AI & ChatGPT searches , social queries for POLYHEDRON
Search references for POLYHEDRON. Phrases containing POLYHEDRON
See searches and references containing POLYHEDRON!
Search references for POLYHEDRON. Phrases containing POLYHEDRON
See searches and references containing POLYHEDRON!POLYHEDRON
Flat-sided three-dimensional shape
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure
Polyhedron
Polyhedron associated with another by swapping vertices for faces
In geometry, every polyhedron is associated with a second dual structure, wherein the vertices of one correspond to the faces of the other and the edges
Dual_polyhedron
Any of the five regular polyhedra
Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical
Platonic_solid
Toroidal polyhedron with 7 faces
Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces. The tetrahedron and the Szilassi polyhedron are the only
Szilassi_polyhedron
Convex polyhedron made from hexagons and pentagons
more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described
Goldberg_polyhedron
Topics referred to by the same term
called a polyhedron; for this concept see n-dimensional polyhedron. Polyhedron may also refer to: Polyhedron (magazine), formerly Polyhedron Newszine
Polyhedron_(disambiguation)
Isogonal polyhedron with regular faces
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows
Uniform_polyhedron
Polyhedron made from triangles that approximates a sphere
A geodesic polyhedron is a convex polyhedron made from triangles which approximates a sphere. They usually have icosahedral symmetry, such that they have
Geodesic_polyhedron
Physical manifestation of a geometric shape
A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less
Polyhedron_model
Polyhedron with regular congruent polygons as faces
regular polyhedron is a polyhedron with regular and congruent polygons as faces. Its symmetry group acts transitively on its flags. A regular polyhedron is
Regular_polyhedron
Edge-joined polygons which fold into a polyhedron
In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane that can be folded (along edges) to become the
Net_(polyhedron)
Point where two or more curves, lines, or edges meet
"sides" meeting at one place. A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges
Vertex_(geometry)
An n-dimensional polyhedron is a geometric object that generalizes the 3-dimensional polyhedron to an n-dimensional space. It is defined as a set of points
N-dimensional_polyhedron
Polyhedron in which all edges are parallel
An orthogonal polyhedron is a polyhedron in which all edges are parallel to the axes of a Cartesian coordinate system, resulting in the orthogonal faces
Orthogonal_polyhedron
Subset of complex n-space bounded by analytic functions
In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space Cn of the form P = { z ∈ D : | f j ( z )
Analytic_polyhedron
Partition of a toroidal surface into polygons
In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) of 1 or greater. Notable examples
Toroidal_polyhedron
dual uniform polyhedron is the dual of a uniform polyhedron. Where a uniform polyhedron is vertex-transitive, a dual uniform polyhedron is face-transitive
Dual_uniform_polyhedron
Type of polyhedron with many holes
Leonardo polyhedron is a polyhedron with a Platonic solid's rotational symmetry and genus g ≥ 2 {\displaystyle g\geq 2} . Here, a polyhedron is the unbounded
Leonardo_polyhedron
Polyhedra in which all vertices are the same
elongated square gyrobicupola or pseudorhombicuboctahedron is an extra polyhedron with regular faces and congruent vertices. Still, it is not generally
Archimedean_solid
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i
List_of_uniform_polyhedra
Non-convex polyhedron with no triangulation
In geometry, a Schönhardt polyhedron is a polyhedron with the same combinatorial structure as a regular octahedron, but with dihedral angles that are non-convex
Schönhardt_polyhedron
Solid with six equal square faces
these edges form six square faces of the same size. It is an example of a polyhedron. It is a special case of a cuboid, a parallelepiped, and a rhombohedron
Cube
Polyhedron resembling a soccerball
truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. The polyhedron can be regarded as a football
Truncated_icosahedron
Solid with 2 parallel n-gonal bases connected by n parallelograms
In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy (rigidly moved without rotation) of the
Prism_(geometry)
3-dimensional geometric figure
In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes
Flexible_polyhedron
Plane tiling corresponding to a polyhedron
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations
Projective_polyhedron
Method of describing higher-order polyhedra
Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified
Conway_polyhedron_notation
Extending the elements of a polytope to form a new figure
stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in n dimensions to form
Stellation
Polyhedron related to sphere packing
discovered around 1990 by the mathematician Steve Waterman. A Waterman polyhedron is created by packing spheres according to the cubic close(st) packing
Waterman_polyhedron
Toroidal polyhedron with 14 triangle faces
geometry, the Császár polyhedron (Hungarian: [ˈt͡ʃaːsaːr]) is a nonconvex toroidal polyhedron with 14 triangular faces. This polyhedron has no diagonals;
Császár_polyhedron
Topological invariant in mathematics
classically defined for the surface of a three-dimensional polyhedron. If the polyhedron has V vertices (corners), E edges, and F faces, then the Euler
Euler_characteristic
Polyhedron with eight triangular faces
In geometry, an octahedron (pl.: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid
Octahedron
Convex polyhedron with regular faces
Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons and that is not a uniform polyhedron. There are 92 such solids: 48 composed
Johnson_solid
Partition of a sphere's surface into polygons
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded
Spherical_polyhedron
Polyhedron sliced by a plane into other polyhedra
In geometry, a composite polyhedron is a convex polyhedron that produces two convex, regular-faced polyhedra when sliced by a plane.[dubious – discuss]
Composite_polyhedron
Flexible polyhedron with 14 triangle faces
In geometry, Steffen's polyhedron is a flexible polyhedron discovered (in 1978) by and named after Klaus Steffen [de]. It is based on the Bricard octahedron
Steffen's_polyhedron
Line segment joining two adjacent vertices in a polygon or polytope
a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. In a polygon, an edge is a line segment
Edge_(geometry)
Polyhedron resulting from the snub operation
a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending
Snub_polyhedron
Polyhedron with 20 faces
icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty'
Icosahedron
Polyhedron with some pattern of nonconvexity
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general
Star_polyhedron
Variously-defined concept in geometry
semiregular polyhedron (or semiregular polytope) is used variously by different authors. In its original definition, it is a polyhedron with regular
Semiregular_polyhedron
Sphere tangent to every edge of a polyhedron
or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform
Midsphere
Shape in hyperbolic geometry
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather
Ideal_polyhedron
Any of 4 regular star polyhedra
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating and faceting the regular convex dodecahedron
Kepler–Poinsot_polyhedron
Planar surface that forms part of the boundary of a solid object
have any dimension. The vertices, edges, and (2-dimensional) faces of a polyhedron are all faces in this more general sense. In elementary geometry, polyhedra
Face_(geometry)
Topics referred to by the same term
Z-polyhedron or Z-polytope may refer to: The set of integer points in convex polyhedron Convex lattice polytope This disambiguation page lists articles
Z-polyhedron
Polyhedron which tiles 3D space
In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections
Space-filling_polyhedron
1997 studio album by Mainliner
Psychedelic Polyhedron is the second album by Mainliner, released in April 1997 by Fractal Records. All lyrics are written by Asahito Nanjo; all music
Psychedelic_Polyhedron
Generalisation of dice with identical faces
dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same
Isohedral_figure
Polyhedron with 7 faces
geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges,
Tetrahemihexahedron
Hungarian mathematician
toroidal polyhedron with seven hexagonal faces, in which each pair of faces share an edge. The new construction was soon dubbed the Szilassi polyhedron. This
Lajos_Szilassi
Self-intersecting uniform polyhedron
polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can
Uniform_star_polyhedron
Polyhedron (formerly Polyhedron Newszine) was a magazine targeting consumers of role-playing games, and originally the official publication of the RPGA
Polyhedron_(magazine)
Polytope or tiling with one type of edge
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal (from Greek τόξον 'arc') or edge-transitive if its symmetries
Isotoxal_figure
Geometric operation which truncates the edges of polyhedra
chamfer or edge-truncation is a topological operator that modifies one polyhedron into another. It separates the faces by reducing them, and adds a new
Chamfer_(geometry)
Compound shape made of identical uniform polyhedra arranged uniformly
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra
Uniform_polyhedron_compound
Outermost stellation of the icosahedron
on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as
Final stellation of the icosahedron
Final_stellation_of_the_icosahedron
Catalan solid with 12 faces
convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron
Rhombic_dodecahedron
Tiling of euclidean or hyperbolic space of three or more dimensions
honeycomb is said to be a space-filling polyhedron. A necessary condition for a polyhedron to be a space-filling polyhedron is that its Dehn invariant must be
Honeycomb_(geometry)
In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra. All omnitruncated
Omnitruncated_polyhedron
an indexed list of the uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger. The book was written as a guide book to building
List of Wenninger polyhedron models
List_of_Wenninger_polyhedron_models
Solid with 12 equal pentagonal faces
regular dodecahedron or pentagonal dodecahedron is a dodecahedron (a polyhedron with 12 faces) composed of regular pentagonal faces, three meeting at
Regular_dodecahedron
Natural number, composite number
alongside other uniform prisms, has 14 faces. The Szilassi polyhedron and its dual, the Császár polyhedron, are the simplest toroidal polyhedra; they have 14
14_(number)
Number of windings of a polytope around its center of symmetry
In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing
Density_(polytope)
Type of polyhedron
uniform star polyhedron with Wythoff symbol of the form 3 | p q or 3/2 | p q are ditrigonal, at least if p and q are not 2. Each polyhedron includes two
Ditrigonal_polyhedron
Polyhedron with four faces
tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices
Tetrahedron
In geometry, a monostatic polytope or unistable polyhedron is a d {\displaystyle d} -polytope which "can stand on only one face". They were described in
Monostatic_polytope
Polytope or tiling whose vertices are identical
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries
Isogonal_figure
Archimedean solid with 8 faces
all of its vertices off, a process known as truncation. The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices
Truncated_tetrahedron
while interior angles in a triangle add up to 180°. However, on a convex polyhedron, the angles of the faces meeting at a vertex add up to less than 360°
Angular_defect
Bipartite non-Hamiltonian polyhedral graph
convex polyhedron), and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. The polyhedron whose
Herschel_graph
Kepler–Poinsot polyhedron
In geometry, the small stellated dodecahedron is a Kepler–Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {5⁄2, 5}. It is one of four
Small_stellated_dodecahedron
Field of mathematics dealing with three-dimensional Euclidean spaces
volumes of various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including truncated) and other solids of revolution
Solid_geometry
Flexible polyhedron
A Kokotsakis polyhedron is a polyhedral surface in three-dimensional space consisting of any polygon as its base and quadrilaterals alternating with triangles
Kokotsakis_polyhedron
Polyhedron with 2 faces
A dihedron (pl. dihedra) is a type of polyhedron, made of two polygon faces which share the same set of n edges. In three-dimensional Euclidean space,
Dihedron
Solid with twenty equal triangular faces
The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from a pentagonal antiprism by attaching two pentagonal
Regular_icosahedron
Polyhedron with 12 faces
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with
Dodecahedron
Prism with a 3-sided base
other polyhedra, examples are some of the Johnson solids and Schönhardt polyhedron. It has a relationship with the honeycombs and polytopes. It can be found
Triangular_prism
Uniform star polyhedron with 204 faces
dirhombidodecahedron, also called Skilling's figure, is a degenerate uniform star polyhedron. Its full realization has 204 faces (120 triangles, 60 squares, and 24
Great disnub dirhombidodecahedron
Great_disnub_dirhombidodecahedron
Isohedral and isogonal polyhedron
A noble polyhedron is one which is isohedral (all faces the same) and isogonal (all vertices the same). They were first studied in any depth by Edmund
Noble_polyhedron
In geometry, the Hessian polyhedron is a regular complex polyhedron 3{3}3{3}3, , in C 3 {\displaystyle \mathbb {C} ^{3}} . It has 27 vertices, 72 3{} edges
Hessian_polyhedron
Sphere tangent to every face of a polyhedron
or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest
Inscribed_sphere
it can be described in terms of geometrically finite groups. A convex polyhedron C in hyperbolic space is called geometrically finite if its closure C
Geometric_finiteness
3D shape made of polyhedra sharing a common center
form a convex polyhedron called its convex hull. A compound is a faceting of its convex hull.[citation needed] Another convex polyhedron is formed by the
Polytope_compound
Type of map projection
projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon
Polyhedral_map_projection
Operation in Euclidean geometry
The rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example
Rectification_(geometry)
peak or (n−3)-face of the polyhedron Edge the ridge or (n−2)-face of the polyhedron Face the facet or (n−1)-face of the polyhedron Vertex the (n−4)-face of
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Solid with eight equal triangular faces
In geometry, a regular octahedron is an eight-sided polyhedron with equilateral triangles as its faces. Known for its highly symmetrical form, the regular
Regular_octahedron
Kepler–Poinsot polyhedron
In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5⁄2, 3}. It is one of four nonconvex regular polyhedra
Great_stellated_dodecahedron
Academic journal
Polyhedron is a peer-reviewed scientific journal covering the field of inorganic chemistry. It was established in 1955 as the Journal of Inorganic and
Polyhedron_(journal)
Polytope whose facets are all simplices
case of a three-dimensional simplicial polytope, known as the simplicial polyhedron, the polytope contains only triangular faces of any type. These polyhedra
Simplicial_polytope
Convex polyhedron whose faces are almost regular polygons
In geometry, a near-miss Johnson solid is a convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely
Near-miss_Johnson_solid
computational geometry, the problem of computing the intersection of a polyhedron with a line has important applications in computer graphics, optimization
Intersection of a polyhedron with a line
Intersection_of_a_polyhedron_with_a_line
Solid with four equal triangular faces
tetrahedron is a polyhedron with four equilateral triangular faces. A regular tetrahedron is a tetrahedron (that is, a four-sided polyhedron) in which all
Regular_tetrahedron
Polyhedron with 8 triangles and 6 squares
A cuboctahedron, rectified cube, or rectified octahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices
Cuboctahedron
Cube with notched surfaces
Chazelle polyhedron is a non-convex polyhedron constructed by removing pieces of wedges from both top and bottom of a cube's sides, leaving the notches
Chazelle_polyhedron
Archimedean solid with 32 faces
In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi-) triangular faces and twelve (dodeca-) pentagonal faces
Icosidodecahedron
Operation that cuts polytope vertices, creating a new facet in place of each vertex
general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation
Truncation_(geometry)
Two pentagonal pyramids fused base-to-base
A pentagonal bipyramid or pentagonal dipyramid is a polyhedron with ten triangular faces. It is constructed by attaching two pentagonal pyramids to each
Pentagonal_bipyramid
Quadrilateral with sides of equal length
dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces. The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi (rhombi
Rhombus
travel, tourism, insurance
POLYHEDRON
POLYHEDRON
POLYHEDRON
POLYHEDRON
POLYHEDRON
POLYHEDRON
POLYHEDRON
POLYHEDRON
POLYHEDRON
travel, tourism, insurance