Sys--Talk » Math

N-Grams

sigurd — Fri, 10 Apr 2009 02:35:20 +0000

You can find some of what I wrote about a decade ago on N-Grams at www.solbakkn.com/math/n-grams.htm . Click on the links for tables to see what N-Grams look like. In my last post I described them as “a collection of diagrams formed using the same mathematical / diagrammatic rules / conventions as the Enneagram, but applied to varying number bases and divisors.” Here are the N-Grams for base ten. Note that the circles for 2, 4, 5, 8 and ten – all with only the prime factors (2 and 5) that make up the base, ten. The circle for 6 is partially gray, because the prime factor 3 generates a repeating pattern, whereas the prime factor 2 does not.

Base Ten N-Grams

I’ve corresponded with Shane Roberts who has taken the same idea in a similar direction, calling his diagrams Rotagrams. Shane has a web page at www.myspace.com/systemlover and said I could share his email, systemlover at hotmail dot com. His diagrams include non-repeating patterns, e.g., .125 for 1/8 (base ten), mine do not.

Multi-Dimensions & Platonic Solids

sigurd — Fri, 10 Apr 2009 02:11:21 +0000

Late last century I wrote about three mathematical ideas that hold particular interest for me. (See yesterday’s Math posting for a web link.) All have to do with the basics of number and structure. I went into some detail about ideas,
* the “balanced base 3″ about which I wrote yesterday,
* and “N-Grams”, a collection of diagrams formed using the same mathematical/diagrammatic rules/conventions as the Enneagram, but applied to varying number bases and divisors. I’ll say more in another post.

This post is about alternate coordinate systems, in addition to the familiar Cartesian system, where each axis is at right angles to every other axis. The terminology I use is xMy where x is the number of axes and y the number of dimensions being measured. Cartesian coordinates are 1M1, 2M2, and 3M3 in 1, 2, or 3 dimensions. There is a 2M1 system, a 3M2 system and a 4M3 system in which all the coordinates add to zero for any point in space. 2M1 is simply a standard Cartesian single dimension paired with its mirror opposite. 3M2 has 3 axes oriented on a plane from the center of a triangle towards its three corners (or towards the centers of its three sides). 4M3 has 4 axes oriented in 3D space from the center of a tetrahedron to its four vertices (or towards the centers of its four faces). Here is a graphic representing the 3M2 system.

3M2 Coordinate System

Coordinates are given in the diagram for points p, q, and y. Point x is precisely between p and q – hence, its coordinates can be determined by averaging the coordinates of p (0,3,-3) and q (2,-3,1). So x is at (1,0,-1).

There are three different versions of 6M3, which I identify using a subscript to the 6, and which I will write here as 6(0)M3, 6(1)M3 and 6(2)M3. The first of these, 6(0)M3 is a doubling of 3M3, each axis pairing with its mirror image. 6(1)M3 has axes oriented from the center of a cuboctahedron (Bucky Fuller calls it a Vector Equilibrium) to its vertices. The 3M3, 4M3, 6(0)M3 and 6(1)M3 can all be easily lined up with the tetrahedron, octahedron, and cube. There are more complex systems, 6(2)M3, 10M3 and 15M3 that line up with the icosahedron and dodecahedron.

Kirby Urner has a different take on what I call 4M3. He named his form quadrays. More info on this can be found at www.grunch.net/synergetics/quadrays.html.

Balanced Base 3

sigurd — Wed, 08 Apr 2009 23:48:32 +0000

The usual way mathematicians discuss number bases 2 and larger (call base “b”) has them with digits that range in value from zero to b-1. Base ten has values 0 to 9; base two has values 0 to 1. Some years ago I hit on the idea that for base 3, using digits with values -1, 0 and +1 (rather than 0, 1, and2) more closely corresponds in some ways to the working of the world. Values are “on target” or to one side or the other. For numbers expressed in this system, there is no difference between rounding and truncating a number.

A few years ago I ran into Abhijit Bhattacharjee of India via the web. He has a site at www.abhijit.info/tristate/tristate.html with much more information on this idea. In his bibliography at http://www.abhijit.info/tristate/biblio.htm Abhijit says of Donald Knuth:

In 1981, in his book “The Art of Computer Programming”, Vol 2: Seminumerical Algorithms. Second Edition. Reading Mass: Addison-Wesley, pp 190-193 calls it the “prettiest number system of all”…

I now realize this is where I first encountered the idea, as this book was text for a programming course I took in the 1970’s.

Here’s are diagrams representing the “standard” and balanced versions of base 3. Each step lower in a diagram represents adding one more digit of precision to a number. In the upper diagram, all numbers that begin with “0.1…” are equal to or larger than 0.1. In the lower diagram, the value 0.1 is the precise middle of all numbers that begin with “0.1…”.

Diagram, two forms of Base 3

In the first form, the descending lines from a single point represent the digits 0, 1, or 2. In the second form, the descending lines represent the digits -1 (to the left), 0 (vertical), or 1 (to the right). To get a feel for how this works, here’s how one would count from -5 to +13 in balanced base 3, using “<” for a digit with value -1, and “>” for a digit with value +1: <>>, 0<<, 0<0, 0<>, 00<, 000, 00>, 0><, 0>0, 0>>, ><<, ><0, ><>, >0<, >00, >0>, >><, >>0, >>>. In these three-digit numbers the first is the 9’s place, the second is the 3’s place, and the third is the 1’s place. So the first number in this sequence, <>> represents (-9)+(+3)+(+1)=-5.

At www.solbakkn.com/math you can find more on this topic. It also includes information on N-Grams, and a pointer to information (in a PDF file) on Dimensionalities, two other math topic of interest to me. Eventually I plan to move and expand information about all three math topics on this site.