Cube (EntityTopic, 20)

(Difference between revisions)

Revision as of 08:08, 24 November 2009

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

l ⇒ length of edges of the cube

total edge length = 12l
surface area = 6l²
volume = l³

[!x,!y,!z] ⇒ square with side length l

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, H₁ = 5Z

2-frame (8 square faces): H₀X = Z, H₁ = 0, H₂ = Z

3-frame (solid cube): H₀X = Z

The cube of side 2 may be segmented into:

Cubic cells are found in these tetrashapes on FGwiki:

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. [z] Cuboid
List of bracketopes

@@ Line 47: / Line 47: @@
 == Homology Groups ==
-Any unstated homology group is the trivial group 0.
+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.
 -frame (8 points):
-H<sub>0</sub> = 8Z
+H<sub>0</sub>X = 8Z
 -frame (12 line segments):
-H<sub>0</sub> = Z, H<sub>1</sub> = 5Z
+H<sub>0</sub>X = Z, H<sub>1</sub> = 5Z
 -frame (8 square faces):
-H<sub>0</sub> = Z, H<sub>1</sub> = 0, H<sub>2</sub> = Z
+H<sub>0</sub>X = Z, H<sub>1</sub> = 0, H<sub>2</sub> = Z
 -frame (solid cube):
-H<sub>0</sub> = Z
+H<sub>0</sub>X = Z
 == Segmentation ==