Sunday, November 14, 2010
Intelligible Arithmetical Geometry
Intelligible Arithmetical Geometry
Some of the most simple ideas of yesterday led to much deeper notions- it is as if learned men around a table drawing hieroglyphics to each other with great big words all grew silent because physics grew so simple even a child could see it and speak it.
Facebook status today: L. Edgar Otto Nature is not in the business of making vacuums. On both sides of some disconnect, real or virtual we observe a change for love and losses as life expands creatively filling the voids.
As I have pointed out before, very young and with nothing else to do at the time I spent an evening late into the night counting- making it to ten thousand I simply counted by thousands, millions and so on with a better idea of large numbers.
Watching the meters on the gasoline pumps I had the ability, before things were changed in grade school with counting by adding place holding zeros and trying to compare apples and oranges take away this or that- I learned to count more closely with my fingers and could no longer add a column of the phone book of the time to which adults had me do then for a half hour add by pencil and paper disputing the sum between them until they found out I was right.
I mention this because how we teach to children counting should be looked at more closely, and in at least these two ways- ways that also command the attention of how we see space. Of course the description can be in pictures or formula and someone proficient in one form or another- even a computer in computation lost in the vision of its stream of numbers but lost more than humans in self awareness in sorting patterns. I know there are formulas for all this number theory- some of which are algebraic obviously. But the discovery as a child I could count the number of stars on the flag by multiplying the x and y.
more later neighbor needs computer:
back. Now in Pascals triangle we have a description of the parts of simplexes (things based on triangles, tetrahedra, five-cells in four space... and so on) if we are to picture the numbers as space structures and the sub-cells that make them. A triangle has three points three edges and one area (and one null) to equal 2 to the nth when we sum them from the expansion of (1+1) to the nth. One can also find the stacking of cannon balls in a given dimension.
If we espand (2 + 1) to the nth we get all the square shapes, square, cube, hypercube in four space... and so on and the sub-cells that compose them, provided we also multiply by the numbers in the 2 to the nth triangle honeycomb.
A level in such Pascal like triangles is a power of things... the sum of one of those lines say 1 cube 6 faces 12 edges 8 points would be (2+1)cubed = 27.
This much is commonly known as a way to see Pascals triangle. But what I have realized is that this also applies in some form to a honeycomb of such triangles or squares stacked in a dimension.
In the simplex or triangle case the numbers from one dimension to another in two space sum to the difference of consecutive cubes, 1,7,19.37,61... and so on
(I had drawings of this but the other computer lost data with its new trojan)
Take the case of a square divided into four quadrants. We have four squares, 12 edges, and 9 points. Any of these can add to consecutive odd number cubes between the dimensions- in that case it would be five cubed.
If we disconnect the 4 sub-squares we also get an intelligible arithmetic across the dimensions. In a sense we are dealing with primitive ideas of what is the nature of continuity and consecutive count and the contiguity of abstract or real entities- for we are talking at a very simple level about what in defining ultimate continuity as Newton did, the notions of solid, liquid and gas and asking as in the idea of two lead atoms smashed in the LHC what is the nature of the quark-gluon plasma. From this arithmetical and geometric analogy it is clear there there be a wider conception of the metaphor of glueon- that is, crazyglue, muscillague and so on... of which our stance toward how we see the possibilities define density in physics and the idea of transfinite s in number theory.
Of course these primitive methods to enable better counting and pictures will also show better the next level of how to picture algebra (which is) as geometry. The laws of factoring taking less counting on our fingers piece meal. We can also image a wider space of the rim-flange, iota point-string, to picture even hyperbolic space.