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New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational
American Journal of Applied Mathematics
Volume 2, Issue 3, June 2014, Pages: 74-78
Received: Mar. 19, 2014; Accepted: May 8, 2014; Published: May 30, 2014
Author
Lorna A. Willis, Department of Physics, Faculty of Science and Technology, University of the West Indies, Mona Campus, Jamaica, West Indies
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Abstract
Historically, mathematicians sought for a unique relationship between a square and a circle of equal area without much success. The ratio of perimeter of a circle to its diameter is known and given as the symbol π. However, π was deemed IRRATIONAL. By using the concept of a TESSELLATION, that is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps, a square is described for the first time, as an equi-edge juxtaposition of eight identical right isosceles triangles. The usual median of a triangle is consistently identified in each of these triangles and is designated SECONDARY MEDIAN in relation to a square. There are eight Secondary Medians in a square. When the size of the Secondary Median of a square matches the size of the radius of a circle, and the two shapes are placed so that their centers are coincident, it is established that the areas of the two shapes are equal, thereby demonstrating the basis for the exact solution to the ancient geometric construction problem- SQUARING THE CIRCLE, with the consequences that; 1) π is, unambiguously a feature of the area of a square, 2) π is rational, has an exact value of 3.2, from any circle, a square of equal area is constructed in finite steps as well as the converse, 3) a square and an ellipse of equal area can be constructed, 4) π is not a feature limited to circles and associated shapes, as has been historically documented, but is a feature of Euclidian Geometry. Exact value of π means formulae featuring π are unchanged qualitatively, but changes slightly, quantitatively.
Keywords
Tessellation, Secondary Median, Exact
Lorna A. Willis, New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational, American Journal of Applied Mathematics. Vol. 2, No. 3, 2014, pp. 74-78. doi: 10.11648/j.ajam.20140203.11
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