Editing Equilateral triangle (section)

== Appearances ==
=== In other related figures ===
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 | image1 = Tiling 3 simple.svg
 | caption1 = The equilateral triangle tiling fills the plane
 | image2 = Sierpinski triangle.svg
 | caption2 = The Sierpiński triangle
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Notably, the equilateral triangle [[Euclidean tilings by convex regular polygons#Regular tilings|tiles]] the [[Euclidean plane]] with six triangles meeting at a vertex; the dual of this tessellation is the [[hexagonal tiling]]. [[Truncated hexagonal tiling]], [[rhombitrihexagonal tiling]], [[trihexagonal tiling]], [[snub square tiling]], and [[snub hexagonal tiling]] are all [[Euclidean tilings by convex regular polygons#Archimedean, uniform or semiregular tilings|semi-regular tessellations]] constructed with equilateral triangles.{{sfnp|Grünbaum|Shepard|1977}} Other two-dimensional objects built from equilateral triangles include the [[Sierpiński triangle]] (a [[Fractal|fractal shape]] constructed from an equilateral triangle by subdividing recursively into smaller equilateral triangles) and [[Reuleaux triangle]] (a [[Circular triangle|curved triangle]] with [[Curve of constant width|constant width]], constructed from an equilateral triangle by rounding each of its sides).{{sfnp|Alsina|Nelsen|2010|p=[https://books.google.com/books?id=mIT5-BN_L0oC&pg=PA102 102&ndash;103]}}

[[File:Octahedron.jpg|thumb|left|200px|The regular octahedron is a [[deltahedron]], as well as a member of the family of [[antiprism]]s.]]
Equilateral triangles may also form a polyhedron in three dimensions. A polyhedron whose faces are all equilateral triangles is called a [[deltahedron]].  There are eight [[convex set|strictly convex]] deltahedra: three of the five [[Platonic solid]]s ([[regular tetrahedron]], [[regular octahedron]], and [[regular icosahedron]]) and five of the 92 [[Johnson solid]]s ([[triangular bipyramid]], [[pentagonal bipyramid]], [[snub disphenoid]], [[triaugmented triangular prism]], and [[gyroelongated square bipyramid]]).{{sfnp|Trigg|1978}} More generally, all [[Johnson solid]]s have equilateral triangles among their faces, though most also have other other [[regular polygon]]s.{{sfnp|Berman|1971}}

The [[antiprism]]s are a family of polyhedra incorporating a band of alternating triangles. When the antiprism is [[Uniform polyhedron|uniform]], its bases are regular and all triangular faces are equilateral.{{sfnp|Horiyama|Itoh|Katoh|Kobayashi|2015|p=[https://books.google.com/books?id=L9WSDQAAQBAJ&pg=PA124 124]}}

As a generalization, the equilateral triangle belongs to the infinite family of <math>n</math>-[[simplex (geometry)|simplexes]], with <math>n = 2</math>.{{sfnp|Coxeter|1948|p=120&ndash;121}}
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=== Applications ===
[[File:Give way outdoor.jpg|thumb|Equilateral triangle usage as a yield sign]]
Equilateral triangles have frequently appeared in man-made constructions and in popular culture. In architecture, an example can be seen in the cross-section of the [[Gateway Arch]] and the surface of the [[Vegreville egg]].{{sfnp|Pelkonen|Albrecht|2006|p=[https://archive.org/details/eerosaarinenshap0000saar/page/160 160]}}{{sfnp|Alsina|Nelsen|2015|p=[https://books.google.com/books?id=2F_0DwAAQBAJ&pg=PA22 22]}} It appears in the [[flag of Nicaragua]] and the [[flag of the Philippines]].{{sfnp|White|Calderón|2008|p=[https://archive.org/details/culturecustomsof00stev/page/3 3]}}{{sfnp|Guillermo|2012|p=[https://books.google.com/books?id=wmgX9M_yETIC&pg=PA161 161]}} It is a shape of a variety of [[traffic sign|road signs]], including the [[yield sign]].{{sfnp|Riley|Cochran|Ballard|1982}}

The equilateral triangle occurs in the study of [[stereochemistry]]. It can be described as the [[molecular geometry]] in which one atom in the center connects three other atoms in a plane, known as the [[trigonal planar molecular geometry]].{{sfnp|Petrucci|Harwood|Herring|2002|p=413–414|loc=See Table 11.1}}

In the [[Thomson problem]], concerning the minimum-energy configuration of <math>n</math> charged particles on a sphere, and for the [[Tammes problem]] of constructing a [[spherical code]] maximizing the smallest distance among the points, the best solution known for <math>n=3</math> places the points at the vertices of an equilateral triangle, [[Circumscribed sphere|inscribed in the sphere]]. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.{{sfnp|Whyte|1952}}