## 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.

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)

## 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 = $l^2$

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

Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth
Hypercube Point Line segment Square Cube Tesseract Penteract Hexeract Hepteract Octeract Enneract Dekeract Hendekeract Dodekeract Tridekeract Tetradekeract Pentadekeract Hexadekeract
Hypersphere Point Line segment Disk Ball Gongol Pentorb Hexorb Heptorb Octorb Enneorb Dekorb Hendekorb Dodekorb Tridekorb Tetradekorb Pentadekorb Hexadekorb
$\{1\}$ $\{2\}$ $\{3\}$ $\{4\}$ $\{5\}$ $\{6\}$ $\{7\}$ $\{8\}$ $\{9\}$ $\{10\}$ ... $\{\aleph_0\}$
Monogon Digon Triangle Square Pentagon Hexagon Heptagon Octagon Enneagon Decagon ... Apeirogon
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