## FANDOM

1,123 Pages

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 $\sqrt{2}$ (also diagonal)

## 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 One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
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
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
Hydrotopes

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

Pentagon

$\{5\}$

Icosahedron

$\{3, 5\}$

Hexacosichoron

$\{3^2, 5\}$

Order-5 pentachoric tetracomb

$\{3^3, 5\}$

Order-5 hexateric pentacomb

$\{3^4, 5\}$

...
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
Cosmotopes

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

Pentagon

$\{5\}$

Dodecahedron

$\{5, 3\}$

Hecatonicosachoron

$\{5, 3^2\}$

Order-3 hecatonicosachoric tetracomb

$\{5, 3^3\}$

Order-3-3 hecatonicosachoric pentacomb

$\{5, 3^4\}$

...
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

$\mathbb B^{\aleph_0}$

$\{0\}$ $\{1\}$ $\{2\}$ $\{3\}$ $\{4\}$ $\{5\}$ $\{\frac{5}{2}\}$ $\{6\}$ $\{7\}$ $\{\frac{7}{2}\}$ $\{\frac{7}{3}\}$ $\{8\}$ $\{\frac{8}{3}\}$ $\{9\}$ $\{\frac{9}{2}\}$ $\{\frac{9}{4}\}$ $\{10\}$ $\{\frac{10}{3}\}$ $\{11\}$ $\{\frac{11}{2}\}$ $\{\frac{11}{3}\}$ $\{\frac{11}{4}\}$ $\{\frac{11}{5}\}$ $\{12\}$ $\{\frac{12}{5}\}$ $\{13\}$ $\{\frac{13}{2}\}$ $\{\frac{13}{3}\}$ $\{\frac{13}{4}\}$ $\{\frac{13}{5}\}$ $\{\frac{13}{6}\}$ $\{14\}$ $\{\frac{14}{3}\}$ $\{\frac{14}{5}\}$ $\{15\}$ $\{\frac{15}{2}\}$ $\{\frac{15}{4}\}$ $\{\frac{15}{7}\}$ $\{16\}$ $\{\frac{16}{3}\}$ $\{\frac{16}{5}\}$ $\{\frac{16}{7}\}$ ... $\{\infty\}$ $\{x\}$ $\{\frac{\pi i}{\lambda}\}$
Zerogon Monogon Digon Triangle Square Pentagon Pentagram Hexagon Heptagon Heptagram Great heptagram Octagon Octagram Enneagon Enneagram Great enneagram Decagon Decagram Hendecagon Small hendecagram Hendecagram Great hendecagram Grand hendecagram Dodecagon Dodecagram Tridecagon Small tridecagram Tridecagram Medial tridecagram Great tridecagram Grand tridecagram Tetradecagon Tetradecagram Great tetradecagram Pentadecagon Small pentadecagram Pentadecagram Great pentadecagram Hexadecagon Small hexadecagram Hexadecagram Great hexadecagram ... Apeirogon Failed star polygon ($x$-gon) Pseudogon ($\frac{\pi i}{\lambda}$-gon)
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
Community content is available under CC-BY-SA unless otherwise noted.