Cube (EntityTopic, 20)

From Hi.gher. Space

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<[#ontology [kind topic] [cats 3D Hypercube] [alt [[freebase:01v_x]] [[wikipedia:Cube]]]]>
{{STS Shape
{{STS Shape
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| image=http://teamikaria.com/dl/ClpnXfceQcCFoY0YuAmG9JyFlFecodMj27oVFuXqYzZp9j-q.png
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| image=<[#embed [hash JNM9PCTD4D70TY5QG5NT0FKDF1] [width 150]]>
| dim=3
| dim=3
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| elements=6, 12, 18
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| elements=6 [[square]]s, 12 [[digon]]s, 8 [[point]]s
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| 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>}
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| pv_circle=~0.3676
| pv_circle=~0.3676
| pv_square=1
| pv_square=1
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| extra={{STS Rotope
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| extra={{STS Matrix|
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| attrib=pure
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4 0
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| notation=111 xyz
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3 1
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| index=5
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1 1}}{{STS Tapertope
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| order=3, 0
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| notation=111
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| index=7
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}}{{STS Toratope
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| expand=[[Cube|111]]
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| notation=III
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| index=2a
}}{{STS Bracketope
}}{{STS Bracketope
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| index=5
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| index=4
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| notation=[III]
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}}{{STS Polytope
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| flayout={{FLD|a4|er|e3}}
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| petrie=6,2
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| altern=[[Tetrahedron]]
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| dual=[[Octahedron]]
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| 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}
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| dynkin=x4o3o, x2x4o, x2x2x
| conway=d[[Octahedron|a]][[Tetrahedron|Y3]]
| conway=d[[Octahedron|a]][[Tetrahedron|Y3]]
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| vfigure=Equilateral [[triangle]], edge √2
| vlayout=[[Square|4]]<sup>3</sup>
| vlayout=[[Square|4]]<sup>3</sup>
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| vfigure=Equilateral [[triangle]], edge sqrt(2)
 
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| bowers=Cube
 
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| kana=ｷｭ
 
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| dual=[[Octahedron]]
 
}}}}
}}}}
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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]].
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A '''cube''' is a special case of a [[prism]] where the base is a [[square]].
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==Coordinates==
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The Cartesian coordinates of a cube with side 2 are:
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== Equations ==
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(±1, ±1, ±1)
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*Variables:
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<blockquote>''l'' ⇒ length of edges of the cube</blockquote>
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*The [[hypervolume]]s of a cube are given by:
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== Equations ==
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*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>
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*The [[planar]] [[cross-section]]s (''n'') of a cube are:
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== Cross-sections ==
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<blockquote>[!x,!y,!z] ⇒ [[square]] with side length ''l''</blockquote>
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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).
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== Segmentation ==
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== Homology groups ==
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The cube of side 2 may be [[segment]]ed into:
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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.
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;0-frame (8 points):H<sub>0</sub>X = 8ℤ
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;1-frame (12 line segments):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = 5ℤ
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;2-frame (8 square faces):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = 0, H<sub>2</sub>X = ℤ
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;3-frame (solid cube):H<sub>0</sub>X = ℤ
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== Dissection ==
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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
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*12× irregular [[tetrahedron]] with sides 3×3<sup>2<sup>-1</sup></sup>, 2×2, 2<sup>2<sup>-1</sup></sup>
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*12× irregular [[tetrahedron]] with sides √3, √3, √3, 2, 2, √2
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== Use ==
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<[#polytope [id 2]]>
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Cubic cells are found in these tetrashapes on FGwiki:
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*[[Tesseract]] (8×, 100%)
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*[[Cubinder]] (1×, 20%)
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*[[Cubic pyramid]] (1×, 14%)
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{{Hypercubes|3}}
{{Trishapes}}
{{Trishapes}}
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{{Rotope Nav|4|5|6|(II)<br>Circle|III<br>Cube|II'<br>Square pyramid|hedra}}
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{{Tapertope Nav|6|7|8|21<br>Cylinder|111<br>Cube|2<sup>1</sup><br>Cone|hedra}}
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{{Bracketope Nav|4|5|6|(xy)<br>Circle|[xyz]<br>Cube|[<xy>z]<br>Cuboid|hedra}}
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{{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}}
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{{Bracketope Nav|3|4|5|(II)<br>Circle|[III]<br>Cube|<nowiki><III></nowiki><br>Octahedron|hedra}}
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[[Category:Regular polyhedra]]
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[[Category:Uniform prismahedra]]
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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:

Incidence matrix

Dual: octahedron

#TXIDVaEa4aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 4a 44 = base of prism: square ;
3 C1a 8126 = cube ;

Usage as facets


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
Retrieved from "http://hi.gher.space/wiki/Cube"

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