unit square
{{Short description|Square with side length one}}
{{distinguish|Square (unit)}}
In mathematics, a unit square is a square whose sides have length {{math|1}}. Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points {{math|(0, 0}}), {{math|(1, 0)}}, {{math|(0, 1)}}, and {{math|(1, 1)}}.{{citation
| last = Horn | first = Alastair N.
| editor-last1 = Crilly | editor-first1 = A. J.
| editor-last2 = Earnshow | editor-first2 = R. A.
| editor-last3 = Jones | editor-first3 = H.
| chapter = IFSs and the Interactive Design of Tiling Structures
| title = Fractals and Chaos
| date = 1991
| publisher = Springer-Verlag
| page = 136
| chapter-url = https://books.google.com/books?id=PZHfBwAAQBAJ&pg=PA136
| doi = 10.1007/978-1-4612-3034-2
| isbn = 978-1-4612-7770-5
}}
Cartesian coordinates
In a Cartesian coordinate system with coordinates {{math|(x, y)}}, a unit square is defined as a square consisting of the points where both {{mvar|x}} and {{mvar|y}} lie in a closed unit interval from {{math|0}} to {{math|1}}.
That is, a unit square is the Cartesian product {{math|I × I}}, where {{mvar|I}} denotes the closed unit interval.
Complex coordinates
The unit square can also be thought of as a subset of the complex plane, the topological space formed by the complex numbers.
In this view, the four corners of the unit square are at the four complex numbers {{math|0}}, {{math|1}}, {{mvar|i}}, and {{math|1 + i}}.
Rational distance problem
{{unsolved|mathematics|Is there a point in the plane at a rational distance from all four corners of a unit square?}}
It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.{{citation
| last = Guy | first = Richard K. |authorlink = Richard K. Guy
| title = Unsolved Problems in Number Theory
| series = Problem Books in Mathematics | volume = 1
| publisher = Springer-Verlag
| edition = 2nd
| year = 1991
| pages = 181–185
| doi = 10.1007/978-1-4899-3585-4
| isbn = 978-1-4899-3587-8 }}
See also
References
{{reflist}}
External links
- {{mathworld | urlname = UnitSquare | title = Unit square}}