Square (EntityTopic, 20)

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A square is a two-dimensional hypercube.

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

The square also forms one of the three regular tilings of two-dimensional space, the square tiling.

Coordinates

The Cartesian cordinates for a square of side 2, centered at the origin, are:

(±1, ±1)

Equations

• Variables:
l ⇒ length of edges of the square
[:x,:y] ⇒ digon of length lsin(45° + (θ % 90°)√2
• The cross sections of a square are:
[!x,!y] ⇒ digon of length l
• The hypervolumes of a square are:
perimeter (total edge length) = 4l
area = l2

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. it can be considered to be represented as <II>.

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.

Dissection

The square of side 2 may be dissected into 4× triangle with sides 2, √2, √2 and angles 45°, 45°, 90°.

Incidence matrix

Dual: Self-dual

 # TXID Type Name Va Ea 0 Va = point ; 1 Ea 2 = digon ; 2 4a 4 4 = square ;

Usage as facets

 Hypercubes point • digon • square • cube • geochoron • geoteron • geopeton

 Notable Dishapes Flat: triangle • square • pentagon • hexagon • octagon • decagon Curved: circle

 2. 2Circle 3. 11Square 4. 11Triangle List of tapertopes

 a. b. 1a. IISquare 1b. (II)Circle 2a. IIICube 2b. (III)Sphere List of toratopes

 1. IDigon 2. [II]Square 3. (II)Circle List of bracketopes

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 Square pyramid