Polyhedra are 3D shapes with flat faces and sharp edges
Call a solid polyhedral when flat polygonal panels meet along straight hinges at sharp corners—Greek “many bases.” Rival definitions quarrel at the edges, yet convex cases agree: each equals the convex hull of its corners. Egyptian pyramids and the Moscow Mathematical Papyrus already chased related volume problems.
The name joins Greek poly, many, with hedron, base or seat, and the word can mean either the solid or just its boundary surface. Counting faces gives the standard names: a tetrahedron has four, a pentahedron five, a hexahedron six. Wrapping the tightest convex skin around the corners of any convex polyhedron recovers it exactly, and doing the same to any finite scatter of points yields a polyhedron, so for convex shapes the competing definitions agree except in degenerate cases.
The ancient world met these shapes early. Egypt’s four-sided pyramids inspired the frustum volume problem in the Moscow Mathematical Papyrus, an Etruscan soapstone dodecahedron turned up on Monte Loffa, and Greek mathematicians studied the Platonic solids.
For polyhedra in general, no definition commands universal agreement, and many books rely on intuition. Some rules exclude self-crossing shapes long counted as polyhedra; others admit solids whose boundaries are not manifolds, and the skew apeirohedra have infinitely many faces. Solid-based definitions may demand a bounded solid with connected interior. O’Rourke in 1993 used a surface approach, gluing finitely many convex polygons so that any two share only a vertex, an edge or nothing. Abstract polyhedra instead treat vertices, edges and faces as a partially ordered set. Some higher-dimensional literature even reserves polyhedron for an intersection of half-spaces and calls the bounded ones polytopes.
Families include space-fillers that pack three-dimensional space, flexible polyhedra that change shape while every face stays rigid, and ideal polyhedra living in hyperbolic space. A paper model of a convex polyhedron can be painted one colour inside and another outside, making it orientable, and some self-crossing forms are coloured the same way yet show regions turned inside out.
Source: Polyhedron
