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Concept
Octahedron
Summary
In geometry, an octahedron () is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations.
An octahedron is the three-dimensional case of the more general concept of a cross polytope.
A regular octahedron is a 3-ball in the Manhattan (l_1) metric.
If the edge length of a regular octahedron is a, the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is
and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is
while the midradius, which touches the middle of each edge, is
The octahedron has four special orthogonal projections, centered, on an edge, vertex, face, and normal to a face. The second and third correspond to the B2 and A2 Coxeter planes.
The octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
An octahedron with edge length can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then
( ±1, 0, 0 );
( 0, ±1, 0 );
( 0, 0, ±1 ).
In an x–y–z Cartesian coordinate system, the octahedron with center coordinates (a, b, c) and radius r is the set of all points (x, y, z) such that
The surface area A and the volume V of a regular octahedron of edge length a are:
Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 rather than 4 triangles).
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Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point.
Polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is a polyhedron that bounds a convex set. Every convex polyhedron can be constructed as the convex hull of its vertices, and for every finite set of points, not all on the same plane, the convex hull is a convex polyhedron. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions.
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato who hypothesized in one of his dialogues, the Timaeus, that the classical elements were made of these regular solids.