Geometry, as we've seen in
previous posts, comes in many different forms. Two that I already explored have
been differential geometry and non -Euclidean geometry, in the context of Einstein's
General Theory of Relativity. In this post I take a look at **projective
geometry**- a whole vast sub-discipline of math- but 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.

http://brane-space.blogspot.com/2010/04/looking-at-groups.html

In Fig. 1, I show 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. We then 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.

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.

**See Also:**

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