So shall we level with grade schoolers, and tell them=20
that real numbers have this spatial interpretation only=20
because we're hiding vectors until later in their careers?
(Many of them will never get to vectors, given the life=20
boat mentality that leads many to hop into the ocean=20
rather than hang out with the math nerds).
If I were into being a deconstructionist, I suppose I'd=20
lecture the students that the negative sign doesn't=20
(repeat, does NOT) "always mean the same thing". The=20
horrible truth about math (something we can only mutter,=20
and not say too loudly, lest we be purged), is that=20
symbols are context-dependent for their meaning. That's=20
obvious in the humanities, but sometimes people want to
make a secret of the fact that math is one of the
humanities -- a liberal art. OK, so I exaggerated about=20
being purged. It's OK to level with students and tell=20
them that this is so.
Remember those little negative signs in the upper left,=20
written all tiny-like? A cultural phenomenon unique=20
to elementary schools -- kinda like training wheels=20
I suppose. I have no problem with introducing and=20
then phasing out symbolic conventions in order to=20
"help make the transition" (whatever that is), but=20
I'm not sure elementary school teachers do enough to=20
explain that this is what they're doing. So kids go
to all the trouble to learn this way of writing "the
negatives" and then, suddenly, they're told this is
just for babies. Betrayal! Why teach us this stuff
if you're just going to take it away later? Some=20
kids never recover I bet -- faith shaken. Teachers=20
no longer seem so wise.
Anyway in geometry we have an obviously useful application
for negativity, because we need symmetry in our spatial=20
affairs. Being able to go right from the origin, but=20
not left, or up, but not down, will never do, is=20
completely at odds with ordinary experience. With=20
the economy of a negative sign, we immediately double=20
all the vectors. From 3 basis vectors we get an=20
additional 3: the 6 spokes of the Cartesian coordinate=20
system, and all is well with the world.
Here's a fun exercise:
Suppose you don't want any negative numbers in your
number line. OK, no problem, just add a tuple=20
instead:
<--*-------------!--------------*-->
(0,3) (0,0) (3,0)
The difference between the two directions is now
handled by placement. Different notation. You
can keep the negative sign as an operator though,
why not? IE:
(0,3) =3D (0,-3) =3D -(3,0)
But keeping a negative in the final expression is
sort of like writing 4/8 instead of 1/2. We like
to reduce to lowest terms ("normalize" is a rough
synonym). So in normal form, we have no negatives.
We think of the negative sign as just a unary=20
operator: reverses a vector by 180 degrees,=20
causing it to point the other way.
Can this way of doing things be generalized to
Euclidean space? Sure. And we don't need to end
up with 6 spokes (the "jack" gizmo) either. We
might instead choose to map a plane with 3 basis
vectors:
(0,0,1)
|
|
|
|
| (0,0,0)
/ \
/ \
/ \
/ \
(1,0,0) (0,1,0)
Sorry if that doesn't look so hot in ASCII -- it's your=20
basic Mercedes logo.
And in volume? You guessed it:=20
Q1=3D (1,0,0,0)
Q2=3D (0,1,0,0)
Q3=3D (0,0,1,0)
Q4=3D (0,0,0,1)
are four basis vectors from the origin to the four corners
of a regular tetrahedron.
With the above apparatus in place, we now have a unique
"lowest terms" (normalized) coordinate address for every
point in Euclidean space. We've got omnidirectional=20
sweepout around the origin (no asymmetry of the kind
addressed by adding the negatives of the 3 basis vectors
in XYZ), AND.... we have no negative numbers (although,
as per the above discussion, we keep the '-' symbol in
play as a vector reversal operator. So now we can show
students a game in which negative numbers are NOT required
to deal with spatial location. Something worth mentioning
at least.
Math is full of fun games like this. Define your rules,
put your operators on the table, and play. Any applications? =20
Actually, in this case, yes. Very simple, whole number=20
coordinates for a lot of common polyhedra is one of the=20
pay-offs.
All of this is developed in great detail, with color and
animated graphics, at my website. If curious, check out the=20
Quadray Papers at http://www.teleport.com/~pdx4d/quadrays.html
Or wait until the video clips come out. I'm working with=20
TV people to make all this stuff available in a very user
friendly format. In a few years (or less -- given this=20
stuff is already on the web), some of your students fresh=20
from K-12 will already know all about quadrays, thanks to=20
the work of your colleagues in the math ed biz.
Kirby
4D Solutions
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