Dual polyhedron, the Glossary
- ️Tue Feb 22 2022
In geometry, every polyhedron is associated with a second dual structure, 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.[1]
Table of Contents
66 relations: Abstract polytope, Apeirogon, Conway polyhedron notation, Cubic honeycomb, Discrete Mathematics (journal), Dodecahedron, Dot product, Dual graph, Dual polygon, Duality (projective geometry), Elongated pyramid, Euclidean space, Face (geometry), Fixed points of isometry groups in Euclidean space, Geometry, Grand stellated 120-cell, Graph theory, Great 120-cell, Hasse diagram, Honeycomb (geometry), Hypercubic honeycomb, Icosahedral honeycomb, Infinite-order apeirogonal tiling, Isogonal figure, Isohedral figure, Isotoxal figure, Java (programming language), Kepler–Poinsot polyhedron, Midsphere, N-skeleton, Order-4 square tiling honeycomb, Order-5 120-cell honeycomb, Order-5 pentagonal tiling, Order-6 hexagonal tiling, Order-6 hexagonal tiling honeycomb, Palindrome, Partially ordered set, PDF, Platonic solid, Point reflection, Pole and polar, Polyhedron, Polytope, Prism (geometry), Projective geometry, Pyramid (geometry), Regular icosahedron, Regular polygon, Regular polytope, Schläfli symbol, ... Expand index (16 more) »
- Self-dual polyhedra
Abstract polytope
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. Dual polyhedron and abstract polytope are polytopes.
See Dual polyhedron and Abstract polytope
Apeirogon
In geometry, an apeirogon or infinite polygon is a polygon with an infinite number of sides.
See Dual polyhedron and Apeirogon
Conway polyhedron notation
In geometry and topology, 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 by various prefix operations. Dual polyhedron and Conway polyhedron notation are polyhedra.
See Dual polyhedron and Conway polyhedron notation
Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells.
See Dual polyhedron and Cubic honeycomb
Discrete Mathematics (journal)
Discrete Mathematics is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications.
See Dual polyhedron and Discrete Mathematics (journal)
Dodecahedron
In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces.
See Dual polyhedron and Dodecahedron
Dot product
In mathematics, the dot product or scalar productThe term scalar product means literally "product with a scalar as a result".
See Dual polyhedron and Dot product
Dual graph
In the mathematical discipline of graph theory, the dual graph of a planar graph is a graph that has a vertex for each face of. Dual polyhedron and dual graph are duality theories.
See Dual polyhedron and Dual graph
Dual polygon
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.
See Dual polyhedron and Dual polygon
Duality (projective geometry)
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. Dual polyhedron and duality (projective geometry) are duality theories.
See Dual polyhedron and Duality (projective geometry)
Elongated pyramid
In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an pyramid to an prism.
See Dual polyhedron and Elongated pyramid
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space.
See Dual polyhedron and Euclidean space
Face (geometry)
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron. Dual polyhedron and face (geometry) are polyhedra.
See Dual polyhedron and Face (geometry)
Fixed points of isometry groups in Euclidean space
A fixed point of an isometry group is a point that is a fixed point for every isometry in the group.
See Dual polyhedron and Fixed points of isometry groups in Euclidean space
Geometry
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.
See Dual polyhedron and Geometry
Grand stellated 120-cell
In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol.
See Dual polyhedron and Grand stellated 120-cell
Graph theory
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
See Dual polyhedron and Graph theory
Great 120-cell
In geometry, the great 120-cell or great polydodecahedron is a regular star 4-polytope with Schläfli symbol.
See Dual polyhedron and Great 120-cell
Hasse diagram
In order theory, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction.
See Dual polyhedron and Hasse diagram
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. Dual polyhedron and honeycomb (geometry) are polytopes.
See Dual polyhedron and Honeycomb (geometry)
Hypercubic honeycomb
In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group for. Dual polyhedron and hypercubic honeycomb are polytopes.
See Dual polyhedron and Hypercubic honeycomb
Icosahedral honeycomb
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space.
See Dual polyhedron and Icosahedral honeycomb
Infinite-order apeirogonal tiling
In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane.
See Dual polyhedron and Infinite-order apeirogonal tiling
Isogonal figure
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 of the figure. Dual polyhedron and isogonal figure are polyhedra and polytopes.
See Dual polyhedron and Isogonal figure
Isohedral figure
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. Dual polyhedron and isohedral figure are polyhedra.
See Dual polyhedron and Isohedral figure
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. Dual polyhedron and isotoxal figure are polyhedra.
See Dual polyhedron and Isotoxal figure
Java (programming language)
Java is a high-level, class-based, object-oriented programming language that is designed to have as few implementation dependencies as possible.
See Dual polyhedron and Java (programming language)
Kepler–Poinsot polyhedron
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
See Dual polyhedron and Kepler–Poinsot polyhedron
Midsphere
In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Dual polyhedron and midsphere are polyhedra.
See Dual polyhedron and Midsphere
N-skeleton
In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the.
See Dual polyhedron and N-skeleton
Order-4 square tiling honeycomb
In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs.
See Dual polyhedron and Order-4 square tiling honeycomb
Order-5 120-cell honeycomb
In the geometry of hyperbolic 4-space, the order-5 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs).
See Dual polyhedron and Order-5 120-cell honeycomb
Order-5 pentagonal tiling
In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane.
See Dual polyhedron and Order-5 pentagonal tiling
Order-6 hexagonal tiling
In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane.
See Dual polyhedron and Order-6 hexagonal tiling
Order-6 hexagonal tiling honeycomb
In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space.
See Dual polyhedron and Order-6 hexagonal tiling honeycomb
Palindrome
A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as madam or racecar, the date "22/02/2022" and the sentence: "A man, a plan, a canal – Panama".
See Dual polyhedron and Palindrome
Partially ordered set
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other.
See Dual polyhedron and Partially ordered set
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems.
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space.
See Dual polyhedron and Platonic solid
Point reflection
In geometry, a point reflection (also called a point inversion or central inversion) is a transformation of affine space in which every point is reflected across a specific fixed point.
See Dual polyhedron and Point reflection
Pole and polar
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
See Dual polyhedron and Pole and polar
Polyhedron
In geometry, a polyhedron (polyhedra or polyhedrons) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. Dual polyhedron and polyhedron are polyhedra.
See Dual polyhedron and Polyhedron
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (faces). Dual polyhedron and polytope are polytopes.
See Dual polyhedron and Polytope
Prism (geometry)
In geometry, a prism is a polyhedron comprising an polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and other faces, necessarily all parallelograms, joining corresponding sides of the two bases.
See Dual polyhedron and Prism (geometry)
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations.
See Dual polyhedron and Projective geometry
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Dual polyhedron and pyramid (geometry) are polyhedra and Self-dual polyhedra.
See Dual polyhedron and Pyramid (geometry)
Regular icosahedron
In geometry, the regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube.
See Dual polyhedron and Regular icosahedron
Regular polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
See Dual polyhedron and Regular polygon
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.
See Dual polyhedron and Regular polytope
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. Dual polyhedron and Schläfli symbol are polytopes.
See Dual polyhedron and Schläfli symbol
Schlegel diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. Dual polyhedron and Schlegel diagram are polytopes.
See Dual polyhedron and Schlegel diagram
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Dual polyhedron and simplex are polytopes.
See Dual polyhedron and Simplex
Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. Dual polyhedron and square tiling are polyhedra.
See Dual polyhedron and Square tiling
Symmetry
Symmetry in everyday life refers to a sense of harmonious and beautiful proportion and balance.
See Dual polyhedron and Symmetry
Tetrahedron
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. Dual polyhedron and tetrahedron are Self-dual polyhedra.
See Dual polyhedron and Tetrahedron
Triangular tiling honeycomb
The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space.
See Dual polyhedron and Triangular tiling honeycomb
Uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other.
See Dual polyhedron and Uniform polyhedron
Uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
See Dual polyhedron and Uniform polytope
Vertex (geometry)
In geometry, a vertex (vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect. Dual polyhedron and vertex (geometry) are polytopes.
See Dual polyhedron and Vertex (geometry)
Vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Dual polyhedron and vertex figure are polyhedra and polytopes.
See Dual polyhedron and Vertex figure
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
See Dual polyhedron and 120-cell
16-cell honeycomb honeycomb
In the geometry of hyperbolic 5-space, the 16-cell honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs).
See Dual polyhedron and 16-cell honeycomb honeycomb
24-cell
In four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
See Dual polyhedron and 24-cell
4-polytope
In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope.
See Dual polyhedron and 4-polytope
5-polytope
In geometry, a five-dimensional polytope (or 5-polytope or polyteron) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.
See Dual polyhedron and 5-polytope
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
See Dual polyhedron and 600-cell
See also
Self-dual polyhedra
- Diminished rhombic dodecahedron
- Dual polyhedron
- Elongated pentagonal pyramid
- Elongated square pyramid
- Elongated triangular pyramid
- Hexagonal pyramid
- Pentagonal pyramid
- Pyramid (geometry)
- Square pyramid
- Tetrahedron
References
[1] https://en.wikipedia.org/wiki/Dual_polyhedron
Also known as Canonical dual, Dorman Luke, Dual (polyhedron), Dual polyhedra, Dual polytope, Dual tessellation, Dual tiling, Geometric dual, Polyhedral dual, Polyhedron dual, Self-dual figure, Self-dual polyhedra, Self-dual polyhedron, Self-dual polytope, Tiling dual.
, Schlegel diagram, Simplex, Square tiling, Symmetry, Tetrahedron, Triangular tiling honeycomb, Uniform polyhedron, Uniform polytope, Vertex (geometry), Vertex figure, 120-cell, 16-cell honeycomb honeycomb, 24-cell, 4-polytope, 5-polytope, 600-cell.