Cube (EntityTopic, 20)
From Hi.gher. Space
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 = 6l^{2}
volume = l^{3}
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)
- H_{0}X = 8ℤ
- 1-frame (12 line segments)
- H_{0}X = ℤ, H_{1}X = 5ℤ
- 2-frame (8 square faces)
- H_{0}X = ℤ, H_{1}X = 0, H_{2}X = ℤ
- 3-frame (solid cube)
- H_{0}X = 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. 2^{1} 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 |