Eine kleine Mathmusik : From Lattice Labyrinths via √-1 to Pythagorean Triples

Since starting this blog I have made a point of “following” twitter accounts maintained by passionate mathematics teachers and many have paid me the complement of following me in return. It is time I thanked them/you for this complement by doing a blog post that ACTUALLY HAS SOME MATHEMATICS IN IT, perhaps even mathematics that might inspire a school lesson or a club meeting. So here it is – a blog on how tessellations of tessellations led me to multiplying number pairs and to a simple way of generating the integral sides of right-angled triangles – the Pythagorean triples.

First, just a reminder of how multiplying two Lattice Labyrinth Tessellations to generate a tessellation of tessellations. Chinese (1,2) x Chinese (5,0) gives a tessellation that repeats with separation parameters (5,10), which contains 125 squares in each repeating part of the pattern, 125 being the multiple of the repeat units, 5 and 25, of its two constituent tessellations.

Now for the algebra of this geometry.

 

So, we have demonstrated how, in a particular case, the multiplication of two integer pairs (a,b) and (c,d)  to give a new integer pair (e,f) implies that the multiple of the sums of their squares, (a²+b²)(c²+d²) is also a sum of squares  (e²+ f²) . We’ve shown this for (1,2) x (3,1). Suppose we reverse the order of the first pair of numbers to (2,1), generating Chinese Lattice Labyrinth (2,1), the mirror image of Labyrinth (1,2). Below is what happens when we multiply (2,1) by (3,1). 

We have shown that (1,2) x (3,1) = (1,7) but that (2,1) x (3,1) = (5,5) and furthermore, as the sum of squares of (2,1) equals the sum of squares of (1,2), we have also found out that 1² + 7² = 5² + 5² = 50.

If instead of multiplying two different number pairs together, we multiply a pair by itself, i.e we SQUARE the number pair and then reversing the order of one pair and doing the multipliaction again, we arrive at a particularly fruitful result. For instance: (2,1) x (2,1) = (3,4)  and (2,1) x (1,2) = (0,5). We have generated the Pythagorean triplet 3,4,5 – the sides of the classic right-angled triangle. Likewise if we square (3,2), we find that (3,2) x (3,2) = (5,12) and (3,2) x (2,3) = (0,13) – we’ve found the triplet 5,12,13.

It looks as if we can generate ALL the Pythagorean triplets this way. (All? – Prove it!) Well done the square root of minus one – the Greeks would have loved you, but perhaps they would have returned gladly to geometrical representations.

Talking of Greeks, if you consider the general case of squaring the number pair (a,b), you will find you have produced Euclid’s formula  but an online search of articles suggests that this elegant way of generating the formula is not widely known.

P.S. It is gratifying to find that the whimsical phrase Eine kleine Mathmusik has been used once before as the title of a suite of music composed by Rachel W. Hall and performed at a none other than a Bridges Conference on Mathematics and the Arts.

Eine Kleine Mathmusik: Six Mathematical Compositions for Bridges Pécs 2010

latticelabyrinths@gmail.com

Here are some links to the inexpensive how-to-do-it  Lattice Labyrinths workbook which is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

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