Cubic pyramid (EntityTopic, 23)

From Hi.gher. Space

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<[#ontology [kind topic] [cats Cube Pyramid 4D Tapertope Polytope]]>
{{STS Shape
{{STS Shape
| name=Cubic pyramid
| name=Cubic pyramid
| dim=4
| dim=4
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| image=<[#embed [hash YBP0RT0PWBX7TAFHCF3JWCMKHH] [width 180]]>
| elements=7, 18, 20, 9
| elements=7, 18, 20, 9
| genus=0
| genus=0
| ssc=[xyz]P
| ssc=[xyz]P
| ssc2=&Ko1
| ssc2=&Ko1
-
| extra={{STS Rotope
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| extra={{STS Tapertope
| order=3, 1
| order=3, 1
| notation=[111]<sup>1</sup>
| notation=[111]<sup>1</sup>
| index=19
| index=19
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}}{{STS Polytope
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| flayout={{FLD|dim=4|left=e3|erev2|a4|line|a4|line2|atbend2|end}}
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| dual=[[Octahedral pyramid]]
}}}}
}}}}
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The '''cubic pyramid''' is a [[CRF polychoron]] among Klitzing's list of [[segmentochora]] (K4.26).
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== Dichoral angles ==
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* Between cube and square pyramid: 45° (exact)
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* Between two adjacent square pyramids: 180° - acos(1/3) ≈ 109.4712206345°
== Equations ==
== Equations ==
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*The [[hypervolume]]s of a cubic pyramid are given by:
*The [[hypervolume]]s of a cubic pyramid are given by:
<blockquote>total edge length = 20''l''<br>
<blockquote>total edge length = 20''l''<br>
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total surface area = ''Unknown''<br>
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total surface area = (6 + 3√3) {{DotHV}}<br>
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surcell volume = ''Unknown''<br>
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surcell volume = (1 + √2) {{DotHV|3}}<br>
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bulk = ''l''<sup>4</sup>4<sup>-1</sup></blockquote>
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bulk = {{Over|1|4}} {{DotHV|4}}</blockquote>
*The [[realmic]] [[cross-section]]s (''n'') of a cubic pyramid are:
*The [[realmic]] [[cross-section]]s (''n'') of a cubic pyramid are:
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== Net ==
== Net ==
The net of a cubic pyramid is a [[cube]] surrounded by six [[square pyramid]]s.
The net of a cubic pyramid is a [[cube]] surrounded by six [[square pyramid]]s.
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==Coordinates==
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<pre>
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apacs<1,1,1> ~ <0>
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<0,0,0,1>
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</pre>
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<[#polytope [id 45]]>
{{Tetrashapes}}
{{Tetrashapes}}
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{{Tapertope Nav|18|19|20|[21]<sup>1</sup><br>Cylindrone|[111]<sup>1</sup>|Cubic pyramid|21<sup>1</sup><br>Cyltrianglinder|chora}}
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{{Tapertope Nav|18|19|20|[21]<sup>1</sup><br>Cylindrone|[111]<sup>1</sup><br>Cubic pyramid|21<sup>1</sup><br>Cyltrianglinder|chora}}

Revision as of 19:32, 27 March 2014

The cubic pyramid is a CRF polychoron among Klitzing's list of segmentochora (K4.26).

Dichoral angles

  • Between cube and square pyramid: 45° (exact)
  • Between two adjacent square pyramids: 180° - acos(1/3) ≈ 109.4712206345°

Equations

  • Variables:
l ⇒ length of edges of the cubic pyramid
total edge length = 20l
total surface area = (6 + 3√3) · l2
surcell volume = (1 + √2) · l3
bulk = 14 · l4
[!x,!y,!z] ⇒ triangle
[!w] ⇒ square

Net

The net of a cubic pyramid is a cube surrounded by six square pyramids.

Coordinates

apacs<1,1,1> ~ <0>
<0,0,0,1>

Incidence matrix

Dual: octahedral pyramid

#TXIDVaVbEaEb3a4aC1aC2aTypeName
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 = square ; lateral: faces
6 C1a 144441 = square pyramid ; sweep: faces
7 C2a 0801206 = base of pyramid: cube ; lateral: cell
8 H4.1a 1881212661 = cubic pyramid ;

Usage as facets

This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.


Notable Tetrashapes
Regular: pyrochoronaerochorongeochoronxylochoronhydrochoroncosmochoron
Powertopes: triangular octagoltriatesquare octagoltriatehexagonal octagoltriateoctagonal octagoltriate
Circular: glomecubinderduocylinderspherindersphonecylindronediconeconinder
Torii: tigertorispherespheritorustorinderditorus


18. [21]1
Cylindrone
19. [111]1
Cubic pyramid
20. 211
Cyltrianglinder
List of tapertopes