Square pyramid (EntityTopic, 23)

From Hi.gher. Space

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{{Shape|Square pyramid|''No image''|3|5, 8, 5|0|N/A|N/A|[[Line (object)|E]][[Square|E]]T|11<sup>1</sup> xy<sup>z</sup>|N/A|N/A|N/A|6|N/A|N/A|pure}}
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<[#ontology [kind topic] [cats Square Pyramid 3D Polytope] [alt [[freebase:048qg3]] [[wikipedia:Square_pyramid]]]]>
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== Geometry ==
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{{STS Shape
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| dim=3
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| elements=1 [[square]], 4 [[triangle]]s, 8 [[digon]]s, 5 [[point]]s
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| sym=[[Stauroaxial symmetry|C<sub>4v</sub>]]
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| genus=0
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| ssc=G4P
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| ssc2=&G4
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| extra={{STS Matrix|
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<i>2 2</i>
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4 0
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1 1}}{{STS Tapertope
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| order=2, 1
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| notation=[11]<sup>1</sup>
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| index=9
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}}{{STS Polytope
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| flayout={{FLD|a4|i|ade|cr|e4}}
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| dual=''Self-dual''
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| bowers=Squippy
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}}}}
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]].
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=== Equations ===
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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.
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==Coordinates==
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The coordinates of a square pyramid with side 2 and a base centered on the origin are:
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<blockquote>(±√2, 0, 0)<br>(0, ±√2, 0)<br>(0, 0, √2)</blockquote>
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These coordinates are obtained from those of an [[octahedron]] by removing one fertex, since joining two square pyramids makes an octahedron.
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== Equations ==
*Variables:
*Variables:
<blockquote>''l'' ⇒ length of edges of the square pyramid</blockquote>
<blockquote>''l'' ⇒ length of edges of the square pyramid</blockquote>
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*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>
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surface area = ''l''<sup>2</sup>(1+2√3)<br>
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surface area = (1 + √3) {{DotHV}}<br>
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volume = ''l''<sup>3</sup>6<sup>-1</sup>√2</blockquote>
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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>
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{{Polyhedra}}
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<[#polytope [id 44]]>
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{{Rotope Nav|5|6|7|III<br>Cube|II'<br>Square pyramid|(III)<br>Sphere|hedra}}
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{{Trishapes}}
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{{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}}

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
total edge length = 8l
surface area = (1 + √3) · l2
volume = √26 · l3
[!x,!y] ⇒ triangle
[!z] ⇒ square

Incidence matrix

Dual: Self-dual

#TXIDVaVbEaEb3a4aTypeName
0 Va = point ; sweep: apex
1 Vb = point ; lateral: vertices
2 Ea 11 = digon ; sweep: vertices
3 Eb 02 = digon ; lateral: edges
4 3a 1221 = triangle ; sweep: edges
5 4a 0404 = base of pyramid: square ; lateral: faces
6 C1a 144441 = square pyramid ;

Usage as facets


Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind


8. 21
Cone
9. [11]1
Square pyramid
10. 111
Triangular prism
List of tapertopes