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.
Hypercube Product[]
The square can be expressed as a product of hypercubes in two different ways:
- (line prism)
- (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 before turning back to a point.
Hypervolumes[]
Subfacets[]
- 1 null polytope (-1D)
- 4 points (0D)
- 4 line segments (1D)
- 1 square (2D)
Radii[]
- Vertex radius:
- Edge radius:
Angles[]
- Vertex angle: 90º
Equations[]
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
Vertex coordinates[]
The vertex coordinates of a square of side 2 are (±1, ±1).
The dual orientatoin of this square, with side length has coordinates:
- (±1,0)
- (0,±1)
Notations[]
- Toratopic notation:
- Tapertopic notation:
Related shapes[]
- Dual: Self dual
- Vertex figure: Line segment, length (also diagonal)
Coordinate System[]
The coordinate system associated with the square is plane cartesian coordinates. This coordinate system has a length element with length and an area element .
See Also[]
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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 (-gon) | Pseudogon (-gon) |
Regular |
Rectified |
Truncated |
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Square | Square | Octagon |
Regular |
Rectified |
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Digon | Digon | Square |
Regular |
Rectified |
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Square | Square | Octagram |