The triaugmented triangular prism, a convex deltahedron

A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta (Δ), which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having four, six, eight, ten, twelve, fourteen, sixteen, and twenty faces. The number of faces, edges, and vertices is listed below for each of the eight convex deltahedra. Image File history File links Triaugmented_triangular_prism. ... Image File history File links Triaugmented_triangular_prism. ... In geometry, the triaugmented triangular prism is one of the Johnson solids (J51). ... Look up Plural on Wiktionary, the free dictionary Plural is a grammatical number, typically referring to more than one of the referent in the real world. ... A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ... A face is a polygonal component of a higher dimensional polytope. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... Majuscules or capital letters (in the Roman alphabet: A, B, C, ...) are one type of case in a writing system. ... Delta (upper case Δ, lower case δ) is the fourth letter of the Greek alphabet. ... In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ... Edge may have <math>one</math> of the following special meanings, in addition to its dictionary definition: wiktionary:edge. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ...

Only three of the deltahedra are Platonic solids (polyhedra in which the number of faces meeting at each vertex is constant). These are: This article may be too technical for most readers to understand. ... Edge may have <math>one</math> of the following special meanings, in addition to its dictionary definition: wiktionary:edge. ... In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the triangular dipyramid is a polyhedron made entirely out of 6 faces, which are all equilateral triangles, 9 edges, and 5 vertexes. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the pentagonal dipyramid is one of the Johnson solids (J13). ... In geometry, the Snub disphenoid is one of the Johnson solids (J84). ... In geometry, the triaugmented triangular prism is one of the Johnson solids (J51). ... In geometry, the gyroelongated square dipyramid is one of the Johnson solids (J19). ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ...

• the 4-faced deltahedron (or tetrahedron), in which three faces meet at each vertex
• the 8-faced deltahedron (or octahedron), in which four faces meet at each vertex
• the 20-faced deltahedron (or icosahedron), in which five faces meet at each vertex

There is one non-convex uniform polyhedron made of all triangles, the great icosahedron. For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ... In geometry, the great icosahedron is a Kepler-Poinsot solid. ...

In the 6-faced deltahedron, some vertices have degree 3 and some degree 4. In the 10-, 12-, 14-, and 16-faced deltahedra, some vertices have degree 4 and some degree 5. These five irregular deltahedra belong to the class of Johnson solids: convex polyhedra with regular polygons for faces. The elongated square gyrobicupola (J37), a Johnson solid In geometry, a Johnson solid is a convex polyhedron, each face of which is a regular polygon, which is not a Platonic solid, Archimedean solid, prism, or antiprism. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...

Deltahedra retain their shape, even if the edges are free to rotate around their vertices so that the angles between edges are fluid. Not all polyhedra have this property: for example, if you relax some of the angles of a cube, the cube can be deformed into a non-right square prism. A cube (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, with three meeting at each vertex. ... See: Prism (geometry) Prism (optics) Prism (band) PRISM is an abbreviation for Probabilistic Symbolic Model Checker PRISM was an aborted RISC processor effort at DEC, see DEC PRISM This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

An example of a non-convex deltahedron is the simplest of the five regular polyhedral compounds: the stella octangula. One of the stellated icosahedra (I₂₀) is also a (non-convex) deltahedron.[1] A polyhedral compound is a polyhedron which is itself composed of several other polyhedra sharing a common centre, the three-dimensional analogs of polygonal compounds such as the hexagram. ... Stella octangula The stella octangula (eight-pointed star), also known as the stellated octahedron, is the polyhedral compound of two tetrahedra. ... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ... An icosahedron [ËŒaıkÉ™sÉ™hiːdrÉ™n] noun (plural: -drons, -dra [-drÉ™]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ...

The deltahedra should not be confused with the deltohedra (spelled with an "o"), polyhedra whose faces are geometric kites. The n-sided trapezohedron or deltohedron is the dual polyhedron of a regular n-sided antiprism. ... A separate article is about kite flying. ...