Tuesday, November 29, 2011

A Mathematical Diversion: Magic Discs

Geometry, as we've seen in previous blogs, comes in many different forms. Two that I already explored have been plane geometry (to do with lines and planes in the context of linear algebra), and non -Euclidean geometry, in the context of Einstein's General Theory of Relativity. In this blog I take a look at projective geometry- a whole vast sub-discipline of math- in the context of "magic discs", which are simple representations of it.

Magic discs are useful because they keep consideration to a finite number of points. One then explores projective "n-spaces" - denote them as P^n(F_q) over some field F. When this is done, one finds that P^n(F_q) has exactly: 1 + q + q^2 + q^3 + .......q^n = (q^(n+1) -1)/(q -1) different points. Magic discs enter because they enable some very elegant constructions. When I first taught these to advanced 2nd formers, during a Peace Corps math teaching stint- I encouraged them to visualize the magic disc by making cardboard cutouts. The cutouts were done for different diameters, which were then numbered with evenly spaced marks around the circumference. Having done this, the students used pins to attach them to firm backboards, and the circles could then be rotated.

The method for enumerating a given cardboard disc was always the same: i.e. mark 1 + q + q^2 equally spaced points around the circumference matching marks on the cardboard circles to the backboard. Then label them in an anti-clockwise direction by the numbers: 0, 1, 2 ...q(1 + q) . Remember at all times that 'q' is a power of a prime. Say, for example, that q = 2, then one will use a clock face that is marked off starting from '0' (on the immediate right of the circle). The total number of equally spaced points to be marked off is computed as:

1 + 2 + 2^2 = 1 + 2 + 4 = 7

Since the first one is always marked at the '0' point, then the others will be: 1, 2, 3, 4, 5 and 6. The next job is to partition this circular field into (1 + q) points so that for q it will be 3 points. One finds that, apart from the 0 point, the other positions will always be such that for any selection of two marked points there is one position of the disc that "works", that is the selected distances end up in points that are coincident with two special points on the disc. The spacings for q = 2 will then be obtained from: 1 + 2 + 4, or more simply 1,2, 4. In other words, starting at the zero point, mark one space over to reach the number 1 on the background, then mark 2 more to reach the number 3 on the background, then mark 4 more to reach the number 2 + 4 = 0 where we began. In many ways, this procedure is similar to what we saw in the earlier blog (last year) to do with groups and "clock face" arithmetic. See, e.g.


In Fig. 1, is shown the magic disc resolution for the case of q = 3. And this leads to q(1 + q) marked off numbers in toto, or 3(1 + 3) = 3 x 4 = 12. And we confirm that the numbers go from 0 - 12 on the clock face. (Or in terms of the physical model, the numbers appearing on the backboard). The number of special points spanning the circle is similarly: 1 + q = 1 + 3 = 4 in all. The trick is then to identify them. The partition that works is by successive spacings of: 1, 2, 6 and 4 in succession, i.e. 1 added to 0, then 2 added to 1 (3), then 6 added to 3 (9) and finally 4 added to that ...bringing us back to 0.

Lastly, in Fig. 2 we have a much larger disc for the case of q = 5. Here, the total numbers marked off will be: q (1 + q) = 5 (1 + 5) = 30, in all. The number of special points to partition the circle will be 1 + q = 1 + 5 = 6, which will yield six partitioned spaces. These will be obtained from: 1, 2, 7, 4, 12 and 5. In other words, 1 added to 0, then 2 added to that (3), then 7 added to that (10), then 4 added to that (14), then 12 added to that (26) and finally 7 added to that ....which takes us back to 0.

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