Cubic pyramid (EntityTopic, 23)

From Hi.gher. Space

(Difference between revisions)
m
 
(14 intermediate revisions not shown)
Line 1: Line 1:
 +
<[#ontology [kind topic] [cats Cube Pyramid 4D Tapertope Polytope]]>
{{STS Shape
{{STS Shape
| name=Cubic pyramid
| name=Cubic pyramid
| dim=4
| dim=4
-
| elements=7, 18, 20, 9
+
| 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
| ssc2=&Ko1
| ssc2=&Ko1
-
| extra={{STS Rotope
+
| extra={{STS Tapertope
-
| attrib=pure
+
| order=3, 1
-
| notation=111<sup>1</sup> xyz<sup>w</sup>
+
| notation=[111]<sup>1</sup>
-
| index=15
+
| 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 18: 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 = ''Unknown''<br>
+
total surface area = (6 + 3√3) {{DotHV}}<br>
-
surcell volume = ''Unknown''<br>
+
surcell volume = (1 + √2) {{DotHV|3}}<br>
-
bulk = ''l''<sup>4</sup>4<sup>-1</sup></blockquote>
+
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] ⇒ [[triangle]]<br>
+
<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}}
-
{{Rotope Nav|14|15|16|IIII<br>Tesseract|III'<br>Cubic pyramid|(IIII)<br>Glome|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
total edge length = 20l
total surface area = (6 + 3√3) · l2
surcell volume = (1 + √2) · l3
bulk = 18 · l4
[!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

#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