## FANDOM

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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)

## 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.

### Subfacets

• 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

• Dual: Self dual
• Vertex figure: Line segment, length $\sqrt{2}$

## 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 $\text{d}A = \text{d}x \text{d}y$.

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

$\{3^9\}$

$\{3^{10}\}$

Tridecahendon

$\{3^{11}\}$

$\{3^{12}\}$

$\{3^{13}\}$

$\{3^{14}\}$

$\{3^{15}\}$

... Omegasimplex

$\{3^{\aleph_0}\}$

Cross

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

Square

$\{4\}$

Octahedron

$\{3, 4\}$

$\{3^2, 4\}$

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

$\{3^{11}, 4\}$

$\{3^{12}, 4\}$

$\{3^{13}, 4\}$

$\{3^{14}, 4\}$

... Omegacross

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

Hydrotopes

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

Pentagon

$\{5\}$

Icosahedron

$\{3, 5\}$

Hexacosichoron

$\{3^2, 5\}$

Order-5 pentachoric tetracomb

$\{3^3, 5\}$

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

$\{4, 3^{11}\}$

$\{4, 3^{12}\}$

$\{4, 3^{13}\}$

$\{4, 3^{14}\}$

... Omegeract

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

Cosmotopes

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

Pentagon

$\{5\}$

Dodecahedron

$\{5, 3\}$

Hecatonicosachoron

$\{5, 3^2\}$

Order-3 hecatonicosachoric tetracomb

$\{5, 3^3\}$

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

$\mathbb B^{13}$

$\mathbb B^{14}$

$\mathbb B^{15}$

$\mathbb B^{16}$

... 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