Cube (EntityTopic, 20)
From Hi.gher. Space
m (elemental naming) |
Username5243 (Talk | contribs) |
||
Line 3: | Line 3: | ||
| image=<[#embed [hash JNM9PCTD4D70TY5QG5NT0FKDF1] [width 150]]> | | image=<[#embed [hash JNM9PCTD4D70TY5QG5NT0FKDF1] [width 150]]> | ||
| dim=3 | | dim=3 | ||
- | | elements=6, 12, 8 | + | | elements=6 [[square]]s, 12 [[digon]]s, 8 [[point]]s |
| sym=[[Staurohedral symmetry|O<sub>h</sub>, BC<sub>3</sub>, [4,3], (*432)]] | | sym=[[Staurohedral symmetry|O<sub>h</sub>, BC<sub>3</sub>, [4,3], (*432)]] | ||
| genus=0 | | genus=0 | ||
Line 23: | Line 23: | ||
}}{{STS Bracketope | }}{{STS Bracketope | ||
| index=4 | | index=4 | ||
+ | | notation=[III] | ||
}}{{STS Polytope | }}{{STS Polytope | ||
| flayout={{FLD|a4|er|e3}} | | flayout={{FLD|a4|er|e3}} | ||
Line 32: | Line 33: | ||
| wythoff=<nowiki>3 | 2 4, 2 4 | 2, or 2 2 2 |</nowiki> | | wythoff=<nowiki>3 | 2 4, 2 4 | 2, or 2 2 2 |</nowiki> | ||
| schlaefli={[[Square|4,]]3}, t{2,4} or tr{2,2} | | schlaefli={[[Square|4,]]3}, t{2,4} or tr{2,2} | ||
+ | | dynkin=x4o3o, x2x4o, x2x2x | ||
| conway=d[[Octahedron|a]][[Tetrahedron|Y3]] | | conway=d[[Octahedron|a]][[Tetrahedron|Y3]] | ||
| vfigure=Equilateral [[triangle]], edge √2 | | vfigure=Equilateral [[triangle]], edge √2 | ||
| vlayout=[[Square|4]]<sup>3</sup> | | vlayout=[[Square|4]]<sup>3</sup> | ||
}}}} | }}}} | ||
- | A '''cube''' is a special case of a [[prism]] where the base is a [[square]]. | + | A '''cube''' is a special case of a [[prism]] where the base is a [[square]]. It is one of the five Platonic solids, containing six square faces joining three to a vertex. It is the only regular polyhedron that can completely tile three-dimensional space. |
== Equations == | == Equations == | ||
Line 58: | Line 60: | ||
The cube of side 2 may be [[dissect]]ed into: | The cube of side 2 may be [[dissect]]ed into: | ||
*6× [[square pyramid]] with base side 2 and height 1 | *6× [[square pyramid]] with base side 2 and height 1 | ||
- | *12× irregular [[tetrahedron]] with sides | + | *12× irregular [[tetrahedron]] with sides √3, √3, √3, 2, 2, √2 |
<[#polytope [id 2]]> | <[#polytope [id 2]]> |
Revision as of 16:26, 25 March 2017
A cube is a special case of a prism where the base is a square. It is one of the five Platonic solids, containing six square faces joining three to a vertex. It is the only regular polyhedron that can completely tile three-dimensional space.
Equations
- The hypervolumes of a cube with side length l are given by:
total edge length = 12l
surface area = 6l2
volume = l3
Cross-sections
The face-first cross-sections of a cube is a set of squares of constant edge length, and the edge-first cross-sections are a set of rectangles of constant width. However, the vertex-first cross-sections are more interesting - they are a set of triangles and hexagons, all regular apart from the non-central hexagons, which have edges of alternating widths (and equal angles).
Homology groups
All homology groups are zero unless stated. Here X is the shape in the given frame, and nℤ is the direct sum of n copies of the group of integers ℤ.
- 0-frame (8 points)
- H0X = 8ℤ
- 1-frame (12 line segments)
- H0X = ℤ, H1X = 5ℤ
- 2-frame (8 square faces)
- H0X = ℤ, H1X = 0, H2X = ℤ
- 3-frame (solid cube)
- H0X = ℤ
Dissection
The cube of side 2 may be dissected into:
- 6× square pyramid with base side 2 and height 1
- 12× irregular tetrahedron with sides √3, √3, √3, 2, 2, √2
Incidence matrix
Dual: octahedron
# | TXID | Va | Ea | 4a | Type | Name |
---|---|---|---|---|---|---|
0 | Va | = point | ; | |||
1 | Ea | 2 | = digon | ; | ||
2 | 4a | 4 | 4 | = base of prism: square | ; | |
3 | C1a | 8 | 12 | 6 | = cube | ; |
Usage as facets
- prism: 8× 1-facets of a geochoron
- pyramid: 1× 1-facets of a cubic pyramid (named lateral: cell)
- 3× 1-facets of a triangular diprism
- 9× 1-facets of a triangular octagoltriate (named antifrustoids)
- 1× 1-facets of a square pyramid prism
- 1× 1-facets of a square biantiprismatic ring
- 32× 1-facets of a D4.11 dual
- 12× 1-facets of a D4.7 dual
- 40× 2-facets of a geoteron
- 3× 2-facets of a triangular triprism (named base)
- 18× 2-facets of a triangular triprism (named sweep)
Hypercubes |
point • digon • square • cube • geochoron • geoteron • geopeton |
Notable Trishapes | |
Regular: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
Direct truncates: | tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |
Mesotruncates: | stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |
Snubs: | snub staurohedron • snub rhodohedron |
Curved: | sphere • torus • cylinder • cone • frustum • crind |
6. 21 Cylinder | 7. 111 Cube | 8. 21 Cone |
List of tapertopes |
1a. II Square | 1b. (II) Circle | 2a. III Cube | 2b. (III) Sphere | 3a. (II)I Cylinder | 3b. ((II)I) Torus |
List of toratopes |
3. (II) Circle | 4. [III] Cube | 5. <III> Octahedron |
List of bracketopes |