A K Dewdeney mentioned
discrete iteration in the
August 1985 issue of Scientific American,
as a way of demonstrating “Attractors”.
That column introduced the Mandlebrot Set to Sci Am readers.
What is discrete iteration? Given a number 'i':
(new) i = i^2 modulo n.
X modulo 'n' is the remainder when X is divided by n.
Let's consider the case when n is 11. We evaluate the recurrence for 11 values of 'i' from 0 to 10:
How the numbers 0-10* go for Mod 11
We observe the connections formed between the numbers:
Iteration Diagram for n=11 based on the table above.
Think of it like Primordal Soup, where every particle is literally attracted to one other particle, or to itself.
*Footnote: We just use 0 to n-1, since n mod n is 0. ... same as zero! I.e., Why bother including '11' — only to have it point to 0?
ATTRACTORS
What are "attractors"?
Attractors are single numbers or closed loops of numbers. Attractors are of the following kinds:
a looper (1-cycle)
an oscillator (2-cycle)
a larger cycle (polygons)
Non attractors are 'attract-ees'. Tree-like basins feed into attractors, but trees themselves are not attractors... they are the pathways to attractors!
All iteration diagrams consist of one or more subgraphs, each having an attractor. Attractors are like black holes for numbers.
Sci AM Diagram
A diagram for 100 was shown in 1985.
Mod 100 is easy - just take last 2 digits of the squared number.
The six components of the iteration diagram for squaring the first 100 integers. From p21, August 1985 Scientific American.
Note: The 4-cycle in the diagram on the left had the arrows going the wrong direction when originally published!
Note: There are six components (subgraphs) and there are six attractors. Each component has one attractor.
Examples from the Diagram:
Take 68 (lower right) - it goes to 24 and then to 76, an attractor.
Take 33 (middle right) - it goes to 89, then into the 21-41-81-61 cycle.
Take 55 (upper right) - it goes to 25, as do all the multiples of 5. (55 squared, reduced modulo 100 is 25) So, 25 is an attractor.
Any questions??
More Neato Examples
Diagram #65, from my birthday series...
65 in PNG image format
The attractors in the 65 are: 26, 40, 1, 16↔61, and 35↔55. Zero is attracted to itself. Zero is not always alone like this - see 100 above.
The silly question is, what if each of the numbers represents one of the 65 years of your Life, can you make any sense of the diagram, including the Attractors?
