A square is the 2-dimensional hypercube. It has the schläfli symbol , 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.
The square can be expressed as a product of hypercubes in two different ways:
- (line prism)
Structure and 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 before turning back to a point.
The coordinate system associated with the square is plane cartesian coordinates. This coordinate system has a length element with length and an area element .
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