Square (EntityTopic, 20)

From Hi.gher. Space

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A '''square''' is a two-dimensional [[hypercube]].
A '''square''' is a two-dimensional [[hypercube]].
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The square of side 2 may be [[dissect]]ed into 4× [[triangle]] with sides 2, 2×2<sup>2<sup>-1</sup></sup> and angles 2×45°, 90°.
The square of side 2 may be [[dissect]]ed into 4× [[triangle]] with sides 2, 2×2<sup>2<sup>-1</sup></sup> and angles 2×45°, 90°.
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== Use ==
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<[#polytope [id -4]]>
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Square faces are found in these trishapes on FGwiki:
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*[[Cube]] (6×, 100%)
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*[[Triangular prism]] (3×, 60%)
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*[[Cuboctahedron]] (6×, 43%)
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*[[Octahedral truncate]] (6×, 43%)
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*[[Square antiprism]] (2×, 20%)
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*[[Square pyramid]] (1×, 20%)
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*[[Cubic snub]] (6×, 16%)
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== See also ==
== See also ==

Revision as of 22:02, 16 February 2014

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
[: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.

Dissection

The square of side 2 may be dissected into 4× triangle with sides 2, 2×22-1 and angles 2×45°, 90°.

Incidence matrix

Dual: Self-dual

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

Usage as facets

See also


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


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