Equilateral triangle in the context of "Deltahedron"

In geometry, an icosahedron (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/) is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι (eíkosi) 'twenty' and ἕδρα (hédra) 'seat'. The plural can be either "icosahedra" (/-drə/) or "icosahedrons".

There are infinitely many non-similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the (convex, non-stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles.

The Star of David (Hebrew: מָגֵן דָּוִד, romanized: Māḡēn Dāvīḏ, [maˈɡen daˈvid] , lit. 'Shield of David') is a symbol generally recognized as representing both Jewish identity and the Jewish people's ethnic religion, Judaism. Its shape is that of a hexagram: the compound of two equilateral triangles.

A derivation of the Seal of Solomon was used for decorative and mystical purposes by Kabbalistic Jews and Muslims. The hexagram appears occasionally in Jewish contexts since antiquity as a decorative motif, such as a stone bearing a hexagram from the arch of the 3rd–4th century Khirbet Shura synagogue. A hexagram found in a religious context can be seen in the Leningrad Codex, a manuscript of the Hebrew Bible from 11th-century Cairo.

In geometry, an octahedron (pl.: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex and non-convex shapes.

The Winter Hexagon is an asterism appearing to be in the form of a hexagon with vertices at Rigel, Aldebaran, Capella, Pollux, Procyon, and Sirius. It is mostly upon the Northern Hemisphere's celestial sphere. On most locations on Earth (except the South Island of New Zealand and the south of Chile and Argentina and further south), this asterism is visible in the evening sky at the equator from approximately December to June, and in the morning sky from July to the end of November, while in the evenings on the northern hemisphere it is less months visible between December and June, and on the southern hemisphere less months between July and November. In the tropics and southern hemisphere, this (then called "summer hexagon") can be extended with the bright star Canopus in the south.

Smaller and more regularly shaped is the Winter Triangle, an approximately equilateral triangle that shares two vertices (Sirius and Procyon) with the larger asterism. The third vertex is Betelgeuse, which lies near the center of the hexagon. These three stars are three of the ten brightest objects, as viewed from Earth, outside the Solar System. Betelgeuse is also particularly easy to locate, being a shoulder of Orion, which assists stargazers in finding the triangle. Once the triangle is located, the larger hexagon may then be found.

A regular tetrahedron is a polyhedron with four equilateral triangular faces.

In geometry, a regular octahedron is a highly symmetrical type of octahedron (eight-sided polyhedron) with eight equilateral triangles as its faces, four of which meet at each vertex. It is a type of square bipyramid or triangular antiprism with equal-length edges. Regular octahedra occur in nature as crystal structures. Other types of octahedra also exist, with various amounts of symmetry.

A regular octahedron is the three-dimensional case of the more general concept of a cross-polytope.

The regular icosahedron (or simply icosahedron) is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

Many polyhedra and other related figures are constructed from the regular icosahedron, including its 59 stellations. The great dodecahedron, one of the Kepler–Poinsot polyhedra, is constructed by either stellation of the regular dodecahedron or faceting of the icosahedron. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron's dual polyhedron is the regular dodecahedron, and their relation has a historical background in the comparison mensuration. It is analogous to a four-dimensional polytope, the 600-cell.

In geometry, an isosceles triangle (/aɪˈsɒsəliːz/) is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case.Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.

The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings.