Cubic pyramid (EntityTopic, 23)
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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 | ||
- | | elements= | + | | image=<[#embed [hash YBP0RT0PWBX7TAFHCF3JWCMKHH] [width 180]]> |
+ | | elements=1 [[cube]], 6 [[square pyramid]]s, 6 [[square]]s, 12 [[triangle]]s, 20 [[digon]]s, 9 [[point]]s | ||
| genus=0 | | genus=0 | ||
| ssc=[xyz]P | | ssc=[xyz]P | ||
- | | extra={{STS | + | | ssc2=&Ko1 |
- | | | + | | extra={{STS Tapertope |
- | | notation=111<sup>1 | + | | order=3, 1 |
- | | index= | + | | notation=[111]<sup>1</sup> |
+ | | index=19 | ||
+ | }}{{STS Polytope | ||
+ | | flayout={{FLD|dim=4|left=e3|erev2|a4|line|a4|line2|atbend2|end}} | ||
+ | | dual=[[Octahedral pyramid]] | ||
+ | | bowers=Cubpy | ||
}}}} | }}}} | ||
+ | The '''cubic pyramid''' is a [[CRF polychoron]] among Klitzing's list of [[segmentochora]] (K4.26). It is bounded by its cubical base and 6 square pyramidal sides. | ||
+ | |||
+ | == Dichoral angles == | ||
+ | |||
+ | * Between cube and square pyramid: 45° (exact) | ||
+ | * Between two adjacent square pyramids: 180° - acos(1/3) ≈ 109.4712206345° | ||
== Equations == | == Equations == | ||
Line 17: | Line 30: | ||
*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> | ||
- | total surface area = | + | total surface area = (6 + 3√3) {{DotHV}}<br> |
- | surcell volume = | + | surcell volume = (1 + √2) {{DotHV|3}}<br> |
- | bulk = | + | bulk = {{Over|1|8}} {{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: | ||
- | <blockquote>[!x,!y,!z] ⇒ [[ | + | <blockquote>[!x,!y,!z] ⇒ [[square pyramid]]<br> |
- | [!w] ⇒ square</blockquote> | + | [!w] ⇒ cube</blockquote> |
+ | |||
+ | == Net == | ||
+ | The net of a cubic pyramid is a [[cube]] surrounded by six [[square pyramid]]s. | ||
+ | |||
+ | ==Coordinates== | ||
+ | The coordinates of a cubic pyramid with side 2 are: | ||
+ | <blockquote>(±1, ±1, ±1, 0)<br>(0, 0, 0, 1)</blockquote> | ||
+ | |||
+ | <[#polytope [id 45]]> | ||
{{Tetrashapes}} | {{Tetrashapes}} | ||
- | {{ | + | {{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}} |
Latest revision as of 16:11, 26 March 2017
The cubic pyramid is a CRF polychoron among Klitzing's list of segmentochora (K4.26). It is bounded by its cubical base and 6 square pyramidal sides.
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
- The hypervolumes of a cubic pyramid are given by:
total edge length = 20l
total surface area = (6 + 3√3) · l2
surcell volume = (1 + √2) · l3
bulk = 1∕8 · l4
- The realmic cross-sections (n) of a cubic pyramid are:
[!x,!y,!z] ⇒ square pyramid
[!w] ⇒ cube
Net
The net of a cubic pyramid is a cube surrounded by six square pyramids.
Coordinates
The coordinates of a cubic pyramid with side 2 are:
(±1, ±1, ±1, 0)
(0, 0, 0, 1)
Incidence matrix
Dual: octahedral pyramid
# | TXID | Va | Vb | Ea | Eb | 3a | 4a | C1a | C2a | 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 | = square | ; lateral: faces | ||||
6 | C1a | 1 | 4 | 4 | 4 | 4 | 1 | = square pyramid | ; sweep: faces | ||
7 | C2a | 0 | 8 | 0 | 12 | 0 | 6 | = base of pyramid: cube | ; lateral: cell | ||
8 | H4.1a | 1 | 8 | 8 | 12 | 12 | 6 | 6 | 1 | = cubic pyramid | ; |
Usage as facets
This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.
Notable Tetrashapes | |
Regular: | pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |
Powertopes: | triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |
Circular: | glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |
Torii: | tiger • torisphere • spheritorus • torinder • ditorus |
18. [21]1 Cylindrone | 19. [111]1 Cubic pyramid | 20. 211 Cyltrianglinder |
List of tapertopes |