Cube (EntityTopic, 20)
From Hi.gher. Space
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Revision as of 17:11, 18 November 2011
A cube is a special case of a prism where the base is a square.
Equations
- Variables:
l ⇒ length of edges of the cube
- The hypervolumes of a cube 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
Segmentation
The cube of side 2 may be segmented into:
- 6× square pyramid with base side 2 and height 1
- 12× irregular tetrahedron with sides 3×32-1, 2×2, 22-1
Use
Cubic cells are found in these tetrashapes on FGwiki:
- Tesseract (8×, 100%)
- Cubinder (1×, 20%)
- Cubic pyramid (1×, 14%)
| 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 | ||
