"structured grids. Quadrilateral elements are usually excluded from being or becoming concave. The basic 3-dimensional element are the tetrahedron, quadrilateral pyramid, triangular prism, and hexahedron. They all have triangular and quadrilateral faces. Extruded 2-dimensional models may be represented entirely by the prisms and hexahedra as extruded triangles and quadrilaterals. In general, quadrilateral faces in 3-dimensions may not be perfectly planar. A nonplanar quadrilateral face can be considered a thin tetrahedral volume that is shared by two neighboring elements. A tetrahedron has 4 vertices, 6 edges, and is bounded by 4 triangular faces. In most cases a tetrahedral volume mesh can"
"Tetragonal trapezohedron The tetragonal trapezohedron, or deltohedron, is the second in an infinite series of face-uniform polyhedra, which are dual to the antiprisms. It has eight faces, which are congruent kites, and is dual to the square antiprism. This shape has been used as a test case for hexahedral mesh generation, simplifying an earlier test case of Rob Schneider in the form of a square pyramid with its boundary subdivided into 16 quadrilaterals. In this context the tetragonal trapezohedron has also been called the cubical octahedron, quadrilateral octahedron, or octagonal spindle, because it has eight quadrilateral faces and is uniquely"
"Tetradecahedron A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces. A tetradecahedron is sometimes called a tetrakaidecahedron. No difference in meaning is ascribed. The Greek word ""kai"" means 'and'. There is evidence that mammalian epidermis cells are shaped like flattened tetrakaidecahedra, an idea first suggested by Lord Kelvin. There are 1,496,225,352 topologically distinct ""convex"" tetradecahedra, excluding mirror images, having at least 9 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to"
"Heptahedron A heptahedron (plural: heptahedra) is a polyhedron having seven sides, or faces. A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven faces form a rudimentary projective plane. No heptahedra are regular. There are 34 topologically distinct ""convex"" heptahedra, excluding mirror images. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles"
"original faces and the incenter. Since the four subtetrahedra fill the volume, we have formula_28. Denote the circumradius of a tetrahedron as ""R"". Let ""a,b,c"" be the length of the three edges that meet at a point, and ""A,B,C"" the length of the opposite edges. Let ""V"" be the volume of the tetrahedron. Then The sum of the areas of any three faces is greater than the area of the fourth face. There exist tetrahedra having integer-valued edge lengths, face areas and volume. One example has one edge of 896, the opposite edge of 990 and the other four edges"
"Tetrahedrally diminished dodecahedron In geometry, a tetrahedrally diminished dodecahedron (also tetrahedrally stellated icosahedron or propello tetrahedron) is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces (4 equilateral triangles and 12 identical quadrilaterals). It has chiral tetrahedral symmetry, and so its geometry can be constructed from pyritohedral symmetry of the pseudoicosahedron with 4 faces stellated, or from the pyritohedron, with 4 vertices diminished. Within its tetrahedral symmetry, it has geometric varied proportions. By Dorman Luke dual construction, a unique geometric proportion can be defined. The kite faces have edges of length ratio ~ 1:0.6325. Topologically, the"
"Hexagonal trapezohedron The hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has twelve faces which are congruent kites. One degree of freedom within D symmetry changes the kites into congruent quadrilaterals with 3 edges lengths. In the limit, one edge of each quadrilateral goes to zero length, and these become bipyramids. Crystal arrangements of atoms can repeat in space with hexagonal trapezohedral cells. If the kites surrounding the two peaks are of different shapes, it can only have C symmetry, order 12. These can be called"
"Deltahedron In geometry, a deltahedron (plural ""deltahedra"") is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta (formula_1), which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra. There are only eight strictly-convex deltahedra: three are regular polyhedra, and five are Johnson solids. In the 6-faced deltahedron, some vertices have degree 3 and some"
"Pentahedron In geometry, a pentahedron (plural: pentahedra) is a polyhedron with five faces or sides. There are no face-transitive polyhedra with five sides and there are two distinct topological types. With regular polygon faces, the two topological forms are the square pyramid and triangular prism. The ""square pyramid"" can be seen as a ""triangular prism"" where one edge of its side edges is collapsed into a point, losing one edge and one vertex, and changing two squares into triangles. Geometric variations with irregular faces can also be constructed. Some irregular pentahedra with six vertices may be called wedges. An irregular"
"tetrahedron. These symmetries can be emphasized by different colorings of the faces. It has eleven arrangements of nets. The octahedron is the dual polyhedron to the cube. The uniform tetrahemihexahedron is a tetrahedral symmetry faceting of the regular octahedron, sharing edge and vertex arrangement. It has four of the triangular faces, and 3 central squares. The following polyhedra are combinatorially equivalent to the regular polyhedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of a regular octahedron. More generally, an octahedron can be any polyhedron with eight faces. The regular octahedron"
"Tetrakis cuboctahedron The tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It a dual of the truncated rhombic dodecahedron. Its name comes from a topological construction from the cuboctahedron with the kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance from the center, while in general octahedral symmetry can be maintain even with the 6 order-4 vertices at a different distance from the center as the other 12. It can also be topologically constructed from the octahedron, dividing each triangular face into"
"have been known since antiquity. In a regular tetrahedron, all faces are the same size and shape (congruent) and all edges are the same length. Regular tetrahedra alone do not tessellate (fill space), but if alternated with regular octahedra in the ratio of two tetrahedra to one octahedron, they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. The following Cartesian coordinates define the four vertices of a tetrahedron with edge"
"Truncated triakis tetrahedron The truncated triakis tetrahedron, or more precisely an order-6 truncated triakis tetrahedron, is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps. It is constructed from taking a triakis tetrahedron by truncating the order-6 vertices. This creates 4 regular hexagon faces, and leaves 12 mirror-symmetric pentagons. A topologically similar equilateral polyhedron can be constructed by using 12 regular pentagons with 4 equilateral but nonplanar hexagons, each vertex with internal angles alternating between 108 and 132 degrees. Topologically, as a near-miss Johnson solid, the four"
"example, any of the six squares that bound a cube is a face of the cube. Sometimes ""face"" is also used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, each sharing two of 8 cubic cells. Some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets (flat polygons formed by coplanar vertices which do not lie in the same face of the polyhedron). Any convex polyhedron's surface has Euler characteristic where ""V"" is the number of vertices,"
"distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) An incomplete list of forms includes: Tetradecahedra having all regular polygonal faces (all exist in irregular-faced forms as well): Tetradecahedra having at least one irregular face: Tetradecahedron A tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces. A tetradecahedron is sometimes called a tetrakaidecahedron. No difference in meaning is ascribed. The Greek word ""kai"" means 'and'. There is evidence that mammalian epidermis cells are shaped like"
"(Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) Hendecahedron A hendecahedron (or undecahedron) is a polyhedron with 11 faces. There are numerous topologically distinct forms of a hendecahedron, for example the decagonal pyramid, and enneagonal prism. Three forms are Johnson solids: augmented hexagonal prism, biaugmented triangular prism, and elongated pentagonal pyramid. Two classes, the bisymmetric and the sphenoid hendecahedra, are space-filling. The name of hendecahedron is based on"
"Tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U. It has six vertices, 12 edges, and seven faces: four triangular and three square. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin diagram is (although this is a double covering of the tetrahemihexahedron). It is the only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 | 2, but that represents a double covering of the tetrahemihexahedron with eight triangles and six squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.)"
"4 triangles by adding mid-edge vertices (an ortho operation). From this construction, all 32 triangles will be equilateral. This polyhedron can be confused with a slightly smaller Catalan solid, the tetrakis hexahedron, which has only 24 triangles, 32 edges, and 14 vertices. Tetrakis cuboctahedron The tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It a dual of the truncated rhombic dodecahedron. Its name comes from a topological construction from the cuboctahedron with the kis operator applied to the square faces. In this construction, all the vertices are assumed to be the same distance"
"Hexadecahedron A hexadecahedron (or hexakaidecahedron) is a polyhedron with 16 faces. No hexadecahedron is regular; hence, the name is ambiguous. There are numerous topologically distinct forms of a hexadecahedron, for example the pentadecagonal pyramid, tetradecagonal prism and heptagonal antiprism. There are 387,591,510,244 topologically distinct ""convex"" hexadecahedra, excluding mirror images, having at least 10 vertices. (Two polyhedra are ""topologically distinct"" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) There are 302,404 self-dual hexadecahedron, 1476"
"Enneahedron In geometry, an enneahedron (or nonahedron) is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular. The most familiar enneahedra are the octagonal pyramid and the heptagonal prism. The heptagonal prism is a uniform polyhedron, with two regular heptagon faces and seven square faces. The octagonal pyramid has eight isosceles triangular faces around a regular octagonal base. Two more enneahedra are also found among the Johnson solids: the elongated square pyramid and the elongated triangular bipyramid. The three-dimensional associahedron, a"
"Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces (8 regular hexagonal and 6 square), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron. It is also the Goldberg polyhedron G(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or ""pack"") 3-dimensional space, as a permutohedron. Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis cube has edge lengths and . A truncated octahedron is constructed from a regular octahedron with"
"regular tetrahedron. This polyhedron represents the vertex figure of a hyperbolic uniform honeycomb, the partially diminished icosahedral honeycomb, pd{3,5,3}, with 12 pentagonal antiprisms and 4 dodecahedron cells meeting at every vertex. Tetrahedrally diminished dodecahedron In geometry, a tetrahedrally diminished dodecahedron (also tetrahedrally stellated icosahedron or propello tetrahedron) is a topologically self-dual polyhedron made of 16 vertices, 30 edges, and 16 faces (4 equilateral triangles and 12 identical quadrilaterals). It has chiral tetrahedral symmetry, and so its geometry can be constructed from pyritohedral symmetry of the pseudoicosahedron with 4 faces stellated, or from the pyritohedron, with 4 vertices diminished. Within its"
"Tetrahedral prism In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices. It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids. The tetrahedral prism is bounded by two tetrahedra and four triangular prisms. The triangular prisms are joined to each other via their square faces, and are joined to the two tetrahedra via their triangular faces. The"
"between edges or faces.) An example of each type is depicted below, along with the number of sides on each of the faces. The images are ordered by descending number of six-sided faces (if any), followed by descending number of five-sided faces (if any), and so on. Heptahedron A heptahedron (plural: heptahedra) is a polyhedron having seven sides, or faces. A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven faces form a rudimentary projective plane. No heptahedra are"
"tetrahedra has been known for hundreds of years. It comes up regularly in the world of origami. Joining the twenty vertices would form a regular dodecahedron. There are both left-handed and right-handed forms, which are mirror images of each other. The square hosohedron is another polyhedron with four faces, but it does not have triangular faces. In numerical analysis, complicated three-dimensional shapes are commonly broken down into, or approximated by, a polygonal mesh of irregular tetrahedra in the process of setting up the equations for finite element analysis especially in the numerical solution of partial differential equations. These methods have"
"Trapezohedron The ""n""-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an ""n""-gonal antiprism. With a highest symmetry, its 2""n"" faces are congruent kites (also called trapezia or deltoids). The faces are symmetrically staggered. The ""n""-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry. The dual ""n""-gonal antiprism has two actual ""n""-gon faces. An ""n""-gonal trapezohedron can be dissected into two equal ""n""-gonal pyramids and an ""n""-gonal antiprism. These figures, sometimes called deltohedra, must not be confused with deltahedra, whose faces are equilateral triangles. In texts describing"
"bimedians are If the four faces of a tetrahedron have the same perimeter, then the tetrahedron is a disphenoid. If the four faces of a tetrahedron have the same area, then it is a disphenoid. The centers in the circumscribed and inscribed spheres coincide with the centroid of the disphenoid. The bimedians are perpendicular to the edges they connect and to each other. Some tetragonal disphenoids will form honeycombs. The disphenoid whose four vertices are (-1, 0, 0), (1, 0, 0), (0, 1, 1), and (0, 1, -1) is such a disphenoid. Each of its four faces is an isosceles"
"Szilassi polyhedron The Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. Each face of this polyhedron shares an edge with each other face. As a result, it requires seven colours to colour all adjacent faces, providing the lower bound for the seven colour theorem. It has an axis of 180-degree symmetry; three pairs of faces are congruent leaving one unpaired hexagon that has the same rotational symmetry as the polyhedron. The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surface of a torus. The tetrahedron"
"pentahedron can be a non-convex solid: Consider a non-convex (planar) quadrilateral (such as a dart) as the base of the solid, and any point not in the base plane as the apex. There is a third topological polyhedral figure with 5 faces, degenerate as a polyhedron: it exists as a spherical tiling of digon faces, called a pentagonal hosohedron with Schläfli symbol {2,5}. It has 2 (antipodal point) vertices, 5 edges, and 5 digonal faces. Pentahedron In geometry, a pentahedron (plural: pentahedra) is a polyhedron with five faces or sides. There are no face-transitive polyhedra with five sides and there"
"This polyhedron can be constructed from 6 great circles on a sphere. It can also be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, and a central point. The edges of the tetrakis hexahedron form 6 circles (or centrally radial lines) in the plane. Each of these 6 circles represent a mirror line in tetrahedral symmetry. The 6 circles can be grouped into 3 sets of 2 pairs of orthogonal circles. These edges can also be seen as a compound of 3 orthogonal"
"dual snub polyhedra and tilings with face configuration V3.3.4.3.""n"". Tetragonal trapezohedron The tetragonal trapezohedron, or deltohedron, is the second in an infinite series of face-uniform polyhedra, which are dual to the antiprisms. It has eight faces, which are congruent kites, and is dual to the square antiprism. This shape has been used as a test case for hexahedral mesh generation, simplifying an earlier test case of Rob Schneider in the form of a square pyramid with its boundary subdivided into 16 quadrilaterals. In this context the tetragonal trapezohedron has also been called the cubical octahedron, quadrilateral octahedron, or octagonal spindle,"
"tetrahedron. It is constructed by a truncated tetrahedron with hexagonal pyramids augmented. Truncated triakis tetrahedron The truncated triakis tetrahedron, or more precisely an order-6 truncated triakis tetrahedron, is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps. It is constructed from taking a triakis tetrahedron by truncating the order-6 vertices. This creates 4 regular hexagon faces, and leaves 12 mirror-symmetric pentagons. A topologically similar equilateral polyhedron can be constructed by using 12 regular pentagons with 4 equilateral but nonplanar hexagons, each vertex with internal angles alternating between"
"12 triangles and 2 triamonds . It may also be described as having a hybrid square-pentagonal antiprismatic core (an antiprismatic core with one square base and one pentagonal base); each base is then augmented with a pyramid. The dissected regular icosahedron is a name for this polytope with the two sets of 3 coplanar faces as trapezoids. This is the vertex figure of a 4D polytope, grand antiprism. It has 10 vertices, 23 edges, and 11 equilateral triangular faces and 2 trapezoid faces. In chemistry, this polyhedron is most commonly called the octadecahedron, for 18 triangular faces, and represents the"