
Elementary Geometry: Stumbling Blocks
Posted:
Jul 8, 2010 5:25 AM


I've been rereading Kiselev's two volume work, designed for students just starting on Euclidean geometry.
The translator, Alexander Givental, provides some interesting analysis at the end of volume 2 (Stereometry) trying to debunk some of the myths that swirl around this subject.
Why do we teach it again?
Is it really true that "formal reasoning" is the royal road?
Don't mathematicians also think heuristically, and don't axioms often come towards the end of the day, more as definitional and unifying than as "self evident truths"?
What is it about "rigor" that we're trying to get across?
Speaking of rigor, one thing I notice about the Euclidean mindset is the notion of congruence teaches us to overlook chirality or handedness.
In the opening pages, the notion is developed that geometric figures may be slid around (translated) and rotated, and that if, by these means, two such figures may be shown to superimpose, such that all features align, then these two figures must be congruent. So far so good.
However, immediately after the isosceles triangle is introduced, things get messy, as we're introduced to symmetrical figures that cannot be made to superimpose by translation or rotation. Anyone who has played Tetris will know what I'm talking about.
A lefthanded L is just not going to do the work of a righthanded L, no matter how ya turn it. Getting the "wrong or right L" can be a game changing experience. Yet in Kiselev, we're told that to pick a figure off the plane temporarily, and to set it down reversed, is a congruence preserving operation.
I can sense students starting to squirm at this point, as the logic hardly seems secure.
Consider the analogous situation in space. Having two lefthanded gloves or rightfooted shoes is a show stopper, in terms of having a proper pair.
The way to turn a left handed glove into a right handed one is to turn it inside out. This is like folding a tetrahedron back to a flat planenet, then creasing the folds the other way, such that what used to be a convexity is now a concavity and vice versa.
Should such a radical operation be considered "congruence preserving"? How can a left shoe and a right shoe be considered "the same" when clearly they ain't?
At least we should empathize with students who see some sleight of hand going on. We can explain that these are just the conventions, evolved over a long period. It's not like we can't imagine alternatives, such as nonEuclidean geometries that really care more about handedness.
On another list, I'm back to discussing Karl Menger's "geometry of lumps" again. He proposed that one way to branch away from Euclidean geometry is precisely in this matter of initial definitions.
Instead of saying a point has no parts, or that a line has no thickness, go ahead and have all your entities made from "clay" ("res extensa" in Cartesian terminology). Take a topological approach: planes are not points because they're flat and thin, not round or pointy. Lines are like sausages, or thinner, but are not infinitely thin (nor infinitely long).
So these objects are no longer distinguished by "dimension number" then. Could a consistent geometry be built in this manner? Menger proposes this challenge in:
'Modern Geometry and the Theory of Relativity', in Albert Einstein: PhilosopherScientist , The Library of Living Philosophers VII, edited by P. A. Schilpp, Evanston, Illinois, pp. 459474.
Here's a web site about Karl: http://www.iit.edu/csl/am/about/menger/about.shtml
The goal, in running by this alternative set of definitions (or axioms if you prefer) is to sensitize students to what is meant by "axioms" in the first place. Jiggering with Euclid's 5th postulate is not the only way to set a branch point, although one may do that too.
Zooming back from the sandy beach, locally flat, the place where Euclideans inscribe their proofs (using lines with thickness, traced with sticks in the sand, a string for a fixed length), we see that said planar surface is actually curved (globally speaking), as are the lines thereon.
Flatness is a local / parochial phenomenon, not something "to infinity". Yet Euclidean constructions are possible in such a setting (or call it "ancient Greece").
And really, what standard Euclidean constructions require "infinity" in the first place? Finite Universe models work just as well (cite Knuth), in terms of setting the stage for those various proofs about triangles and parallelograms etc. So again, we have our nonEuclidean alternatives.
Givental takes issue with tautological / dogmatic expositions of geometry as the worst way to approach the subject. Taking Euclideanism with a grain of salt, rather than with worshipful obedience, is probably a good way to inspire critical thinking and a better appreciation for what real mathematics is like. It's not about developing rigid / set beliefs about the "one way" it has to be.
Math is not a "my way or the highway" kind of discipline, much as some authoritarian types would like it to be.
Challenging the primacy or inevitability of Euclidean geometry is a useful exercise, not just for advanced students, but for those just starting out. Why? Because we don't want them to feel intimidated by some onerous "one truth".
This subject of handedness is pretty hot by the way, features in this debate about what counts as the most primitive spacefiller (a pretty basic question, accessible to laypersons). Hexahedra and pentahedra don't count because they're topologically too complicated. All the candidates are tetrahedra, but which one is the winner?
You get some Archimedeans pointing to the orthoscheme of the cube, otherwise known as "the characteristic tetrahedron" or 1/48th of a cube. This is their champion in the ring.
The problem with this selection is obvious: said characteristic tetrahedron is either left or right handed and will only fill space in complement with its mirror. Ergo, we're really talking about a *pair* of tetrahedra, not a single spacefiller.
Joining two such characteristic tetrahedra, to form another tetrahedron, is arguably to generate the true minimum spacefiller, a tetrahedron that fills space with identical copies of itself, no worries about handedness. This was D.M.Y. Sommerville's approach and his argument is used by Senechal to explain why some Aristotle fans have tended to bristle over the oft stated claim that Aristotle was wrong about tetrahedra filling space (see Math World under space filling): they most certainly do (he never said "regular").
Which Tetrahedra Fill Space? by Marjorie Senechal Mathematics Magazine, Vol. 54, No. 5 (Nov., 1981), pp. 227243.
If this debate resolves the way I think it should, then we have all the more reason for advancing our "minimum tetrahedron" or Mite in more of the literature.
Sommerville identified this shape in his research, as well as some of the shapes it makes in turn, additional spacefilling tetrahedra (e.g. the disphenoid tetrahedron and the monorectangular tetrahedron). It even assembles the characteristic tetrahedron. Indeed, every spacefilling tetrahedron on Guy Inchbald's useful chart may be assembled from our Minimum Tetrahedron (mite).
Useful chart: http://www.steelpillow.com/polyhedra/AHD/chart_lg.gif
(note handed characteristic tetrahedron at the top  the "tap root" of Archimedean thinking  with our champion, the Mite on the 2nd row, the three Sytes below that: Lite, Bite and Rite respectively. Four mites will make the characteristic tetrahedron so it needn't be top row if we wish to organize our thinking differently (no reason we shouldn't)).
These amateur geometry teachers discover the disphenoid tetrahedron (rite) for themselves in this oft cited Math Forum paper, but don't seem to realize it's composed of two Sommerville tetrahedra (or we could call them Aristotle's?).
An Amazing, Space Filling, Nonregular Tetrahedron by Joyce Frost and Peg Cagle http://mathforum.org/pcmi/hstp/resources/dodeca/paper.html
These threads (handedness and spacefilling tessellations) are linked, in that Euclidean geometry may desensitize students from appreciating handedness. Its notion of "congruence" obscures it. Having a poor grasp of handedness might lead students to think the socalled "characteristic tetrahedron" is capable of filling space without pairing up with its mirror image.
An interesting hypothesis will test whether our intro duction of nonEuclidean definitions helps students grasp the importance of handedness. We have other reasons for introducing Menger's geometry of lumps. For example, in a ray tracing context, it makes little sense to define anything "dimensionless" as any object needs to reflect light, or might as well not be there at all. This goes for points, lines, planes and socalled "solids".
For those of you not familiar with Kiselev's Geometry, these two volumes were at the heart of the Russian school system, starting in the Tsarist period and on through the opening decades of the USSR. For a time, they were considered THE geometry textbooks of that nation. They have only recently been translated into English (2008) by A. Givental, based at UC Berkeley.
Kirby Urner 4dsolutions.net

