Square (EntityTopic, 20)

From Hi.gher. Space

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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.

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

#TXIDVaEaTypeName
0 Va= point ;
1 Ea2= digon ;
2 4a44= square ;

Usage as facets

See also


Hypercubes
pointdigonsquarecubegeochorongeoterongeopeton


Notable Dishapes
Flat: trianglesquarepentagonhexagonoctagondecagon
Curved: circle


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


a. b. 1a. II
Square
1b. (II)
Circle
2a. III
Cube
2b. (III)
Sphere
List of toratopes


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

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