Cube (EntityTopic, 20)
From Hi.gher. Space
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*The [[planar]] [[cross-section]]s (''n'') of a cube are: | *The [[planar]] [[cross-section]]s (''n'') of a cube are: | ||
<blockquote>[!x,!y,!z] ⇒ [[square]] with side length ''l''</blockquote> | <blockquote>[!x,!y,!z] ⇒ [[square]] with side length ''l''</blockquote> | ||
| + | |||
| + | == Homology Groups == | ||
| + | Any unstated homology group is the trivial group 0. | ||
| + | |||
| + | 0-frame (8 points): | ||
| + | H<sub>0</sub> = 8Z | ||
| + | |||
| + | 1-frame (12 line segments): | ||
| + | H<sub>0</sub> = Z, H<sub>1</sub> = 5Z | ||
| + | |||
| + | 2-frame (8 square faces): | ||
| + | H<sub>0</sub> = Z, H<sub>1</sub> = 0, H<sub>2</sub> = Z | ||
| + | |||
| + | 3-frame (solid cube): | ||
| + | H<sub>0</sub> = Z | ||
| + | |||
== Segmentation == | == Segmentation == | ||
Revision as of 07:41, 24 November 2009
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
- The planar cross-sections (n) of a cube are:
[!x,!y,!z] ⇒ square with side length l
Homology Groups
Any unstated homology group is the trivial group 0.
0-frame (8 points): H0 = 8Z
1-frame (12 line segments): H0 = Z, H1 = 5Z
2-frame (8 square faces): H0 = Z, H1 = 0, H2 = Z
3-frame (solid cube): H0 = 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 |
| 4. (xy) Circle | 5. [xyz] Cube | 6. [ Cuboid |
| List of bracketopes | ||
