Regular polyhedra are the solid analogies to regular polygons in the plane. Regular polygons are defined as having equal (congruent) sides and angles. In analogy, a solid is called regular if its faces are congruent regular polygons and its polyhedral angles (angles at which the faces meet) are congruent. This concept has been generalized to higher-dimensional (coordinate) Euclidean spaces.
Whereas in the plane there exist (in theory) infinitely many regular polygons, in three-dimensional space there exist exactly five regular polyhedra. These are known as the Platonic solids: the tetrahedron, or pyramid, with 4 triangular faces; the cube, with 6 square faces; the octahedron, with 8 equilateral triangular faces; the dodecahedron, with 12 pentagonal faces; and the icosahedron, with 20 equilateral triangular faces. (See animation.)
In four-dimensional space there exist exactly six regular polytopes, five of them generalizations from three-dimensional space. In any space of more than four dimensions there exist exactly three regular polytopes, the generalizations of the tetrahedron, the cube, and the octahedron.
The-figure-illustrates-the-three-basic-theorems-that-triangles-areThe figure illustrates the three basic theorems that triangles are congruent (of equal shape and …[Credits : Encyclopædia Britannica, Inc.]
Proof-that-the-sum-of-the-angles-in-a-triangleProof that the sum of the angles in a triangle is 180 degrees.[Credits : Encyclopædia Britannica, Inc.]
The-formula-in-the-figure-reads-k-is-to-lThe formula in the figure reads k is to l as m is to n if and only if …[Credits : Encyclopædia Britannica, Inc.]
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Thales-of-Miletus-is-generally-credited-with-giving-the-firstThales of Miletus (fl. c. 600 bc) is generally credited with giving the first proof that for any …[Credits : Encyclopædia Britannica, Inc.]
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Bridge-of-AssesBridge of Asses.[Credits : Encyclopædia Britannica, Inc.]
Quadrilateral-of-Omar-Khayyam-Omar-Khayyam-constructed-the-quadrilateral-shownQuadrilateral of Omar Khayyam[Credits : Encyclopædia Britannica, Inc.]
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