Chaos: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 Relativityexplained 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 treatment. 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. 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 ReferenceAbsolute frame of reference in the physics community.Free pdf file of the book:, 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 inRelativity TrailRelativity 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