August 2011 - mathfun@mailman.xmission.com

Re: [math-fun] Rational Parametric Polys vs Conic Sections vs (Un)?Solvable Quintics
by Dan Asimov 05 Aug '11

05 Aug '11

P.P.S. Fred wrote: << A circle is rationally parameterised by [w,x,y] = [1+t^2, 1-t^2, 2t], the Cartesian coordinates of the conic then being [x/w, y/w]. >> Just now I'm realizing that the [w,x,y] probably represent *projective* coordinates, which I apologize for not grasping at first. --Dan Sometimes the brain has a mind of its own.

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Re: [math-fun] Rational Parametric Polys vs Conic Sections vs (Un)?Solvable Quintics
by Dan Asimov 05 Aug '11

05 Aug '11

Chiming in with Fred, I didn't understand the question, either. The general planar conic section is expressible in the implicit form: A x^2 + B xy + C y^2 + Dx + Ey + F = 0 (where ABC <> 0). Maybe you could phrase your question by using mathematical notation so that it's crystal clear just what you're asking. --Dan P.S. (For t in R u {oo}) the parametric equations (x(t),y(t)) = ((t^2 - 1)/(t^2 + 1), 2t/(t^2 + 1)) yield a circle in the plane. But (x(t),y(t),z(t)) = (t^2 - 1, 2t, t^2 + 1)) define the space curve that's the intersection of the cone x^2 + y^2 = z^2 with the parabolic cylinder x = (1/4)y^2 - 1, with all coordinates unbounded. I'm not sure what that's called. << AIUI, in order to form conic setion curves using rational parametric polynomials such that the (Cartesian) coordinates of the loci, the radii and the coefficients are all in Q, the polymonials have to be at least quintic. Otherwise, at least one of the loci, radii or coefficients are radicals. Presuming that is true, is there any relation between that and the fact that quintics are the lowest degree polys which are not always solvable in terms of radicals (given coefficients in Q)? >> Sometimes the brain has a mind of its own.

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[math-fun] Earth's 2 Moons crashed into one another
by Henry Baker 03 Aug '11

03 Aug '11

FYI -- But were they dancing cheek-to-cheek ? ;-) http://www.huffingtonpost.com/2011/08/03/earth-two-moons-theory_n_917464.ht… Earth's 2 Moons? It's Not Lunacy, But New Theory SETH BORENSTEIN 08/ 3/11 07:11 PM ET WASHINGTON Â In a spectacle that might have beguiled poets, lovers and songwriters if only they had been around to see it, Earth once had two moons, astronomers now think. But the smaller one smashed into the other in what is being called the "big splat." The result: Our planet was left with a single bulked-up and ever-so-slightly lopsided moon. The astronomers came up with the scenario to explain why the moon's far side is so much more hilly than the one that is always facing Earth. The theory, outlined Wednesday in the journal Nature, comes complete with computer model runs showing how it might have happened and an illustration that looks like the bigger moon getting a pie in the face. Outside experts said the idea makes sense, but they aren't completely sold yet. This all supposedly happened about 4.4 billion years ago, long before there was any life on Earth to gaze up and see the strange sight of dual moons. The moons themselves were young, formed about 100 million years earlier when a giant planet smashed into Earth. They both orbited Earth and sort of rose in the sky together, the smaller one trailing a few steps behind like a little sister in tow. The smaller one was a planetary lightweight. The other was three times wider and 25 times heavier, its gravity so strong that the smaller one just couldn't resist, even though it was parked a good bit away. "They're destined to collide. There's no way out. ... This big splat is a low-velocity collision," said study co-author Erik Asphaug, a planetary scientist at the University of California, Santa Cruz. What Asphaug calls a slow crash is relative: It happened at more than 5,000 mph, but that's about as slow as possible when you are talking planetary smashups. It's slow enough that the rocks didn't melt. And because the smaller moon was more than 600 miles wide, the crash took a while to finish even at 5,000 mph. Asphaug likened the smaller moon to a rifle bullet and said, "People would be bored looking at it because it's taking 10 minutes just for the bullet to bury itself in the moon. This is an event if you were looking at, you'd need a big bag of popcorn." The rocks and crust from the smaller moon would have spread over and around the bigger moon without creating a crater, as a faster crash would have done. "The physics is really surprisingly similar to a pie in the face," Asphaug said. And about a day later, everything was settled and the near and far sides of the moon looked different, Asphaug said. Co-author Martin Jutzi of the University of Bern in Switzerland said the study was an attempt to explain the odd crust and mountainous terrain of the moon's far side. Asphaug noticed it looked as if something had been added to the surface, so the duo started running computer simulations of cosmic crashes. Earth had always been an oddball in the solar system as the only planet with a single moon. While Venus and Mercury have no moons, Mars has two, while Saturn and Jupiter have more than 60 each. Even tiny Pluto, which was demoted to dwarf status, has four moons. The theory was the buzz this week in Woods Hole, Mass., at a conference of scientists working on NASA's next robotic mission to the moon, said H. Jay Melosh of Purdue University. "We can't find anything wrong with it," Melosh said. "It may or may not be right." Planetary scientist Alan Stern, former NASA associate administrator for science, said it is a "very clever new idea," but one that is not easily tested to learn whether it is right. A second moon isn't just an astronomical matter. The moon plays a big role in literature and song. And poet Todd Davis, a professor of literature at Penn State University, said this idea of two moons Â one essentially swallowing the other Â will capture the literary imagination. "I'll probably be dreaming about it and trying to work on a poem," he said.

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[math-fun] quintic question
by Bill Gosper 03 Aug '11

03 Aug '11

I believe the saying goes: A quintic with rational coeffs is solvable in radicals iff it factors or the Lagrange resolvent sextic (discussed here with JHC decades ago) has a rational root. What about if the quintic has radical coeffs? E.g., the center disk in the D_5 packing if 31 disks has radius a satisfying (43875 + 37180*I) + 74360*(-1)^(1/10) - 97460*(-1)^(3/10) - 70760*(-1)^(2/5) + 70760*(-1)^(3/5) + 23100*(-1)^(7/10) + ((47063255 + 40030096*I) + 80060192*(-1)^(1/10) - 104803824*(-1)^(3/10) - 76142960*(-1)^(2/5) + 76142960*(-1)^(3/5) + 24743632*(-1)^(7/10))*a + ((-4765010 - 4048824*I) - 8097648*(-1)^(1/10) + 10603816*(-1)^(3/10) + 7702560*(-1)^(2/5) - 7702560*(-1)^(3/5) - 2506168*(-1)^(7/10))*a^2 + ((1630 + 336*I) + 672*(-1)^(1/10) - 1584*(-1)^(3/10) - 1040*(-1)^(2/5) + 1040*(-1)^(3/5) + 912*(-1)^(7/10))*a^3 + ((-225 + 1116*I) + 2232*(-1)^(1/10) - 2964*(-1)^(3/10) + 200*(-1)^(2/5) - 200*(-1)^(3/5) + 732*(-1)^(7/10))*a^4 - 5*a^5 Can we say that such a quintic is solvable in radicals iff the resolvent has a root expressible in radicals? If not can we at least say that no resolvent solution implies no quintic solution? -rwg I have the resolventin Mma (LeafCount 3029) and Macsyma, if anyone wants. OLEO-GUM-RESIN NUMEROLOGIES

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[math-fun] flowsnake bathroom tile
by Bill Gosper 02 Aug '11

02 Aug '11

(to go with the Penrose toilet paper) Neil, Corey, and I (over two sessions) worked out how, with tiles of two solid colors, to map a fill of the flowsnake (Gosper) curve onto a hex grid,<http://gosper.org/hexflo.png> a staggered square grid <http://gosper.org/staggerflo.png>, and a square grid <http://gosper.org/gridflo.png> wherein squares are considered joined at their NE and SW corners but not the other two. The illustrations are of three flowsnakes joined in a triangle to create a closed curve. A few weeks(?) ago I had a burning desire to find this hex version in conjunction with either a math-fun thread or a project with the kids, but da*ned if I can remember what it was. Maybe it was just to emphasize to Erez Lieberman that hex grids and square grids aren't all that different for spacefilling purposes. --rwg

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Re: [math-fun] Hausdorff dimension of Julia sets
by Dan Asimov 01 Aug '11

01 Aug '11

Thanks, Bill, for that interesting information. Incidentally, I just found an abstract online (at < http://www.springerlink.com/content/n4cqr7e18e81m284/ >) that claims certain quadratic polynomials indeed do have Julia sets of Hausdorff dimension 2. --Dan Sometimes the brain has a mind of its own.

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[math-fun] Hausdorff dimension of Julia sets
by Dan Asimov 01 Aug '11

01 Aug '11

Has anyone seen a graph {(c, H(c)) in R^3 | c in C=R^2} of the Hausdorff dimension H(c) of the Julia set of f(z) = z^2 + c, c in C ? --Dan Sometimes the brain has a mind of its own.

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Re: [math-fun] "France" boundary dimension
by Dan Asimov 01 Aug '11

01 Aug '11

RWG wrote: << Each subFrance shares 1/6th of its boundary with each neighbor, leaving 6/2 = 3 to comprise the outer boundary, which must be sqrt 7 times bigger than a subFrance boundary by area consideration. So scaling the boundary stuff by sqrt 7 gets you three times as much. This was in the original Gardner article. Or is this the way you were doing it >> Yes, exactly. Sometimes the brain has a mind of its own.

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[math-fun] "France" boundary dimension
by Bill Gosper 01 Aug '11

01 Aug '11

DanA> Thanks for the links. The angle used to normalize what I call the map-of-France is exactly what I proposed: Divide the unnormalized set by 2 + exp(2pi*i/6). What I asked is whether there are other ways of doing it as well. I am extremely familiar with the definition of Hausdorff measure (and dimension). What I don't know is how to compute it for even slightly complicated actual examples like the map-of-France. --Dan Dan, look at http://www.mathcurve.com/fractals/gosper/gosper-pavage3.gif Each subFrance shares 1/6th of its boundary with each neighbor, leaving 6/2 = 3 to comprise the outer boundary, which must be sqrt 7 times bigger than a subFrance boundary by area consideration. So scaling the boundary stuff by sqrt 7 gets you three times as much. This was in the original Gardner article. Or is this the way you were doing it? --rwg Also I forgot to mention that the rotating gridflo<http://gosper.org/gridrot.gif>is slightly different from the first one I mentioned <http://gosper.org/gridflo.png>, but they both mess with your eyes the same way. Actually, it's probably a simple optical effect rather than a hairy perceptual illusion. We should be able to simulate astigmatism with a simple anisotropic blurring filter.

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Re: [math-fun] flowsnake bathroom tile
by Bill Gosper 01 Aug '11

01 Aug '11

DanA > I love the fractal I call the map-of-France (but which RWG can probably> correct my terminology as he may have done in the past): [...]> > BUT there is a lot of choice in how 6 copies of each rosette are> placed around itself, so this method can give a plethora of> different curves in the limit. (Probably continuum many, given> a countable number of discrete choices.) Yow, I clean forgot those. And every one has its own flowsnakoid spacefill <http://gosper.org/flopnoflop.png>. joerg> > I recently spammed some short routines, one of them would> produce what is an p.94 of the fxtbook:>> bool bit_dragon3_turn(ulong &x)> [...]>> For the picture one has to replace each move by 120 degs> by two moves by 60 degs.>> The picture you made certainly reminded me of it. Not to mention http://www.tweedledum.com/rwg/7posies.bmp . rwg> > Neil, Corey, and I (over two sessions) worked out how, with tiles of two>> solid colors,>> to map a fill of the flowsnake (Gosper) curve onto a hex>> grid,<http://gosper.org/hexflo.png>>> Nice!>>> a staggered square grid <http://gosper.org/staggerflo.png>,>> and a square grid <http://gosper.org/gridflo.png> wherein squares are>> considered joined at their NE and SW corners but>> not the other two. The illustrations are of three flowsnakes joined in a>> triangle to create>> a closed curve. If you have normal vision, you probably don't perceive gridflo.png as a flowsnake unless you squint or stand far away. And if you're astigmatic like me, you may see zebra stripes instead! I made an animated gif <http://gosper.org/gridrot.gif> (2 MB) of the thing continuously rotating. Borrow someone's cylindrically correcting specs (or remove your own). The alternating effect is strong, at least for me. --rwg

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