equations - graphs - program code
Eight equations generating exotic behavior,
along with the program code and graphical output.
The mathematical intrigues on these pages have, I suppose,
about 1/10,000 the sophistication and complex behavior as
found in such things as Mandelboxes and other 3D fractal
endeavors which are popular today. But this is the level
of mathematical endeavor I'm capable of, and I do enjoy
creating my own mathematical entities, doing my own
explorations, and making my own discoveries. If you like
the stuff 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 own creations. The rest
of the sections I merely added my own investigations to,
especially the logistic map and Sierpinski.
Iteration of a balloting paradigm, using random number
generator. It grew out of my analysis of the 2000
Gore - Bush Florida balloting. Very surprising patterns
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 top of
this page and click "Simple trig" to see many more.)
3. Logistic map (LM)
X = r * X * (1 - X)
Iterating this equation produces regions of distinct
values, involving period doubling, as well as regions
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.
Mathematician Paul Stein called the complexity of
this iterated equation "frightening".
4. 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. Called a
5. Sine and pi (no link called for)
X = r * Sin (pi * X)
Iterating this equation generates a bifurcation diagram
visually identical to the one generated by the above
logistic difference equation (LDF), though the actual
values are much different. With LDF, 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 LDF, one can generate segments of
a parabola by plotting X vs previous X.
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
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 amazingly it does nothing.
Also called The Sierpinski Gasket.
end chaos web page
Special Relativity explained in absolute terms -
eliminates the twin paradox, shows Einstein's clock sychronization
diagrammed in absolute terms, and ends all confusion regarding
relative frames of reference. Completely compatible with, and in
fact subsumes, Einstein's relativity. Not Lorentzian relativity.
Reveals what is transpiring behind the scenes of Einstein's
Relativity in Absolute Terms.
My most comprehensive online document. A concise overview
of why special relativity must be diagrammed in absolute terms.
Twin Paradox Animation on youtube.
Light rays and traveling twins are charted in absolute terms,
free of the misleading space-time diagram.
Twin Paradox Animation.
Expanded text, and animation of the twin paradox. (Embedded youtube animation.)
Twin Paradox Explained.
A similar discussion of the failure of spacetime diagrams.
Twin Paradox Animation.
Alternative text, and animation of the twin paradox. (Embedded youtube animation.)
Absolute Frame of Reference
Absolute frame of reference in the physics community.
Free pdf file of the book:
Relativity Trail, free pdf format, 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.
Excerpts from the book Relativity Trail with included images.
Einstein explained in excerpts from Relativity Trail.
Diagrams and derivations from the book Relativity Trail.
home page: rogerluebeck.com