[My ancient post (December 17, 2006) from a now defunct blog on Blogger: ]
In his comment on one of my previous posts, Euclid’s Elements, Interactive, Rick Booth brought my attention to GeoGebra, a free interactive geometry package. GeoGebra is nice, simple, intuitive to use, and could be a great help at school.
For some years, I was an occasional user of a commercial package, Cinderella. I decided to run a quick comparision of the two packages on a problem about the radical axis of two circles — I briefly mention the problem in my book. The problem is easy to solve with some help from Pythagoras:
Prove that the set of points P in the plane such that the tangents from P to two given non-intersecting circles are equal, is a straight line (it is called the radical axis of the two circles).
The problem is even easier when the two circles intersect – in that case, the radical axis is exactly the line through the two points of intersection. And in the case of tangent circles it is even easier — in that case the radical axis is the common tangent line.
It is interesting to compare the behaviour, in Cinderella and GeoGebra, of a simple interactive diagram: two interesecting circles of varying radii and the straight line determined by their points of intersection. In GeoGebra, when you vary the radii or move the centers of the circles and make the circles non-intersecting, the line through the points of intersection disappears — exactly as one should expect. In Cinderella, the line does not disappear, it moves following the movements of the circles, always separating them; when circles touch each other and start to intersect again, the line happens to be, again, the tangent line or the line through the points of intersection. And it is exactly the radical axis of the two circles!
I have not read documentation for GeoGebra, but the behaviour of this diagram suggests that, in GeoGebra (at least in the default mode), the underlying mathematical structure is the honest real Euclidean plane. In Cinderella, the underlying structure is the complex projective plane; what we see on the screen is just its tiny fragment, a real affine part. The radical axis of two non-intersecting circles is the real part of the complex line through two complex points of intersection; since the intersection points of two real circles are complex conjugate, the line is invariant under complex conjugation and therefore is real and shows up on the real Euclidean plane.
The Help Index of GeoGebra says about complex numbers:
GeoGebra does not support complex numbers directly, but you may use points to simulate operations with complex numbers.
However, even if one cannot do in GeoGebra the complex tricks of Cinderella, it is not a serious drawback — one should work hard to find a problem, of a secondary school level, where the difference in the nature of real and complex mathematical engines becomes visible. Such a problemis likely to involve circles or other conics, that is, quadratic equations — but remember, even the distance function is already involving a quadratic form, therefore complex numbers can pop up in rather unexpected places.
[I can add that another problem:
find the points of kiss of three mutually tangential cirles with centers at three given points
is linear, not quadratic, and can be solved by affine tools, say, with the help of a ruler with two parallel edges. ]
But still, when assessing software for mathematics teaching, it is useful to check the power, both computational and theoretical, of the mathematical engine. Unfortunately, most reviews of software for mathematics teaching never discuss this point.
To finish on a note more suitable for the festive season, I wish to mention that, in my book, radical axes appear in a solution of the following problem:
The Holes in the Cheese Problem. A big cubic piece of cheese has some spherical holes inside (like Swiss Emmental cheese, say). Prove that you can cut it into convex polytopes in such way that every polytope contains exactly one hole.
The previous discussion is the strongest hint for its solution.
And finally, a quote from Cheese.com:
“Emmental [...] is considered to be one of the most difficult cheeses to be produced because of its complicated hole-forming fermentation process.”
Is it realy surprising that compex numbers naturally appear in the cutting the wheel of cheese?