Square

Square
Dimensionality	2
Schläfli Symbol	\(\{4\}\)
Symmetry	Square dihedral symmetry (D4)
Coxeter–Dynkin diagram
Bowers acronym	square
Subfacets
Vertices	4 points
Edges	4 line segments
Faces	1 square
Hypervolumes
Vertex Count	\(4\)
Edge Length	\(4l\)
Surface Area	\(l^2\)

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

• vertex count = \(4\)
• edge length = \(4l\)
• surface area = \(l^2\)

Subfacets

• 1 null polytope (-1D)
• 4 points (0D)
• 4 line segments (1D)
• 1 square (2D)

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

• 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 \(\text{d}A = \text{d}x \text{d}y\).

See Also

Dimensionality	Negative One	Zero	One	Two	Three	Four	Five	Six	Seven	Eight	Nine	Ten	Eleven	Twelve	Thirteen	Fourteen	Fifteen	Sixteen	Seventeen	...	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 \(\{3^9\}\)	Dodecadakon \(\{3^{10}\}\)	Tridecahendon \(\{3^{11}\}\)	Tetradecadokon \(\{3^{12}\}\)	Pentadecatradakon \(\{3^{13}\}\)	Hexadecatedakon \(\{3^{14}\}\)	Heptadecapedakon \(\{3^{15}\}\)	Octadecapedakon \(\{3^{16}\}\)	...	Omegasimplex
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 \(\{3^{14}, 4\}\)	Heptadekacross \(\{3^{15}, 4\}\)	...	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 \(\{4, 3^{11}\}\)	Tetradekeract \(\{4, 3^{12}\}\)	Pentadekeract \(\{4, 3^{13}\}\)	Hexadekeract \(\{4, 3^{14}\}\)	Heptadekeract \(\{4, 3^{15}\}\)	...	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 \(\mathbb B^{13}\)	Tetradekorb \(\mathbb B^{14}\)	Pentadekorb \(\mathbb B^{15}\)	Hexadekorb \(\mathbb B^{16}\)	Heptadekorb \(\mathbb B^{17}\)	...	Omegaball \(\mathbb B^{\aleph_0}\)

\(\{0\}\)	\(\{1\}\)	\(\{2\}\)	\(\{3\}\)	\(\{4\}\)	\(\{5\}\)	\(\{\frac{5}{2}\}\)	\(\{6\}\)	\(\{7\}\)	\(\{\frac{7}{2}\}\)	\(\{\frac{7}{3}\}\)	\(\{8\}\)	\(\{\frac{8}{3}\}\)	\(\{9\}\)	\(\{\frac{9}{2}\}\)	\(\{\frac{9}{4}\}\)	\(\{10\}\)	\(\{\frac{10}{3}\}\)	\(\{11\}\)	\(\{\frac{11}{2}\}\)	\(\{\frac{11}{3}\}\)	\(\{\frac{11}{4}\}\)	\(\{\frac{11}{5}\}\)	\(\{12\}\)	\(\{\frac{12}{5}\}\)	\(\{13\}\)	\(\{\frac{13}{2}\}\)	\(\{\frac{13}{3}\}\)	\(\{\frac{13}{4}\}\)	\(\{\frac{13}{5}\}\)	\(\{\frac{13}{6}\}\)	\(\{14\}\)	\(\{\frac{14}{3}\}\)	\(\{\frac{14}{5}\}\)	\(\{15\}\)	\(\{\frac{15}{2}\}\)	\(\{\frac{15}{4}\}\)	\(\{\frac{15}{7}\}\)	\(\{16\}\)	\(\{\frac{16}{3}\}\)	\(\{\frac{16}{5}\}\)	\(\{\frac{16}{7}\}\)	\(\{17\}\)	\(\{\frac{17}{2}\}\)	\(\{\frac{17}{3}\}\)	\(\{\frac{17}{4}\}\)	\(\{\frac{17}{5}\}\)	\(\{\frac{17}{6}\}\)	\(\{\frac{17}{7}\}\)	\(\{\frac{17}{8}\}\)	...	\(\{\infty\}\)	\(\{x\}\)	\(\{\frac{\pi i}{\lambda}\}\)
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	Heptadecagon	Tiny heptadecagram	Small heptadecagram	Heptadecagram	Medial heptadecagram	Great heptadecagram	Giant heptadecagram	Grand heptadecagram	...	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