Jim Muth's "...most surprising discovery of all [his] fractal explorations."
############################################################## In Jim Muth's FotD for September 19th, 2006, named "Rectangle variation" he says:
One day quite a few years ago, I stumbled upon this rectangle, which still remains the most surprising discovery of all my fractal explorations. The rectangle is impressive even in its pure form...
I agree with Jim. This is a really extraordinary pattern to find -- even in the often strange fractal world. This fractal really struck me when I first saw it in 1998 and I have remembered it ever since. I was very pleased to be reminded of it again by Jim and to be able to recalculate it at much higher resolution on my more recent computer, as well as zoom into the rectangle's pattern. In addition to the object's clearly being a gently-curving rectangle of a single color, it has another really astounding feature. It has an embedded fractal pattern, like a pattern on a rug. The pattern is of a fractal tree composed of narrow triangles branching at right angles to one another. A version of this tree-like shape is included in Mandelbrot's "The Fractal Geometry of Nature." I'm not able to locate my copy right now as I have just moved also, but if memory serves, Mandelbrot's version was created by a substitution process similar to the ones used to create Hilbert curves and Koch snowflakes. I remember being struck at the time by the fact that a particular fractal formed by a simple substitution process could also be created by a complex formula's 'finding' it hidden at a very specific location in the four-dimensional Julibrot. This tree-like shape appears to be one of the 'self avoiding' branching fractal patterns that, no matter how often limbs branch off, always seem to 'know' where other branches are and carefully avoid them. Perhaps someone who has a copy of the book near at hand can locate the image and confirm the reference and give the page number for me. The rectangle in this image also seems to be suspended in space by a delicate 'railroad switching yard' pattern prominent in Jim's September 17th, 2006 FotD. Zooming into the rectangle reveals that each of the narrow triangles that form the tree-like pattern is covered with the 'switching yard' pattern. At the resolution of Jim's original rectangle image this 'railroad track' pattern on the narrow triangles only appears as a dusting of a few random pixels. If you remove both of Jim's two degree rotations by changing params=88/88/ to params=90/90/ the rectangle loses its slight curve and becomes very rectangle-like. However, the rectangle's tree-like pattern disappears and zooming cannot locate it. If you increment both of Jim's two degree rotations by by two more degrees by changing params=88/88/ to params=86/86/ the rectangle becomes a comet complete with tail. The rectangle's/comet's tree-like pattern remains in this version. Below my signature is Jim's October 7th, 2003 FOTD, which is a repeat of the January 1st, 1998 "Classic F.O.T.D." - Hal Lane ######################### # hallane@earthlink.net # ######################### -------------------------------------------------------------- Date: Tue, 07 Oct 2003 08:18:17 -0400 From: Jim Muth <jamth@mindspring.com> Subject: [Fractint] Classic FOTD 01-01-98 (Projective Plane) To: fractint@mailman.xmission.com Cc: philofractal@lists.fractalus.com Message-ID: <1.5.4.16.20031007081910.0d5759b0@pop.mindspring.com> Content-Type: text/plain; charset="us-ascii" Classic F.O.T.D., January 01, 1998 (Projective Plane) Fractal visionaries: The world of higher dimensions is filled with objects which are impossible in our three-space, and can be represented here in only a distorted way. The Klein Bottle is one such object. In three dimensions it appears as a closed figure which intersects itself and joins itself in such a manner that despite having no breaks it has only one side. The inside is also the outside. But this is a distortion of the true object, which can exist only in spaces of four or more dimensions. In four dimensions, the Klein Bottle is constructed by taking a rubber sheet, curling it and connecting one pair of edges so that a tube results, then bending the tube and joining the open ends into a doughnut shaped object. But before joining the edges, and with no cutting, the tube is given a half-twist and turned inside- out, so that without self-intersection, the resulting doughnut- shaped object has only one side. Its inside is also its outside. The Klein Bottle is difficult enough to visualize, but the Projective Plane is even more difficult. In fact it is difficult to even describe. In this case, the sheet of rubber is given a half-twist into a kind of Moebius Strip tube before being curled and given a second twist before the open ends are joined to each other, forming the Projective Plane. In this case, even a distorted model is nearly impossible in three-space. Well, if an accurate model of a Projective Plane is impossible in three dimensions, one could never hope to illustrate the monster on a two-dimensional screen. I named today's fractal "Projective Plane" only because that's what I thought of when I saw the image. Actually, it is a picture of a curious feature that appears at Z=0.00019,0.07388 C=-1.7435,0.0 in the Z^2.003 Julibrot figure. This object is extremely thin and exists only very near the Julia orientation, where it appears as a near- perfect rectangle. To create today's image, I gave the object a 2-degree double rotation from the Julia orientation, which distorted the rectangle into the curved shape in the picture. A little playing with the colors produced the effect of a flying sheet of rubber. The flying plane has landed at Paul's web site at: http://home.att.net/~Paul.N.Lee/FotD/FotD.html Tomorrow, I'll have another interesting FOTD. At this time I have no idea what it will be, but something will turn up -- as it always does. Until then, take care, and keep finding those fractal gems. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER-FORMULA FILE================================ Projective_Plane { ; 3-1/2 min on a P200 at SF5 reset=1960 type=formula formulafile=multirot.frm formulaname=multi20031 function=flip/ident/ident\ /flip passes=1 center-mag=-0.00037327503160875/\ +0.00003799627771399/514.8005/1/25 params=88/88/0.00022/0.0755/-1.74308/0 float=y maxiter=1800 bailout=25 inside=253 logmap=yes symmetry=none periodicity=10 colors=000QVZ<2>PXZ\ PYZP_Z<7>PlZPnZRt_<4>PmZPkZOhZ<6>MTZMRZMQZ<12>I9\ ZI8ZJ6a<19>I8OI8NH9KH9IH9HH9IH9IH9HH9FH9FH9H<11>\ HA9HA9JCB<2>OIGQJHRMJ<7>dgVejWgkY<3>nncooeqnf<3>\ wshxuixwjyylyzm<4>zwizvhzugzugztgzthzshzsh<10>pl\ `ol_mkZ<7>YfTWfSVdR<3>RZOQXNPYN<6>IUHHTGGUG<3>I_\ IJ`IJ`I<38>asSXrT<19>stPttPvvJvuM<3>oqWmpZmqZ<3>\ moZnoZopZskZwzZzwZzwZ } frm:multi20031 {; Jim Muth, best=ifif, fiif, fifi, iffi a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994, z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p2, c=cos(b)*fn3(real(pixel))+cos(a)*fn4(imag(pixel))+p3: z=z^2.003+c, |z| <= 100 } END PARAMETER-FORMULA FILE================================== ############################################################## -- No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.1.405 / Virus Database: 268.12.5/450 - Release Date: 9/18/06
Hal, Jim's Sept. 17th FOTD: FractalOf_Tomorrow: params=45/45/... Jim's Sept. 19th FOTD: RectangleVariation params=91.81/90.94/91.93/89.48/... You say: ---------------------------------- If you remove both of Jim's two degree rotations by changing params=88/88/ to params=90/90/ the rectangle loses its slight curve and becomes very rectangle-like. However, the rectangle's tree-like pattern disappears and zooming cannot locate it. If you increment both of Jim's two degree rotations by by two more degrees by changing params=88/88/ to params=86/86/ the rectangle becomes a comet complete with tail. The rectangle's/comet's tree-like pattern remains in this version. ------------------------------------------ I can't find any reference in either of these images that you refer to with params=88/88/ Am I missing something?? Lee
##############################################
I can't find any reference in either of these images that you refer to with params=88/88/
Oops. Apparently I failed to make it clear that I was referring to modifying the parm 'file' contained *within that email*... Sorry to have you go to the trouble of looking for it in the wrong places. Oh! Also note that there was also an error in the date that Jim quoted as the original date the fractal enclosed in that email was posted. I duplicated that error in my previous email. The fractal was actually posted Jan 6th, 1998 rather than Jan 1st, 1998 that Jim erroneously said in his 7th Oct, 2003 FOTD post. I sent a follow-up email to [Philofractal] and [Fractint] revisiting this date issue. Just to be extremely clear, my previous email's subject was: Jim Muth's "...most surprising discovery of all [his] fractal explorations." Here's a snippet from that previous email showing the "params=88/88/" in question: ======= Quoted text =========== Until then, take care, and keep finding those fractal gems. Jim Muth jamth@mindspring.com jimmuth@aol.com START PARAMETER-FORMULA FILE================= Projective_Plane { ; 3-1/2 min on a P200 at SF5 reset=1960 type=formula formulafile=multirot.frm formulaname=multi20031 function=flip/ident/ident\ /flip passes=1 center-mag=-0.00037327503160875/\ +0.00003799627771399/514.8005/1/25 params=88/88/0.00022/0.0755/-1.74308/0 float=y ==== End of Quoted text ============ I hope you're still willing to try those changes to the parameters. The results show a little of what can happen as things are rotated in small increments in the 4th dimension. Fortunately the fractals calculate relatively rapidly. Changing the parameters from 90/90 to 88/88 and then to 86/86 nicely reveal the trend of what happens to a geometric shape -- a rectangle -- as it gets warped in aesthetic ways. We don't often have the luxury of a geometric shape to start with to gauge its amount any type of distortion as a formula's parameters are varied, so I found this an opportunity I couldn't pass up. Actually, I should be using the evolver to look at many more of the possibilities in the Julibrot. It can effectively show four dimensions at once -- the two regular real and imaginary and whatever two additional parameters you choose to vary in the formula, say, rotations about two different multi-dimensional 'axes.' What a powerful way to explore! --------------------------------------------------- In Jim's FotD from September 19, 2006, named "Rectangle variation" he states that the fractal is derived from 'the original rectangle fractal.' That reference is what got me looking for the rectangle fractal that I remembered all these years. The thing that stuck with me was that a particular fractal (the tree-like pattern in the rectangle) formed by a simple substitution process (shown in Mandelbrot's book) could also be created by a complex iterated formula in a rotated slice from the four-dimensional Julibrot. I'm still amazed by that. In "Rectangle variation" you can still make out the 'rectangle' as the warped confetti-colored 'door mat' in the center of the action. The 'railroad switching yard' pattern prominent in Jim's September 17th, 2006 FotD is also visible in the periphery of "Rectangle variation." But the other material in "Rectangle variation" has apparently rotated into the image after being hidden in other dimensions... - Hal Lane ######################### # hallane@earthlink.net <mailto:hallane@earthlink.net> # #########################
-----Original Message----- From: Lee H. Skinner [mailto:skinner@thuntek.net] Sent: Tuesday, September 19, 2006 2:33 PM To: philofractal@lists.fractalus.com; Fractint and General Fractals Discussion Subject: Re: [philofractal] Jim Muth's "...most surprising discovery of all [his] fractal explorations."
Hal,
Jim's Sept. 17th FOTD: FractalOf_Tomorrow: params=45/45/...
Jim's Sept. 19th FOTD: RectangleVariation params=91.81/90.94/91.93/89.48/...
You say: ---------------------------------- If you remove both of Jim's two degree rotations by changing params=88/88/ to params=90/90/ the rectangle loses its slight curve and becomes very rectangle-like. However, the rectangle's tree-like pattern disappears and zooming cannot locate it.
If you increment both of Jim's two degree rotations by by two more degrees by changing params=88/88/ to params=86/86/ the rectangle becomes a comet complete with tail. The rectangle's/comet's tree-like pattern remains in this version. ------------------------------------------ I can't find any reference in either of these images that you refer to with params=88/88/
Am I missing something??
Lee ##############################################
-- No virus found in this outgoing message. Checked by AVG Free Edition. Version: 7.1.405 / Virus Database: 268.12.5/450 - Release Date: 9/18/06
participants (2)
-
Hal Lane -
Lee H. Skinner