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Topic: Elementary Geometry: Stumbling Blocks
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kirby urner

Posts: 2,502
Registered: 11/29/05
Elementary Geometry: Stumbling Blocks
Posted: Jul 8, 2010 5:25 AM
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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

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

A left-handed L is just not going to do the work
of a right-handed 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 left-handed gloves or right-footed 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
plane-net, 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 non-Euclidean
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: Philosopher-Scientist ,
The Library of Living Philosophers VII, edited
by P. A. Schilpp, Evanston, Illinois, pp. 459-474.

Here's a web site about Karl:

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

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 non-Euclidean 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 space-filler
(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 ortho-scheme 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 space-filler.

Joining two such characteristic tetrahedra, to
form another tetrahedron, is arguably to generate
the true minimum space-filler, 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. 227-243.

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 space-filling tetrahedra (e.g. the
disphenoid tetrahedron and the mono-rectangular
tetrahedron). It even assembles the characteristic
tetrahedron. Indeed, every space-filling tetrahedron
on Guy Inchbald's useful chart may be assembled
from our Minimum Tetrahedron (mite).

Useful chart:

(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, Non-regular Tetrahedron
by Joyce Frost and Peg Cagle

These threads (handedness and space-filling tessellations)
are linked, in that Euclidean geometry may de-sensitize
students from appreciating handedness. Its notion of
"congruence" obscures it. Having a poor grasp of
handedness might lead students to think the so-called
"characteristic tetrahedron" is capable of filling space
without pairing up with its mirror image.

An interesting hypothesis will test whether our intro-
duction of non-Euclidean 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
so-called "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

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