Cube (EntityTopic, 20)

From Higher Dimensions Database

A cube is a special case of a prism where the base is a square.

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
pointdigonsquarecubegeochorongeoterongeopeton


Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind


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

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