At 19:55 17/05/2002, David Jones wrote:
Besides, the Julibrot itself can be treated as a slice of and index to an uncoubtable infinity of other higher-dimensional structures.
That could be true, beyond my knowing.
Now that's defeatism talking. I'm sure if you put your mind to it you could suss something out. Think of it as an opportunity for self-education. This doesn't look like much of a guided tour - it looks more like Billy Connolly in one of his more manic routines, wildly pointing in all directions at once. Complex numbers can be regarded as a special case of the quaternions - if a quaternion is written as (r,i,j,k) with r,i,j,k being real numbers, the complex numbers are those where j=k=0. The basics of quaternion algebra are in the Fractint documentation, but the thing to note is that if (a,b,0,0) and (w,x,0,0) are complex numbers (written and treated as quaternions), then their sum and product are likewise complex numbers (i.e. when written as quaternions their last two terms are 0). The mapping that generates the quaternion Mandelbrot set and Julia sets is q->q^2+c, just as for the complex Mandelbrot set. Since there's nothing being done there except multiplication and addition, any points in the plane where j=k=0 remain in that plane, with the practical upshot being that the two-dimensional cross-section of the quaternion Mandelbrot set through that plane reveals the classic complex Mandelbrot set. Ditto for Julia sets: slicing quaternion Julia sets along the j=k=0 plane reveals the corresponding complex Julia set. The quaternion M-set serves as an index to the quaternion J-sets in exactly the same way as in the complex case. A noteworthy feature of this indexing is that each point of the M-set (and its environs) is a duplicate of the centre point of the corresponding J-set. That point can be looked upon as the one where the space in which the J-set lives intersects with the one where the M-set lives. In the complex case the two spaces are both planes, while for quaternions they're both four- dimensional. (In four dimensions, two intersecting planes generally do so at a single point - e.g. the only point the planes (w,0.3,y,3.14159) and (-4/3,x,3,z) have in common is the one at (-4/3, 0.3, 3, 3.14159).) That's the idea behind the (complex) Julibrot - a plane passing through each point of the M-set's plane carries the J-set for that point; a two- dimensional stack of two-dimensional structures. The result is a four- dimensional structure: two coordinates from the M-set's plane (looking up the J-set in the index) and the other two for the J-set itself. The quaternion Julibrot is built up the same way (This isn't to be confused with Fractint's using the Julibrot code to render three- dimensional slices of quaternion J-sets). With each point of the 4D quaternion M-set providing an index to the 4D quaternion J-sets. That gives us eight independent coordinates to throw around, and so the quaternion Julibrot lives in 8-space. Quaternions aren't the only way to extend complex numbers. Fractint also provides the other simple four-dimensional alternative, known there as "hypercomplex" numbers. "Simple" is a regrettable lapse into jargon - it would take us too far afield to explain here, but I feel obliged to use it as I just now cooked up a "non-simple" group of four-dimensional numbers. Where hypercomplex numbers have (see Fractint's docs): ij=ji=k, jk=kj=-i, ki=ik=-j, ii=jj=-kk=-1, ijk=1 and quaternions have ij=-ji=k, jk=-kj=i, ki=-ik=j, ii=jj=-kk=-1, ijk=-1 These ones I just made have ij=ji=k, jk=kj=i, ki=ik=j, ii=-jj=-kk=-1, ijk=1 Each one has a corresponding eight-dimensional Julibrot. The embedding can continue. A common extension of quaternions are known as "octonions"; so-called because each number has eight components (as opposed to the quaternion's four or the complex's two). Needless to say, these can be used to construct a 16-dimensional Julibrot. A Google search should turn up their definition. When I first came across them I was a bit troubled - the way they're described made me wonder how the interconnections could remain consistent without imploding into "0=0"-like triviality. I stared at it for a bit until a flickering 60W bulb came on. "Block diagrams!" I thought to myself, that being another branch of mathematics which deals with exactly the sort of interrelations I was looking at. With that I promptly cooked up decanions and pentadecanions - ten- and fifteen- dimensional numbers; with respectively twenty- and thirty-dimensional Julibrots. All of these classes of numbers up till now - complex, quaternion, hypercomplex, (unnamed 4D number), octonion, decanion, pentadecanion - can have their structures described in terms of group theory. There are an infinite number of groups (even simple ones), some being suitable for extending complex numbers. Each one has associated Mandelbrot, Julia, and Julibrot sets. Even those that aren't suitable have such sets, but they don't grow out of the Mandelbrot set. Heading off in the direction of block diagrams; just as quaternions can be embedded in octonions, decanions and pentadecanions by means of a suitable block diagram, the other classes of number we've seen can themselves be embedded in larger classes by means of a larger block diagram. Each of _these_ also yields the sets we're looking at; each one can be built up from the traditional Mandelbrot. The extension can be made in other directions completely. Take the z->z^2+c of the basic quadratic map. c is the (two-dimensional) parameter that indexes each Julia set. But what about z->z^(d+2)+c? Now _two_ complex numbers are needed before the Julia set is specified; the Julibrot is now six-dimensional - that four-dimensional Julibrots is just a single slice through this larger beast. And of course there's nothing sacred about z->z^(d+2)+c, either. Any function that has z->z^2+c as a special case can provide a home. And each one has a Julibrot of some number of dimensions. Most will be boring: z->z^(d+e+2)+c is just a warped z->z^(d+2)+c. But that still leaves, as I said, an uncountable infinity of distinct cases. Something I said some time ago on this list: "No matter how many knobs and buttons are added to Fractint, there'll always be something else that someone wants to twiddle." Arthur Cayley, one of the originators of the of such fields as n- dimensional geometry, group theory, matrix theory, and the notion of invariance put it thus: "It is difficult to give an idea of the vast extent of modern mathematics. The word 'extent' is not the right one: I mean extent crowded with beautiful detail - not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood, and flower." Morgan L. Owens "Windswept and interesting."