Square (EntityTopic, 20)

From Hi.gher. Space

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{{Shape|Square|''No image''|2|4, 4|0|{4}|N/A|[[Line (shape)|E]]E|11 xy|[[Line (shape)|Line]], length √2|N/A|''Self-dual''|2|[xy]|2|pure}}
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<[#ontology [kind topic] [cats 2D Hypercube] [alt [[freebase:030jx3]] [[wikipedia:Square_(geometry)]]]]>
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{{STS Shape
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| dim=2
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| elements=4 [[digon]]s, 4 [[point]]s
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| ssc=[xy] or G4
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| ssc2=G4
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| pv_circle=<sup>2</sup>⁄<sub>π</sub> ≈ 0.6366
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| pv_square=1
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| extra={{STS Matrix|
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4 0
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1 1}}{{STS Tapertope
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| order=2, 0
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| notation=11
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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
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}}{{STS Bracketope
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| index=2
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| notation=[II]}}{{STS Polytope
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| bowers=Square
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| dual=''Self-dual''}}{{STS Uniform polytope
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| schlaefli={4}
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| dynkin=x4o, x2x
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| vfigure=[[Digon]], length √2
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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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== Geometry ==
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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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=== Equations ===
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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)
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== Equations ==
*Variables:
*Variables:
<blockquote>''l'' ⇒ length of edges of the square</blockquote>
<blockquote>''l'' ⇒ length of edges of the square</blockquote>
*The [[radial slice]]s ''θ'' of a square are:
*The [[radial slice]]s ''θ'' of a square are:
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<blockquote>[:x,:y] ⇒ [[line segment]] of length ''l''sin(45° + (''θ'' % 90°)√2</blockquote>
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<blockquote>[:x,:y] ⇒ [[digon]] of length ''l''sin(45° + (''θ'' % 90°)√2</blockquote>
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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 ==
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Any unstated homology group is the trivial group 0.
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;0-frame (four points) :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> = ℤ
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== Diamond ==
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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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The diamond should not be confused with the [[rhombus]].
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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.
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== Brick ==
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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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== Dissection ==
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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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<[#polytope [id -4]]>
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== See also ==
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*[[Tetragon]]
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{{Hypercubes|2}}
{{Dishapes}}
{{Dishapes}}
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{{Rotope Nav|1|2|3|I<br>Line segment|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>Line segment|[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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{{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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