; Date: Wed, 18 Feb 2004 10:05:14 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 18-02-04 (Where is the Seahorse [5]) ; Id: <1.5.4.16.20040218100751.298f875e@pop.mindspring.com> ; --------- ; ; 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: ; ; ; ; 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 PARAMETER 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 PARAMETER FILE================================== ;