: 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.

[3] In all of these definitions, a polyhedron is typically understood as a three-dimensional example of the more general polytope in any number of dimensions.

The Platonic solids are the five ancientness polyhedrons—tetrahedron, octahedron, icosahedron, cube, and dodecahedron—classified by Plato in his Timaeus whose connecting four classical elements of nature.

For more complicated shapes, the Euler characteristic relates to the number of toroidal holes, handles or cross-caps in the surface and will be less than 2.

For polyhedra defined in these ways, the classification of manifolds implies that the topological type of the surface is completely determined by the combination of its Euler characteristic and orientability.

All of these choices lead to vertex figures with the same combinatorial structure, for the polyhedra to which they can be applied, but they may give them different geometric shapes.

It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with the same volumes and Dehn invariants can be cut up and reassembled into each other.

[52] Polycubes are a special case of orthogonal polyhedra that can be decomposed into identical cubes, and are three-dimensional analogues of planar polyominoes.

Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron.

By forgetting the face structure, any polyhedron gives rise to a graph, called its skeleton, with corresponding vertices and edges.

Such figures have a long history: Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione, and similar wire-frame polyhedra appear in M.C.

A polyhedron has been defined as a set of points in real affine (or Euclidean) space of any dimension n that has flat sides.

Polyhedra appeared in early architectural forms such as cubes and cuboids, with the earliest four-sided Egyptian pyramids dating from the 27th century BC.

An early commentator on Euclid (possibly Geminus) writes that the attribution of these shapes to Plato is incorrect: Pythagoras knew the tetrahedron, cube, and dodecahedron, and Theaetetus (circa 417 BC) discovered the other two, the octahedron and icosahedron.

[69] The 9th century scholar Thabit ibn Qurra included the calculation of volumes in his studies,[70] and wrote a work on the cuboctahedron.

[71] As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian Renaissance.

[73] Toroidal polyhedra, made of wood and used to support headgear, became a common exercise in perspective drawing, and were depicted in marquetry panels of the period as a symbol of geometry.

[74] Piero della Francesca wrote about constructing perspective views of polyhedra, and rediscovered many of the Archimedean solids.

Leonardo da Vinci illustrated skeletal models of several polyhedra for a book by Luca Pacioli,[75] with text largely plagiarized from della Francesca.

A marble tarsia in the floor of St. Mark's Basilica, Venice, designed by Paolo Uccello, depicts a stellated dodecahedron.

[78] As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Dürer and others also depicted polyhedra of increasing complexity, many of them novel, in imaginative etchings.

[79] In the same period, Euler's polyhedral formula, a linear equation relating the numbers of vertices, edges, and faces of a polyhedron, was stated for the Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico.

[80] René Descartes, in around 1630, wrote his book De solidorum elementis studying convex polyhedra as a general concept, not limited to the Platonic solids and their elaborations.

One of its contributions was Descartes' theorem on total angular defect, which is closely related to Euler's polyhedral formula.

[81] Leonhard Euler, for whom the formula is named, introduced it in 1758 for convex polyhedra more generally, albeit with an incorrect proof.

[83] The core concepts of this field, including generalizations of the polyhedral formula, were developed in the late nineteenth century by Henri Poincaré, Enrico Betti, Bernhard Riemann, and others.

[84] In the early 19th century, Louis Poinsot extended Kepler's work, and discovered the remaining two regular star polyhedra.

[86] Meanwhile, the discovery of higher dimensions in the early 19th century led Ludwig Schläfli by 1853 to the idea of higher-dimensional polytopes.

[87] Additionally, in the late 19th century, Russian crystallographer Evgraf Fedorov completed the classification of parallelohedra, convex polyhedra that tile space by translations.

[92] In the second part of the twentieth century, both Branko Grünbaum and Imre Lakatos pointed out the tendency among mathematicians to define a "polyhedron" in different and sometimes incompatible ways to suit the needs of the moment.