Jim Muth <jamth@mindspring.com> wrote:

Today's image takes us to the 'quat' formula, which draws slices of the supposedly 4-dimensional Mandelbrot set created by 4-part quaternion numbers. I have several problems with this formula. To start, the parameter screen of the formula is puzzling. While spaces exist to enter four variable parameters, only two are used, as though the formula were still being developed. And a little bit of investigation will show that only one parameter is needed to draw all the images that the formula can create. There must be a reason why the two unused parameters exist, or more importantly, why there are not six parameters, as there are in the Julia version of the same formula, which is far richer than this Mandelbrot version.

I agree with your assessment, and suspect that the formula is insufficient to show all the views. OTOH, it is a slicing formula and it may be that the quaternion M-set has 4-D symmetry such that all visual variations are covered, much like there wouldn't be much point in slicing a sphere with all possible distinct planes-- you get circles any possible way you slice it. I doubt this is the case, but the only way we'll know is to write a formula that does all the slices. Six parameters seems like the minimum sufficient to vary over them all; the way I look at this problem, which is the same for all the 2-D view slicing 4-D space, is this: We have four degrees of freedom to choose a point in the 4-space. At that point, the vector from the 4-space origin to the point uniquely determines a 3-space which is a subspace of the 4-space (the point is the origin of the 3-space) every nonorigin point of which represents a vector orthogonal to the first vector. Now, within any particular one of these 3-spaces we have two degrees of freedom to choose a direction; any plane passing through the local origin will be determined by that, its orthogonal direction (probably a pair of angles). So six does seem to be right for the minimum number of parameters. The devil however, does seem to be in the details. The only way to use exactly six that I've come up with is the following. First choose the orthogonal vector in 4-space; the easiest and sloppiest is just enter its (x,y,z,w) coordinates as parameters (sloppy because you cannot specify the direction of vectors with 0 length this way, which eliminates planes passing through the origin. To include them you seem to need to use 4-D spherical coordinates, 3 angles and a magnitude, but this complicates calculation significantly). Next, find three basis vectors for the orthogonal 3-space: normalize the originating vector and take the "ternary cross product" with possible pairs of basis vectors of the 4-space. The sloppy approach is, for an originating vector (x,y,z,w) to form the ternary cross product with {(1,0,0,0),(0,1,0,0)} and {(0,0,1,0),(0,0,0,1)} to get unnormalized basis vectors (-y,x,0,0) and (0,0,-w,z) which are obviously orthogonal to the originator and each other (the caveat is of course that it doesn't work if either x==y==0 or w==z==0, but again it saves a lot of logic). Normalize them and use as the x' and y' unit vectors of the 3-space. Now take ternary cross product( normalized originator,x',y') to get the z' unit vector for the 3 space. Finally, use the fifth and sixth parameters as theta, omega angles in sperical coordinates to generate the normal direction to the plane in the 3-space (passing through its local origin, the head of the originating vector) over which the fractint "pixel" variable parameterizes the surface. That defines the slice view.

With two 4-part numbers included in its calculation, the quaternion Julibrot figure must be 8-dimensional Agree so far... , so there must be six directions perpendicular to any image on the screen,

Don't agree with that conclusion. The quat math is different from a vector with 4 real components and I don't think you can just make that assumption without a more rigorous examination of what you actually mean by slices and dimensions.

as is the case in the Julia version, it should be possible to move the illustrated slice in any of these six directions. I can't escape the feeling that, as it is written, the formula short-changes the user. (The hypercomplex Mandelbrot formula has the same parameter limitation, though in that case I have written my own formula that includes some of the missing parameters and draws some very curious and interesting images.)

By all means please post it!

Actually, I wonder whether the quaternion Mandelbrot set is 4-dimensional at all. It appears to me that we are examining a figure with only three dimensions, which is the 3-dimensional figure that would be traced out if the 2-dimensional Mandelbrot set were to be rotated around its x-axis.

Maybe that's true, but I think it's hard to form that conclusion by looking only at 2-D slices of a 4-D object, and apparently a limited selection of those. A cube is visually quite different from the polygons one gets from slicing it in 2-D; relative placement of the otherwise boring slices makes a difference. A series of 3-D perspective slices of the QM-set would probably be more informative as to whether your conjecture is true or not. There might even be a way with a fractint .frm to generate a series of slices parallel (in the two normal directions to the slice) to it and use color to visualize the two extra dimensions. ---Hiram