A hexahedron (PL: hexahedra or hexahedrons) or sexahedron (PL: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex.
There are seven topologically distinct convex hexahedra, [1] one of which exists in two mirror image forms. There are three topologically distinct concave hexahedra. 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.
Quadrilaterally-faced hexahedron (Cuboid) 6 faces, 12 edges, 8 vertices | ||||||
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Cube (square) | Rectangular cuboid (three pairs of rectangles) | Trigonal trapezohedron (congruent rhombi) | Trigonal trapezohedron (congruent quadrilaterals) | Quadrilateral frustum (apex-truncated square pyramid) | Parallelepiped (three pairs of parallelograms) | Rhombohedron (three pairs of rhombi) |
Oh, [4,3], (*432) order 48 | D2h, [2,2], (*222) order 8 | D3d, [2+,6], (2*3) order 12 | D3, [2,3]+, (223) order 6 | C4v, [4], (*44) order 8 | Ci, [2+,2+], (×) order 2 |
Convex | |||||
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Triangular bipyramid | Tetragonal antiwedge. Chiral – exists in "left-handed" and "right-handed" mirror image forms. | Pentagonal pyramid | |||
36 Faces 9 E, 5 V | 4.4.3.3.3.3 Faces 10 E, 6 V | 4.4.4.4.3.3 Faces 11 E, 7 V | 5.35 Faces 10 E, 6 V | 5.4.4.3.3.3 Faces 11 E, 7 V | 5.5.4.4.3.3 Faces 12 E, 8 V |
There are three further topologically distinct hexahedra that can only be realised as concave figures:
Concave | ||
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4.4.3.3.3.3 Faces 10 E, 6 V | 5.5.3.3.3.3 Faces 11 E, 7 V | 6.6.3.3.3.3 Faces 12 E, 8 V |
A digonal antiprism can be considered a degenerate form of hexahedron, having two opposing digonal faces and four triangular faces. However, digons are usually disregarded in the definition of non-spherical polyhedra, and this case is often simply considered a tetrahedron and the four remaining triangular faces considered to compose the full solid.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.
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, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces, 36 edges, and 24 vertices.
In geometry, the triangular bipyramid is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.
In geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid.
In geometry, a pentahedron is a polyhedron with five faces or sides. There are no face-transitive polyhedra with five sides and there are two distinct topological types.
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, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex. Conway called it a quadrille.
In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
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.
In geometry, the tetrakis cuboctahedron is a convex polyhedron with 32 triangular faces, 48 edges, and 18 vertices. It is a dual of the truncated rhombic dodecahedron.
In geometry, an enneahedron 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.
In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathematical literature, it was rediscovered and popularized by M. C. Escher, who used it in the central image of his 1948 woodcut Stars.
In geometry, an octadecahedron is a polyhedron with 18 faces. No octadecahedron is regular; hence, the name does not commonly refer to one specific polyhedron.
A mesh is a representation of a larger geometric domain by smaller discrete cells. Meshes are commonly used to compute solutions of partial differential equations and render computer graphics, and to analyze geographical and cartographic data. A mesh partitions space into elements over which the equations can be solved, which then approximates the solution over the larger domain. Element boundaries may be constrained to lie on internal or external boundaries within a model. Higher-quality (better-shaped) elements have better numerical properties, where what constitutes a "better" element depends on the general governing equations and the particular solution to the model instance.
In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces.