hedra

Hedra is the plural form used in geometry to refer to polyhedra—three-dimensional solids with flat faces that meet along edges. In ordinary usage, specific solids are named with the -hedron suffix (for example, tetrahedron, cube or hexahedron, octahedron, dodecahedron, icosahedron), and their plural forms end in -hedra: tetrahedra, hexahedra, octahedra, dodecahedra, icosahedra. The generic plural polyhedra is also common when speaking about polyhedral shapes as a class.

Etymology and naming conventions stem from the Greek root hedra, meaning a base or seat, applied to solids built from faces. The plural -hedra is used when listing several solids of the same family, while polyhedra serves as a general term for polyhedral shapes.

Hedra encompass a wide range of solids, from regular (Platonic) hedra to more complex forms. The five Platonic hedra are regular solids with congruent faces and identical vertices: tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron. Beyond these, there are Archimedean (semi-regular) hedra with multiple face types, as well as numerous Johnson solids—convex polyhedra with regular polygonal faces that are not uniform.

Key properties include faces, edges, and vertices connected by Euler’s formula for convex polyhedra: V − E + F = 2. Many hedra have duals, obtained by exchanging faces and vertices (the cube and the octahedron are duals, as are the dodecahedron and icosahedron).

Hedra are central in geometry, with applications in modeling, architecture, and computer graphics, and they underpin much of the study of polyhedral symmetry, tilings, and three-dimensional structure.

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