Editing Square pyramid (section)

=== Equilateral square pyramid ===
[[File:J1 square pyramid.stl|thumb|3D model of an equilateral square pyramid]]
{{anchor|Equilateral square pyramid}}If all triangular edges are of equal length, the four triangles are [[Equilateral triangle|equilateral]], and the pyramid's faces are all [[regular polygon]]s, it is an ''equilateral square pyramid.''{{sfnp|Hocevar|1903|p=[https://books.google.com/books?id=0OAXAAAAYAAJ&pg=PA44 44]}} The [[dihedral angle]]s between adjacent triangular faces are <math display="inline">\arccos \left(-1/3 \right) \approx 109.47^\circ </math>, and that between the base and each triangular face being half of that, <math display="inline">\arctan \left(\sqrt{2}\right) \approx 54.74^\circ </math>.{{sfnp|Johnson|1966}} A [[Convex set|convex]] polyhedron in which all of the faces are [[regular polygons]] is called a [[Johnson solid]]. The equilateral square pyramid is among them, enumerated as the first Johnson solid <math>J_1</math>.{{sfnp|Uehara|2020|p=[https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62 62]}}

Because its edges are all equal in length (that is, <math> b = l </math>), its slant, height, surface area, and volume can be derived by substituting the formulas of a right square pyramid:{{sfnmp
 | 1a1 = Simonson | 1y = 2011 | 1p = [https://books.google.com/books?id=Ws6-DwAAQBAJ&pg=PA123 123]
 | 2a1 = Berman | 2y = 1971 | 2loc = see table IV, line 21
}}
<math display="block">
\begin{align}
 s = \frac{\sqrt{3}}{2}l \approx 0.866l, &\qquad h = \frac{1}{\sqrt{2}}l \approx 0.707l,\\
 A = (1 + \sqrt{3})l^2 \approx 2.732l^2, &\qquad V = \frac{\sqrt{2}}{6}l^3 \approx 0.236l^3.
\end{align}
</math>

Like other right pyramids with a regular polygon as a base, a right square pyramid has [[pyramidal symmetry]]. For the square pyramid, this is the symmetry of [[cyclic group]] <math>C_{4\mathrm{v}}</math>: the pyramid is left invariant by rotations of one-, two-, and three-quarters of a full turn around its [[axis of symmetry]], the line connecting the apex to the center of the base; and is also [[mirror symmetric]] relative to any perpendicular plane passing through a bisector of the base.{{sfnp|Johnson|1966}} It can be represented as the [[wheel graph]] <math> W_4 </math>, meaning its [[Skeleton (topology)|skeleton]] can be interpreted as a square in which its four vertices connects a vertex in the center called the [[universal vertex]].{{sfnp|Pisanski|Servatius|2013|p=[https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA21 21]}} It is [[Self-dual polyhedron|self-dual]], meaning its [[dual polyhedron]] is the square pyramid itself.{{sfnp|Wohlleben|2019|p=[https://books.google.com/books?id=rEpjDwAAQBAJ&pg=PA485 485–486]}}

An equilateral square pyramid is an [[elementary polyhedron]]. This means it cannot be separated by a plane to create two small convex polyhedrons with regular faces.{{sfnmp
 | 1a1 = Hartshorne | 1y = 2000 | 1p = [https://books.google.com/books?id=EJCSL9S6la0C&pg=PA464 464]
 | 2a1 = Johnson | 2y = 1966
}}