Cube (EntityTopic, 20)
From Hi.gher. Space
(Difference between revisions)
m |
(→Homology groups) |
||
| Line 54: | Line 54: | ||
;0-frame (8 points):H<sub>0</sub>X = 8ℤ | ;0-frame (8 points):H<sub>0</sub>X = 8ℤ | ||
| - | ;1-frame (12 line segments):H<sub>0</sub>X = ℤ, H<sub>1</sub> = 5ℤ | + | ;1-frame (12 line segments):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = 5ℤ |
| - | ;2-frame (8 square faces):H<sub>0</sub>X = ℤ, H<sub>1</sub> = 0, H<sub>2</sub> = ℤ | + | ;2-frame (8 square faces):H<sub>0</sub>X = ℤ, H<sub>1</sub>X = 0, H<sub>2</sub> = ℤ |
;3-frame (solid cube):H<sub>0</sub>X = ℤ | ;3-frame (solid cube):H<sub>0</sub>X = ℤ | ||
Revision as of 04:30, 25 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
All homology groups are zero unless stated. Here X is the square 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, H2 = 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 | |||||
| 4. (xy) Circle | 5. [xyz] Cube | 6. [ Cuboid |
| List of bracketopes | ||
