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."  James Gleick
Called a strange attractor.
5. 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 identical to 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.
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 amazingly it does nothing.
Also called The Sierpinski Gasket.
 Roger Luebeck end chaos web page
=========================================
Special Relativity explained in absolute terms 
eliminates the twin paradox, shows Einstein's clock sychronization
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Reveals what is transpiring behind the scenes of Einstein's
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Documents:
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.
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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
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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.

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