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Emmental squares are tasty

Published online by Cambridge University Press:  08 February 2018

Tony Crilly ,
Stan Dolan  and
Colin R. Fletcher
Tony Crilly
Affiliation:
10 Lemsford Road, St Albans AL1 3PB e-mail: t.crilly@btinternet.com
Stan Dolan
Affiliation:
126A Harpenden Road, St Albans AL3 6BZ e-mail: stan@standolan.co.uk
Colin R. Fletcher
Affiliation:
Atalaya, Lon Glanfred, Llandre, Aberystwyth SY24 5BY e-mail: Atalaya1@btinternet.com
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Extract

A shape is called equable if its area and perimeter are numerically equal relative to some given system of units. For example, if a square is equable, then its side, a, must satisfy 4a = a2. So there is only one equable square, and it has side 4.

It is easy to investigate this idea for other shapes. Though not connected with this problem, the work of Imre Lakatos suggested a generalisation to us. Lakatos showed that Euler's classical formula V + F = E − 2 for polyhedra could be extended when the notion of tunnels was introduced [1].

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 553 , March 2018 , pp. 13 - 22
Copyright
Copyright © Mathematical Association 2018 

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References

1. Lakatos, I., Proofs and refutations: the logic of mathematical discovery, Cambridge University Press (1976).CrossRefGoogle Scholar
2. Conway, J. H., ‘Mrs. Perkins’ Quilt’, Mathematical Proceedings of the Cambridge Philosophical Society, 60 (1964), pp. 363–368.CrossRefGoogle Scholar
3. Tutte, W. T., Squaring the square, Scientific American 199 (5) (November 1958) pp. 136–142.Google Scholar
4. Erdős, P., Graham, R. L., On packing squares with equal squares. Journal of Combinatorial Theory, Series A, 19 (1975), pp. 119–123.CrossRefGoogle Scholar
5. Dolan, S., Thoughts on a conjecture of Erdős, Math. Gaz. 101 (November, 2017) pp. 449–457.CrossRefGoogle Scholar