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)
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)
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
Zeroth | First | Second | Third | Fourth | Fifth | Sixth | Seventh | Eighth | Ninth | Tenth | Eleventh | Twelfth | Thirteenth | Fourteenth | Fifteenth | Sixteenth | ... | Omegath | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Simplex | Point | Line segment | Triangle | Tetrahedron | Pentachoron | Hexateron | Heptapeton | Octaexon | Enneazetton | Decayotton | Hendecaxennon | Dodecadakon | Tridecahendon | Tetradecadokon | Pentadecatradakon | Hexadecatedakon | Heptdecapedakon | ... | Omegasimplex |
Hypercube | Point | Line segment | Square | Cube | Tesseract | Penteract | Hexeract | Hepteract | Octeract | Enneract | Dekeract | Hendekeract | Dodekeract | Tridekeract | Tetradekeract | Pentadekeract | Hexadekeract | ... | Omegeract |
Cross | Point | Line segment | Square | Octahedron | Hexadecachoron | Pentacross | Hexacross | Heptacross | Octacross | Enneacross | Dekacross | Hendekacross | Dodekacross | Tridekacross | Tetradekacross | Pentadekacross | Hexadekacross | ... | Omegacross |
Hyperball | Point | Line segment | Disk | Ball | Gongol | Pentorb | Hexorb | Heptorb | Octorb | Enneorb | Dekorb | Hendekorb | Dodekorb | Tridekorb | Tetradekorb | Pentadekorb | Hexadekorb | ... | Omegaball |
... | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Monogon | Digon | Triangle | Square | Pentagon | Hexagon | Heptagon | Octagon | Enneagon | Decagon | ... | Apeirogon |
Regular | Rectified | Truncated |
---|---|---|
Square | Square | Octagon |
Regular | Rectified | Truncated |
---|---|---|
Digon | Digon | Square |
Regular | Rectified | Truncated |
---|---|---|
Square | Square | Octagram |