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jehovajah
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« Reply #60 on: January 02, 2010, 10:34:16 AM » |
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In a fractal i would expect discontinuities at boundaries where one region juxtaposes with another. The mathematical horror which called these things monsters must now give way to the realisation that their beauty and artistic worth allied with computers means the computational swamp no longer need fill us with dread.
I like your train of thought.  Regarding your "vx" formula, I have also wandered down rabbit holes thinking about similar approaches. If we boil the 2D Mandelbrot down to its core, it is a process where do this, over and over: z := z 2z += z 0I tend to think about things like "what does it mean, conceptually, to 'square' a number?" It's simple enough with reals: 1) Take a (1D) line of length x. 2) Take another line of length x and lay it perpendicular to the original line. 3) Take those two lines as sides of a square, and the area of that geometric shape is the "square" of x. Extending that concept to 3D is quite natural (just form a "cube" instead of a "square"), but complex numbers aren't quite as easy to visually "square." Squaring a complex number amounts to a rotation about the origin and a scaling of magnitude. The term "square" doesn't have such a nice geometric analog any more. Looking at the number being squared in polar form is sometimes more revealing... ^2 = r^2 e^{2i\theta}) In two dimensions, what it's saying is: 1) Take the distance to the point from the origin and square it (using the word 'square' here in the real sense). 2) Take the angle to the point measured from the +x axis and double it. 3) At the point described by this new distance and angle is your new, "squared," complex point. Now we immediately run into the big question... How do we extend that concept into three dimensions? Things like this keep my mind very busy. Your train of thought prompted me to explain that the identity  =  Is a logarithmic transformational one which euler derived based on a naperian construction process for logarithms. Napers construction for logarithms is by definition 2d. To extend it to 3d therefore you would need to define a construction of logarithms in 3d . You may also need to derive a 3d equivalent to the sine function. In any case it is an interesting project and may lead to a better understanding of so calle 3d |
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #61 on: January 02, 2010, 11:24:53 AM » |
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David you said that the cut down quaternion would lathe and i think paul shows such a rendering, so can you explain what is happening in these renderings? The first is a 3d representation of z=z2 +c. The second and third if it shows are
z= z2- 2yzij +c which should be the same as
real(z)2-imag(z)2-imaj(z)2 +2*real(z)*imag(z)*i+2*real(z)*imaj(z)*j+c
which is <Quoted Image Removed>.
I don't see how "-2yzj" is the same as "+2xzj" ? Did you remember to avoid variable corruption ? I mean you can't do: zri = zri^2+cri zj = 2*real(zri)*zj + cj You have to do (for example) zj = 2*real(zri)*zj + cj zri = zri^2 + cri I put the formulae in as described in my post. Also i have used a standard expansion with i*j not defined by myself but by the programme. This is not your programme it is by terry w glintz. |
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« Last Edit: January 04, 2010, 07:26:16 AM by jehovajah »
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #62 on: January 04, 2010, 07:38:40 AM » |
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@ David
can you write z^2+c in terms of real(zri), zj and presumably zi ?
Is zj = imag?(zri) and zi = imag(zri) ? in your programme?
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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David Makin
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« Reply #63 on: January 04, 2010, 12:58:21 PM » |
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@ David
can you write z^2+c in terms of real(zri), zj and presumably zi ?
Is zj = imag?(zri) and zi = imag(zri) ? in your programme?
In my program complex z^2+c would simply be zri^2+c, however for the triplex zri (complex) is the r and i part of the triplex and zj (real) is the j part, so the (-sine) Mandelbulb (trig version) in UF code could be simply: magn = sqrt(|zri| + sqr(zj)) th = power*atan2(zri) ph = power*asin(zj/magn) magn = magn^power zri = magn*cos(ph)*(cos(th) + flip(sin(th))) + cri zj = -magn*sin(ph) + cj where |zri| is x^2+y^2 if zri is x+i*y. |
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jehovajah
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« Reply #64 on: January 05, 2010, 05:34:50 AM » |
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Thanks dave.
I see what is happening in terry glintz programme now. Although he uses Quaternion or hypernion math to shape the mandlebulb i can write a bespoke formula for the iteration. I am not sure how he plots the result but i do not get a lathed mandlebulb for the above equation.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #65 on: January 08, 2010, 03:34:41 AM » |
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so i got the lathed mandelbrot finally! Beginners mistake really. So here are some 3d ones using terry glintz programme.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #66 on: January 08, 2010, 04:06:39 AM » |
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Some more variations.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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vector
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Posts: 14
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« Reply #67 on: January 09, 2010, 07:25:35 PM » |
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hi, i found it useful to take the geometrical view:
in the picture, it is shown, how we can interpret the 2d and 3d mandelbrot algorithm, means, the squaring of a respective number z1. I used this to write some programs, varying the way, how a z1 will be "squared". If we preserve in one plane the pattern, shown at top left, we will get in this plane the original 2d set. Its trajectorial field consists of logarithmic spirals, working together with the constant vctors added, as shown at the right above. Now, what is the best corresponding field of trajectories in 3d? The respective field when squaring quaternions is rotating the complex plane around the x-axis until z1 will lay in this oblique plane, then it will be squared in the rotated plane as usual. The resulting trajectorial field looks in some direction not so rich, therefore the quaternion-M-set is a little bit disappointing. Other approaches are shown below: in "gedatou", (implemented in QUASZ by T.Gintz) i took one point (zero)on the line, representing the north pole in the coordinate system, which represents a flattened uni circle in the green rectangle, squaring is done by doubling the distance from zero.(in the shown ccord-system, the z-axis will represent the log(|z|), so doubling from zero will cause squaring of the distance(in the "real coordinate system)). I also wrote "rings of fire"(implemented in QUASZ as well about 10 years or so ago), which uses the way, which is used in the wonderful mandelbulb(great work-congratulations), it used preferably only the squaring-mode,(was experimenting with higher multiplications of angles, because f.e. tripling the angles, the north pole would not be projected to the equator, as it will be by squaring, but to the opposite pole-but i usually did not triple .. the distance-and-in my beloved basic-programs-fast and furious-the resulting objects did look like the multiples we get when doing higher powers in the 2d-sets-but i was not far from it! Anyway, the mandelbulb objects are looking great, i wonder how they are producing the fine structures? In the mandelbulb-algorithm a squaring(or respective taking to higher powers) is done by doubling in the shown("flattened") coord.-system the distances of any points z1 from a point on the equator. While in "gedatou"(the name a combination of german"gehirn"-(brain), because often the resulting objects resemble to brains(and this not just happening, i wrote some posters and papers about fractal neural nets, using the rich connectivity of fractals) and Fatou-), the z1-points will be sent from one pole on a journey around the meridians, while these are rotating around the polar axis, the points in the mandelbulb-algo will be sent from the equator alon the meridians, which are as well rotating around the polar axis. best regards (I am not right sure, whether the picture will be shown, -hope best)
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« Last Edit: January 09, 2010, 07:29:02 PM by vector »
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vector
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« Reply #68 on: January 09, 2010, 07:43:18 PM » |
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some might enjoy a picture, made with "gedatou"-seems to be too large to be uploaded here or to the gallery-as said in the reply above, it is related to the mandelbulb-as well as to the quaternion-squaring-done with chaos pro, the (very provisionary-you have to scroll down little bit)link: http://trajektorulm.tr.funpic.de/index.htmthe formula as: a1 = x1 * x1 + y1 * y1 + z1 * z1; //a2 = a1;//: REM new distance if squaring only a1 = sqrt(a1); loa1=log(a1); a2=exp(power*loa1); yioi1 = y1 * y1 + z1 * z1; yioi1 = sqrt(yioi1) ; //alpha1 = angle(x1, yioi1); alpha1 = arg(x1+1i* yioi1);(would be in C++ alpha1=atan2(x1,yioi1))// if not, then atan2(yioi1,x1) gamma1 = arg(y1+1i* z1); alpha2 = power*alpha1; gamma2 =power* gamma1 ; x2 = cos(alpha2) * a2 ; yioi2 = sin(alpha2) * a2 ; y2 = (yioi2) *cos(gamma2); z2 = (yioi2) * sin(gamma2) ;  |
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jehovajah
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« Reply #69 on: January 10, 2010, 12:47:41 AM » |
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Hi thomas and welcome. I am trialling terry glintz programme and am impressed with its usefulness. Your diagram enlarges ok when clicked on. Terry says he is bringing out an updated version with the mandlebulb formulae in built soon. For me the ability to explore variations of formulae quickly is the most useful aspect of his programme. I do not understand yet how zplot works but it seems to give very good renditions of the mandlebulb just in 3d using i and j unit vectors. Anyway your exposition is welcome.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #70 on: January 10, 2010, 06:01:33 AM » |
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Here is gedatou from terry glintz excellent quasz trial version.
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« Last Edit: January 10, 2010, 06:15:12 AM by jehovajah »
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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jehovajah
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« Reply #71 on: January 10, 2010, 06:46:16 AM » |
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Using
z= x^2-y^2-z^2+2y(x-z)i+2zxj,z=z^7+c
and also
z= x^2-y^2-z^2+2yxi+2z(x-y)j,z=z^7+c
Gives the following.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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vector
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« Reply #72 on: January 10, 2010, 07:38:54 PM » |
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hi jehovajah, thank you for your reply, the pictures are loking very interesting, will later try to understand, what is happening in your formula. Found, on can download the pictures attached in original size, by clicking on the blue link left below, so i only attach the additional trajectorial fields, shown for points on the unit sphere, for programs quaternions, gedatou and mandelbulb. Would be interesting to see, how the trajectories will wander throughout the unit sphere in the formulas, you showed. In gedatou, multiplying "alpha"(latitude, corresponds to phi in mandelbulb) with higher values gives a mandelbulb-like aspect, more than if we change alpha(identical with "theta" in mandelbulb-longitude). The journey around the poles enables the vectors, to stay around zero, because the points z^2 will lie often on the opposite half of the unit sphere, so adding the orriginal vector will end up in most cases within the unit sphere and then taking the distance to a high power will end up very near to zero, so the next addition has a great chance to stay within the unit sphere. This seem to be a basic mechanism how the mandelbulb structures evolve. With power 2 the distances will not be so small after squaring the numbers within the unit sphere, so they may leave more often the uni sphere after addition of the original vector c. (How can one change the inbuilt formulas in quasz? And btw-trying to upload a picture to fractalforums, i often get the message: error 500, internal server error-anybody knows, how to manage that? The large picture with the combined trajectorial fields is therefore for afficionados available at a provisionary site: please scroll down a little bit: http://trajektorulm.tr.funpic.de/index.htm |
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jehovajah
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« Reply #73 on: January 12, 2010, 07:08:04 AM » |
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some of your spirals? This is twinbee formula in quasz trial at .0001 really small value and 12 iterations bailout 6.
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May a trochoid of 去逸 iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
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vector
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« Reply #74 on: January 12, 2010, 10:25:34 PM » |
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this is gedatou with power 3 (odd powers will save he complex plane as in the mandelbrot-al gorithm it is iterations 7, with higher iterations not looking really great, nevertheless interesting (your pictures show complete fractals, interesting how the rectangular grid you got, what is the really small number/value(power n in z^n ? The spirals are ways, we could imagine, that points will wander, if not taking z=z ^2 + c directly, bt going continuously from z to z ^1.1, z ^1.2, z ^1.3 etc. the fields of spirals, we get by his give us a good impression an information, whether the fractal resulting will be interesting)  |
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