Square

A **square** is the 2 dimensional hypercube. It has the schläfli symbol

{
4
}

{\displaystyle \{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

{\displaystyle \{\}^2}

 (line prism)
* {
4
}

{\displaystyle \{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

2

{\displaystyle \sqrt{2}}

 before turning back to a point.

### Hypervolumes

* vertex count =

4

{\displaystyle 4}
* edge length =

4
l

{\displaystyle 4l}
* surface area =

l
2

{\displaystyle l^2}

### Subfacets

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

### Radii

* Vertex radius:

2

2

l

{\displaystyle \frac{\sqrt{2}}{2}l}
* Edge radius:

1
2

l

{\displaystyle \frac{1}{2}l}

### 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

{\displaystyle \max(x^2,y^2) = 1}

A square rotated by 45º, with side

2

{\displaystyle \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

2

{\displaystyle \sqrt{2}}

 has coordinates:

* (±1,0)
* (0,±1)

### Notations

* Toratopic notation:

{\displaystyle ||}
* Tapertopic notation:

11

{\displaystyle 11}

### Related shapes

* Dual: Self dual
* Vertex figure: Line segment, length

2

{\displaystyle \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

d

s
2

=
d

x
2

d

y
2

{\displaystyle \text{d}s^2 = \text{d}x^2 + \text{d}y^2}

 and an area element

d
# A
d
x
d
y

{\displaystyle \text{d}A = \text{d}x \text{d}y}

.

## See Also

t
0

{
4
}

{\displaystyle t\_0 \{4\}}

t
1

{
4
}

{\displaystyle t\_1 \{4\}}

t

0
,
1

{
4
}

{\displaystyle t\_{0,1} \{4\}}

t
0

{
2
}

{\displaystyle t\_0 \{2\}}

t
1

{
2
}

{\displaystyle t\_1 \{2\}}

t

0
,
1

{
2
}

{\displaystyle t\_{0,1} \{2\}}

t
0

{

4
3

}

{\displaystyle t\_0 \{\frac{4}{3} \}}

t
1

{

4
3

}

{\displaystyle t\_1 \{\frac{4}{3} \}}

t

0
,
1

{

4
3

}

{\displaystyle t\_{0,1} \{\frac{4}{3} \}}