Unit square: Difference between revisions
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{{distinguish|Square (unit)}} |
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[[Image:Unit Square.svg|thumb|300px|The unit square in the [[Euclidean geometry|real plane]]]] |
[[Image:Unit Square.svg|thumb|300px|The unit square in the [[Euclidean geometry|real plane]]]] |
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In [[mathematics]], a '''unit square''' is a [[square (geometry)|square]] whose sides have length {{math|1}}. Often, ''the'' unit square refers specifically to the square in the [[Cartesian coordinate system#Cartesian coordinates in two dimensions|Cartesian plane]] with corners at the four points {{math|(0, 0}}), {{math|(1, 0)}}, {{math|(0, 1)}}, and {{math|(1, 1)}}. |
In [[mathematics]], a '''unit square''' is a [[square (geometry)|square]] whose sides have length {{math|1}}. Often, ''the'' unit square refers specifically to the square in the [[Cartesian coordinate system#Cartesian coordinates in two dimensions|Cartesian plane]] with corners at the four points {{math|(0, 0}}), {{math|(1, 0)}}, {{math|(0, 1)}}, and {{math|(1, 1)}}.<ref>{{citation |
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| last = Horn | first = Alastair N. |
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| editor-last1 = Crilly | editor-first1 = A. J. |
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| editor-last2 = Earnshow | editor-first2 = R. A. |
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| editor-last3 = Jones | editor-first3 = H. |
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| chapter = IFSs and the Interactive Design of Tiling Structures |
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| title = Fractals and Chaos |
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| publisher = Springer-Verlag |
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| page = 136 |
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| url = https://books.google.com/books?id=PZHfBwAAQBAJ&pg=PA136 |
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| doi = 10.1007/978-1-4612-3034-2 |
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}}</ref> |
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==Cartesian coordinates== |
==Cartesian coordinates== |
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Revision as of 01:53, 13 October 2023
In mathematics, a unit square is a square whose sides have length 1. Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points (0, 0), (1, 0), (0, 1), and (1, 1).[1]
Cartesian coordinates
In a Cartesian coordinate system with coordinates (x, y), a unit square is defined as a square consisting of the points where both x and y lie in a closed unit interval from 0 to 1.
That is, a unit square is the Cartesian product I × I, where 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 0, 1, i, and 1 + i.
Rational distance problem
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.[2]
See also
References
- ^ Horn, Alastair N., "IFSs and the Interactive Design of Tiling Structures", in Crilly, A. J.; Earnshow, R. A.; Jones, H. (eds.), Fractals and Chaos, Springer-Verlag, p. 136, doi:10.1007/978-1-4612-3034-2
- ^ Guy, Richard K. (1991), Unsolved Problems in Number Theory, vol. 1 (2nd ed.), Springer-Verlag, pp. 181–185, doi:10.1007/978-1-4899-3585-4