SysMath, eBooki, matematyka
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Foundations
of
Number, Structure, & Pattern
Preface
(Edited 1999 July 3)
The theme that unifies this work is one of looking for foundations, of number, structure, and
pattern (in space and in time—process and rhythm). I build on foundations set by past sear-
chers for pattern—meditators (zero and infinity), Arabs (zero in base ten), Greeks (Platonic
solids), Buckminster Fuller (geodesics, synergetics), John Bennett and those from whom he
learned (systematics), Stafford Beer (his varied concepts and Ashby’s requisite variety).
These words preface a work that’s been in process for well over a year. An inspiration
triggered this work. It planted a seed that seems to grow, to bear new fruit, with
(at-)tending. The process of producing this work is teaching me, as it comes into clearer
focus,
volumes
about the birth, growth, and dissemination of ideas. It has focused my
attention and given me a passion to
express
that is sorely needed to balance my retentive
and generally uncommunicative nature. The words of Gurdjieff and his followers
1
on “the
work” describe it as difficult and occasionally painful. Though I can see the truth of this, I
also have experienced the joy of discovery that can come from engaging in the process of
growth and transformation; that makes the effort
more
than worthwhile. In retrospect, in
terms of “the work” and its rewards, my inspiration may have been, in part, fruit from the
decades I’ve pondered a shape that Bucky Fuller dubbed “Vector Equilibrium” (or VE)
2
.
Acknowledgments
I’m indebted first of all to the source that brings inspiration. Also to Tony Blake for his
encouragement of and fodder for my mathematical explorations and for events and books he
has helped bring into being, to Bucky Fuller for his ideas and as a role-model for dedication
to the pursuit of meaning, and to Saul Kuchinsky for including some of my words and
pictures in publications of UniS
3
.
I couldn’t have come to write what’s here without the love, support, and ideas of my family,
my wife & best friend Bonnie, our soul-friends Stephen & Susan, & others.
I’m expanding and revising this document for 1998, trying to keep it moving towards
greater clarity and fuller expression.
G. I. Gurdjieff lived during the first half of the 20
th
century. He brought a wisdom to the West—through
dance and other physical, mental, and spiritual exercises; his personal way of being; his writings—that has
changed the lives of many individuals.
2
For R. Buckminster Fuller this term relates to a shape with six square and eight triangular sides, whose
mass is composed of eight tetrahedrons and six half-octahedrons (pyramids), and whose vertices are in the
spatial relation of the centers of closest-packed spheres around an equal-sized central sphere. Until recently,
I paid little heed to Bucky’s demonstration of the transformation between VE and icosahedron (Platonic
solid with twenty triangular sides). That transformation is a key part of the idea that inspired this work.
3
UniS Institute, P.O. Box 6615, Bridgewater, NJ 08807, USA
Foundations of Number, Structure
,
and Pattern
Page 1
Copyright © 1998, Sigurd L. Andersen, Jr.
1
N-Grams
(Pattern in Number)
&
Dimensionalities
(Structure in Geometric Form)
to
Foundations of
Number,
Structure,
& Pattern
One reason for the change is to make a title that could be the seed for a
Mind Map
, as Tony
Buzan (author of many books) uses the term. There’s a central word (Foundations)
connected to a few key organizing ideas/concepts (Number, Structure, & Pattern).
In
Number
, I discuss zero, a new approach to base 3 (and other odd-number bases, using
zero as a central value), and what I call
N-Grams
. I originally wrote the
N-Grams
material
(to which I may add the patterns formed in zero-centered odd bases) because I knew that I
could from my understanding of the numeric basis for the enneagram, and thought others
might come to understand those concepts better through having their structure drawn …
and “spelled out.” The “spelling out” has given me new insights, too. The
N-Grams
material
may also be the basis for some new writing in
Pattern
. In
Structure
is the material from
Dimensionalities
, revised. In
Pattern
, I want to expand on “informing” and “reflecting”
patterns. I may also attempt to say something about rhythm as the expression of pattern
through time. The various coordinate systems discussed in
Dimensionalities
may have
relationships similar to musical notes, and/or which exhibit fractal properties.
There are a few key ideas to me that seem fresh, and worth exploring—ways of considering
the topic-at-hand other than the usual.
First is the concept of Vector Measure, particularly as a way of bringing-to-light zero-sum-
ming coordinate systems (2M
1
, 3M
2
, 4M
3
). I think my mathematical intuition is yelling in
my deaf ear that this is important, but I can’t discount that it may be just an empty ringing.
I have yet to explore how angular/polar coordinates might fit with Vector Measures.
Another concept identifies two families of (3D, or M
3
) Vector Measures, and how the trans-
ition from one family to the other may tie to the 4-5 enneagram transition. I envision this as
a step from the mineral world to that of life, of spirit, of potential for transformation.
A third concept is one spurred to paper in response to interest in a “triadic computer.”
Foundations of Number, Structure
,
and Pattern
Page 2
Copyright © 1998, Sigurd L. Andersen, Jr.
Number: Zero
Introduction
My mathematical musings tend to include zero in unusual ways.
The way I approach base 3 has zero as the central value.
In what I call N-Grams the top point is special. It tends to fade into the background, yet ties
the patterns of all N-Grams together through being part of each. Buckminster Fuller also
offers a brief, intriguing idea about this point.
One of the features of some of the Vector Measures I study is that the coordinates add to
zero. I have recently learned that these may be what are called Barycentric coordinates.
Zero appeals to my mystical bent as a numeric analogue to the void,
sunyata
, nothingness,
the sleep of Brahma. This is one aspect of supreme reality, which also is behind all the being
we experience, but in the background, out of focus, not directly visible.
Beyond Zero
The philosophical underpinnings of my exploration are not far from those of Plato and other
Greeks, nor from John Bennett’s Systematics. The common underlying belief, as I would
express it, is that there are certain fundamental patterns from and into which complexity is
likely to be formed. From this I surmise that the observable world will often be found to
echo these patterns in its structure and process. What I write here is of my attempts to
explore portions of the vast domain of mathematics, looking at/for/into fundamentals.
Foundations of Number, Structure
,
and Pattern
Page 3
Copyright © 1998, Sigurd L. Andersen, Jr.
Number: Triadic Base 3
Introduction
Numbers as we know them are a sort of fractal, in the sense that as we change scale, the
patterns formed remain similar, if not identical, to those observed at other scales. Two
added to three is five, singly or by the millions. A million divided by 7 is 142,857—with one
left over to begin the fraction formed from the same pattern of digits
4
, repeated for as far as
one wants to extend the calculation.
This section contains ovelapping/redundant writings gathered together.
This section describes a measuring stick whose center is zero. The centrality of zero is
possible only when using an odd-numbered base. In bases 5, 7, 9, etc. the sides “weigh
more” than the center. In base 5, for instance, there are two negative values, two positive
values, and zero by itself at the center. With base 3, the “mass” of the center is equal to the
“mass” of its left and right (- and +) sides. Each of the three parts (-, 0, and +) holds an
equal fraction of the whole.
Bases other than ten
We are accustomed to using and thinking with base ten, though other numeric bases are part
of our everyday world as well. We have a dozen and a gross (base twelve), minutes and sec-
onds (base sixty, whether for time or angular measure). The computer’s binary (base two)
numbers and the derivative octal and hexadecimal have made the concept familiar to others.
In any base the number of distinct digits is equal to the base. Base two uses zero and one,
base ten uses zero through nine. In numbers with two or more digits, the digits to the left of
the initial digit indicate multiples of the base to some power. In base two, two is expressed
as 10—one times two (to the first power) plus zero times one. Four is 100, eight is 1000. In
any base
x
, the value
x
is expressed as 10—one times
x
plus zero times one. The number
100 represents the base to the second power, 1000 is the base to the third power. Digits to
the right of the units digit (separated by a “base point”) represent fractions.
Base Two:
Binary (base 2) numbers have only two digits, 0 and 1. Left from the “binary
point” are the one's, two's, four's, eight's, etc. positions. Going to the right of the “binary
point” are the half's, fourth's, eighth's, etc. The binary number 1101.101 represents 8 + 4 + 0
+ 1 + 1/2 + 0/4 + 1/8, or 13.625 (1*10 + 3 + 6/10 + 2/100 + 5/1000) in decimal form.
The fraction 1/7 is 0.142857
142857
…, where the underlining indicates a pattern repeating infinitely.
Foundations of Number, Structure
,
and Pattern
Page 4
Copyright © 1998, Sigurd L. Andersen, Jr.
4
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