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Topic: Multitudes, magnitudes, and multiplication
Replies: 67   Last Post: Mar 21, 2010 7:54 PM

 Messages: [ Previous | Next ]
 kirby urner Posts: 3,690 Registered: 11/29/05
Re: Multiplication in R arises from ADDITION
Posted: Mar 16, 2010 2:48 PM

On Mon, Mar 15, 2010 at 8:40 AM, Joe Niederberger
<niederberger@comcast.net> wrote:

<< SNIP >>

> So, the evidence as it stands, lends (strongly) towards the view that
> addition is basic, multiplication derived.  It says you can always
> correctly view it that way if you choose to.
>
> Joe N.
>
> p.s. Pam, this is a bit related to the little intuition you sent me in your
> last email.
>
>

Somehow I thought we were done with this question.

We don't only care about Real Numbers.

Sometimes, we don't care about Real Numbers at all.

Matrices multiply. Adding them may be pretty worthless.

Vectors add, but don't multiply (with each other), although quaternions do.

We preview these meanings of "multiply" with younger students,
not be drilling them in all kinds of arcane algorithms, by by giving
them a heads up as to the multiplicity of types in the "math objects
zoo".

Your nomenclature may vary, but the basic concepts are as
traditional and conservative as you need them to be.

At more advanced levels, it pays to show the family resemblances
between the different kinds of addition and multiplication.

At these levels a more abstract algebra approach enters in, and
we share about closure, inverse, neutral element (identity element),
associativity (matrices are associative, not commutative, w/r to
that binary operation we call "multiplication" w/r to them).

Using clever little logic games, encouraging free play and
discovery with finite groups is not out of reach. I'm not going to
make extravagant claims for their pedagogical value, but nor
am I going to sneer and jeer like some know-it-all pundit.

Back to my iconoclastic / radical agenda, here are pictures of
icosahedra held together by strong springs:

http://www.flickr.com/photos/17157315@N00/4436951027/in/photostream/
http://www.flickr.com/photos/17157315@N00/4436951019/in/photostream/

(flextegrity by Sam Lanahan, sculpture by Glenn Stockton)

Of course no classroom has such a thing, as we're looking at a
prototype fresh from the factory, woven to suit. With 4-frequency,
you first get a nucleus.

The edge and face centers are easy to apprehend. 4-frequency
means five members along an edge, plastic icosahedra in this
case. So there's a middle one that's easy for students to grasp.

Having something heavy and robust adds measurably to the quality
of the kinesthetics upon passing this around -- makes a lasting
impression.

This is not salesmanship as I can't offer any for sale. More like
(obviously) and consider open source in large degree i.e. I'm
not the gatekeeper (nor bottleneck). The truth is out there.

What might be a real world application for getting into polyhedral
numbers ala sphere packing? The tetrahedral numbers? The
triangular numbers?

Well, here's a write-up of a recent nanotechnology lecture we got
in Portland, Oregon recently, where I think the connections are
pretty clear (note Coxeter paper for further reading, if you want
to go deeper into this stuff):

http://worldgame.blogspot.com/2010/03/towards-nanoscience.html

Kirby

Date Subject Author
3/7/10 Bill Marsh
3/8/10 Joshua Fisher
3/8/10 Joe Niederberger
3/9/10 Bill Marsh
3/10/10 kirby urner
3/9/10 Bill Marsh
3/9/10 Paul A. Tanner III
3/10/10 Joe Niederberger
3/10/10 Joe Niederberger
3/10/10 kirby urner
3/11/10 Joe Niederberger
3/11/10 Joe Niederberger
3/11/10 kirby urner
3/11/10 Paul A. Tanner III
3/11/10 Joe Niederberger
3/11/10 Joshua Fisher
3/11/10 Michael Paul Goldenberg
3/11/10 Joe Niederberger
3/11/10 Paul A. Tanner III
3/12/10 Joe Niederberger
3/12/10 Paul A. Tanner III
3/12/10 Joe Niederberger
3/12/10 Joshua Fisher
3/12/10 Michael Paul Goldenberg
3/12/10 Joe Niederberger
3/12/10 Michael Paul Goldenberg
3/12/10 Pam
3/12/10 Michael Paul Goldenberg
3/13/10 Joe Niederberger
3/12/10 Joshua Fisher
3/12/10 Paul A. Tanner III
3/12/10 Joe Niederberger
3/13/10 Joe Niederberger
3/13/10 Michael Paul Goldenberg
3/12/10 Paul A. Tanner III
3/13/10 Paul A. Tanner III
3/14/10 Joe Niederberger
3/14/10 Joe Niederberger
3/15/10 Paul A. Tanner III
3/15/10 Joe Niederberger
3/15/10 Michael Paul Goldenberg
3/15/10 Joe Niederberger
3/15/10 Joe Niederberger
3/15/10 Joshua Fisher
3/15/10 Joe Niederberger
3/15/10 Michael Paul Goldenberg
3/15/10 Joshua Fisher
3/16/10 Joe Niederberger
3/16/10 Michael Paul Goldenberg
3/16/10 Jonathan Groves
3/16/10 Joe Niederberger
3/16/10 Joe Niederberger
3/16/10 Michael Paul Goldenberg
3/16/10 Joe Niederberger
3/16/10 Michael Paul Goldenberg
3/16/10 kirby urner
3/16/10 Paul A. Tanner III
3/16/10 Joe Niederberger
3/17/10 Joe Niederberger
3/18/10 Michael Paul Goldenberg
3/19/10 Paul A. Tanner III
3/16/10 Paul A. Tanner III
3/16/10 kirby urner
3/16/10 Joshua Fisher
3/16/10 Paul A. Tanner III
3/16/10 Joshua Fisher
3/17/10 Joshua Fisher
3/21/10 Bill Marsh