hemicube is the Petrie dual to the regular tetrahedron, with the four vertices, six edges of the tetrahedron, and three Petrie polygon quadrilateral faces. The faces can be seen as red, green, and blue edge colorings in the tetrahedral graph:
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
In geometry, a regular icosahedron 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.
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.
In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It is the 4-simplex (Coxeter's polytope), the simplest possible convex 4-polytope, and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The 5-cell is a 4-dimensional pyramid with a tetrahedral base and four tetrahedral sides.
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of dimension ≤ n.
In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,4}. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called C16, hexadecachoron, or hexdecahedroid.
In mathematics, an abstract polytope is an algebraic partially ordered set (poset) which captures the combinatorial properties of a traditional polytope without specifying purely geometric properties such as angles or edge lengths.
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 diagram is (although this is a double covering of the tetrahemihexahedron).
In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least four vertices. The interior surface of such a polygon is not uniquely defined.
In geometry, demihypercubes are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n(n−1)-demicubes, and 2n(n−1)-simplex facets are formed in place of the deleted vertices.
A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. 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 the hemisphere into three equal parts.
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 dividing the hemisphere into three equal parts.
A hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron.
A hemi-cuboctahedron is an abstract polyhedron, containing half the faces of a semiregular cuboctahedron.