Unit square

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 $I \times I$ , 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

Unsolved problem in mathematics:

Is there a point in the plane at a rational distance from all four corners of a unit square?

(more unsolved problems in mathematics)

It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.^[2]

References

^ 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