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 D_{4}, 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
- 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
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
Regular polygons | ... | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Monogon | Digon | Triangle | Square | Pentagon | Hexagon | Heptagon | Octagon | Enneagon | Decagon | Hendecagon | Dodecagon | Tridecagon | Tetradecagon | Pentadecagon | Hexadecagon | ... | Apeirogon | |
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 |