polyhedron

Also known as 3-polytope, polyhedra, polyhedrons

In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term polyhedron is often used to refer implicitly to the whole structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices.

AI overview

A polyhedron is a three-dimensional shape made up of flat polygonal faces connected by straight edges and sharp corners (called vertices). Polyhedra are fundamental objects in geometry that help us understand and describe the structure of three-dimensional space and solid forms.

AI-generated from the Wikipedia summary — may contain errors.

Key facts

.title: Examples of polyhedra
.label1: Definition
.data1: A three-dimensional example of the more general polytope in any number of dimensions.
.label2: Characteristics
.data2: number of faces, topological classification and Euler characteristic,duality,vertex figures,surface area and volume,lines as in geodesics and diagonals,Dehn invariant,symmetry group.

via Wikipedia infobox

Wikidata facts

Image: Revolução de poliedros 01.jpg

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Commons category: Polyhedra
equivalent class: dbpedia.org/ontology/Polyhedron
Commons gallery: Polyhedra
Stack Exchange tag: or.stackexchange.com/tags/polyhedra

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Article

43 sections

Contents

Definition
General characteristics
Number of faces
Topological classification
Duality
Vertex figures
Surface area and lines inside polyhedra
Volume
Dehn invariant
Net
Symmetries
By elements of polyhedron
By point group in three dimensions
Convex polyhedra
Other families of polyhedra
Space-filling polyhedra
Flexible polyhedra
Ideal polyhedron
Lattice polyhedron
Polyhedral compound
Zonohedron
Orthogonal polyhedron
Generalisations
Apeirohedra
Complex polyhedra
Curved polyhedra
Higher-dimensional polyhedra
History
Before the Greeks
Ancient Greece
Ancient China
Medieval Islam
Renaissance
17th–19th centuries
20th–21st centuries
See also
Notes
References
External links
General theory
Lists and databases of polyhedra
Free software
Resources for making physical models

There are many definitions of polyhedra, not all of which are equivalent. Under any definition, polyhedra are typically understood to generalize two-dimensional polygons and to be the three-dimensional specialization of polytopes (a more general concept in any number of dimensions). Polyhedra have several general characteristics that include the number of faces, topological classification by Euler characteristic, duality, vertex figures, surface area, volume, interior lines, Dehn invariant, and symmetry. A symmetry of a polyhedron means that the polyhedron's appearance is unchanged by a transformation, such as rotating and reflecting.