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A square is the 2 dimensional hypercube. It has the schläfli symbol \{4\}, as it is a four-sided polygon. Other names of square are called tetragon or tetrasquaron (Using Googleaarex's polytope naming system). Its Bowers acronym is also "square". Under the elemental naming scheme it is called a geogon, aerogon, or staurogon.

Squares are one of the three regular polygons that tile the plane. The others are the equilateral triangle and regular hexagon. The tiling is called a square tiling, and has four squares around each vertex.

The reason why squares can tile the plane is that the interior angle of a square is (1/n) * 360 degrees, where n is a whole number. If n is not a whole number, then you cannot tile the plane.

The symmetry group of a square is D4, since there are four possible reflections that will leave the square unchanged: through the two lines joining the midpoints of opposite edges, and through the two lines joining the opposite vertices of the square.

Four squares can fit between a vertex, at least in Euclidian geometry.

Hypercube Product

The square can be expressed as a product of hypercubes in two different ways:

  • \{\}^2 (line prism)
  • \{4\} (square)

Symbols

A square can be given several Dynkin symbols and their extensions, including:

  • x4o (fully regular)
  • x x (rectangle)
  • qo oq&#zx (rhombus)
  • xx&#x (trapezoid)
  • oqo&#xt (kite)
  • oooo&#xr (generic quadrilateral)

Structure and Sections

Sections

The square can be thought of as infinitely many line segments stacked on each other in the y direction, or a prism with a line segment as the base. As such, when viewed from a side, the sections are identical lines. It is composed of two pairs of parallel line segments.

When viewed from a vertex, the point will expand into a line of length \sqrt{2} before turning back to a point.

Hypervolumes

Subfacets

Radii

  • Vertex radius: \frac{\sqrt{2}}{2}l
  • Edge radius: \frac{1}{2}l

Angles

  • Vertex angle: 90º

Equations

All points on the surface of a square with side length 2 can be given by the equation

\max(x^2,y^2) = 1

A square rotated by 45º, with side \sqrt{2}, can be given by the equation

|x|+|y| = 1

Vertex coordinates

The vertex coordinates of a square of side 2 are (±1, ±1).

The dual orientatoin of this square, with side length \sqrt{2} has coordinates:

  • (±1,0)
  • (0,±1)

Notations

  • Toratopic notation: ||
  • Tapertopic notation: 11

Related shapes

Coordinate System

The coordinate system associated with the square is plane cartesian coordinates. This coordinate system has a length element with length \text{d}s^2 = \text{d}x^2 + \text{d}y^2 and an area element \text{d}A = \text{d}x \text{d}y.

See Also

Dimensionality Negative First Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth ... Omegath
Simplex

\{3^{n-1}\}

Null polytope

\emptyset

Point

()
\mathbb{B}^0

Line segment

\{\}
\mathbb{B}^1

Triangle

\{3\}

Tetrahedron

\{3^2\}

Pentachoron

\{3^3\}

Hexateron

\{3^4\}

Heptapeton

\{3^5\}

Octaexon

\{3^6\}

Enneazetton

\{3^7\}

Decayotton

\{3^8\}

Hendecaxennon

\{3^9\}

Dodecadakon

\{3^{10}\}

Tridecahendon

\{3^{11}\}

Tetradecadokon

\{3^{12}\}

Pentadecatradakon

\{3^{13}\}

Hexadecatedakon

\{3^{14}\}

Heptadecapedakon

\{3^{15}\}

... Omegasimplex

\{3^{\aleph_0}\}

Cross

\{3^{n-2},4\}

Square

\{4\}

Octahedron

\{3, 4\}

Hexadecachoron

\{3^2, 4\}

Pentacross

\{3^3, 4\}

Hexacross

\{3^4, 4\}

Heptacross

\{3^5, 4\}

Octacross

\{3^6, 4\}

Enneacross

\{3^7, 4\}

Dekacross

\{3^8, 4\}

Hendekacross

\{3^9, 4\}

Dodekacross

\{3^{10}, 4\}

Tridekacross

\{3^{11}, 4\}

Tetradekacross

\{3^{12}, 4\}

Pentadekacross

\{3^{13}, 4\}

Hexadekacross

\{3^{14}, 4\}

... Omegacross

\{3^{\aleph_0}, 4\}

Hydrotopes

\{3^{n-2}, 5\}

Pentagon

\{5\}

Icosahedron

\{3, 5\}

Hexacosichoron

\{3^2, 5\}

Order-5 pentachoric honeycomb

\{3^3, 5\}

Hypercube

\{4, 3^{n-2}\}

Square

\{4\}

Cube

\{4, 3\}

Tesseract

\{4, 3^2\}

Penteract

\{4, 3^3\}

Hexeract

\{4, 3^4\}

Hepteract

\{4, 3^5\}

Octeract

\{4, 3^6\}

Enneract

\{4, 3^7\}

Dekeract

\{4, 3^8\}

Hendekeract

\{4, 3^9\}

Dodekeract

\{4, 3^{10}\}

Tridekeract

\{4, 3^{11}\}

Tetradekeract

\{4, 3^{12}\}

Pentadekeract

\{4, 3^{13}\}

Hexadekeract

\{4, 3^{14}\}

... Omegeract

\{4, 3^{\aleph_0}\}

Cosmotopes

\{5, 3^{n-2}\}

Pentagon

\{5\}

Dodecahedron

\{5, 3\}

Hecatonicosachoron

\{5, 3^2\}

Order-3 hecatonicosachoric honeycomb

\{5, 3^3\}

Hyperball

\mathbb B^n

Disk

\mathbb B^2

Ball

\mathbb B^3

Gongol

\mathbb B^4

Pentorb

\mathbb B^5

Hexorb

\mathbb B^6

Heptorb

\mathbb B^7

Octorb

\mathbb B^8

Enneorb

\mathbb B^9

Dekorb

\mathbb B^{10}

Hendekorb

\mathbb B^{11}

Dodekorb

\mathbb B^{12}

Tridekorb

\mathbb B^{13}

Tetradekorb

\mathbb B^{14}

Pentadekorb

\mathbb B^{15}

Hexadekorb

\mathbb B^{16}

... Omegaball

\mathbb B^{\aleph_0}

Regular polygons \{1\} \{2\} \{3\} \{4\} \{5\} \{6\} \{7\} \{8\} \{9\} \{10\} \{11\} \{12\} \{13\} \{14\} \{15\} \{16\} ... \{\aleph_0\}
\{\frac{n}{1}\} Monogon Digon Triangle Square Pentagon Hexagon Heptagon Octagon Enneagon Decagon Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon ... Apeirogon
\{\frac{n}{2}\} N/A N/A Triangle (retrograde) Degenerate Pentagram Degenerate Heptagram Degenerate Enneagram Degenerate Small hendecagram Degenerate Small tridecagram Degenerate Small pentadecagram Degenerate ... N/A
\{\frac{n}{3}\} N/A N/A N/A Square (retrograde) Pentagram (retrograde) Degenerate Great heptagram Octagram Degenerate Decagram Hendecagram Degenerate Tridecagram Tetradecagram Degenerate Small hexadecagram ... N/A
\{\frac{n}{4}\} N/A N/A N/A N/A Pentagon (retrograde) Degenerate Great heptagram (retrograde) Degenerate Great enneagram Degenerate Great hendecagram Degenerate Medial tridecagram Degenerate Pentadecagram Degenerate ... N/A
\{\frac{n}{5}\} N/A N/A N/A N/A N/A Hexagon (retrograde) Heptagram (retrograde) Octagram (retrograde) Great enneagram (retrograde) Degenerate Grand hendecagram Dodecagram Great tridecagram Great tetradecagram Degenerate Hexadecagram ... N/A
\{\frac{n}{6}\} N/A N/A N/A N/A N/A N/A Heptagon (retrograde) Degenerate Degenerate Degenerate Grand hendecagram (retrograde) Degenerate Grand tridecagram Degenerate Degenerate Degenerate ... N/A
\{\frac{n}{7}\} N/A N/A N/A N/A N/A N/A N/A Octagon (retrograde) Enneagram (retrograde) Decagram (retrograde) Great hendecagram (retrograde) Dodecagram (retrograde) Grand tridecagram (retrograde) Degenerate Great pentadecagram Great hexadecagram ... N/A
Regular
t_0 \{4\}
Rectified
t_1 \{4\}
Truncated
t_{0,1} \{4\}
Square Square Octagon
Regular
t_0 \{2\}
Rectified
t_1 \{2\}
Truncated
t_{0,1} \{2\}
Digon Digon Square
Regular
t_0 \{\frac{4}{3} \}
Rectified
t_1 \{\frac{4}{3} \}
Truncated
t_{0,1} \{\frac{4}{3} \}
Square Square Octagram