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 LDF and Sierpinski.

1.    Balloting

      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 
      amidst chaos.

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 Difference Equation

      X = r * X * (1 - X)

      Iterating this equation produces regions of distinct 
      values, involving period doubling, as well as regions 
      of chaos.  

      On the LDF 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 unseen 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 
      strange attractor.

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 
      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.


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 
the rest state 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.


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