From Osher Doctorow

The 3-dimensional generalization of a cardioid is something like G. Taubin's (IBM Thomas J. Watson Center, Yorktown Heights, N.Y.) Heart Surface in his "An accurate algorithm for rasterizing algebraic curves and surfaces," IEEE Graphics Appl. 14, 14-23, 1994. Rasterizing is representing or converting graphical information into or with pixels. The equation of the Heart Surface, which looks like a 3-dimensional cardioid with a cusplike part of the surface on top (in the positive z direction), is the sextic equation: 1) [x^2 + ((3/2)y)^2 + z^2 - 1]^3 - x^2 z^3 - [(3/2)y]^2 z^3/20 = 0 If we use spherical coordinates after changing (3/2)y to a new y coordinate which could be labelled y' (not the derivative of y!), we get an expression for Dz(rho) where rho is the spherical radius coordinate whose square is x^2 + y'^2 + z^2 which resembles the last derivative for the cardioid in the previous posting. In other words, Dz(rho) is an expression with (rho - 1)^2 in the denominator. Mathematical physicists are immediately reminded of a black hole singularity in this context, especially with a chaotic Heart Surface. Although there were various theoretical reasons for various quantum theorists to consider that the Big Bang was not exactly a black hole, quantum theorists have not exactly kept up with analysis of Existence and Change in their own field, and if we add the fact that a similarity to a black hole doesn't imply that something is exactly a black hole, there is plenty of room for theorizing. A Big Bang and a Singularity are rather odd events even for Growth Events/Processes. Most Growth is not Big Bang type or Singularity type, and the fact that division is involved for cardioids and Heart Surfaces suggests that a major split has occurred between additive-subtractive Growth as in exponential and logistic and Riccati differential equations, on the one hand and chaotic-related Mandelbrot and Heart Surface growth with division, on the other hand. This division corresponds to a division between Rare Events and Fairly Frequent Events - Rare Events involve additive-subtractive Growth, Fairly Frequent Events involve division-multiplication in or out of Growth although division is primary. Very Frequent Events appear to be on the borderline between Rare and Fairly Frequent Events with regard to Growth. If you follow most of my postings to various cited places over the past few years, including some publications, you'll probably understand this most rapidly. Also look at researchmathematics@yahogroups.com. (Some of my postings have a remarkable tendency to disappear after several years - for example, I posted to anzap-l (an applied probability forum/mailing list in New Zealand-Australia) for several years, only to have the list owner close down the list with the rather cryptic remark which in essence amounted to the list not going in the directed that he had wanted (whatever that was - I'm still not clear about it). Superstringtheory.com is currently closed in the main forum on Strings/M Theory- Duality to which I had posted for quite a long time, apparently due to a deliberate virus by somebody.) Speaking of viruses, there is an indication that Rare Event Growth, which is the most important type of Growth except possibly for Singularities, is more Lifelike than Deathlike or Virus-like. A good model for Rare Event Change and Existence, in other words, is certain lifelike processes and events such as Knowledge (or Knowledge-Information), Perception, Memory, Consciousness. The other types of Change and Existence appear to bring in the Deathlike or Virus- like or Singularity-like regimes, and to be associated with events whose probabilities are greater than .05. The intuitive concept of an "over- bombarded" Existence with higher frequency Events/Processes in central to this. It isn't Knowledge that is the culprit at least in its simple expressions, but more like Complication and perhaps even Imitation and Mediocrity. Are Fractals and Chaos then signs of the breakdown of "beneficial" Change and Existence? We always assumed that Chaos at least involves the breakdown of predictability, but I emphasized the difference between Knowledge as Semantics or Meaning and "Information" as Syntactics or Form of messages. In some sense, becoming less predictable increases Information and Entropy but decreases Knowledge as Semantics or Meaning. To understand the Meaning of something is closely related to our ability to predict something. Rare Events seem to increase in Knowledge, whereas more common Events seem to increase in Entropy and Information. So in a deep sense, Existence and Change near the Big Bang involved a clash between Meaning and Form. The Mandelbrot and Julia sets demarcate the boundaries between the two pairs of clashing tendencies, Meaning vs Form, Existence vs Change. Existence and Change seem to be more compatible in Growth than Meaning and Form, Semantics and Syntactics. Osher Doctorow Ph.D.