FOTD 15-05-02 (Mandelia [6])
FOTD -- May 15, 2002 (Rating 6) Fractal visionaries and enthusiasts: As has been obvious for some time, I am fascinated by things that cannot be comprehended by the human mind. When these things are irrational, some on the philofractal list feel that I am indulging in fantasy, but none can dispute that the Mandelbrot set and its associated Julia sets comprise a single four-dimensional object that in a sense most certainly does exist, yet cannot be comprehended by the human mind. This 4-dimensional object is sometimes called the Julibrot figure. Depending on the direction and location in which this 4-D figure is sliced, either Julia sets or perturbed Mandelbrot sets will appear. But in addition to producing Julia and Mandelbrot sets, the Julibrot can be sliced in other directions which produce hybrid sets that are neither Julia nor Mandelbrot sets, but combine the features of both. Today's image is one of these hybrid sets. I could have named the image "Julibrot", but that word has become hackneyed, so I named the image "Mandelia", a name which combines the names of both families of sets and also has a nice sound to it. I rated the image a 6. To produce today's image, I began with the Julia set of Seahorse Valley, which normally appears as a chain of empty bays, but I did not slice the Julibrot in exactly the Julia direction. Instead, I double rotated the slice 1/100 degree and 1 degree from the Julia direction. This tiny rotation produces a tremendous difference in the resulting image. (Curiously, the values of the rotation are quite critical this close to the Julia direction.) The overall shape of the fractal is that of the Julia set of Seahorse Valley. But the chain of bays is no longer empty. Instead of a dull, blank inside, we see an inside filled with detail quite unlike any normally seen in Julia or Mandelbrot sets. The valley itself appears at the exact center of the image, and the great spirals that lie in the area are also in evidence, appearing where the many unadorned valleys of the pure Julia set would normally lie. I have tilted the image to give it a more dynamic feeling, and colored it with the basic equal-iteration-band method. I used the basic coloring method to assure that the unusual effects in the image are actually a part of the fractal and not a result of a fancy coloring system that distorts and sometimes totally hides the original fractal. Due to the unusually high maxiter and the periodicity setting of 15, both of which are necessary, the parameter file is a rather slow one, taking 25 minutes on my machine. But Paul and Scott are here to save such frustration, and will soon have the completed GIF image file posted to their web sites at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> and at: <http://sdboyd.dyndns.org/~sdboyd/fotd/index.html> from where it may be downloaded with ease. The fractal weather Tuesday was chilly (64F 18C) and very windy, with heavy threatening clouds that finally released a damaging rain squall at 7pm. When the squall passed, a glorious double rainbow appeared. (All rainbows are double, but the second bow, which lies outside the main bow, is usually too faint to be seen.) Then the sun set peacefully. The dynamic duo, who were still shell-shocked from the hailstorms of Sunday and Monday, dashed for cover when the wind started roaring and things started flying around again, and didn't reappear for several hours. Due to a power failure Sunday evening, I am a bit behind on the FOTD CD. But a half-hour extra work today will make up the shortfall. And it's now time to get busy. Until next time, take care, and it all depends on which direction you view your fractals. Jim Muth jamth@mindspring.com START 20.0 PAR-FORMULA FILE================================ Mandelia { ; time=0:25:37.91--SF5 on a P200 reset=2002 type=formula formulafile=allinone.frm formulaname=multirot-XY-ZW-new function=flip/ident passes=t center-mag=3.19189e-016/6.66134e-016/0.99\ 3388/1/55/3.88578058618804789e-016 params=89/89.99\ /2/0/0/0/-0.75/0 float=y maxiter=75000 logmap=yes inside=0 periodicity=15 colors=000N7PL8SL9SKGSJ9TI\ CUHDVGEWEFXCGYAH_8L_6P_4Wa2_a0ca0ha0nc3tc9ycDzcFzf\ HzfJzfLzfNzfPzfRzfUzfWzfUz_UzUUzNRzHRzBRz5_z0fv0mn\ 0tf0zY0zR0zJ0zD0zH0tL0fN0RR1FWB3YJ7cFBjBDn9Hv5Jz3N\ z0Pz0Uz0Wz0Yz0Yz5YzB_zJ_yP_tWrnBzj0zh0zz0zz0zz0zz3\ zv7zrBzmFzhJzcNz_RzWWzR_zNczJfzHhzHhzHhzHhzHjzHjzH\ jyHjvHmtHmrHmnHmmDjjBhj7fh5ch3af0_f0_c0Ya0Wa0U_0R_\ 0PY0NY0Nj5YyUhzttzzzzzyzzrzzjzzcvzYrzRmzchzlcyjlwj\ tvjyumzsnzqrznrzktzitzfvycvtaymayh_za_zY_z__aa_Hc_\ 0f_0hY0jW0mU0nT0rS9tRJvQUyPPzOLzRHzUDzXBz_7vb3re0n\ h0jk0fn0cq0_t0Ww0Uz0Nz0Hy0Dv07t01r00n00m00j00h00f0\ 0c00a00_00YLYWzWUyURrRPmWPhYPc_P_aPWfRUhRPjRLmRHrU\ FtUBvU7yU3zU1nW3fY3YY3P_3Ha39a3J_FUYPcWanUmyRyfYvR\ cvBjt0rt0yr0zr0nh0ea1gUBhNNhHYc9jU3vK0zA9z0Jz0Uz3c\ zDnz9mz5mtAmjHmaLmWOjNRjFUj9Xj1_j0bh0eh0hh0kh0nh0q\ j0tm0wn0zr0zt0zv0zy0zz0zz0zz0zz0zz0zt0zm0zf3zf3zf } frm:multirot-XY-ZW-new {; draws 6 planes and rotations ;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J ;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994, z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p3, c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p4: z=z^(p2)+c, |z| <= 36 } END 20.0 PAR-FORMULA FILE==================================
On 15 May 2002, at 10:49, Jim Muth wrote:
am indulging in fantasy, but none can dispute that the Mandelbrot set and its associated Julia sets comprise a single four-dimensional object that in a sense most certainly does exist, yet cannot be comprehended by the human mind. This 4-dimensional object is sometimes called the Julibrot figure.
Hmm, I disagree. I believe the 4D Julibrot can be *comprehended* by the human mind - it just cannot be *visualized* by the human mind. David gnome@hawaii.rr.com
At 18:35 16/05/2002, David Jones wrote:
On 15 May 2002, at 10:49, Jim Muth wrote:
am indulging in fantasy,
No, just not that good at it.
but none can dispute that the
Mandelbrot set and its associated Julia sets comprise a single four-dimensional object that in a sense most certainly does exist, yet cannot be comprehended by the human mind. This 4-dimensional object is sometimes called the Julibrot figure.
Hmm, I disagree. I believe the 4D Julibrot can be *comprehended* by the human mind - it just cannot be *visualized* by the human mind.
David gnome@hawaii.rr.com
No, Jim finds he can't comprehend it, and tends to mistake his own abilities and opinions for universal constants. What's so impossible about comprehending a two-dimensional stack of two-dimensional structures? It's like comprehending the graph of y=x^3+ax+b as a and b are varied independently. It could take more or less work, but comprehension always does. How much work depends in part on how much practice and thought is undertaken. Remember the hassles involved in learning to read? Besides, the Julibrot itself can be treated as a slice of and index to an uncoubtable infinity of other higher-dimensional structures. Morgan L. Owens "That's not mathematics, that's computation. If you want computation go get a computer."
On 17 May 2002, at 12:16, Morgan L. Owens wrote:
At 18:35 16/05/2002, David Jones wrote:
On 15 May 2002, at 10:49, Jim Muth wrote:
am indulging in fantasy,
No, just not that good at it.
but none can dispute that the
Mandelbrot set and its associated Julia sets comprise a single four-dimensional object that in a sense most certainly does exist, yet cannot be comprehended by the human mind. This 4-dimensional object is sometimes called the Julibrot figure.
Hmm, I disagree. I believe the 4D Julibrot can be *comprehended* by the human mind - it just cannot be *visualized* by the human mind.
David gnome@hawaii.rr.com
No, Jim finds he can't comprehend it,
Personally, I suspect he comprehends it better than I do. I just poke around and see how they look. Jim actually has some idea of what a formula will give him.
and tends to mistake his own abilities and opinions for universal constants. What's so impossible about comprehending a two-dimensional stack of two-dimensional structures? It's like comprehending the graph of y=x^3+ax+b as a and b are varied independently. It could take more or less work, but comprehension always does. How much work depends in part on how much practice and thought is undertaken. Remember the hassles involved in learning to read?
Nope, did it too long ago. ;-)
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. David gnome@hawaii.rr.com
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."
From: "Morgan L. Owens" <packrat@nznet.gen.nz>
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
I just had to take a look! The well known Julia iteration znew = z^2 + c, implemented in, well, Owens Numbers... I hope you like it. Gerald K. Dobiasovsky ------------------------- OWENS4D.PAR ------------------------- Peek_1 { ; ...a peek into Owens Space ; reset=2002 type=formula formulafile=arch4d4.frm formulaname=RotOwensJulZiZjk corners=-1.6/1.6/-1.2/1.2 params=-0.75/0.75/30/0/0.25/0.5/0/0/50/253 float=y maxiter=1000000000 outside=summ periodicity=0 colors=@altern.map } Peek_2 { ; ...and another one ; reset=2002 type=formula formulafile=arch4d4.frm formulaname=RotOwensJulZiZjk corners=-1.6/1.6/-1.2/1.2 params=-0.5/0.5/30/90/0.25/0.5/-0.25/0.25/50/253 float=y maxiter=1000000000 outside=summ periodicity=0 colors=@altern.map } comment { For a (slightly) different point of view replace the group of variables at the start of the formula by one of the three following groups: HPixZ = (0.965925826289068,0.0) HPixZ2 = -0.258819045102521 VPixZ = (-0.0449434555275477,0.984807753012208) VPixZ2 = -0.16773125949652 NZ = (0.254887002244179,-0.17364817766693) NZ2 = 0.951251242564197 HPixZ = (0.866025403784439,0.0) HPixZ2 = -0.5 VPixZ = (-0.25,0.866025403784439) VPixZ2 = -0.43301270189222 NZ = (0.43301270189222,0.5) NZ2 = 0.75 HPixZ = -0.258819045102521 HPixZ2 = (0.965925826289068,0.0) VPixZ = -0.16773125949652 VPixZ2 = (-0.0449434555275477,0.984807753012208) NZ = 0.951251242564197 NZ2 = (0.254887002244179,-0.17364817766693) } frm:RotOwensJulZiZjk {;4d-algebra by Morgan L. Owens ; ;periodicity=no, outside=summ ;maxit > mymaxit*(stepnum+1) --> mymaxit: max. iterations per slice, ; stepnum: number_of_slices - 1 ; ;Params: p1r: depth of far clipping plane ; p1i: depth of near clipping plane ; p2r: rotation angle of 3d slice around zi ; p2i: rotation in jk-plane ; p3r: cr ; p3i: ci ; p4r: cj ; p4i: ck ; p5r: mymaxit ; p5i: stepnum ; HPixZ = (1.0,0.0) HPixZ2 = 0.0 VPixZ = (0.0,1.0) VPixZ2 = 0.0 NZ = (0.0,0.0) NZ2 = 1.0 ; bailout = 4 ; stepnum = imag(p5) delta = (real(p1)-imag(p1))/stepnum zz = NZ*imag(p1) + HPixZ*real(pixel) + VPixZ*imag(pixel) zz2 = NZ2*imag(p1) + HPixZ2*real(pixel) + VPixZ2*imag(pixel) dzz = NZ*delta, dzz2 = NZ2*delta ang = real(p2)*pi/180 si = sin(ang), co = cos(ang) tmp = real(dzz)*co + dzz2*si + flip(imag(dzz)) dzz2 = dzz2*co - real(dzz)*si dzz = tmp zz0 = real(zz)*co + zz2*si + flip(imag(zz)) zz2 = zz2*co - real(zz)*si zz = zz0 ang = exp(flip(imag(p2)*pi/180)) zz2 = zz20 = zz2*ang, dzz2 = dzz2*ang j = m = i = 0: tmp = sqr(zz) + real(zz2)*zz2 + imag(zz2)*flip(zz2) + p3 zz2 = 2*(real(zz)*zz2+imag(zz)*flip(zz2)) + p4 zz = tmp IF (bailout >= |zz|+|zz2|) i = i + 1 ELSE i = 0 m = m + 1 zz = zz0 = zz0 + dzz zz2 = zz20 = zz20 + dzz2 ENDIF z = m - j j = j + 1 stepnum >= m && p5 >= i } ------------------------- End of .PAR -------------------------
At 07:48 20/05/2002, Gerald K. Dobiasovsky wrote:
From: "Morgan L. Owens" <packrat@nznet.gen.nz>
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
I just had to take a look!
The well known Julia iteration znew = z^2 + c, implemented in, well, Owens Numbers...
Gee, I'm blushing... :-)
I hope you like it.
The depth code in the formula alone makes them worth it. For the pictures themselves, I found myself reminded of photos of asteroids - partly the mood of the pieces, and partly for the air of discovery about them. Morgan L. Owens "There's a world to explore; tales to tell back on shore."
From: "Morgan L. Owens" Sent: Monday, May 20, 2002 9:39 AM
The depth code in the formula alone makes them worth it. For the pictures themselves, I found myself reminded of photos of asteroids - partly the mood of the pieces, and partly for the air of discovery about them.
The original depth code stems from a time the formula parser didn't have the IF...THEN...ELSE logic implemented ==> nearly unreadable code and many hours of calculation on my then AMD486DX2/80MHz (not that my present machine is *that* much faster - I'm only marginally ahead of Jim Muth :-)). Interestingly, no one seems to have come up with the idea of *shading* those depth images - selfmade or Fractint's Julibrot types. After all, in a Fractint "3d transform with lightsource" (without x- or y-axis rotation and no perspective) the source depth image would just have the function of a z-buffer. Although, for a satisfying quality of the shaded picture one needs more available depth values in the source image, which would mean .POT output... Regards, Gerald
On 18 May 2002, at 13:55, Morgan L. Owens wrote:
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.
Then lets phrase it as, "Beyond my available time." ;-) I taught myself topology while in high school a few many decades ago, so I could probably learn it it I set my mind to it. David gnome@hawaii.rr.com
participants (4)
-
David Jones -
Gerald K. Dobiasovsky -
Jim Muth -
Morgan L. Owens