Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Non-Euclidean Arithmetic
Replies: 108   Last Post: Sep 13, 2012 3:39 PM

 Messages: [ Previous | Next ]
 kirby urner Posts: 3,690 Registered: 11/29/05
Re: Non-Euclidean Arithmetic
Posted: Sep 5, 2012 4:13 PM

On Wed, Sep 5, 2012 at 12:16 PM, Joe Niederberger
<niederberger@comcast.net> wrote:
> Kirby says:
>>What I have been showing is "repeated addition" will take us a long way, if we extend it to include what others call "scaling" -- we're repeatedly adding length.
>
> I see it a bit differently. We have pre-mathematical (perhaps proto-mathematical) intuitions about the world.
> Replication, and growing (or shrinking) are quite different from that viewpoint. We can connect both of these as abstracted concepts to what we learn about the formal symbolic manipulations we call multiplication (pick your numbers N - C) -- which have course evolved over a long period. Its perhaps a bit surprising that two seemingly dissimilar concepts should both map to mult.
> Think of babies: replicating babies, growing babies, rather different. Once we have found a connection, we can amuse ourselves by finding deeper connections, such as the fact that growing babies involves replication of cells, which is something that people long ago didn't know for sure. Perhaps all growing involves replication in this physical world.
>
> Joe N

Yes, it is easy to connect these two (scale and replication),
especially in volume, where when we talk about "a growing supply of
grain" we mean more and more grain particles.

Cell growth: the body gets bigger as there are more and more of them.
Exponential growth important. Relation between linear, areal, and
volumetric growth as powers 1,2,3 also important.

Studying this on a growing / shrinking sphere, variously divided
(surface-wise, volume-wise) is a focus (as Polyhedra are), in a
STEM-worthy K-12 math/geometry/geodyssey. Calculus gets a foothold
here, in terms of the differences, the rates of change, in these
growth numbers.

The sequence/pattern 1, 12, 42, 92... is one I keep harping on (a 2nd
power function): a nuclear ball surrounded by 12, in the form of six
in a plane, 3 on top, 3 below, such that it could be viewed as four
intersecting hexagons (parallel to the 4 planes of the regular
tetrahedron). A next layer of balls has 42, then 92 and so on.

http://wikieducator.org/File:Cuboctahedralpacking.jpg (from Linus
Pauling's personal collection, Oregon State University)

density and is a basic "holodeck" for chemistry, where hexagons are
more common than squares, 60 and 120 degree angles more common than 90
degree (although this sphere packing matrix has 90 degrees also).

In STEM (which includes architecture-engineering) we spiral back into
the above packing matrix in the form of the octet truss, a commonly
used space frame originally pioneered by Alexander Graham Bell (who
didn't patent it) and Bucky Fuller (who did). This was before
were getting there, through Barlow etc.

http://www.barlowgenealogy.com/england/famous/williamcrystal.htm

The whole number particulate nature of multiplication ("be fruitful
and replicate") goes to fractions Q because we can easily see
fractions as "wholes" in their own right.

I go back to the English preposition "of" as when we say "half of this
dozen eggs belongs to you, the other half to me". The "of" gets to be
"multiply".

In STEM, we want to deal with big numbers, like Avogadro's, like the
giant primes needed for RSA. We also have extended precision to play
with. These are discrete math tools.

A bit of a change in subject. Just agreeing that replication and
scaling are connectable, even intuitively, especially through volume.

Kirby

Date Subject Author
9/1/12 Jonathan J. Crabtree
9/1/12 Paul A. Tanner III
9/2/12 kirby urner
9/3/12 Paul A. Tanner III
9/3/12 kirby urner
9/3/12 Paul A. Tanner III
9/4/12 kirby urner
9/4/12 Paul A. Tanner III
9/4/12 kirby urner
9/5/12 Paul A. Tanner III
9/5/12 Robert Hansen
9/6/12 kirby urner
9/1/12 kirby urner
9/1/12 Joe Niederberger
9/1/12 Wayne Bishop
9/1/12 Joe Niederberger
9/2/12 Robert Hansen
9/3/12 Paul A. Tanner III
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Paul A. Tanner III
9/4/12 Joe Niederberger
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 kirby urner
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Joe Niederberger
9/6/12 Joe Niederberger
9/8/12 Robert Hansen
9/7/12 Jonathan J. Crabtree
9/8/12 kirby urner
9/8/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Joe Niederberger
9/8/12 Jonathan J. Crabtree
9/9/12 kirby urner
9/8/12 Clyde Greeno @ MALEI
9/8/12 Jonathan J. Crabtree
9/8/12 Jonathan J. Crabtree
9/8/12 Joe Niederberger
9/8/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/10/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/9/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Wayne Bishop
9/9/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/10/12 Paul A. Tanner III
9/10/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/13/12 Paul A. Tanner III
9/13/12 kirby urner
9/13/12 Paul A. Tanner III
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 kirby urner
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 israeliteknight
9/11/12 Joe Niederberger
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/11/12 Jonathan J. Crabtree