# Square pyramid (EntityTopic, 23)

### From Hi.gher. Space

A **square pyramid** is a special case of a pyramid where the base is a square.

The square pyramid with equilateral triangular sides is the first of the 92 Johnson solids, and is referred to as J_{1}. It can be obtained from an octahedron by cutting it in half.

## Coordinates

The coordinates of a square pyramid with side 2 and a base centered on the origin are:

(±√2, 0, 0)

(0, ±√2, 0)

(0, 0, √2)

These coordinates are obtained from those of an octahedron by removing one fertex, since joining two square pyramids makes an octahedron.

## Equations

- Variables:

l⇒ length of edges of the square pyramid

- The hypervolumes of a square pyramid are given by:

total edge length = 8l

surface area = (1 + √3) ·l^{2}

volume =^{√2}∕_{6}·l^{3}

- The planar cross-sections (
*n*) of a square pyramid are:

[!x,!y] ⇒ triangle

[!z] ⇒ square

## Incidence matrix

Dual: *Self-dual*

# | TXID | Va | Vb | Ea | Eb | 3a | 4a | Type | Name |
---|---|---|---|---|---|---|---|---|---|

0 | Va | = point | ; sweep: apex | ||||||

1 | Vb | = point | ; lateral: vertices | ||||||

2 | Ea | 1 | 1 | = digon | ; sweep: vertices | ||||

3 | Eb | 0 | 2 | = digon | ; lateral: edges | ||||

4 | 3a | 1 | 2 | 2 | 1 | = triangle | ; sweep: edges | ||

5 | 4a | 0 | 4 | 0 | 4 | = base of pyramid: square |
; lateral: faces | ||

6 | C1a | 1 | 4 | 4 | 4 | 4 | 1 | = square pyramid |
; |

## Usage as facets

- 12× 1-facets of a cubic bipyramid
- 6× 1-facets of a cubic pyramid (named
*sweep: faces*) - 2× 1-facets of a square pyramid bipyramid
*prism*: 2× 1-facets of a square pyramid prism- 4× 1-facets of a gyrated octahedral prism
- 4× 1-facets of a K4.13 dual
- 4× 1-facets of a K4.8
- 4× 1-facets of a K4.8 dual
- 8× 1-facets of a (dual of square biantiprismatic ring)
- 4× 1-facets of a square biantiprismatic ring
- 288× 1-facets of a (dual of truncated snub demitesseract)
- 8× 1-facets of a (dual of bilunabirotunda pseudopyramid)
- 4× 1-facets of a bilunabirotunda pseudopyramid
- 3× 1-facets of a D4.16
- 120× 1-facets of a D4.3.2 dual
- 240× 1-facets of a D4.3.2 dual
- 8× 1-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 8× 1-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 4× 1-facets of a bitrigonal diminished pyrocantichoron
- 8× 1-facets of a (4D analog of J37)
- 16× 1-facets of a (4D analog of J37)
- 6× 1-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 3× 1-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 6× 1-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 3× 1-facets of a triangular hebesphenorotunda pseudopyramid (named
*tola to roof*) - 12× 1-facets of a triangular hebesphenorotundaeic rhombochoron (named
*oobpp*) - 12× 1-facets of a triangular hebesphenorotundaeic rhombochoron (named
*ooggb*) - 6× 1-facets of a triangular hebesphenorotundaeic rhombochoron (named
*roooo*) - 6× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named
*roooo*) - 12× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named
*ooggb*) - 12× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named
*oobpp*) - 12× 1-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named
*tola to roof*) - 6× 1-facets of a D4.7
- 24× 1-facets of a D4.7 dual
- 24× 1-facets of a D4.7 dual

Notable Trishapes
| |

Regular:
| tetrahedron • cube • octahedron • dodecahedron • icosahedron |

Direct truncates:
| tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |

Mesotruncates:
| stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |

Snubs:
| snub staurohedron • snub rhodohedron |

Curved:
| sphere • torus • cylinder • cone • frustum • crind |

8. 2^{1}Cone | 9. [11]
^{1}Square pyramid | 10. 11^{1}Triangular prism |

List of tapertopes |