Square (EntityTopic, 20)

From Hi.gher. Space

Revision as of 17:09, 18 November 2011

A square is a two-dimensional hypercube.

Squares are the most common base for two-dimensional manifolds and polyominoes.

Equations

• Variables:

l ⇒ length of edges of the square

• The radial slices θ of a square are:

[:x,:y] ⇒ digon of length lsin(45° + (θ % 90°)√2

Homology groups

Any unstated homology group is the trivial group 0.

0-frame (four points) 

H₀ = 4ℤ

1-frame (four line segments) 

H₀ = ℤ, H₁ = ℤ

2-frame (solid square) 

H₀ = ℤ

Diamond

A diamond is the dual of a square when orientation is preserved. In other words, it's a square rotated around the origin by 45 degrees.

The diamond should not be confused with the rhombus.

Brick

The diamond is a brick with unique points at (1,0) and (0,1). It represents the tegum product and the SUM function.

Brick

The square is a brick with one unique point at (1,1). It represents the Cartesian product and the MAX function.

Segmentation

The square of side 2 may be segmented into 4× triangle with sides 2, 2×2^2-1 and angles 2×45°, 90°.

Use

Square faces are found in these trishapes on FGwiki:

• Cube (6×, 100%)
• Triangular prism (3×, 60%)
• Cuboctahedron (6×, 43%)
• Octahedral truncate (6×, 43%)
• Square antiprism (2×, 20%)
• Square pyramid (1×, 20%)
• Cubic snub (6×, 16%)

See also

• Tetragon

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