March 2011 - mathfun@mailman.xmission.com

[math-fun] The hunt for the "true" 3D Mandelbrot
by David Makin 02 Mar '11

02 Mar '11

Hi all, Am currently on a Mac Mini and have no software to render 3D+ fractals on here at the moment but thinking about the 3D Mandy I did come up with a couple of ideas using the reduced (3D) quaternion method: Method 1: Per iteration: t = x; x = x^2 - y^2 - z^2; y = 2*t*y; z = 2*t*z Now rotate (y,z) by atan2(pixely,pixelx) (or minus the angle) then add the constant. Method 2: As above but instead rotate (y,z) by atan2(oldy,oldx) (or minus the angle) Method 3: As above but instead rotate (y,z) by atan2(y,z) (or minus the angle) I don't think method 3 will produce the desired result but I'm really interested to see the results of 1 and 2 :) bye Dave

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Re: [math-fun] Conformal mapping in R3+
by Dan Asimov 02 Mar '11

02 Mar '11

I wrote: << I don't know why fractals in 3-space would need to be based on something so very analogous to the Mandelbrot or Julia sets in the plane. There are plenty of other mappings one might experiment with, like the real polynomial mappings (x,y,z) -> (P(x,y,z), Q(x,y,z)). I don't think a great deal is known about these even if P and Q are only quadratic polynomials. >> but this should have read: << I don't know why fractals in 3-space would need to be based on something so very analogous to the Mandelbrot or Julia sets in the plane. There are plenty of other mappings one might experiment with, like the real polynomial mappings (x,y,z) -> (P(x,y,z), Q(x,y,z), R(x,y,z)). I don't think a great deal is known about these even if P, Q, R are only quadratic polynomials. >> --Dan Those who sleep faster get more rest.

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Re: [math-fun] What is "geometric" about a geometric progression?
by James Propp 02 Mar '11

02 Mar '11

Could the use of the term "geometric progression" have anything to do with Vi Hart's elephants? See http://vihart.com/blog/doodling-infinity/ Jim Propp

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Re: [math-fun] Conformal mapping in R3+
by Dan Asimov 02 Mar '11

02 Mar '11

David wrote: << This is a question relating to the elusive "holy grail" of a perfect Mandelbrot Set in 3D. As I understand it the problems with respect to getting "whipped cream" in non-linear pure-3D fractals is related to the facts that there are no complete algebraic fields in R3 and that (by Louiville's (conformality) theorem) conformal mappings are restricted to mobius transforms only. >> I don't know why fractals in 3-space would need to be based on something so very analogous to the Mandelbrot or Julia sets in the plane. There are plenty of other mappings one might experiment with, like the real polynomial mappings (x,y,z) -> (P(x,y,z), Q(x,y,z)). I don't think a great deal is known about these even if P and Q are only quadratic polynomials. << With respect to Louiville's theorem: http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29 The proof shows that any Rn (n>2) mapping/transform that is not a mobius transform will not be conformal. >> Just to be clear, conformal mappings R^n -> R^n already must satisfy the conclusions of this theorem even for n = 2. What makes this theorem of Liouville striking is that it's even true when the domain of the conformal mapping is any open *subset* of R^n. (Since, there is a huge profusion of conformal mappings U -> R^2 where U is a proper open subsets of R^2, but not for n > 2.) << My question (due to my own ignorance/lack of understanding) is: Does Louiville's theorem also discount the possibility of an Rn non-mobius mapping/transform being conformal in Rx space (x<n) if only Rx of the Rn space is considered ? >> Yes, such possibilities are already excluded by the theorem, since [compositions of] translations, similarities, rotations on any affine subspace of R^x of R^n can be extended to the same type of composition on all of R^n. (Inversions would be excluded since they're not continuous on all of R^x.) << For example is it not possible that if we have Rn (n>3) then a non-Mobius mapping/transform in Rn could be such that all non-conformality is restricted to a particular Rx of Rn where x>=1 and x<n-2 ? >> This is possible in many ways. E.g., (x,y,z) -> (x^2 - y^2, 2xy, z) is conformal except along the z-axis. (But isn't this the opposite of an example of your previous question?) --Dan Those who sleep faster get more rest.

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[math-fun] A little introduction
by quad 02 Mar '11

02 Mar '11

Hello all, Mr. Schroeppel suggested I introduce myself. My name is Robert Smith, in St. Paul, Minnesota, USA. I'm highly interested in computer algebra and symbolic computation, and implementations thereof. I am currently (slowly!) writing a book "Computer Algebra and Algebraic Programming", and have been designing a computer algebra system for a couple years (takes lots of reading and thinking!). I consider myself well versed in C, Lisp, and Standard ML. If I have to choose a favorite area of computer algebra, it'd be around those calculus algorithms: summation, integration, etc. With that said, I have a strong preference for computable -- especially algebraic -- objects. My blog is at <http://www.symbo1ics.com/blog> if you're interested. It contains a good amount of JavaScript to render mathematical expressions, so don't worry if your browser goes into overdrive for a second while it renders. It's unfortunate, but the resulting math is beautifully typeset and resizable. I recommend you have the STIX fonts installed so you don't get raster/image fonts. Cheers! -Robert (or 'quad' for something more unique) P.S. That genius Bill Gosper is the proxy through which I learned about this list. :)

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[math-fun] Conformal mapping in R3+
by David Makin 02 Mar '11

02 Mar '11

HI all, This is a question relating to the elusive "holy grail" of a perfect Mandelbrot Set in 3D. As I understand it the problems with respect to getting "whipped cream" in non-linear pure-3D fractals is related to the facts that there are no complete algebraic fields in R3 and that (by Louiville's (conformality) theorem) conformal mappings are restricted to mobius transforms only. With respect to Louiville's theorem: http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29 The proof shows that any Rn (n>2) mapping/transform that is not a mobius transform will not be conformal. My question (due to my own ignorance/lack of understanding) is: Does Louiville's theorem also discount the possibility of an Rn non-mobius mapping/transform being conformal in Rx space (x<n) if only Rx of the Rn space is considered ? For example is it not possible that if we have Rn (n>3) then a non-Mobius mapping/transform in Rn could be such that all non-conformality is restricted to a particular Rx of Rn where x>=1 and x<n-2 ? If so then I think the holy grail 3D Mandelbrot view is still a possibility albeit with the actual algebra being Ry (y>3). Please forgive me if I sound completely ignorant or totally insane :) Or if you don't think this is "math fun" ;) bye Dave

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[math-fun] It has an e, an i, and a pi in it
by Bill Gosper 01 Mar '11

01 Mar '11

Even a -1. Why isn't it beautiful? 43 %i %pi --------- 3/4 1/4 1/8 48 1 (8 %i - 9 2 - 6 2 ) %e gamma(-) - %pi 4 (d120) eta(%i %e ) = - ------------------------------------------------ 3/8 3/4 2 2 %pi (c121) EXPAND(BFLOAT(%)); (d121) 1.9655177733257701608b-2 %i + 8.7954174363898772974b-1 = 1.9655177733257701608b-2 %i + 8.7954174363898772974b-1 --rwg

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