Square (EntityTopic, 20)

From Hi.gher. Space

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(Added homology groups. This might look better as a table. Also how do I get blackboard bold and the circle plus in this wiki?)
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<[#ontology [kind topic] [cats 2D Hypercube] [alt [[freebase:030jx3]] [[wikipedia:Square_(geometry)]]]]>
{{STS Shape
{{STS Shape
| dim=2
| dim=2
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| elements=4, 4
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| elements=4 [[digon]]s, 4 [[point]]s
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| genus=0
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| ssc=[xy] or G4
| ssc=[xy] or G4
| ssc2=G4
| ssc2=G4
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| extra={{STS Matrix|
| extra={{STS Matrix|
  4 0
  4 0
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  1 1}}{{STS Rotope
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  1 1}}{{STS Tapertope
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| attrib=pure
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| order=2, 0
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| notation=11 xy
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| notation=11
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| index=2
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| index=3
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}}{{STS Toratope
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| expand=[[Square|11]]
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| notation=II
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| index=1a
}}{{STS Bracketope
}}{{STS Bracketope
| index=2
| index=2
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}}{{STS Uniform polytope
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| notation=[II]}}{{STS Polytope
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| bowers=Square
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| dual=''Self-dual''}}{{STS Uniform polytope
| schlaefli={4}
| schlaefli={4}
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| dynkin=x4o, x2x
| vfigure=[[Digon]], length √2
| vfigure=[[Digon]], length √2
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| dual=''Self-dual''
 
}}}}
}}}}
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A '''square''' is a two-dimensional [[hypercube]].
A '''square''' is a two-dimensional [[hypercube]].
Squares are the most common base for two-dimensional [[manifold]]s and [[polyomino]]es.
Squares are the most common base for two-dimensional [[manifold]]s and [[polyomino]]es.
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The square also forms one of the three regular tilings of two-dimensional space, the [[square tiling]].
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==Coordinates==
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The Cartesian cordinates for a square of side 2, centered at the origin, are:
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(±1, ±1)
== Equations ==
== Equations ==
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<blockquote>[:x,:y] ⇒ [[digon]] of length ''l''sin(45° + (''θ'' % 90°)√2</blockquote>
<blockquote>[:x,:y] ⇒ [[digon]] of length ''l''sin(45° + (''θ'' % 90°)√2</blockquote>
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== Homology Groups ==
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*The cross sections of a square are:
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<blockquote>[!x,!y] ⇒ [[digon]] of length ''l''</blockquote>
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*The hypervolumes of a square are:
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<blockquote>perimeter (total edge length) = 4''l''<br>
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area = ''l''<sup>2</sup></blockquote>
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== Homology groups ==
Any unstated homology group is the trivial group 0.
Any unstated homology group is the trivial group 0.
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0-frame (four points):
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;0-frame (four points) :H<sub>0</sub> = 4ℤ
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H<sub>0</sub> = 4Z
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;1-frame (four line segments) :H<sub>0</sub> = ℤ, H<sub>1</sub> = ℤ
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;2-frame (solid square) :H<sub>0</sub> =
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1-frame (four line segments):
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== Diamond ==
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H<sub>0</sub> = Z
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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>.
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H<sub>1</sub> = Z
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2-frame (solid square):
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The diamond should not be confused with the [[rhombus]].
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H<sub>0</sub> = Z
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=== Brick ===
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The diamond is a brick with unique points at (1,0) and (0,1). It represents the [[tegum product]] and the SUM function.
== Brick ==
== Brick ==
The square is a brick with one unique point at (1,1). It represents the [[Cartesian product]] and the MAX function.
The square is a brick with one unique point at (1,1). It represents the [[Cartesian product]] and the MAX function.
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== Segmentation ==
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== Dissection ==
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The square of side 2 may be [[segment]]ed into 4× [[triangle]] with sides 2, 2×2<sup>2<sup>-1</sup></sup> and angles 2×45°, 90°.
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The square of side 2 may be [[dissect]]ed into 4× [[triangle]] with sides 2, √2, √2 and angles 45°, 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 ==
*[[Tetragon]]
*[[Tetragon]]
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*[[Diamond]]
 
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{{Hypercubes}}
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{{Hypercubes|2}}
{{Dishapes}}
{{Dishapes}}
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{{Rotope Nav|1|2|3|I<br>Digon|II<br>Square|I'<br>Triangle|gons}}
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{{Tapertope Nav|2|3|4|2<br>Circle|11<br>Square|1<sup>1</sup><br>Triangle|gons}}
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{{Bracketope Nav|1|2|3|x<br>Digon|[xy]<br>Square|<xy><br>Diamond|gons}}
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{{Toratope Nav A||1|2|||II<br>Square|(II)<br>Circle|III<br>Cube|(III)<br>Sphere|gons}}
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[[Category:Regular polygons]]
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{{Bracketope Nav|1|2|3|I<br>Digon|[II]<br>Square|(II)<br>Circle|gons}}

Latest revision as of 11:12, 26 March 2017

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

#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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