Square pyramid (EntityTopic, 23)
From Hi.gher. Space
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- | {{Shape| | + | <[#ontology [kind topic] [cats Square Pyramid 3D Polytope] [alt [[freebase:048qg3]] [[wikipedia:Square_pyramid]]]]> |
- | == | + | {{STS Shape |
+ | | dim=3 | ||
+ | | elements=5, 8, 5 | ||
+ | | sym=[[Stauroaxial symmetry|C<sub>4v</sub>]] | ||
+ | | genus=0 | ||
+ | | ssc=G4P | ||
+ | | ssc2=&G4 | ||
+ | | extra={{STS Matrix| | ||
+ | <i>2 2</i> | ||
+ | 4 0 | ||
+ | 1 1}}{{STS Tapertope | ||
+ | | order=2, 1 | ||
+ | | notation=[11]<sup>1</sup> | ||
+ | | index=9 | ||
+ | }}{{STS Polytope | ||
+ | | flayout={{FLD|a4|i|ade|cr|e4}} | ||
+ | | 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]]. | ||
- | + | == Equations == | |
*Variables: | *Variables: | ||
<blockquote>''l'' ⇒ length of edges of the square pyramid</blockquote> | <blockquote>''l'' ⇒ length of edges of the square pyramid</blockquote> | ||
Line 9: | Line 27: | ||
*The [[hypervolume]]s of a square pyramid are given by: | *The [[hypervolume]]s of a square pyramid are given by: | ||
<blockquote>total edge length = 8''l''<br> | <blockquote>total edge length = 8''l''<br> | ||
- | surface area = | + | surface area = (1 + √3) {{DotHV}}<br> |
- | volume = | + | volume = {{Over|√2|6}} {{DotHV|3}}</blockquote> |
*The [[planar]] [[cross-section]]s (''n'') of a square pyramid are: | *The [[planar]] [[cross-section]]s (''n'') of a square pyramid are: | ||
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[!z] ⇒ square</blockquote> | [!z] ⇒ square</blockquote> | ||
- | {{ | + | <[#polytope [id 44]]> |
- | {{ | + | |
+ | {{Trishapes}} | ||
+ | {{Tapertope Nav|8|9|10|2<sup>1</sup><br>Cone|[11]<sup>1</sup><br>Square pyramid|11<sup>1</sup><br>Triangular prism|hedra}} |
Revision as of 22:08, 2 March 2014
A square pyramid is a special case of a pyramid where the base is a square.
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 |