My favorite mind-bending phenomenon to mention to future high school teachers is the so-called(adsbygoogle = window.adsbygoogle || []).push({}); Euler-Cramer paradox. The excellent and readable textbook by C. G. Gibson,Elementary Geometry of Curves, Cambridge University Press, 1998, does briefly discuss this (as do some more advanced textbooks such as Hartshorne), but not by name, and I know of no algebraic geometry textbooks which take the time to give what I feel would be appropriate emphasis to it.

This "paradox" is IMO so interesting that I'll take the trouble to try to explain it.

In the first pass I'll deliberately omit something--- see if you can spot it!

Stirling proved in 1717 that a set of [itex]n(n+3)/2[/itex] points in [itex]CP^2[/itex] (the complex projective plane) determine auniquedegree n curve, namely the unique degree n curve passing through all [itex]n(n+3)/2[/itex] points. (Stirling's theorem is not hard to "prove" by considering the equation of the curve.)

So a set of 9 points determines a unique cubic curve. In other words, loosely speaking, the information needed to specify a cubic curve is at most the information needed to specify 9 points.

McLaurin proved in 1720 that a degree m curve intersects a degree n curve in [itex]m \, n[/itex] points. This is known asBezout's theoremeven though Bezout's proof was neither first nor correct! (This theorem is not hard to "prove" either.)

So a pair of cubic curves intersects in 9 points. In other words, loosely speaking, the information needed to specify 9 points is less than the information needed to specify a cubic curve.

McLaurin noticed that these results contradict one another! The first says 9 points determine a unique cubic curve, while the second says a pair of distinct cubics determines nine points--- yet onlyonecubic is supposed to pass through these nine.

Similarly, 14 points determine a unique quartic curve, but a pair of distinct quartic curves intersects in 16 points. Even worse! Is it [itex]I \leq 14[/itex] or [itex]I > 16[/itex]?

Around 1750, Cramer and Euler independently noticed the same problem, and were apparently the first to suggest that the problem is cured by adding the phrase "pointsin general position" to the statement of Stirling's theorem and adding the phrase "agenericpair of distinct curves" to the statement of the McLaurin-Bezout theorem.

Here, "general position" is a famously slippery phrase, but the idea is that a "generic" set of points won't satisfy any algebraic conditions which might mess up the desired conclusion, so you should define "GP" to mean whatever you need it to mean in order to eliminate special cases!

Plucker proved in 1828 that if you delete any one point from a set of [itex]n \, (n+3)/2[/itex] points in GP, then an entire pencil of degree n curves passes through the remaining points, and any pair of these will intersect in [itex]n^2[/itex] points. In other words, the remaining points determine [itex](n-1)(n-2)/2[/itex] additional points, all lying on the original cubic curve, which form anon-genericset of [itex]n^2[/itex] "pinch points" which common to a pencil of degree n curves.

So: a pair of cubic curves intersect in 9 points, but these arenongeneric. Agenericset of 9 points determines auniquecubic, but omitting any one of these allows us to pass a one-parameter family (pencil) of cubics through the remaining 8. These then determine auniqueadditional point, forming anongenericset of 9 pinch points common to the entire pencil; that is, any pair of curves from the pencil intersect in precisely these 9.

Similarly, a pair of quartic curves intersect in 16 points, but these arenongeneric. Agenericset of 14 points determines auniquequartic, but omitting any one allows us to pass a one-parameter family (pencil) of quartics through the remaining 13 points. These 13 points determineuniquely3 additional points, forming anon-genericset of 16 pinch points common to the entire pencil; that is, any pair of curves from the pencil intersect in precisely these 16. These 16 pinch points are in factmultiply-special, in the sense that no subset of 14 is in GP!

(Actually, even these two statements might overlook some additional "genericity" requirements--- this stuff can get rather tricky!)

Turning things around, we can say that the [itex]n^2[/itex] points determined in the McLaurin-Bezout theorem satisfy a special relationship, which we might call the Cramer-Euler-McLaurin-Plucker (CEMP) property. Too bad Cramer was not I, eh? .oO

This kind of reasoning lead by the end of the nineteenth century to the development by Hilbert, Noether, and others, of what we now callhomological algebra, which provides powerful algebraic tools capable of expressing various senses in which a finite set of points can benon-generic.

Take (almost) any cubic curve, choose (almost) any nine points on this curve, and delete any one of these nine. The remaining eight determines a unique CEMP nonet lying on the original curve (but "almost certainly" not the same as the original nonet). So CEMP nonets abound, and similarly for higher degrees.

Suppose we have a CEMP [itex]n^2[/itex]-tuple. If we apply a projective transformation, the result will be a new CEMP [itex]n^2[/itex]-tuple. So this relation is a projective invariant.

According to John Baez (keep your eye out for forthcoming Weeks), the key to understanding Kleinian geometry is to consider q-ary relations which are invariant under some group action--- in this case, the projective group [itex]PGL(3,C)[/itex] in its natural action on [itex]CP^2[/itex]. I have often decried the effort expended these days on quantizing gravity, compared to that spent on explicating probability. I have suggested seeking an algebraic theory in which probability emerges from the notion of "generic" (think "probability one", and compare the notion of "spin networks").

Clearly, there's much more to arrangements of points in the plane than most people realize!

Maybe others here also have candidates for fun topics they'd like to see better known among high school math teachers?

(Note: I originally posted this at the end of another thread, but I have moved it to a new thread because I don't want it to be buried.)

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# The Cramer-Euler-McLaurin Paradox

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