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)
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
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.
- Vertex radius:
- Edge radius:
- Vertex angle: 90º
All points on the surface of a square with side length 2 can be given by the equation
A square rotated by 45º, with side , can be given by the equation
The vertex coordinates of a square of side 2 are (±1, ±1).
The dual orientatoin of this square, with side length has coordinates:
- Toratopic notation:
- Tapertopic notation:
- Dual: Self dual
- Vertex figure: Line segment, length
The coordinate system associated with the square is plane cartesian coordinates. This coordinate system has a length element with length and an area element .
|N/A||N/A||Triangle (retrograde)||Degenerate||Pentagram||Degenerate||Heptagram||Degenerate||Enneagram||Degenerate||Small hendecagram||Degenerate||Small tridecagram||Degenerate||Small pentadecagram||Degenerate||...||N/A|
|N/A||N/A||N/A||Square (retrograde)||Pentagram (retrograde)||Degenerate||Great heptagram||Octagram||Degenerate||Decagram||Hendecagram||Degenerate||Tridecagram||Tetradecagram||Degenerate||Small hexadecagram||...||N/A|
|N/A||N/A||N/A||N/A||Pentagon (retrograde)||Degenerate||Great heptagram (retrograde)||Degenerate||Great enneagram||Degenerate||Great hendecagram||Degenerate||Medial tridecagram||Degenerate||Pentadecagram||Degenerate||...||N/A|
|N/A||N/A||N/A||N/A||N/A||Hexagon (retrograde)||Heptagram (retrograde)||Octagram (retrograde)||Great enneagram (retrograde)||Degenerate||Grand hendecagram||Dodecagram||Great tridecagram||Great tetradecagram||Degenerate||Hexadecagram||...||N/A|
|N/A||N/A||N/A||N/A||N/A||N/A||Heptagon (retrograde)||Degenerate||Degenerate||Degenerate||Grand hendecagram (retrograde)||Degenerate||Grand tridecagram||Degenerate||Degenerate||Degenerate||...||N/A|
|N/A||N/A||N/A||N/A||N/A||N/A||N/A||Octagon (retrograde)||Enneagram (retrograde)||Decagram (retrograde)||Great hendecagram (retrograde)||Dodecagram (retrograde)||Grand tridecagram (retrograde)||Degenerate||Great pentadecagram||Great hexadecagram||...||N/A|
|Regular ||Rectified ||Truncated ||Square||Square||Octagon|
|Regular ||Rectified ||Truncated ||Digon||Digon||Square|
|Regular ||Rectified ||Truncated ||Square||Square||Octagram|