Square pyramid

Square pyramid
(3D model) (OFF file)
Rank	3
Type	Segmentotope
Space	Spherical
Notation
Bowers style acronym	Squippy
Coxeter diagram	ox4oo&#x
Tapertopic notation	[11]1
Elements
Faces	4 triangles, 1 square
Edges	4+4
Vertices	1+4
Vertex figures	1 square, edge length 1
	4 isosceles triangles, edge lengths 1, 1, √2
Measures (edge length 1)
Circumradius	$\frac{\sqrt2}{2} ≈ 0.70711$
Volume	$\frac{\sqrt2}{6} ≈ 0.23570$
Dihedral angles	3-3: $\arccos\left(-\frac13\right) ≈ 109.47122°$
	3-4: $\arccos\left(\frac{\sqrt3}{3}\right) ≈ 54.73561°$
Heights	Point atop square: $\frac{\sqrt2}{2} ≈ 0.70711$
	Dyad atop trig: $\frac{\sqrt6}{3} ≈ 0.81650$
Central density	1
Related polytopes
Army	Squippy
Regiment	Squippy
Dual	Square pyramid
Conjugate	None
Abstract properties
Flag count	26
Net count	8
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B2×I, order 8
Convex	Yes
Nature	Tame

The square pyramid, or squippy, is a pyramid with a square base and 4 triangles as sides. The version with equilateral triangles as sides is the first of the 92 Johnson solids (J₁). In what follows, unless otherwise specified, this is what will be meant by a "square pyramid", even though other variants with isosceles triangles as sides exist.

Two square pyramids can be joined together at their square base to form a regular octahedron. As such it can also be thought of as a diminished octahedron.

In addition to being a point atop a square, the square pyramid has a second representation as a segmentohedron, as a dyad atop a triangle, obtained from removing a vertex from the octahedron.

Abstractly, the square pyramid is the simplest non-regular polytope without digonal faces overall.

It is one of three regular polygonal pyramids to be CRF. The others are the regular tetrahedron (triangular pyramid) and the pentagonal pyramid.

The orthoplecial pyramids, a family of Blind polytopes, are one way to generalize the square pyramid to any number of dimensions.

Vertex coordinates

Coordinates for a square pyramid of edge length 1 are given by:

• $\left(±\frac{\sqrt2}{2},\,0,\,0\right),$
• $\left(0,\,±\frac{\sqrt2}{2},\,0\right),$
• $\left(0,\,0,\,\frac{\sqrt2}{2}\right).$

Representations

A square pyramid has the following Coxeter diagrams:

• ox4oo&#x (full symmetry)
• ox ox&#x (rectangle pyramid)
• oxx&#x (isosceles trapezoid pyramid)

General variant

For the general square pyramid with base edges of length b and lacing edges of length l, its height is given by $\sqrt{l^2-\frac{b^2}{2}}$, its circumradius by $\frac{l}{2\sqrt{1-\frac{b^2}{2l^2}}}$, and its volume is given by $\frac13b^2\sqrt{l^2-\frac{b^2}{2}}$.

A square pyramid with base edges of length 1 and side edges of length $\sqrt2$ appears as the vertex figure of the octahedral prism, and with side edges of length $\sqrt3$ as the vertex figure of the truncated hexadecachoron.

Variations

Various other tetragonal pyramids with less symmetry exist, differentiated by their bases. These include:

• Rectangular pyramid - 1 rectangle and 2 pairs of isosceles tiranlges
• Rhombic pyramid - 1 rhombus and 4 identical scalene triangles
• Isosceles trapezoidal pyramid - 1 isosceles trapezoid, 2 separate isosceles triangles and 2 different scalene triangles
• Kite pyramid - 1 kite and 2 pairs of scalene triangles
• Irregular tetragonal pyramid - no symmetry

Related polyhedra

A cube can be attached to the base of a square pyramid to form the elongated square pyramid. If a square antiprism is attached instead, the result is the gyroelongated square pyramid.

The dihedral angle between two adjacent triangles is a supplementary angle to that of the regular tetrahedron. Thus, augmenting one of the triangular faces with a tetrahedron produces coplanar faces, disqualifying the resulting polyhedron from being a Johnson solid.

External links

• Bowers, Jonathan. "Batch 1: Oct and Co Facetings" (#3 under oct).

• Klitzing, Richard. "squippy".

• Quickfur. "The Square Pyramid".

• Weisstein, Eric W. "Square Pyramid" ("Johnson solid") at MathWorld.

• Wikipedia Contributors. "Square pyramid".
• McCooey, David. "Square Pyramid"

• Hi.gher.Space Wiki Contributors. "Square pyramid".