chaos  complexity  order
Eight equations generating exotic behavior.
Algorithms, program code and graphical output
by Roger Luebeck © 2000, 2017

The mathematical intrigues on these pages don't rival
the sophistication of Mandelboxes and other 3D fractal
endeavors which are popular today. If you like the
things you see here, and have yet to discover the world
of fractals, just do a YouTube search on Mandelboxes.
The first two sections below are my creations. The rest
of the sections I merely added my investigations to,
especially the logistic map.
Sections 1, 2, 3 and 4 are by far the most interesting.
1. Balloting Florida 2000 Gore vs. Bush
Iteration of a balloting algorithm, using random
number generator. It grew out of my analysis of
the 2000 Gore  Bush Florida balloting. Surprising
patterns amidst chaos. Includes commentary on the
media, the Supreme Court and lawyers.
2. Simple trig
I played a hunch and came up with a pair of repeating
trig equations, with offsetting scalar multipliers
for each equation. Unbelievable patterns.
(See side panel and click "Simple trig" to see many more.)
3. Logistic map (LM)
X = r * X * (1  X)
Mathematician Paul Stein called the complexity of
this iterated equation "frightening".
Iterating this equation produces regions of distinct
values, involving period doubling, as well as regions
of chaos.
On the LM page, you'll find bifurcation diagrams
at various scales showing the depth of this equation.
You'll also find the equation graphed parabolically,
superimposed onto a straight line graph and the
bifurcation diagram. The LRLRR patterning and
L:R ratios are examined.
Finally, you'll find circular plots of the equation,
revealing patterns amidst the chaos.
4. Sine and pi (no link called for)
X = r * Sin (pi * X)
Here's what my investigations turned up:
Profoundly 
Iterating this equation generates a bifurcation diagram
visually indistinguishable from the one generated by the
above logistic map (LM), though the actual values are much
different. With LM, the valid values for r range from
0 to 4 and valid values for initial X range from 0 to 1.
With the sine pi equation, r can range from 0 to
infinity, though there is only chaos beyond r = 57.29578.
Initial X can range from 0 to infinity, though the graph
flips upside down when initial X exceeds 57.29578. It
flips right side up again when initial X is double that
value, with the flipping occuring at each multiple of
57.29578, which equals 360/(2 pi). And just as with
the LM, one can generate segments of a parabola by
plotting X vs previous X.
5. The Attractor of Henon
X = ((previousY + 1)  (1.4 * previousX ^ 2))
Y = (0.3 * previousX)
"Iterating this pair of equations produces a strange
simple set of curved lines that is poorly understood
by mathematicians. As thousands, then millions of
points appear, more and more detail emerges. What
appear to be single lines prove, on magnification,
to be pairs, then pairs of pairs, and so on to
infinity. Whether any two successive points appear
nearby or far apart is unpredictable."  James Gleick
Called a strange attractor.
6. Number doubling
Choose an initial value between 0 and 1.
Double it. Drop the integer part. Repeat.
Values occur in discreet bunches. Bunches
occur in multiples of five, depending on the
number of decimal places used. Repeating
patterns occur within the bunches.
7. Barnsley's Fern
Randomly selected parts of a four part algorithm
produces a realistic looking fern. Invented by
Michael Barnsley.
8. The Sierpinski Triangle
Randomly selected parts of a three part algorithm
produces a triangular lattice of triangles. I tried
doing this with an orderly, alternating selection of
the parts of the algorithm, and got a partial gasket.
Also called The Sierpinski Gasket.
 Roger Luebeck end chaos web page
home page: https://oldendays.me
The preprint for my relativity journal article is at:
rogcad.com/nature.physics.preprint/special_relativity.pdf
My book on relativity in absolute terms
at the U of M, East Bank Campus:
Relativity in absolute terms
There are only three sane sections in the Wikipedia article
on the twin paradox of special relativity. I authored those
sections in 2011, and they've been there continuously since that time:
Here's the link to the three sections below.
4. A non spacetime approach
5. The equivalence of biological aging and clock timekeeping
12. No twin paradox in an absolute frame of reference
On several fundamental levels, Relativity Trail eclipses the works
of Poincare, Lorentz and Einstein regarding the underpinnings and
kinematics of special relativity:
link: The originality and uniqueness of Relativity Trail
The twin paradox of special relativity cannot be resolved without
acknowledging a hierarchy of clock rates dependent on a hierarchy
of inertial motion. © 2022 relativitytrail.com
Relativity Trail explains the time differential between reunited clocks,
eliminates the twin paradox, diagrams Einstein's clock sychronization
in absolute terms, and ends all confusion regarding relative frames of
reference. It's completely compatible with, and in fact subsumes,
Einstein's relativity. It reveals what is transpiring behind the scenes
of Einstein's treatment.
Spacetime is shown to be dependent on Einstein's clock synchronization
method, and is properly relegated to a geometrical construct which
comes up short as a physical reality. © 2008 relativitytrail.com
Relativity Trail, with 192 pages, 65 diagrams and 75 illustrations,
will provide you with complete detailed algebraic derivations of all the
kinematical effects of special relativity. Everything is charted out in
absolute terms against a system at rest with respect to the totality of
the universe for perfect clarity as well as soundness of theoretical basis.
It is the totality of the universe that imparts the inertial properties of
clock rates and lengths which generate the effects of relativity.
This is explained in detail in Relativity Trail
Introductory document:
Relativity in absolute terms (Twin Paradox in Relativity)
(can be read with comprehension in twelve minutes)
Absolute versions of Einstein's postulates
(a snippet from the above document  can be read
with comprehension in three minutes)
Twin Paradox Animation on youtube.
Light rays and traveling twins are charted in absolute terms,
free of the misleading spacetime diagram.
home page: https://oldendays.me (copy and paste into your browser)
Updated 12/19/2023
