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Rational polygons

Published online by Cambridge University Press:  09 April 2009

T. K. Sheng
T. K. Sheng
Affiliation:
University of MalayaKuala Lumpur
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A polygon is said to be rational if all its sides and diagonals have rational lengths. I. J. Schoenberg has posed the interesting problem, “Can any polygon be approximated as closely as we like by a rational polygon?” Besicovitch [2] proved that right-angled triangles and parallelograms can be approximated in Schoenberg's sense, the proofs were improved by Daykin [5]. Mordell [7] proved that any quadrilateral can be approximated by a rational quadrilateral. By adapting Mordell's proof, Almering [1] generalised Mordell's result by showing that, if A, B, C are three distinct points with the distances AB, BC, CA all rational, then the set of points P for which PA, PB, PC are rational is everywhere dense in the plane that contains ABC. Daykin [4] extended the results of Besicovitch [3] and Mordell [7] by adding the requirement that the approximating quadrilaterals have rational area.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 4 , November 1966 , pp. 452 - 459
Copyright
Copyright © Australian Mathematical Society 1966

References

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