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Petrial Muoctahedron
In topological graph theory, the Petrie dual of an embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie polygons of the first embedding as its faces. The Petrie dual is also called the Petrial, and the Petrie dual of an embedded graph G may be denoted G^\pi. It can be obtained from a signed rotation system or ribbon graph representation of the embedding by twisting every edge of the embedding. Properties Like the usual dual graph, repeating the Petrie dual operation twice returns to the original surface embedding. Unlike the usual dual graph (which is an embedding of a generally different graph in the same surface) the Petrie dual is an embedding of the same graph in a generally different surface. Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations. Regular polyhedra Applying the Petrie dual to a regular polyhedron produces a regular map. The number of skew ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 regular dodecahedron. is a dodecahedron composed of regular polygon, regular pentagonal faces, three meeting at each Vertex (geometry), vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity. The regular dodecahedron is a truncated trapezohedron because it is the result of Truncation (geometry), truncating axial vertices of a pentagonal trapezohedron. It is also a Goldberg polyhedron because it is the initial polyhedron to construct new polyhedrons by the process of chamfering. It has a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hemi-cube (geometry)
In abstract geometry, a hemicube is an abstract, regular polyhedron, produced by cutting a cube in half with a plane that passes through 2 opposite corners and the midpoints of 2 edges. A hemicube is also sometimes called a square hemiprism. Realization 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 ''K''4 (the complete graph with four vertices) on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petrial Icosahedron
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. A regular polyhedron is identified by its Schläfli symbol of the form , where ''n'' is the number of sides of each face and ''m'' the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In addition, there are five regular compounds of the regular polyhedra. The regular polyhedra There are five convex regular polyhedra, known as the Platonic solids; four regular star polyhedra, the Kepler–Poinsot polyhedra; and five regular compou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron 4 Petrie Polygons
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex and non-convex shapes. Combinatorially equivalent to the regular octahedron The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it: * Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral. * Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube 4 Petrie Polygons
A cube or regular hexahedron is a three-dimensional solid object in geometry, which is bounded by six congruent square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It is a type of parallelepiped, with pairs of parallel opposite faces, and more specifically a rhombohedron, with congruent edges, and a rectangular cuboid, with right angles between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron. The cube can be represented in many ways, one of which is the graph known as the cubical graph. It can be constructed by using the Cartesian product of graphs. The cube is the three-dimensional hypercube, a family of polytopes also including the two-dimensional square and four-dimensional tesseract. A cube with unit side length is the canonical unit of volume in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron 3 Petrie Polygons
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces. Reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face Of Petrial Icosahedron
The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the psyche adversely. Structure The front of the human head is called the face. It includes several distinct areas, of which the main features are: *The forehead, comprising the skin beneath the hairline, bordered laterally by the temples and inferiorly by eyebrows and ears *The eyes, sitting in the orbit and protected by eyelids and eyelashes * The distinctive human nose shape, nostrils, and nasal septum *The cheeks, covering the maxilla and mandible (or jaw), the extremity of which is the chin *The mouth, with the upper lip divided by the philtrum, sometimes revealing the teeth Facial appearance is vital for human recognition and communication. Facial muscles in humans allow expression of emotions. The face is itself a highly sensitive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
