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

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

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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