Calcyman


« on: March 22, 2010, 10:21:37 PM » 

On the thread about Gimbal lock, it was suggested that different topologies, not just the sphere, should be explored. An obvious starting point is the onehole torus, or hollow doughnut. It has a number of desirable properties, including simplicity and smoothness. The coordinates on (the surface of) a torus can easily be specified by two angles. These could be designated theta and phi, altitude and azimuth, longitude and latitude, or even Mary and Bob (indeed, whatever you want to call them!) Another (more exciting) topology, which isn't so well known, is Klein's quartic. Despite the complicatedsounding name, it is essentially a threeholed torus. It is of special interest because it contains the group PSL(2,7) as its symmetries, which means that it has 336 automorphisms including reflections. In fact, the Klein quartic is the most symmetrical threeholed torus possible. This compares with the 120 automorphisms of the dodecahedron, which is the maximum finite symmetry on the surface of a sphere (excluding dihedral groups, which are pretty boring). An accessible, yet detailed, description of Klein's quartic is here: http://math.ucr.edu/home/baez/klein.htmlThis is of particular interest, since a Riemann surface (like Klein's quartic) is just a complex plane, embedded in threedimensional space. And since there are so many fractals on the complex plane, it should be easy to migrate to these exotic surfaces. Even better, the quartic has a simple formula to embed it in 3D space: x³y + y³z + z³x = 0



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reesej2
Guest


« Reply #1 on: March 22, 2010, 10:43:26 PM » 

Exciting! These could produce some really interesting fractals. One thing, thoughspheres are convenient because every point in 3D space is on the surface of a unique sphere centered on the origin. It'd be tough to arrange a similar system for a torus or a Klein quartic. Unless, of course, we look only at the surface of a fixed version of one of these and designate a "point at infinity" by which to measure escape? (I could be misunderstanding, is that what you meant already?) I'll have to take a look at this possibility.



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kram1032


« Reply #2 on: March 22, 2010, 11:02:22 PM » 

Looks like a challenge. I read somewhere on the linked page, it's impossible to exactly render the shape into euclidean 3D space? Why is that? Topologically, it's a 3holed torus but it does not allow an undistorted view in 3D space for some reason?



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jehovajah


« Reply #3 on: March 23, 2010, 05:56:08 AM » 

Thanks for the link to john Baez. He is a good communicator. I hope he does not give up the playful approach under the pressure of his new venture. By the way, the representation on this 3 holed torus or any general Riemann space will suffer from the 2d/3d drawing problem.It may be something that has to be realised on 3d printer technology. The animation though impressive is not intuitive. As Klein himself says : Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.  Felix Klein
Which i take to mean those who get their hands dirty sticking and gluing and using pastry have inevitably a better sense of the object than those who view it symbolically.
What i mean by the animation not being intuitive is that i would not intuit that shape or rotational flexibility,however if i had built the thing as per Klein's instructions my intuition would be formulated and enhanced. Like i say playing around is the best way to enjoy and comprehend iterative thinkingalso called math.



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May a trochoid of ¥h¶h 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!



Timeroot


« Reply #4 on: March 24, 2010, 01:12:35 AM » 

This sounds pretty sweet! I can think of at least one good of way implementing the torus: say Mary is the angle of the "large" circle, relative to the xaxis, and Bob is the angle from a point on the large circle relative to the xy plane (or, the angle to the origin). Then do the same thing, essentially: Mary=Mary*n, Bob=Bob*n, Radus=Radus^n.
Personally, I would really like to call the angles Mary and Bob, and call the distance to the origin Radus. This nomenclature would be so utterly strange that all confusion through ambiguous words for angles / radii of other objects would be completely lost! I think this naming system should also be applied to the regular Mandelbulb. Mary=Longitude, Bob=Latitude. All in favor say "Aye".
I agree Jehovajah, playing around with something is the best way to learn about it. I spent literally days going back to the "Rhombic Dodecahedron" page on Wikipedia, trying to understand from the picture and it's duality how it would look. Then I built one and... Ohhhh!



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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.



David Makin


« Reply #5 on: March 24, 2010, 01:36:20 AM » 

I'm almost certain someone already posted a version using a torus, but I forget who and don't have time to dig up the post now....maybe someone else's memory is better than mine



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Timeroot


« Reply #7 on: March 24, 2010, 03:25:46 AM » 

Hmm... a similar definition, except I would also try it with Radus being the distance to the origin, in lo of the distance to the large circle. Another option is the distance to the surface  that version of polar coordinates, except radius=ralpha. I viewed the Kelin quartic ( http://www.wolframalpha.com/input/?i=Klein+quartic&a=*C.Klein+quartic_*Surface), and it doesn't look anything at all torodial...



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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.



jehovajah


« Reply #8 on: March 24, 2010, 08:54:01 AM » 

Try here: http://www.fractalforums.com/theory/toroidalcoordinates/for toroidal coordinates. In the meantime some sculptures inspired by x ^{3}y+y ^{3}z+xz ^{3}=0



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May a trochoid of ¥h¶h 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!



jehovajah


« Reply #9 on: March 24, 2010, 09:14:19 AM » 

I am afraid Wolfram Alpha is still at a very basic stage, as sophisticated as it is.



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May a trochoid of ¥h¶h 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!



jehovajah


« Reply #10 on: March 24, 2010, 10:37:22 AM » 

some more sculptures.



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May a trochoid of ¥h¶h 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!



kram1032


« Reply #11 on: March 24, 2010, 06:13:48 PM » 

Timeroot: It looks like what Wolfram Alpha shows is just one or so solution if you solve it for z, however it's supposed to be a quartic surface, so probably 4 kinds of solutions need to be considered. (For simpler geometries like spheres, that works just fine, though.) Also, there actually where more than one version of torodial coordinates as far as I remember. (Or at least threads about them...) They where disrearded because they basically did the same as the Mbulb but with torodial shapes rather than spheres. However, while I don't think, this will be "the true" Mbulb, the results could still be interesting. Actually, it's generally very interesting how iterative, algebratic and geometric maths combined in several threads here to form totally new results and one based on toroids with three holes, rather than one, could be pretty neat


« Last Edit: March 24, 2010, 06:20:20 PM by kram1032 »

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jehovajah


« Reply #12 on: March 25, 2010, 03:32:37 AM » 

Some more sculptures showing the change twixt > and >=0. I am afraid ==0 does not compute! If you can imagine the knickers this fat lady would have to wear you will get some idea of the kleinian quartic range of shapes spanned by this sculpture! This range of knickers could be called Quartic Kleins!


« Last Edit: March 25, 2010, 03:48:01 AM by jehovajah »

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May a trochoid of ¥h¶h 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!



Calcyman


« Reply #13 on: March 25, 2010, 08:47:59 AM » 

I viewed the Kelin quartic, and it doesn't look anything at all torodial... This may take a lot of explaining. The Klein quartic is a curve of complex numbers, which is equivalent to a surface of real numbers embedded in 3D real space. The quartic can be folded into a 3holed torus, in which case it occupies a finite space. However, the curve x³y + y³z + z³x = 0 is equivalent to a Klein quartic with some points at infinity (namely the vertices of an octahedron). In other words, it is 'open', rather than 'closed'. It would be toroidal if you added an extra 'point at infinity' to transform R³space into a 3sphere, in the same way that the rectangular hyperbola is equivalent to a circle.



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