Cube (EntityTopic, 20)
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| + | <[#ontology [kind topic] [cats 3D Hypercube] [alt [[freebase:01v_x]] [[wikipedia:Cube]]]]> | ||
{{STS Shape | {{STS Shape | ||
| - | | image= | + | | 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)]] | ||
| genus=0 | | genus=0 | ||
| ssc=[xyz], [x<sup>3</sup>] or {G4<sup>3</sup>} | | ssc=[xyz], [x<sup>3</sup>] or {G4<sup>3</sup>} | ||
| Line 11: | Line 13: | ||
4 0 | 4 0 | ||
3 1 | 3 1 | ||
| - | 1 1}}{{STS | + | 1 1}}{{STS Tapertope |
| - | | | + | | order=3, 0 |
| - | | notation=111 | + | | notation=111 |
| - | | index= | + | | index=7 |
| + | }}{{STS Toratope | ||
| + | | expand=[[Cube|111]] | ||
| + | | notation=III | ||
| + | | index=2a | ||
}}{{STS Bracketope | }}{{STS Bracketope | ||
| - | | index= | + | | index=4 |
| + | | notation=[III] | ||
}}{{STS Polytope | }}{{STS Polytope | ||
| - | | flayout={{FLD|a4| | + | | flayout={{FLD|a4|er|e3}} |
| petrie=6,2 | | petrie=6,2 | ||
| altern=[[Tetrahedron]] | | altern=[[Tetrahedron]] | ||
| dual=[[Octahedron]] | | dual=[[Octahedron]] | ||
| + | | bowers=Cube | ||
}}{{STS Uniform polytope | }}{{STS Uniform polytope | ||
| 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 | + | | 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]]. 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 in the [[cubic honeycomb]]. | ||
| - | + | ==Coordinates== | |
| + | The Cartesian coordinates of a cube with side 2 are: | ||
| - | + | (±1, ±1, ±1) | |
| - | + | ||
| - | + | ||
| - | *The [[hypervolume]]s of a cube are given by: | + | == Equations == |
| + | *The [[hypervolume]]s of a cube with side length ''l'' are given by: | ||
<blockquote>total edge length = 12''l''<br> | <blockquote>total edge length = 12''l''<br> | ||
surface area = 6''l''<sup>2</sup><br> | surface area = 6''l''<sup>2</sup><br> | ||
volume = ''l''<sup>3</sup></blockquote> | volume = ''l''<sup>3</sup></blockquote> | ||
| - | + | == Cross-sections == | |
| - | + | The face-first [[cross-section]]s of a cube is a set of [[square]]s 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 [[triangle]]s and [[hexagon]]s, all regular apart from the non-central hexagons, which have edges of alternating widths (and equal angles). | |
| - | == | + | == Homology groups == |
| - | The cube of side 2 may be [[ | + | 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):H<sub>0</sub>X = 8ℤ | ||
| + | ;1-frame (12 line segments):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = 5ℤ | ||
| + | ;2-frame (8 square faces):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = 0, H<sub>2</sub>X = ℤ | ||
| + | ;3-frame (solid cube):H<sub>0</sub>X = ℤ | ||
| + | |||
| + | == Dissection == | ||
| + | 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]]> | |
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
| + | {{Hypercubes|3}} | ||
{{Trishapes}} | {{Trishapes}} | ||
| - | {{ | + | {{Tapertope Nav|6|7|8|21<br>Cylinder|111<br>Cube|2<sup>1</sup><br>Cone|hedra}} |
| - | {{Bracketope Nav|4|5 | + | {{Toratope Nav A|1|2|3|II<br>Square|(II)<br>Circle|III<br>Cube|(III)<br>Sphere|(II)I<br>Cylinder|((II)I)<br>Torus|hedra}} |
| - | + | {{Bracketope Nav|3|4|5|(II)<br>Circle|[III]<br>Cube|<nowiki><III></nowiki><br>Octahedron|hedra}} | |
| - | + | ||
| - | + | ||
Latest revision as of 11:50, 26 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 in the cubic honeycomb.
Coordinates
The Cartesian coordinates of a cube with side 2 are:
(±1, ±1, ±1)
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 | ||
