FOTD -- February 18, 2004 (Rating 5) Fractal visionaries and enthusiasts: Since yesterday, I have reworked the Julibrot rotation formula and given it a new name. The only change is that the angles of rotation are now entered as degrees rather than fractions of pi. This makes it easier for me to work with. It's hard enough trying to think in four dimensions without doing the conversion math. The new formula is included in the parameter file at the bottom of this letter, but the [frm:] that I have inserted in front of the formula name must be removed before the formula can be copied into a formula file. Using the new formula, today's image takes us to the area we know and love so well -- Seahorse Valley of the Mandelbrot set. But where are the seahorses? Actually, they are in the picture, but they are distorted to such a degree that they have become totally unrecognizable. The thing we know as Seahorse Valley is actually nothing more than a particular part of a two-dimensional slice down the center of the four-dimensional Z^2+C Julibrot figure. This slice is known as the Mandelbrot set. But Seahorse Valley is not limited to two dimensions. It actually extends off into two more dimensions, both of which are perpendicular to the familiar Mandelbrot aspect of the valley, making the entire valley a true four-dimensional object. These two extra dimensions, when displayed on the screen, create the familiar Julia set associ- ated with Seahorse Valley. A slightly distorted version of this Julia set may be seen by making one full outzoom from today's image. The Julia set is distorted because it is not a true Julia set. The orientation of the image on the screen has been slightly rotated in four different directions from the true Julia orientation. The spirals appearing on the tips of the valleys are Julia features which are familiar enough, but what is that straight- edged border cutting diagonally through the scene? It is not part of a Julia set, nor is it a part of the Mandelbrot set. It is the edge of something entirely new, which I call a bridge, a true 4-D object totally impossible for mere 3-D beings to visualize. These straight edges are one of the most common features of the odd slices of the Julibrot, and make up an entire world in themselves. We will be seeing many more of them as we delve deeper into the hyperspace of four dimensions. Today's image, which consists mostly of gaudily colored spirals and a straight edge, has been named "Where is the Seahorse". The name was inspired by the fact that, though the scene is Seahorse Valley, no seahorses are recognizable. The comb-like fringes throughout the image are true features, and not artifacts. These fringes are also very common in the odd slices of the Julibrot, especially in the areas associated with Mandelbrot valleys. Sometimes these odd slices are rather unpleasant appearing, as illustrated by today's image, which I could rate at only a 5. Taking the render time of 23 minutes into consideration gives an overall value of 21. Today's hyper-image may be seen by starting the parameter file and sitting back to watch the fun, or by downloading the finished GIF image from Paul's web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> thereby saving time but missing the fun. The temperature reached 36F 2C and the clouds increased threateningly on Tuesday here at Fractal Central, but the forecasted snow never arrived. The cats not only approved of the situation, they actually found the courage to endure over 1/2 hour in the yard at the warmest time of the day. And when evening came, no treat was needed. Today is starting sunny and milder. It should be an even better day for the local cats. For me it will be an average day. Most days are. And when I start pondering the fourth dimension, that will also be part of an average day. Until next FOTD, take care, and where does one look to find this fourth dimension we hear so much about? Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ WhereIsTheSeahorse { ; time=0:23:49.82--SF5 on a P200 reset=2003 type=formula formulafile=allinone.frm formulaname=SliceJulibrot passes=t center-mag=-1.11565/-0.0388942/7.803134/1/-35.0000\ 000000000142/3.85524945301085609e-014 params=86/89/89/89/-0.75/0.03/0/0 float=y maxiter=65000 inside=0 logmap=yes periodicity=10 colors=000bOScNQdMPeMNfLMgKLhKJiJIjIGkIFlHEmGCnGBo\ F9pE8qD6rE7qF8oG9mG9kHAiIBgJCeJCcKDaLE_LEYMFWNGUOH\ TOHRPIPQJNRKLRKJSLHTMFUKGTMGSNGROGQPGPQGPRGOSGNTGM\ UGLVILWKKUMKSOLUQMWSMXUNZWN`YOa_OcaPdcPfeQhgQiiRkk\ QkmRloSmpTnnUolVojWpiXqgYreZsc_sa`t_auYbvWawXcvXev\ YgvYhvZjvZlvZmv_ov_qv`sv`tvavvaxv`zyayvbxscxpdwmev\ jfvgfudgtahtZisXjsUkrRkqOlqLmpInoFooCpn9pn7okAohDo\ eFobIoRKoQNoPPoNSoMUoKYoIaoFbo9coIcnNblRajU_hWZfYY\ d_WbaV`cUZeVXgWVgWTgXRfXRfYTfYUfZVe_Xe_Ye`Ze``daad\ abdbddceccfcdhcdiceicejceigZhkTfpMetFey8dx9dwAdvAd\ uBdtBdsCdrCdqDdpDdoEdnEdmFdlFclGckGcjHciHchIcgIcfJ\ ceJcdKccKcbLcaLc`Mc`MdaNdaOdaPebQebRebSecSfcTfcUfc\ VfdWgdXgdXgeYgeZhe_he`hfahfaifbigcigdigejgfjhfjhgj\ hikikkimkioljqkiskhukhwkgxkgzkfzjezjezjdzjdzjczjbz\ ibziaziazi`zi_zi_zhZzhZzhYzhXzhXzhWzgWzgVzgUzgUzgT\ zeSzgTzhTziUzkQzmQzmPzmOz } frm:SliceJulibrot {; draws most slices of Julibrot pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1*0.0055555555555556), b=pi*imag(p1*0.0055555555555556), g=pi*real(p2*0.0055555555555556), d=pi*imag(p2*0.0055555555555556), ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g), sg=sin(g), cd=cos(d), sd=sin(d), p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd), q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd), r=u*sg+v*ca*sb*cg, s=v*sin(a), c=p+flip(q)+p3, z=r+flip(s)+p4: z=sqr(z)+c |z|<=9 } END 20.0 PAR-FORMULA FILE==================================