Cube (EntityTopic, 20)

From Hi.gher. Space

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 3×3^2-1, 2×2, 2^2-1

Use

Cubic cells are found in these tetrashapes on FGwiki:

• Tesseract (8×, 100%)
• Cubinder (1×, 20%)
• Cubic pyramid (1×, 14%)

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