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Discrete version of the Pythagorean theorem

Published online by Cambridge University Press:  08 February 2018

Matúš Harminc  and
Lucia Janičková
Matúš Harminc
Affiliation:
Institute of Mathematics, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic e-mail: [email protected]
Lucia Janičková
Affiliation:
Institute of Mathematics, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic e-mail: [email protected]
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Extract

The following observations are motivated by the facts that the area of a planar figure displayed on a screen can be expressed by a certain number of pixels; and if the figure is drawn by a plotter, then its area can be characterised by the total length of a line which fills it in.

The generalisations of the Pythagorean theorem are of three kinds. Firstly, the squares on the sides of the right triangle are substituted by other geometrically similar planar figures (Euclid's Elements Book VI, Proposition 31 [1]). Secondly, the assumption of the right angle is omitted (the law of cosines), or both of these generalizations occur simultaneously (Pappus’ area theorem [2], see also H. W. Eves [3]). Thirdly, mathematical spaces other than the plane are considered (for example, de Gua-Faulhaber theorem about trirectangular tetrahedra [3], further generalised by Tinseau [4], Euclidean n-spaces, Banach spaces [5], see also [6]).

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 553 , March 2018 , pp. 77 - 88
Copyright
Copyright © Mathematical Association 2018 

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References

1. Joyce, D. E., Euclid's Elements, Book VI. Clark University, Massachusetts (2002), available at http://aleph0.clarku.edu/∼djoyce/elements/bookVI/propVI31.html Google Scholar
2. Sefrin-Weiss, H., Pappus of Alexandria: Book 4 of the Collection, Springer Verlag (2010).Google Scholar
3. Eves, H. W., Great moments in mathematics (Before 1650), Mathematical Association of America (1983).Google Scholar
4. Weisstein, E. W., de Gua's Theorem. MathWorld-A Wolfram Web Resource, accessed August 2017 http://mathworld.wolfram.com/deGuasTheorem.html Google Scholar
5. Rynne, B. P., Youngson, M. A., Linear functional analysis, an introduction to metric spaces, Hilbert spaces, and Banach algebras, Springer Verlag (2008).Google Scholar
6. Alvarez, S. A., Note on an n-dimensional Pythagorean theorem, Boston College, Massachusetts accessed August 2017 http://www.cs.bc.edu/∼alvarez/NDPyt.pdf Google Scholar
7. Deza, E., Deza, M. M., Figurate Numbers, Word Scientific, Singapore (2012).Google Scholar