From Osher Doctorow Ph.D.

The renowned Professor Peebles of Princeton University in 1993 or 1994 devoted a few pages of his volume on Astrophysics to showing that a fractal model of the Universe appeared to disagree with various radiation results under the most common model. However, since the Universe now appears to be divided into Rare Event/Process, Fairly Frequent Event/Process, and Very Frequent Event/Process regimes or parts, which were not considered in Peebles' work, it's probably time to look at fractal and chaotic models of the Universe again. The Mandelbrot set in polar coordinates has cardioids of type: 1) r = a1(1 - cos(u)) where r can be intuitively thought of for our purposes as the distance from the origin or pole to a point on the cardioid and u is the angle which r regarded as a line segment makes with the positive x-axis or polar axis. I don't know how familiar the different members of Fractint are with calculus, so I'll assume that nobody knows calculus and just gives the result from calculus, letting v be the velocity or speed of u with respect to time at each instant of time and a be its acceleration with regard to time, and "speed of r" refers to its speed of change in time, etc.: 2) speed of r = a1vsin(u) 3) acceleration of r = a1asin(u) + a1(v)^2 cos(u) Here adjacent letters mean multiplication, so for example a1vsin(u) means a1 times v times sin(u), and ^2 means square so that (v)^2 is the speed squared. There is a space between (v)^2 and cos(u) to make reading easier, but they're actually multiplied. Equation (3) is a Riccati differential equation growth control type equation in v. Nowdays, most people learn sine and cosine functions in algebra courses already, so if anybody isn't familiar with them, just look up keywords like graphs of trigonometric functions on the internet. They're basically up and down wavy lines like hills followed by valleys followed by hills followed by valleys followed etc. I'm going to ignore the third dimension of space and concentrate on a two- dimensional model for simplicity, although a "round cardioid" in 3-dimensional space may be more applicable to the universe. The above equations tell us that the main results of astrophysics and cosmology such as the expanding universe and the recently (as geological time goes so to speak) accelerating universe can be modelled by the above equations provided that the velocities and accelerations of the angle u are properly chosen in certain parts or intervals of time. For much of the history of the Universe, there doesn't have to be much "control" of v or a, but periodically sine and cosine (abbreviated sin( ), cos( )) graphs become negative, and since two negatives multiplied make a positive, in those intervals of time v and/or a has to be made negative. This doesn't mean that the Universe slows down or decelerates, only that the angle u with which it traces out a cardioid in the Mandelbrot set decreases or decelerates in time. A slight problem concerns the interpretation of r as both the line segment from the pole to the cardioid and the interpretation of r as the "principal radius" of the Universe or for short the radius of the Universe. We could make r ultimately extremely large by choosing a1 in r = a1(1 - cos(u)) to be very large - far larger than we're likely to observe with current astronomical technology. Some people might argue that a1 should change in time, at least a little over long intervals, and in that case we get what might be called a time- generalized cardioid. So in a sense we can have our Chaos and eat it too. :>) Osher Doctorow