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The
Fractal Theory of Counting and Its Implications
An examination of fractals in the context of number theory http://www.sinc.sunysb.edu/Stu/nkilb/fractals/IChing.html
This paper will demonstrate a
method for or organizing numbers that relies on a simple recursive
algorithm. The process outlined herein is based upon Shao Yung’s
(approx. 1000 AD) (Patrick Grim, Lecture #8 Notes, State
University of New York at Stony Brook: PHI.365, Fall 1998)
ordering of the hexagrams of the I Ching (an ancient
Chinese fortune-telling method). By demonstration of this method,
it will become clear to the reader that number systems may be
viewed as fractals. A fractal is an image or set that has been
created by applying a simple, recursive rule to a mathematical
set. The image/set will be self-similar on ascending or descending
scales. This may explain why many mathematical and logical
equations yield fractal images such as the Sierpinski Triangle
when modeled (However, this matter will not be regarded here, as
its complexity merits a study of its own). My intent is to
illustrate the technique used to generate ordered sets of numbers,
and give examples of a few of its computational abilities.
First, let’s look at the idea of bases in number systems.
A "base" is the number of digits that are contained in
any number system. For example: A "base-3" number system
contains 3 digits (0, 1, 2). So, the first few numbers in base-3
would be 0, 1, 2, 10 (worth "3" in out number system),
11 (worth "4"), etc. We use a base-10 number system,
since there are 10 elementary digits in it (0 through 9).
There
are a few systems with other bases that have been commonly used in
history. One of these is the Phoenician number system (a base-6
number system). Its influence can still be found in degrees (units
of angle measure), as well as telling time (that’s why there are
60 minutes in an hour, for example).
A Brief description of Shao Yung’s method: The
hexagrams of the I Ching are composed of a series of six
lines (fig. 1b), which may either be broken or unbroken (fig. 1a).
In order to arrange the hexagrams, Shao Yung generates them
one line at a time. First, we write two lines.
2b:step
2,part 1
- -
- -
-
- Next, we duplicate each figure in the previous step,
and
we add another layer on top, alternating between broken and
unbroken.
We repeat that process until we have generated 64
hexagrams.
Now,
assume that the broken lines represent 0, and the unbroken lines
represent
The four binary numbers shown in step 2, part 2 are
equivalent to 0, 1, 2,
Therefore, by taking a mathematical set (any real number
system with a base of "n"), and applying the simple
recursive rule, it cranks out a Yung-Kilb set for a given number
of digits of that system, depending on how many generations you
carry out. In addition, each generation is similar to both the
previous generation and the first generation (since it is composed
of those two, and since it is an ordered series of numbers).
Compare this description to the definition of a fractal given in
the introduction, and you will see that it appears counting, in
any base, may be reduced to a fractal.
One possible application of this theory is for its use in
base-conversion. Leibniz, when constructing his binary series,
spent considerable time converting numbers from decimal (base-10)
notation to binary (Redshaw). However, using my method, it’s
simply a matter of following columns. Tables 3a-3c show how to
convert between binary and decimal notation. First, we construct a
Yung-Kilb series for decimal notation (Table 3a; Note only the
first generation of the decimal system is used, since the second
generation of base-10 yields a Yung-Kilb set consisting of 100
places!). Then we generate a Yung-Kilb series of approximately the
same size for binary notation (Table 3b). Finally, we combine the
two series’ into a single table (Table 3c). Each column within
the table contains numbers of equivalent values. This process, of
course, may be used to convert from any base to any other base.
An
investigation into fractals in nature, science, and mathematics in
relation to the Fractal Theory of Counting could also yield many
discoveries. Therefore, this introduction of the theory is just
the beginning…in time, who knows how far-reaching its
implications will be? bibliography Brethren
of Lemurian and Dolphin Logic. "Leibniz’ Life
(1698-1710)." November 1996. http://www.threeweb.ad.jp/~infoindo/leibniz/dl1698.htm
(October 25, 1998) |
