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86 relations: Alfredo Andreini, Alternation (geometry), Architectonic and catoptric tessellation, Branko Grünbaum, Cartesian coordinate system, Chaim Goodman-Strauss, Convex uniform honeycomb, Coxeter group, Coxeter notation, Coxeter–Dynkin diagram, Crystal structure, Crystal system, Cube, Cubic crystal system, Cubic honeycomb, Cuboctahedron, Cuboid, Dual polyhedron, Expansion (geometry), Face (geometry), Fibrifold, Geombinatorics, Harold Scott MacDonald Coxeter, Hexagon, Hexagonal crystal family, Honeycomb (geometry), Hyperbolic space, Hypercubic honeycomb, Icosahedron, Isogonal figure, Isohedral figure, Isotoxal figure, John Horton Conway, List of regular polytopes and compounds, Monoclinic crystal system, Octagon, Octagonal prism, Octahedral symmetry, Octahedron, Order-5 cubic honeycomb, Orthorhombic crystal system, Parallelepiped, Perovskite (structure), Point group, Pyramid, Quasiregular polyhedron, Regular polytope, Regular Polytopes (book), Rhombicuboctahedron, Schläfli symbol, ..., Skew apeirohedron, Snub cube, Snub square tiling, Space group, Square, Square antiprism, Square pyramid, Square tiling, Tessellation, Tesseract, Tetragonal crystal system, Tetragonal disphenoid honeycomb, Tetrahedral-octahedral honeycomb, Tetrahedron, Three-dimensional space, Trapezohedron, Triangle, Triangular cupola, Triangular prism, Triangular tiling, Triclinic crystal system, Trihexagonal tiling, Truncated cube, Truncated cuboctahedron, Truncated octahedron, Truncated square tiling, Uniform coloring, Uniform polyhedron, Uniform tiling, Vertex arrangement, Vertex configuration, Vertex figure, Voxel, Wedge (geometry), Wythoff construction, 4-polytope. Expand index (36 more) »
Alfredo Andreini
Alfredo Andreini (born on July 27, 1870 in Florence and died on December 11, 1943 in Lippiano) was an Italian physician and entomologist.
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Alternation (geometry)
In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.
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Architectonic and catoptric tessellation
In geometry, John Horton Conway defines architectonic and catoptric tessellations as the uniform tessellations (or honeycombs) of Euclidean 3-space and their duals, as three-dimensional analogue of the Platonic, Archimedean, and Catalan tiling of the plane.
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Branko Grünbaum
Branko Grünbaum (ברנקו גרונבאום; born 2 October 1929) is a Yugoslavian-born mathematician and a professor emeritus at the University of Washington in Seattle.
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Cartesian coordinate system
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.
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Chaim Goodman-Strauss
Chaim Goodman-Strauss (born June 1967 in Austin TX) is an American mathematician who works in convex geometry, especially aperiodic tiling.
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Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
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Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
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Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups.
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Coxeter–Dynkin diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).
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Crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material.
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Crystal system
In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals.
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Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
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Cubic crystal system
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube.
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Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells.
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Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.
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Cuboid
In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube.
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Dual polyhedron
In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.
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Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements (vertices, edges, etc.). Equivalently this operation can be imagined by keeping facets in the same position but reducing their size.
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Face (geometry)
In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.
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Fibrifold
In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds.
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Geombinatorics
Geombinatorics is a quarterly mathematical journal founded by Alexander Soifer and published by the University of Colorado, United States, since 1991.
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Harold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.
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Hexagon
In geometry, a hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a six-sided polygon or 6-gon.
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Hexagonal crystal family
In crystallography, the hexagonal crystal family is one of the 6 crystal families, which includes 2 crystal systems (hexagonal and trigonal) and 2 lattice systems (hexagonal and rhombohedral).
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Honeycomb (geometry)
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps.
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Hyperbolic space
In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.
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Hypercubic honeycomb
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensions with the Schläfli symbols and containing the symmetry of Coxeter group Rn (or B~n-1) for n>.
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Icosahedron
In geometry, an icosahedron is a polyhedron with 20 faces.
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Isogonal figure
In geometry, a polytope (a polygon, polyhedron or tiling, for example) is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure.
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Isohedral figure
In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when all its faces are the same.
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Isotoxal figure
In geometry, a polytope (for example, a polygon or a polyhedron), or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges.
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John Horton Conway
John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.
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List of regular polytopes and compounds
This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces.
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Monoclinic crystal system
In crystallography, the monoclinic crystal system is one of the 7 crystal systems.
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Octagon
In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon.
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Octagonal prism
In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps.
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Octahedral symmetry
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation.
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Octahedron
In geometry, an octahedron (plural: octahedra) is a polyhedron with eight faces, twelve edges, and six vertices.
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Order-5 cubic honeycomb
The order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.
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Orthorhombic crystal system
In crystallography, the orthorhombic crystal system is one of the 7 crystal systems.
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Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning).
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Perovskite (structure)
A perovskite is any material with the same type of crystal structure as calcium titanium oxide (CaTiO3), known as the perovskite structure, or XIIA2+VIB4+X2−3 with the oxygen in the edge centers.
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Point group
In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed.
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Pyramid
A pyramid (from πυραμίς) is a structure whose outer surfaces are triangular and converge to a single point at the top, making the shape roughly a pyramid in the geometric sense.
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Quasiregular polyhedron
In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex.
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Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.
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Regular Polytopes (book)
Regular Polytopes is a mathematical geometry book written by Canadian mathematician Harold Scott MacDonald Coxeter.
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Rhombicuboctahedron
In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces.
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Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.
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Skew apeirohedron
In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.
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Snub cube
In geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid with 38 faces: 6 squares and 32 equilateral triangles.
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Snub square tiling
In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane.
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Space group
In mathematics, physics and chemistry, a space group is the symmetry group of a configuration in space, usually in three dimensions.
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Square
In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted.
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Square antiprism
In geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
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Square pyramid
In geometry, a square pyramid is a pyramid having a square base.
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Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane.
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Tessellation
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.
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Tesseract
In geometry, the tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.
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Tetragonal crystal system
In crystallography, the tetragonal crystal system is one of the 7 crystal systems.
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Tetragonal disphenoid honeycomb
The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells.
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Tetrahedral-octahedral honeycomb
The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space.
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Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners.
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Three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).
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Trapezohedron
The n-gonal trapezohedron, antidipyramid, antibipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism.
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Triangle
A triangle is a polygon with three edges and three vertices.
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Triangular cupola
In geometry, the triangular cupola is one of the Johnson solids (J3).
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Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides.
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Triangular tiling
In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane.
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Triclinic crystal system
Triclinic (a ≠ b ≠ c and α ≠ β ≠ γ) In crystallography, the triclinic (or anorthic) crystal system is one of the 7 crystal systems.
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Trihexagonal tiling
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons.
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Truncated cube
In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid.
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Truncated cuboctahedron
In geometry, the truncated cuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron.
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Truncated octahedron
In geometry, the truncated octahedron is an Archimedean solid.
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Truncated square tiling
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex.
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Uniform coloring
In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive.
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Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other).
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Uniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.
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Vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions.
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Vertex configuration
In geometry, a vertex configuration by Walter Steurer, Sofia Deloudi, (2009) pp.
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Vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
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Voxel
A voxel represents a value on a regular grid in three-dimensional space.
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Wedge (geometry)
In solid geometry, a wedge is a polyhedron defined by two triangles and three trapezoid faces.
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Wythoff construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling.
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4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.
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3-cube honeycomb, Alternated cantitruncated cubic honeycomb, Alternated omnitruncated cubic honeycomb, Cantellated cubic cellulation, Cantellated cubic honeycomb, Cantitruncated cubic cellulation, Cantitruncated cubic honeycomb, Cubic cellulation, D3 lattice, Omnitruncated cubic cellulation, Omnitruncated cubic honeycomb, Quarter oblate octahedrille, Rectified cubic cellulation, Rectified cubic honeycomb, Regular cubic honeycomb, Runcic cantitruncated cubic honeycomb, Runcicantellated cubic honeycomb, Runcinated cubic cellulation, Runcinated cubic honeycomb, Runcinated cubic honycomb, Runcitruncated cubic cellulation, Runcitruncated cubic honeycomb, Simo-square prismatic cellulation, Snub rectified cubic honeycomb, Snub square prismatic honeycomb, Square quarter pyramidille, Tomo-square prismatic cellulation, Triangular pyramidille, Truncated cubic cellulation, Truncated cubic honeycomb, Truncated square prismatic honeycomb.
References
[1] https://en.wikipedia.org/wiki/Cubic_honeycomb
