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(table of contents follows ...)

More Triple-Cross Products

NOTE: Recent theoretical work on the possibilities of Quadrupple (4-tupple) Cross Products has been suspended due to a lack of funding. Data processin continues - un-abated. AxB (v) :: C -[ SC x SP (Earth) :: (expressed via) ART -> Eco Psychology, etc]- See also: The name re-makes the thing (HUM x SCI (word) :: EXP as JAZ). -^_6 On this page: {Intro} {Quick Explosion} {Bifurcation} {The ButterFly Effect} {Chaos Theory} {Quantum Theory} {Randomness} {Catastrophic Systems} {Fractionalist} {Obliteration/Creation: Collage} Note: Determinism is treated here: -[scientist entry]- (physics, etc) -[spiritualist entry]- (philosohical, etc)


The idea of the fractal was first investigated on the planet Earth by ??name?? Mandelbrot, but probably not investigated for the first time in "The Universe of Discourse" by that person before; of.
Start again. Fractals deal with the fractionation of math forms. For example, one of the early examples of a fractal (but, to my knowledge, un-proved AS a fractal) was the so-called "Cantor Set". We take a line segment, conveniently 0 to 1 inclusive (written usually as [0..1]), and remove the middle third, thus we get a set of two line segments: ------ ------- 0 1/3 2/3 1 We then divide that by the same means: -- -- -- -- 0 1/3 2/3 1 The interim pieces using multiples of one sixth as their boundaries. (I would type in 1/sixth, but the "x" (six) key doesn't work on my keyboard). The idea is that this process can continue indefinitely (and with the help of Maxwell's Demon working infinitely fast for an infinite amount of time, the task is easily accomplished). The result of the set of line segments is commonly refered to as "Cantor Dust". {Jump down to FRACTALIST entry} In the same way, geometric things can be created by not necessarily excluding the "cut away pieces", but by changing the direction of the inclusion. For example, ----- | | | | | | ------------- cut to show: ---- ---- which in turn is cut to show: -- | | - - | | - - | | --- --- (or something like that, but hopefully, you get the idea). NOTE: We must not confuse the actually infnite processes of the MATHEMATICAL fractals with their corresponding counter-parts in the PHYSICAL universe. By the same manner that we can calculate an integral wich "sums up" an INFINITE number of infinitely small pieces to give (sometimes) a finite result; eg, the infinite series: 1/2 + 1/4 + 1/8 + ... sums to the limit value (AT infinity) of 1 -- note that i place NO DECIMAL point after the 1, indicating that it is in fact the WHOLE COUNTING NUMBER "1". But, in the physical world, the ideal, mathematical concept of the integral is replaced by a COMPUTATIONAL *SUM* using decimal fractions to a given (and limited) precision). Thus, we might PROGRAM a computer to perform the sumattion and end up with a value of 0.99999999994 instead of the MATHEMATICALLY theoretical value of exactly 1 Thus, fractals give us yet another MATHEMATICAL tool for examining the befaviour of our physical world; ie, Realtity Structure Three. START AGAIN Let us begin (again? for the first time? what is time? More importantly what is the colour of time?) START AGAIN Let us look a way that we can make fractals (or at least start trying to think in a fractalist manner). We recall some of the properties of fractals: 1) They are self similar -- that is, by "breaking off one part, we can (or not) re-create the rest of the structure. The simplest example of this is a candy bar. It is a rectangle. We can break off small pieces of it, to create (about) 8 smaller rectangles of chocolate that are the same SHAPE (but obviously smaller). Let's look at a simple example involving a "regular triangle" -- commonly refered to as the "equilateral triangle" (Greek: Equal sides). A simple triangle, pointing upward, the base is parallel to the \ Now we can find the "midpoint" of each side which we then use to draw the "next" triangle. Thus, we get: two new triangles (the same size as the original)
have now been stacked underneath the first. Thus, they 
form a I have left a bit of space to show the idea, but we could scruntch them up so that the apex (top points) of the lower two triangles will overlap the bottom (left and right) points of the top (original) trianlge. Then, they would look like the sub-divided triangle above. Naturally, they would enclose a space 4 times as large -- re replicated the triangle instead of sub-dividing it, and by "trapping" the space between the three triangles, four times the area was found. Now it's important to note that this "dividing up" (returning to our original sub-divided triangle) is NOT just because we have chosen a regular (equilateral triangle) -- the process bears repetiion (pun slightly intneded); eg, a right (30-sixty-90) triangle divided similarly alt="a some-what badly drawn obtuse triangle (140-25-15) similarly divided"> (My "drafting skills" are a bit "better" in the organic rather than the geometric graphic domains) You get the idea right? Meanwhile, back at our equilateral (and much easier to draw) triangle, we can again sub-divide one of the triangles: i subdivided the lower left one into 
4 new triangles -- now even smaller ones Now, we should point out that i arbitrarily chose to sub-divide the lower left triangle. We could have chosen any (most ostensibly the one in the centre) to divide. Also note, that we could have decided to divide ALL of the triangles, thusly: The 4 triangle version has now been subdivided 
completely into sub-portions (each one creating 4 new,
smaller ones Another VERY IMPORTANT POINT is that this "complete sub-dividing" of a triangle doesn't really lead to fractals as such (or at least we might think of them as sort of "level 0" or "zeroth order" fractals -- that is, a sort of limiting case. The point is, that if WE do CONSISTENTLY sub-divide a fractal (alternatively, we should investigate ALL possible sub-divisions), a pattern will emmerge. This pattern is the way in which the "density" of the triangles shows up. This pattern leads ot a thing called a "Serinpenski Curve" (after the person who studied such things). As it turns out (and here's the FRACTAL stuff), the way in which this simple process "does" things leads to un-expected side effects. The properites of the process (see below), "should" be simple and obvious at all levels -- like our "complete" sub-divisioning which turns out to be easy to predict (eg, the number of triangles created upon subsequent complete sub-divisions is given by the sequence: 1, 4, sixteen, 4, ...). But, as it turns out the properties aren't all that obvious. FOr example, where the "densest part" of the triangles is at any time seems to "float around", spiralling inward. And the area covered, the lenghts of the outlines of the triangles and the shortest path along the vertices turns out to "surprise" us as well.

Fractals - Quick Explosiion

There are several (at least THREE known on Earth (see map) types of "fractal-like" structures: Fractals (both mathematical and physical) Catatostrophic Systems Chaotic Systems Quantum Systems Random Systems Note: I have tried to arrange these hierarchically, but catastrophic systems don't really fit into the above structure any more than do deterministic systems. Deterministic systems are dealt with in the scientist
]- section, but then so is quantum mechanics]-. Now we are ready (or not) to start thinking "fractal-wise". But, first. We need to make SURE that when we set off the hydrogen bomb that a chain reaction doesn't occur that causes ALL of the nitrogen in the Earth's ENTIRE atmosphere to ignite. (An actual problem encountered during the so-called "Manhattan Project" during the construction of the first atomic bomb). That is, we MUST limit (quinch) the expansion of ideas (temporarily) otherwise an idea explosion might well occur; technically, this is refered to as "information cascade phenomenon". We NOTE briefly (and here formally) that the mathematical structure called "fractals" is but a small portion of a much greater "calculus" (in the same way that "naieve set theory" is a small portion of "set theory", "the laws of form", "various meta-logical systems", etc). Thus, we IGNORE (again, temporarily) the physical manifestations of fractals (eg, most notably the "lenghts of coastlines"), other physical systems; eg, quantum, chaotic (as well as mathematical chaotic systems), catastrophe theory, entropy, etc. Thus, we may now begin. Having thus limited ourselves, we begin to break those restrictions. This is where the fun begins.

Fractals (quick explosion) -- Botany

Have you ever noticed the patterns of leaves on trees? If you haven't then you obviously aren't a painter. One of the earliest "impressionist" painters, Paul Ceszane said, "Until i sit down to draw or paing something, i find that i haven't really ever looked at it before." See also: -[Artist]- The idea is that the PATTERN of leaves on a tree (or shrub, or an amoeba, etc), varies from species to species (we'd expect that). But. There are patterns of how the leaves arrange themselves. Some fan out in an almost spherical pattern (eg, the cherry blossom), others tend to droop down (weeping willow), others seem almost random and scraggly (eg, a "live oak") Note we EXCLUDE toparary ??sp? or other attempts to "sculpt" them, in the same way when studying the shapes of glaciers (how they grow, how the shift in sub-zero temperatures, how they melt in above-zero temperatures, etc) from "ice sculptures". Notice two that an icicle is NOT a glacier -- especially in terms of shape, change, etc. As it turns out the patterns (many are possible, few are attempted as John Cage said about musical patterns) are determined not only by random chance, but by the constrictions of geometry, the physics of sunlight, rain, and wind, etc. Also the ways that the branches (and leaves fork). If you get a chance to look at the leaves of a "nandina" or "herperidium corrillio or even two different types of oak or pine leaves, then you sill see that there are an amazing variety of different patterns. Well not really, amazing; after all, it IS fractal now isn't it? Now for the explosion: Fractalist means breaking down into smaller and smaller things (cataloging, physics, and even art does that), but it is the way that the breaking down goes that makes it fractalist. We might take a word list and create a "dictionary" (techncially a concordance) by breaking down and alphabetizing every word by every possible letter in it; thus, we might have: Accident Apple cAt regArd rAbbit ... Cat Clean aCcident acCident aCquiesce ... This would be our "complete" sub-dividing again. But, fractals go beyond that wince they breaking down things into components and conentrating on SIMILARITIES. But, if we looked at our dictionary ideas, we might think about things like "take away". hearth earth ear a Forming words by dropping (taking away letters). But, what if we take away "written strokes"?? HEARTH EARTH FART (the bottom of the "E" has been subtracted) ART (the entire letter "R" has been subtracted) AT if we allowed re-arrangments, we could have: ART AIR (the top of the "T" has been subtracted to make an "I") Technically, we are FRACTIONATING not FRACTALISING. But, the other part of fractalism is the BUILDING UP. We all know the addage of no two snow flakes being the same. This is based on the regular shapes (six-fold symmetries) of the snow flakes. If there was ever a candidate for being a fractal, then it's gotta be a snow flake. But. Has anyone actually checked to see if there are TWO identical snow flakes? Oddly enough, the cartoonist/philosopher Carol Lay in one of her "Story Minute" comic strips created a character that looks for (and finds) two identical snow flakes. Thus, she created a quite interesing story based on FRACTAL thinking. See here site here --[www.waylay.com]-- ??link?? to page??? As romantic as it might seem, we might just as well say "there are no two identical leaves". And of course the aburdity of people when seeing identical (not really, even though probably "nearly so") twins and wanting to say, "Yes, but you're the smart one, and you're the creative one". One might "twist" one of the tau's thru the laws of form and state: Do not create a distinguishing mark where none exists. See: -[Laws of Form]- (in SCI: Maths-) Adding absurdly, unless of course it's a tuesday and a certain duck -(see map)- is available. So: Fractalism has beaking down (systematically) and building up (systematically, but not necessarily the same systematic rule).


Bifurcation - to cut into two branches. In this section: {
Intro} {Binary, Trinary, ... N-ary Splits} {Un-equal Parts} {Multiple Fractures} {Anti- and Quasi- Rules} {Rule Fractioning} {Symbolic Cutting} {Hyper Cuts}


Literally "bifuraction" just means cutting something into two branches. This method has been used for some time in programming and optimisation techniques; eg, "The Binary Tree", "Decision Trees". -[
Mathematical Trees]- (in scientist) In fact the idea (formally called "choice by inclusion/exclusion" - the simplest example of which is "20 Questions". Odd as it may seem, the old "Is it bigger than a bread-box?" giveing yes/no answers can usually break down a search fairly quickly. Note that the use of TWO choices (treu/false, yes/no, etc) is a fundamental model under-laying all decision processes (there is a maths theorem that states that any deterministic decision process can be adequately "covered" by the yes/now model; read this as "the capabilities of a binary system are necessary and sufficient to "encapsulate" any deterministic by the simple "binary" choice model"). And of course, we're all aware of the "Lineaus Classification" system which is used to classify species; particularly the old "genus species" idea. ALso, note that "non-deterministic" systems would of course bring either randomness, quantum, chaotic, etc considerations into the frame-work. Also, note the power of "powers of two". In 20 questions, at the end of the 20th question, about 1 million items have been "sorted" by the process of elmination. Also, in the early days (and still to some extent) many AI (Artificial Intellegence) systems used the binary bifurcation method to classify and distinguish between objects - giving rise to thngs like "A dolphin is an animal, that is a mamal, that swims, that lives in the ocean and is not a fish". Of course, as *new* objects are brought into the *classification system* new TEST RULES (questions) will be needed. Thus, we would need to have specific rules/questions to distinguish between a dolphn and a porpoise. One problem with the bifucation process is that most things have more than just binary attributes; eg, a rock or an mineral might have more than one "state"; eg, graphite and diamond are different (allotropic) forms of carbon. And then of course, we get into things like "BuckyBalls" (named after futurist/philosopher R. Buckminister Fuller of geo-desic dome fame) in which case the carbon atoms can form "mega-complexes" sort of like molecules but consisting of possibly hundreds of carbon molecules joined together - extending vastly the structure of diamond. Note that in this case carbon (graphite) is formed by flat, hexogonal arrays of carbons that form planes (ie, 2-dimensional) that can slide back and forth over each other - giving rising to the "greasy feel" and hence lubricating properties. In the case of the diamond, it forms a 3-dimensional structure and the uniformity of all of the bonds is what gives it its characteristic strength. Another problem with bifurcation is that it has difficulty with what we might call "semantic trees" or "symbolic trees", where the items in the tree are things like words, ideas, etc. (see the section below on "Symbolic Cutting"). In order to classify (split off, distinguish, group) items, we use CHOICE RULES.

Binary, Trinary, ... N-ary Splits

In-order, pre-order, post-order (constant, binary bifurcations, but the node splitting/parenting function changes

Un-equal Parts

Now *normally* human nature would mean two equal cuts. But, what happens if the rule(s) we use for the cutting create clearly un-equal branches? A good example of this is the so-called "Dedikind Cut" - named after the maths prof Richard Dedikind who discovered it. With it he quite tilted maths on its head. Traditionally the types of numbers had been split into various groups: the counting numbers (1, 2, 3, ...), the natural numbers (0, 1, 2, 3, ...), the integegers ( ... -3, -2, -1, 0, 1, 2, 3 ...} the rationals { f | f = p/q } (where p, q are integegrs and q .not-equal. 0) the reals { all decimal fractions, including infinite, non-repeating decimals) etc (complex, surreal, etc numbers) The thing was that irrationals are those numbers which can NOT be expressed as the quotient of two integers; eg, .square-root-of. (2), pi, e, etc. But, he was able to show that (for example), pi was really just a "ratio" where we *cut* the real number axis at pi - thus dividing it into two parts - numbers less than pi and numbers greater than pi. Thus, pi was "just another kind of ratio". eg of Earth's atmosphere Nitrogen / Oxygen

Multiple Fractures

Once a fraction has been taken, parts of that branch can be broken up eg 1 - 20 n-ary split (here a binary tree) 2 4 6 8 10 12 14 16 18 20 (rule: Evens) 1 \ 3 5 7 9 11 13 15 17 19 (rule odds) \ \ \ \ \ \ \ extracting rule (alternate fracture) rule: primes

Anti- and Quasi- Rules

Constructiong one fractal tree and then another using the first tree's anti-rule. Also, contrapostivie, opposite, the negation, the excluiding and/or including principle, etc.

Rule Fractioning

A set of cuts follows one rule, but the rule used to fractionate may change - a list of fractionating rules list. Of course, in terms of learning systems and such (esp ai or hi/artistic/drug-induced) the list might well be infinite or as great as make no odds.

Symbolic Cutting

We now need to look at words, symbols and ideas. I take as read the ideas of Ferdinand de Sausure and Jacques Derrida -- although i make no pretense at having come anywhere near anything resembling an in-depth understanding of their ideas. Let us go back to how we "diagram a sentence" (i will modify it at bit for ease of typing). The big dog barked at the blue car. dog | barked / at ------------------------- \t \b \ \h \i \ car \e \g -------- \t \b \h \l \e \u \e Of course in AI and CS (Computer Science) we can form binary pairs using RELATIONAL OPERATORS (REL OP) as per the following (i'm not being real exact here, btw): RELATIONAL OPERATORS Binary relational rules are of the form (a, REL OP, b) is_doing is_a_noun is_a_verb is_a has_the_property_of Thus, our K-base (Knowledge base - database, tables of info, etc, or what the AI (or HI - Human Intellegence) "knows") is something like: (dog, is_a, animal) (dog, is_doing, barked) - we'd need a REL OP "was_doing" (dog, has_the_property_of, "the") - hmm, that dog over there, no, not that one, that one (etc) (compare with "All dogs hate cats" that is, "all" vs "the") (dog, has_the_property_of, "big") (barked, is_a verb) -- And at the "lowest level", we have: (animal, is_atom) - for simplicity, we'll pretend that "animal" is an un-defined term; as in geometry, when we say that a point, line, etc were un-defined. If we didn't do this, we end up with circular definitions. Note that any dictionary must resort to this eventually. The idea is that it devolves to an "escape clause" something like "everyone knows what an animal is", etc. (noun, is_atom) - of course we'd either have to (verb, is_atom) define the properties of all of (preposition, is_atom) these or *program* them into the classification system. Well, hopefully that gives you a feel for how these things work. The main thing to note here is that as long as we are comparing the same kinds of things, and thus using CHOICE RULES

Hyper Cuts

cuts that then entangle other cuts cuts that are not neceessarily complet - drop points crossing fractional boundaries/dimensions/

The ButterFly Effect

Chaos Theory

Tutorial links

freeuk entry!]-

Quantum Theory - A fractal look

See also: -[
scientist entry]- In this section: {Intro} {<><>} We begin with the idea that the fractal is breaking down and yet somehow retaining order, and yet at the same time creating new orders. In the quantum world, we break things down to the lowest level and at that level, we find everything connected to everything else and nothing may be done without affecting something else. While in Jazz, we take it as read that we eventually must return to the starting point, as such it seems a tad shy of either fractal or quantum effects. I would propose that jazz is somewhere between the fractal and the quantum (ultimate vs no ordering principles). In one sense, the guiding rules of the fractal world impose such absolute boundaries that the quantum is completely excluded. And yet, so much of our world seems governed by both the fractal and the quantum; eg, branching patterns in trees, and atomic interactions - respectively.

The Folder Problem

Consider for a brief moment our old friend the "folder problem" where we need to sort things out into folders. But, we end up creating so many un-used folders (thinking even in just binary terms, we might create a folder "living things" and "non living things", and then of course comes the "clasification problem" as to what we do with something like "ghosts" - and we being the good libarians that we are, go slowly mad. On the one hand, an item can of course be referenced by several folders; eg, just look at the problem of handling something "simple" like "relativity" in a very limited folder problem, namely the iconosphere zix42. And as both Asimov and Clarke (among others) the indexing and cross-referencing becomes a greater problem than the data stored itself. And we can look


See also: -[
scientist entry]- (maths) The main discussions of randomness occur in the above link. The point here is the role of randomness (whether its exactly random or not) and its relationship to how sequences of actions (painting, danse, and other texts)

Catastrophic Systems



This section deals with the idea that ANY thing an be built up out of fractions (or portions if you will). Note that this borders on the concepts of {
Collage} which is dealt with below. It is important at this point to leave the strictly mathematical (or even its closer physical manifestations) of fractals and look at the closely related ideas of "fractionism". We saw earlier that {Cantor's Set (above)"} would allow us to create fractal-like things by systematically dividing up a line in smaller and smaller portions. We now want to think of things in terms of these fractions. A common example is the "cut away" used in films and books. That is we have one "line" in the story and it reaches some sort of dramatic "point", and then we cut away to either another line in the story or a "close up" or a "long shot". The well known plot narrative "As she closed the door hehind her, keeping the light off she was certain no one had observed her. Indeed no one had. Mostly." This of course builds "tension" in the mind of the viewer/reader since we "know" something that the person in the story/film doesn't. Note that this method was discovered early on in story telling and (to me personally) has always seemed a cheap trick. A more direct approach (most common in jokes) is to lay in certain extra facts and let the surprise ending hinge on one of the previously disclosed plot elements that did not AT THE TIME seem important. For example, there is a certain story that hinges on the fact that a warior is left handed, but that most people are right handed. In the same way, the WAY that we focus on a problem and how to solve it may end up being easier to solve if we DO fractionate it. Finding a USEFULL way to fractionate the problem is of course the crux of the matter. In tradtional geometry, there is a thing called "tile-ing". We commonly see this in floor tile consisting of square tiles. As it turns out, there are a multitude of ways that things can be tiled. Using the SAME tiles, we can do this using triangles, squares and hexagons. Otherwise, we have to use non-standard shapes created to tile together. Of course we could use circles and fill the joints between them with mortar to make a pattern of polka-dots, etc. However, if we LET the patterns over-lap for example, try taking octagons (most commonly the shape of the famiilar red "Stop" sign at road intersections) and tiling, letting them overlap as they will. This gives rise to OVER-LAPPING. Now imagine that on the edges of each stop sign we write things. And we then join the edges together (if this is with card stock, we could use scotch tape). This arrements are called "nets" or "meshes". These show up in the oddest places. If we "un-fold" a cube, we get it's 2-d "net", thus: a central square wiwh one square on each side of it
these are the As it turns out, the net of a cube can be used to TILE a fllor (two dimensional, flat plane) perfectly. You can exoperiment with this by cutting out a pattern from graph paper and then using that as a template to cut out more robust pieces from cardstock. As it turns out, we can construct the "five platonic solids" using nets. Actually, we can construct ANY three dimensional structure out of a two dimensional net. The only problem comes if the structure is composed of separate parts. We could imagine a comples "3-d puzzel of say The Lourvre museum, then fold it all out onto the table (they actually make such 3-d puzzles). Of course for the fountain and the glass sculpture created by I.M. Pei in the middle we would need separate nets -- unless the interior concret area becomes a connecting net. Thus, we might have for the tetrahedron: a central regular triangle, bounded on all three
sides by triangles sharing a common line of joining (here
shown separately a bit apart). As though the tetrahedron
(triangle-based pyramid) had been This pattern two will perfectly tile a 2-d plane. But, as it turns out the dodecahedron (Pronunskiated: Doh-deh-kah-hee-drawn) will OVERLAP. Obviosly most nets will. we could imagine nets made with lines that are made upo of things like string, rope, or even with wire we can have 3-d sculptures, or the way that vines either trace along a tree (1-d) or a wall (2-d), the way water lillies (2-d) cluster. THese "un-foldings" are but one way to build up 3 dimensional things using either one or two dimensional (mixtures) of things. Then the way that the things written on each one "plays against" the one beneath or above or near it ccan be used as an idea generator suing the "juxtaposition of ideas" process already common in literary thinking; eg, compare and contrast, synthesis (using things like "cross product generation", or Hegel's familiar "Thesis/Antithesis-Thesis" formula), etc. Als, consider the idea of fractionalist ideas or words that are then "played against" each other. A common "refriderator magnet" decoration consists of words on each tile (rectangles of varying lenghts) from which you can tack them together to make sentences, poems, or non-sense. Thus, by using fractionalisation as well as nets, overlapping, and juxtaposition (imagine juxtaposing two different nets next to each other) we can create "higher order" patterns. Consider for a moment that a pair of dice when rolled in a game of "craps" is actually the justaposition of the 3-d form of a six-faced cube the TOP face of the PAIR of which forms a pattern that we call "the roll of the dice". That this was examined by Pascal (??date??) systemacially and gave rise to the entire field of probablility and statistics is not a co-incidence. Thus, the seemingly insurmountable problem of betting (and why some people were better at it than others), turned out to be a simple fractal problem (in the "restricted" fractionalised form) of two cubes, evenly weighted. And it turns out that by SYSTEMATICALLY listing all possible combinations of the dices ("die", technically) definite PATTENS emerged. Thus, we NOW "know" that the most common rolled total of the two dice is seven. So, if i bet on 7 and you bet on 5 -- in the long term, i must ALWAYS beat you (actually there ARE such things as "runs of luck", but for the most part beat based on the RANDOM odds). That this fact is OBVIOUS is not so obvious when it comes to things like the game of "Keno" (played with THREE dice), the lottery (played with varying numbers of balls), etc. An important point here is that while we talk about "the odds" of an event and we compute fractions (eg, the odds are 1 in 9 that you will roll a "five"), we are actually looking at what is technically a deterministic event (probablity) space. The "roll of the dice" thus results in a MATRIX (the p-space) fromed as the juxtaposition of two cube's top face, added together. In a similar (but totally different) way, the analytical cubist painters would take an object and present a re-geometricisation of it and combine this with "multiple" and "simultaneous" views of the object. Of course, the end product of THAT process is fractalisation or in the case of art forms, COLLAGE -[see artistjazz]-


Enter the Quantum

The Random


Fractals in and of themselves are formed by a consistent set of rules that are "almost" normal. If they were completely "normal", then we get a deterministic system. For example, a circle is a circle. A circle is an elipse or an oval of a special kind), but a oval is not a circle. Neither is a square. Or a duck. We can "put our ducks in a row", or we can arrange them in a circle. Oddly enough, ducks don't need us to do that. If you have ever observed ducks, the patterns that they form when settling for the night is very circle-like. Ah, yes, sez the skeptic: "But it's not a PERFECT cirlce!". Do, i have to become "God the Geometer of the Universe" IMG GOES HERE For here is my tool: a rather fuzzy picture of what appears to be 
set of callipers, perhaps a draftsperson's compass. 
For some BIZZARE reason the image is entitled 
Somewhere a cartoonist appreciates both Robert Dinerro and Tuesday Weld. But, on Wednesday (next), i must present my latest film work. Ah, the vagarities of a thing called time.