Editing Square pyramid (section)

== Applications ==
{{multiple image
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 | image1 = All Gizah Pyramids.jpg
 | caption1 = The [[Egyptian pyramids]] are examples of square pyramidal buildings in architecture.
 | image2 = Piramide Chichen-Itza - panoramio (2).jpg
 | caption2 = One of the [[Mesoamerican pyramids]], a similar building to the Egyptian, has flat tops and stairs at the faces
}}
In architecture, the [[Egyptian pyramids|pyramids built in ancient Egypt]] are examples of buildings shaped like square pyramids.{{sfnp|Kinsey|Moore|Prassidis|2011|p=[https://books.google.com/books?id=fFpuDwAAQBAJ&pg=RA1-PA371 371]}} [[Pyramidology|Pyramidologists]] have put forward various suggestions for the design of the [[Great Pyramid of Giza]], including a theory based on the [[Kepler triangle]] and the [[golden ratio]]. However, modern scholars favor descriptions using integer ratios, as being more consistent with the knowledge of Egyptian mathematics and proportion.<ref>{{harvtxt|Herz-Fischler|2000}} surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle. See {{harvtxt|Rossi|2004}}, pp. [https://archive.org/details/architechture-and-mathematics-in-ancient-egypt-corianna-rossi-2003/page/67/ 67–68], quoting that "there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to <math>\varphi</math>, and <math>\varphi</math> itself as a number, do not fit with the extant Middle Kingdom mathematical sources"; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56. See also {{harvtxt|Rossi|Tout|2002}} and {{harvtxt|Markowsky|1992}}.</ref> The [[Mesoamerican pyramids]] are also ancient pyramidal buildings similar to the Egyptian; they differ in having flat tops and stairs ascending their faces.{{sfnmp
 | 1a1 = Feder | 1y = 2010 | 1p = [https://books.google.com/books?id=RlRz2symkAsC&pg=PA34 34]
 | 2a1 = Takacs | 2a2 =  Cline | 2y =  2015 | 2p = [https://books.google.com/books?id=SPcvCgAAQBAJ&pg=PA16 16]
}} Modern buildings whose designs imitate the Egyptian pyramids include the [[Louvre Pyramid]] and the casino hotel [[Luxor Las Vegas]].{{sfnmp
 | 1a1 = Jarvis | 1a2 = Naested | 1y = 2012 | 1p = [https://books.google.com/books?id=NWzsz8vioZwC&pg=PA172 172]
 | 2a1 = Simonson | 2y =  2011 | 2p = [https://books.google.com/books?id=Ws6-DwAAQBAJ&pg=PA122 122]
}}

In [[stereochemistry]], an [[atom cluster]] can have a [[square pyramidal molecular geometry|square pyramidal geometry]]. A square pyramidal molecule has a [[main-group element]] with one active [[lone pair]], which can be described by a model that predicts the geometry of molecules known as [[VSEPR theory]].{{sfnp|Petrucci|Harwood|Herring|2002|p=[https://books.google.com/books?id=EZEoAAAAYAAJ&pg=PA414 414]}} Examples of molecules with this structure include [[chlorine pentafluoride]], [[bromine pentafluoride]], and [[iodine pentafluoride]].{{sfnp|Emeléus|1969|p=[https://books.google.com/books?id=9SkSBQAAQBAJ&pg=PA13 13]}}

[[File:Tetrakishexahedron.jpg|thumb|upright=0.6|[[Tetrakis hexahedra]], a construction of polyhedra by augmentation involving square pyramids]]

The base of a square pyramid can be attached to a square face of another polyhedron to construct new polyhedra, an example of [[Augmentation (geometry)|augmentation]]. For example, a [[tetrakis hexahedron]] can be constructed by attaching the base of an equilateral square pyramid onto each face of a cube.{{sfnp|Demey|Smessaert|2017}} Attaching [[Prism (geometry)|prisms]] or [[antiprisms]] to pyramids is known as [[Elongation (geometry)|elongation]] or [[gyroelongation]], respectively.{{sfnp|Slobodan|Obradović|Ðukanović|2015}} Some of the other Johnson solids can be constructed by either augmenting square pyramids or augmenting other shapes with square pyramids: [[elongated square pyramid]] <math> J_8 </math>, [[gyroelongated square pyramid]] <math> J_{10} </math>, [[elongated square bipyramid]] <math> J_{15} </math>, [[gyroelongated square bipyramid]] <math> J_{17} </math>, [[augmented triangular prism]] <math> J_{49} </math>, [[biaugmented triangular prism]] <math> J_{50}</math>, [[triaugmented triangular prism]] <math> J_{51} </math>, [[augmented pentagonal prism]] <math> J_{52} </math>, [[biaugmented pentagonal prism]] <math> J_{53} </math>, [[augmented hexagonal prism]] <math> J_{54} </math>, [[parabiaugmented hexagonal prism]] <math> J_{55} </math>, [[metabiaugmented hexagonal prism]] <math> J_{56} </math>, [[triaugmented hexagonal prism]] <math> J_{57} </math>, and [[augmented sphenocorona]] <math> J_{87} </math>.<ref>{{harvtxt|Rajwade|2001}}, pp. [https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84 84–89]. See Table 12.3, where <math> P_n </math> denotes the {{nowrap|<math>n</math>-sided}} prism and <math> A_n </math> denotes the {{nowrap|<math>n</math>-sided}} antiprism.</ref>