; Date: Fri, 11 Oct 2013 16:01:00 -0400 ; From: Jim Muth ; Subject: [Fractint] FOTD 11-10-13 (Vision in the Sky [A-9,M-6]) ; Id: <1.5.4.16.20131011160217.2bcf6780@earthlink.net> ; --------- ; ; FOTD -- October 11, 2013 (Rating A-9,M-6) ; ; Fractal visionaries and enthusiasts: ; ; Those who have no interest in the fourth dimension should skip ; to the start of the fractal stuff 12 paragraphs below. The ; image is pretty good, at least in my humble opinion. ; ; The fourth spatial dimension, where the Julibrots live, holds ; many things that defy common sense when they are squeezed into ; our everyday three dimensional space. One of the most curious ; is the strange motion known as double rotation, in which a 4-D ; object such as a hypersphere may rotate around two absolutely ; perpendicular planes at the same time, with each plane rotating ; on itself independently of the motion of the other plane. ; ; In 3-D space, when two rotations are applied to an object, the ; rotations combine into a single resultant rotation, but a ; double-rotating 4-D hyper-object is in an entirely different ; condition. This strange motion is impossible to visualize in ; 3-D space, but with some effort I have worked up a crude ; visualization of a hypersphere subject to double rotation. ; Hopefully, my description will communicate at least some of my ; visualization. ; ; The trick is to rotate the view in 4-D space. I begin by ; visualizing a 3-D sphere like the earth. On the curved 2-D ; surface of this 3-D sphere, it is possible to move away from the ; equator at a right angle in only two directions. On the earth ; these directions are straight north and south, and following ; these two geodesics one will reach the polar points of the ; rotating sphere. ; ; But on the curved 3-D surface of a 4-D hypersphere it is ; possible to move away from the equatorial circle at a right ; angle in any direction in a plane perpendicular to the ; equatorial circle. Regardless of the direction one chooses, if ; he sets out at 90 degrees to the equator and follows a straight ; geodesic line on the surface of the hypersphere, he will ; intersect the polar circle. The direction in which he sets out ; determines the point of the polar circle he will intersect. ; ; I now drop a dimension and rotate the view 90 degrees. The ; equatorial circle vanishes except for the point at which the ; explorer is standing. He now sees himself as standing at a pole ; of the sphere. The great circle he sees circling the spherical ; slice of the hypersphere at 90 degrees from his location is the ; actual polar circle. This is as close as I can come to ; visualizing a hypersphere. I do it one 3-D slice at a time. ; ; Rotating back so that the entire equatorial circle is again in ; view, the explorer sees that the visible part of the hypersphere ; is rotating exactly as the earth does, with the equatorial ; circle rotating on itself. He now is observing a 3-D slice of a ; hypersphere in a state of a single rotation, and though he ; cannot see it, he knows that every point of the the polar ; axis-plane is remaining stationary while it turns in place. Now ; the fun begins. ; ; There is nothing to prevent the polar axis from rotating on ; itself also, with the equatorial circle as its axis, and when it ; does, the two rotations are totally independent. The resulting ; motion of the hypersphere is called double rotation and like ; everything else in 4-D space, it is impossible to visualize. ; ; But to make an effort at visualization, let's assume that, just ; as on earth, the equatorial circle is rotating on itself once ; every 24 hours. So far, so good. But now let's assume that the ; polar circle is rotating on itself once every hour. While ; standing very near the equator, the explorer finds his location ; on the hypersphere rotating around the closest point of the ; equator while being carried forward with the equator and tracing ; out a kind of helix. ; ; The explorer sees the point where he stands rotating around the ; equator 24 times every time the equator makes one revolution on ; itself, but the point is not tracing out a simple 3-D helix ; looped into a ring, only a close approximation. The farther ; from the equatorial circle he wanders, the stranger the motion ; becomes. If the explorer were standing very close to the polar ; circle, he would find the point rotating around the entire ; hypersphere 24 times before it rotated around the polar circle ; once, tracing out something like a twisted garden hose, and if ; he stationed himself at a point halfway between the two axis ; circles, the point would trace out something that could be ; visualized only by a being with 4-D vision. The incredible part ; is that the hypersphere remains rigid while subject to double ; rotation, undergoing no strain or distortion. ; ; A vague idea of the motion of double rotation can be found by ; observing the 3-D shadow of a double-rotating skeletal 4-D ; hypercube. Such projections are probably on the internet, ; though I have not yet searched. ; ; The fun does not end here however. In six-dimensional space, ; triple rotation is possible, but that's a story for another day. ; ; Now let's get to today's fractal. ; ; To create today's image I cut the portions of Z of yesterday's ; formula in half. This keeps the critical point unchanged. The ; resulting parent Mandeloid is larger than yesterday's, with more ; prominent 'wings'. ; ; Today's scene lies near a minibrot northeast of the north wing ; of its parent. This minibrot is connected to the main body of ; the parent by a broken filament, but unlike almost all ; minibrots, which are connected to the main body of the parent at ; their East Valleys, this minibrot is connected at the southern ; branch of its Seahorse Valley. Today's scene lies in this ; strange Seahorse Valley connection. ; ; I named the image "Vision in the Sky". Something about it ; reminds me of a dream I had long ago, where I saw a similar ; object in the sky. The math interest rates only a 6, but the ; artistic value is a superior 9. ; ; The rather lengthy calculation time of 4-1/2 minutes is a ; drawback. The web sites will remove this problem however. ; ; Enjoy freedom from calculation pains by viewing the finished ; image at one of the web sites at: ; ; ; ; ; ; ; ; ; ; After a night that brought 3.25 inches, 8cm of rain and some ; minor flooding, today dawned cloudy with light rain still ; falling. The rain ended before noon and the sky brightened in ; the afternoon, but the sun failed to appear. The temperature of ; 64F 18C gave no cause for complaint. The fractal cat, who does ; not like water, spent the day checking the outside conditions ; and sleeping under the chair cover. The humans were thankful ; that the heavy rain resulted in only a few small puddles in the ; fractal basement. ; ; The next FOTD fractal will be posted in the proper time, October ; 13 is the best guess. Until whenever, take care, and don't be ; concerned if you fail to understand what I have written say ; about the fourth dimension and double rotation. I don't ; understand it myself. ; ; ; Jim Muth ; jimmuth@earthlink.net ; ; ; START PARAMETER FILE======================================= Vision_in_the_Sky { ; time=0:04:30.00 SF5 at 2000MHZ reset=2004 type=formula formulafile=basicer.frm formulaname=MandelbrotMix3a function=ident center-mag=+1.075076292720345/+2.073914299796522/\ 3.727144e+011/1/18.75/0 params=0.5/1/0.25/2/0.1666\ 666666666667/3/0.125/4/-0.9999999999999999/0 float=y maxiter=750 inside=0 logmap=339 periodicity=6 mathtolerance=0.05/1 colors=00050`50_50Z50Y50X50W50U50S50Q50O50M40L30K2\ 0I10G00F00C00A008006004002FjXDdSBZO9TK7NG5HC3B8154\ VqVQiQMZMHMHDDDzzzzzzzzzosxfovYitQcrIYpKSnIQlGOjEM\ hDLfCKdBJbAI`9HZ8GX7FV6ET5DR5CP4AN49L38I37F26C2491\ 35123011xkUcWKKGAa2XZ1UW1ST1PR1NO1LL1IJ1GG0ED0BA09\ 807504202X2LU1JR1HP1GM1EK1CH1BF09C08A06704503201up\ FqlEmhDieCeaBaZAYV9US8RO7NL6JH5FE4BA3772331OuGGaA8\ J5U4IM3DF29714lM4hK3eJ3bH3ZG2WE2TD2QB2MA1J81G71D51\ 9406203102Fd1Ea1DZ1CX1BU1AS19P18N17K06H05F04C03A02\ 70150023EP2BK18F15A025h3N_2IR1DI199048mZ7hV6cS5_P5\ VM4RJ3MF2IC2D9196043z4Zr3Uj3Qb2LV2HN1DF18704EZYAQP\ 7HH38820R10N10K10G10D00A006003`z8Tm6M`4EP37C11gXTk\ GNaCHS9BJ6593_d0W`0TX0QT0MQ0JM0GI0DE09B06703307ZG6\ WE5TD5QC4OB4L9q9mqApqBsqCvpCnpcgpc_pcTpcMrcKtcIzcG\ zcFzcSzwdzsazo_zkYzgWzdTz`RzzPzzNzzKzzIzzGzzEzzCzz\ FzzIzzLzzOzzRzzUzzXzzZCDa } frm:MandelbrotMix3a {; Jim Muth z=real(p5), c=fn1(pixel), a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=real(p3), h=imag(p3), j=real(p4), k=imag(p4), l=imag(p5)+100: z=(a*(z^b))+(d*(z^f))+(g*(z^h))+(j*(z^k))+c, |z| <=l } ; END PARAMETER FILE========================================= ; ; ; ; _______________________________________________ ; Fractint mailing list ; Fractint@mailman.xmission.com ; http://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint ;