Square pyramid (EntityTopic, 23)
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{{STS Shape | {{STS Shape | ||
| dim=3 | | dim=3 | ||
- | | elements= | + | | elements=1 [[square]], 4 [[triangle]]s, 8 [[digon]]s, 5 [[point]]s |
+ | | sym=[[Stauroaxial symmetry|C<sub>4v</sub>]] | ||
| genus=0 | | genus=0 | ||
| ssc=G4P | | ssc=G4P | ||
Line 16: | Line 17: | ||
| flayout={{FLD|a4|i|ade|cr|e4}} | | flayout={{FLD|a4|i|ade|cr|e4}} | ||
| dual=''Self-dual'' | | dual=''Self-dual'' | ||
+ | | bowers=Squippy | ||
}}}} | }}}} | ||
A '''square pyramid''' is a special case of a [[pyramid]] where the base is a [[square]]. | 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<sub>1</sub>. 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: | ||
+ | <blockquote>(±√2, 0, 0)<br>(0, ±√2, 0)<br>(0, 0, √2)</blockquote> | ||
+ | These coordinates are obtained from those of an [[octahedron]] by removing one fertex, since joining two square pyramids makes an octahedron. | ||
== Equations == | == Equations == | ||
*Variables: | *Variables: |
Latest revision as of 14:33, 26 March 2017
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 J1. 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) · l2
volume = √2∕6 · l3
- 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. 21 Cone | 9. [11]1 Square pyramid | 10. 111 Triangular prism |
List of tapertopes |