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| Description: next attempt, like in wonderful mandelbulb, angles are measured, theta around the equator, phi through the poles. phi is measured in multiples(or parts) of pi, the respective value afterwards squared. Theta is doubled. z1 is determined as a point in a coordinate system, theta along the x-axis, phi along y-axis. the distance from zero on the z-axis, in logarithmic scale. the unit sphere will be mapped onto a rectanglenear the origin. the mandelbulb algo is done by choosing a point on the equator, from this point doubling the distance to any point z1, so we will reach z2. Taking the point on the line, which is representing the north-pole, we will get algorithms (similar to the squaring of quaternions,), they are implemented in T. Gintzīs Quasz as "gedatou" and "ventri".(Squaring quaternions is done by doubling phi, measured from the pole, theta remaining unchanged. We can imagine a rubber-ribbon around the equator. Measuring phi from it, we get mandelbulb("rings of fire" in Quasz-but only for power 2), sliding the ribbon to the north-pole, the mandelbulb-algo will change to "gedatou"(if the ribbon is shrinked to one point on the line, representing the pole, if the ribbon is extended along this pole-line, it will change to the squaring of quaternions, so we may see the connection between squaring of quaternions and hypercomplex numbers). Using this coordinate system, we get a certain freedom to develop new algorithms, even sometimes preserving the mandelbrot set in one plane. In this picture still same angles phi are mapped to points with same angles phi2, maybe we could change this to get more harmonic structures in all(?) directions?
Stats: Total Favorities: 0 View Who Favorited Filesize: 89.19kB Height: 675 Width: 1264 Keywords: mandelbrot set threedimensional 3d mandelbulb gedatou ventri rings of fire Posted by: vector  January 03, 2010, 08:31:59 PMRating:  by 3 members.Image Linking Codes
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Comments (2)  |
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vector
Forums Freshman  Posts: 14 
| January 07, 2010, 09:34:19 AM okay, hen or chicken might be more suitable, but i thought the mass of the corpus(often the mandelbrot sets are somehow flattened, using loxodromic functions or related algorithms) and the spiralic trunk-like structure below at the right,and to remember it one year later, i preferred to memorize elephant more than hen. |
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bib
Global Moderator Fractal Senior  Posts: 2070 At the borders...
| January 05, 2010, 11:09:32 PM Interesting! But where do you see an elephant? I can see a flying hen maybe  |
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