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 | ||
| Line 10: | Line 12: | ||
| notation=[111]<sup>1</sup> | | notation=[111]<sup>1</sup> | ||
| index=19 | | 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 == | ||
*Variables: | *Variables: | ||
| - | + | <blockquote>''l'' ⇒ length of edges of the cubic pyramid</blockquote> | |
*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> | |
| - | 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] ⇒ [[square pyramid]]<br> | |
| - | [!w] ⇒ | + | [!w] ⇒ cube</blockquote> |
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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. | ||
| + | |||
| + | ==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}} | {{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 | ||
