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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]}

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.

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*.

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]}

**^**Horn, Alastair N. (1991), "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, ISBN 978-1-4612-7770-5**^**Guy, Richard K. (1991),*Unsolved Problems in Number Theory*, Problem Books in Mathematics, vol. 1 (2nd ed.), Springer-Verlag, pp. 181–185, doi:10.1007/978-1-4899-3585-4, ISBN 978-1-4899-3587-8

- Weisstein, Eric W. "Unit square".
*MathWorld*.