Hedron
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') 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.
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.
Convex polyhedra are a well-defined class of polyhedra with several equivalent standard definitions. Every convex polyhedron is the convex hull of its vertices, and the convex hull of a finite set of points is a polyhedron. Many common families of convex polyhedra, with cubes and pyramids are familiar examples of such.
There exist many miscellaneous families of polyhedra. Space-filling polyhedra are those that can be packed together with copies of themselves or with other different types of polyhedra in three-dimensional space. Flexible polyhedra are the polyhedra that can change their overall shape while preserving the shapes of their faces; some have such. Ideal polyhedron is a convex polyhedron defined in three-dimensional hyperbolic space. Lattice polyhedra are the convex polyhedra that can be constructed with integers coordinates. Orthogonal polyhedra are polyhedra where all edges are orthogonal, parallel to all three axes of Cartesian coordinate system. Copies of polyhedra can share a centre, which is known as polyhedral compounds. Polyhedra can be generalized into infinitely many faces called apeirohedra, the underlying space of which is a complex Hilbert space known as complex polyhedra, as well as allowing curved faces and edges.
The origin of polyhedra dates back to the ancient era. Ancient Egypt's four-sided Egyptian pyramids are known for the pyramidal structure, with the study of calculating their volume, specifically the volume of a frustum, in Moscow Mathematical Papyrus. In Ancient Greek, the Etruscan dodecahedron was discovered at Etruscan civilization made of soapstone on Monte Loffa, and Platonic solids were discovered and studied by Ancient Greek mathematicians, with Plato describes its association to the natures for each in his Timaeus, later soon treatment studied in Euclid's Elements. In Renaissance, toroidal polyhedra were used for sketching on polyhedral's perspective views, skeletal models, and nets appearance. Leonhard Euler worked on the polyhedral characteristics and the solution for the Seven Bridges of Königsberg's problem, underlying the field of topology. Johannes Kepler discovered two non-convex regular polyhedra, extended by Louis Poinsot prepending two more remaining polyhedra, known as Kepler–Poinsot polyhedra. Many results on polyhedral concepts, like Hilbert's third problem, Steinitz's theorem, and stellation of Platonic solids. Polyhedra are used in many fields, as well as appearing in biological creatures, nature, and modern computational geometry.
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