New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational

Lorna A. Willis

doi:doi:10.11648/j.ajam.20140203.11

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New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational

Lorna A. Willis

Published in American Journal of Applied Mathematics (Volume 2, Issue 3)

Received: 19 March 2014 Accepted: 8 May 2014 Published: 30 May 2014

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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.

Published in	American Journal of Applied Mathematics (Volume 2, Issue 3)
DOI	10.11648/j.ajam.20140203.11
Page(s)	74-78
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Tessellation, Secondary Median, Exact

References

[1]	Tapson, Frank, “Oxford Mathematics Study Dictionary”, p.46, Oxford University Press, 2006
[2]	Biderman, Arthur, “Lexicon Universal Encyclopedia”, pp.287, Lexicon Publications Inc. New York, NY, USA, 1989
[3]	King, Robert H, “The World Book Encyclopedia” Volume 4, P. 434, World Book Inc., 1982
[4]	Orfan, Lucy J, Vogeli, Bruce R, “Mathematics”, pp.262-263, 334-335, 342-343,Silver, Burdett and Ginn Inc., 1987
[5]	The definition of Pi. (2014, February 6). Retrieved from http://en.wikipedia.org/wiki/Pi
[6]	Understanding Pi and Squaring the Circle. (2014, February 6). Retrieved from http://www.docstoc.com/docs/117937307/Understanding-Pi-and-Squaring-the-Circle
[7]	History of Pi and Squaring the Circle. (2014, February 6). Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/Squaring_the_circle.html
[8]	Layne, C.E, Bostock, L, Shephard, A, Ali FW, Chandler, S, Smith, E, “STP Caribbean Mathematics for CXC Book 4” p.226, Nelson Thomas Ltd. UK, 2005
[9]	Goldberg, Nicholas, Cameron- Edwards, Nova, “Oxford Mathematics for the Caribbean”, p. 41, Oxford University Press, UK
[10]	Bremigan, Elizabeth G, Mathematics for Secondary School Teachers ISBN 978-0-88385-773-1, Mathematical Association of America (n.d).

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Lorna A. Willis. (2014). New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational. American Journal of Applied Mathematics, 2(3), 74-78. https://doi.org/10.11648/j.ajam.20140203.11

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Lorna A. Willis. New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational. Am. J. Appl. Math. 2014, 2(3), 74-78. doi: 10.11648/j.ajam.20140203.11

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AMA Style

Lorna A. Willis. New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational. Am J Appl Math. 2014;2(3):74-78. doi: 10.11648/j.ajam.20140203.11

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@article{10.11648/j.ajam.20140203.11,
  author = {Lorna A. Willis},
  title = {New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {3},
  pages = {74-78},
  doi = {10.11648/j.ajam.20140203.11},
  url = {https://doi.org/10.11648/j.ajam.20140203.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140203.11},
  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.},
 year = {2014}
}

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TY  - JOUR
T1  - New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational
AU  - Lorna A. Willis
Y1  - 2014/05/30
PY  - 2014
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JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 74
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PB  - Science Publishing Group
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AB  - 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.
VL  - 2
IS  - 3
ER  -

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Author Information

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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APA Style

Lorna A. Willis. (2014). New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational. American Journal of Applied Mathematics, 2(3), 74-78. https://doi.org/10.11648/j.ajam.20140203.11

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ACS Style

Lorna A. Willis. New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational. Am. J. Appl. Math. 2014, 2(3), 74-78. doi: 10.11648/j.ajam.20140203.11

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AMA Style

Lorna A. Willis. New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational. Am J Appl Math. 2014;2(3):74-78. doi: 10.11648/j.ajam.20140203.11

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@article{10.11648/j.ajam.20140203.11,
  author = {Lorna A. Willis},
  title = {New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational},
  journal = {American Journal of Applied Mathematics},
  volume = {2},
  number = {3},
  pages = {74-78},
  doi = {10.11648/j.ajam.20140203.11},
  url = {https://doi.org/10.11648/j.ajam.20140203.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20140203.11},
  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.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - New Parameter for Defining a Square: Exact Solution to Squaring the Circle; Proving π is Rational
AU  - Lorna A. Willis
Y1  - 2014/05/30
PY  - 2014
N1  - https://doi.org/10.11648/j.ajam.20140203.11
DO  - 10.11648/j.ajam.20140203.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 74
EP  - 78
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20140203.11
AB  - 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.
VL  - 2
IS  - 3
ER  -

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