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[math-fun] Re: Ovaltine Dreams
by James Propp 22 Feb '07

22 Feb '07

>For the Ovaloid where |P - F1| + k|P - F2| = c... > >I think it's the "simplest," even though it has fourth >powers in its equation. But, you know, scare quotes. > >I have a hard time believing it never received interest >before. The questions below are phrased as if it were a >new thing. > >Has anyone thought of a workable mechanical construction >to plot it, analogous to the string around two pins? For suitable values of k, couldn't you just modify the usual string-construction by having the string wrap multiple times? I think I've seen this described before in print (but I forget where). Jim Propp

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[math-fun] Simplest Ovals
by Henry Baker 22 Feb '07

22 Feb '07

In bicycle/car races, the vehicles have a maximum acceleration potential. For simplicity's sake, let's assume that the max acceleration in any direction is capped at a certain constant. When going around corners, these vehicles "go wide", so that they don't go beyond their maximum acceleration when turning the corner. On the other hand, their ability to "go wide" is limited by the width of the road/track. So, one way to produce an egg-shape is to consider a racing track of a certain width, whose overall shape is in the form of an isosceles triangle. If the triangle is equilateral, and the width is wide enough, then the curve is circular. However, when we start reducing the width, we get a curve which starts getting pinched in 3 places. If we then lengthen the altitude of the triangle so that it is no longer equilateral, then the curve starts looking egg-shaped. I assume that curves such as these have been studied. Does anyone know what their equations are? Can anyone point me to any references?

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Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
by Daniel Asimov 22 Feb '07

22 Feb '07

An interesting family of ovoids (a word I prefer to oval, cuz oval has a deeply ingrained casual meaning) is given by applying the transformation (r,theta) -> (r^c, theta) (c > 1) to the circle r= cos(theta), i.e., with radius = 1/2 and center at (1/2,0). The resulting curve has the equation r = cos(theta)^c When c = p/q for integers p,q > 0 with q odd, then the curve is given by the polynomial equation (x^2+y^2)^((q+1)/2) = x^p. E.g., p/q =3/1 gives a reasonable ovoid. Still, "Moss's oval" ( http://mathworld.wolfram.com/MosssEgg.html ) looks to me more like an egg than does the above curve for any values of p and q. --Dan

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Re: [math-fun] Ovaltine Dreams
by Fred lunnon 21 Feb '07

21 Feb '07

On 2/21/07, Steve Witham <sw(a)tiac.net> wrote: > For the Ovaloid where |P - F1| + k|P - F2| = c... > ... > If you make pictures before I do, I wouldn't mind if > you posted them somewhere on the web. > The following script may be pasted directly into the window of Richard Morris' Algebraic Surface Solver at http://www.javaview.de/services/algebraic/index.html Reset resolution (coarse) from 8 to 32, click on "calculate". ((x^2 + y^2 + (z-a*w)^2) + c^2*(x^2 + y^2 + (z+a*w)^2) - 4*b^2*w^2)^2 - 4*c^2*(x^2 + y^2 + (z-a*w)^2)*(x^2 + y^2 + (z+a*w)^2); a = 1; b = 3/2; c = 4/3; w = 3/4; The resulting solid "egg" is the surface of rotation of Steve Gray's weighted oval about a vertical axis, substituting y -> z and x^2 -> x^2+y^2 in the original equation, and introducing a homogenising variable w to make rescaling easier. As before, 2a = focal distance, 2b = string length, c = Gray weighting (k). These may now be tinkered with at will within the range 1 <= c <= b/a; but if c > b/a, the curve becomes a limacon, and the bottom of the egg must be truncated [by adjusting the domain, increasing the scale factor 1/w, or slicing off with a teaspoon] to view the interior lobe. Fred Lunnon

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Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
by Eugene Salamin 21 Feb '07

21 Feb '07

----- Original Message ---- From: R. William Gosper <rwg(a)osots.com> To: math-fun(a)mailman.xmission.com Sent: Monday, February 19, 2007 10:31:59 PM Subject: Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces) >> Still, "Moss's oval" ( http://mathworld.wolfram.com/MosssEgg.html ) Ach, that's what I was calling van Zwolle's. Here's a truncated Fourier expansion which is very close, yet infinitely differentiable: sqrt(2) cis(6 t) sqrt(2) cis(4 t) cis(3 t) (3 sqrt(2) - 1) %pi cis(t) - ---------------- + ---------------- - -------- + -------------------------- 15 6 3 2 sqrt(2) cis(- 2 t) sqrt(2) cis(- 4 t) + cis(- t) - ------------------ + ------------------ 3 10 Some of these terms can probably be Remezed out without visible damage. (I once sent this list a note on complex Remez.) Note the surprising presence of counterrotors, presumably due to my lazy choice of traversal speeds: constant dtheta/dt on all four arcs, meaning instant ac/deceleration at each arc boundary. More generally, these "four point eggs" are determined by the radii of the two endcaps (r1 and r2), and the angles they span (t1 and t2), which determine the radius and span of the arc joining them. The fully general Fourier series is then oo ==== '' \ - 2 (r2 - r1) ( > cis(n t) (sin((n - 1) t1) sin(t2) / ==== n = -oo n - (- 1) sin(t1) sin((n - 1) t2))/((n - 1) n))/(sin(t2) - sin(t1)) 2 (r2 - r1) (t1 cos(t1) sin(t2) - %pi cos(t1) sin(t2) - sin(t1) t2 cos(t2)) + --------------------------------------------------------------------------- sin(t2) - sin(t1) - 2 cis(t) (((r2 - r1) t1 - %pi r2) sin(t2) + (r2 - r1) sin(t1) t2 + %pi r1 sin(t1))/(sin(t2) - sin(t1)), where sum'' means skip n=0 and n=1. Note the (n-1)n in the term denominator. Gene Salamin once convinced me that there are alternate speed functions capable of putting arbitrarily high order polynomials in that denominator, hence arbitrarily rapid convergence of the Fourier series. But this does *not* mean you need arbitrarily few terms! The actuality is that the higher degree you seek, the longer the series diddles around before "flooring it". ------- If the periodic continuous function f(t) has continuous r-th derivative, then the Fourier series coefficients are O(1/n^(r+2)). Given a closed curve C in the complex plane, and nonnegative integer r, there is a function f(t) whose graph is C and which has continuous r-th derivative. ------- Are there smooth speed functions with no counterrotors (negative harmonics)? --------- A simple closed curve is the boundary of a simply connected region R. By the Riemann mapping theorem, there is a conformal map f from the interior of the unit disk onto R, and f(z) = sum(a[n] z^n, n>=0) since the map is analytic in the unit disk. The curve C is traced when z = exp(it), and so there are no counterrotors. A nonsimple closed curve can often be obtained by deforming a simple curve onto sheets of a Riemann surface. A conformal mapping from the unit disk should still exist, and so the conclusion holds for this case as well. I'm not sure how pathological the curve must be before this construction breaks down. Gene --------- --rwg PS, This problem ought to be "trivial" with kappa(s) (curvature(arclength)) notation. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun ____________________________________________________________________________________ Finding fabulous fares is fun. Let Yahoo! FareChase search your favorite travel sites to find flight and hotel bargains. http://farechase.yahoo.com/promo-generic-14795097

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[math-fun] Life on the Edge
by franktaw＠netscape.net 21 Feb '07

21 Feb '07

I have discovered an interesting cellular automaton. This automaton is defined on the edges of a square grid: each edge has six neighbors, three at each end. Like Conway's Life, each edge can be in one of two states: on or off; and its state in the next generation depends only on the current state of the edge itself and how many of its neighbors are on. The rule for this automaton is that an edge that was off becomes on when exactly two of its neighbors are on; and an edge that was on remains on only if exactly one of its neighbors is on. Or, more simply put, an edge is on in the next generation iff exactly two of the edges in its seven edge neighborhood - including the edge itself - are on. Small patterns with this automaton are a bit more fragile than in Life. For example, a row of two edges (you'll need a fixed-width font to properly view these pictures): -.- goes in the next step to _|_ .|. which then dies out. The alternator is a simple period 2 pattern, alternating between |.| and ._ ._ It is difficult to construct stable patterns, as they must be composed of pairs of lines, with their growth blocked off. I believe the smallest is ...|..... _|.|.|_.. .._....._ ...|.|.|. .....|... However, the really interesting part is what happens when we start with 3 sides of a square: ..._ ..|_ This goes in the next generation to: ._|_. ._._| ..|.. which looks like an airplane (which is what I call this pattern). The next generation it seems to fall apart: ....|_ |...._ ....|. but the next generation is ..._|_. ..._._| ....|.. and it flies! Can anyone find other interesting patterns? It would be especially interesting if someone can find an "airplane gun", which generates airplanes at regular intervals. Any other patterns that move, and any other periodic patterns - especially with a period greater than two - would also be of interest. Franklin T. Adams-Watters P.S. Does anyone know of a forum where cellular automata are being actively discussed? I see that Yahoo has a group, but it has been inactive for several years. ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.

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RE: [math-fun] Life on the Edge
by Eric Angelini 21 Feb '07

21 Feb '07

Hello Frank, > I have discovered an interesting cellular automaton. ... nice idea ! ... could you test one by one, starting with zero, the fate of the natural numbers (up to 1000) ? You could design them like this (fixed font): _ _ _ _ _ _ _ | _| _| |_| |_ |_ | |_| |_| | |_ _| | _| |_| | |_| _| Have a look here for "Game of Life Integers": http://www.cetteadressecomportecinquantesignes.com/GOLife.htm Best, É.

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[math-fun] Ovaltine Dreams
by Steve Witham 21 Feb '07

21 Feb '07

For the Ovaloid where |P - F1| + k|P - F2| = c... I think it's the "simplest," even though it has fourth powers in its equation. But, you know, scare quotes. I have a hard time believing it never received interest before. The questions below are phrased as if it were a new thing. Has anyone thought of a workable mechanical construction to plot it, analogous to the string around two pins? I have been designing things with levers and pulleys and holes through the paper with no success. I think maybe a pantograph is called for... Has anyone thought of an interesting optical property, analogous to the ones ellipses and parabolas have? Maybe something about materials with different refractive indices, or coupling between two parallel eggs with different speeds of sound. Or, a dog jumps swims from island to shore with the frisbee, drops off the frisbee, and then swims faster (no frisbee drag) to the second island-- and it takes the same time no matter what point on the shore he swims to. When k = -1 it's a hyperbola. Do other negative numbers give mismatched-pair hyperbola-oids? Do they go asymptotic to straight lines, asymptotic to something else, or what? What is it a cross-section of? Does it correspond to an orbit in a strange force field? If so, is there an analog of the equal-area rule? If you make pictures before I do, I wouldn't mind if you posted them somewhere on the web. --Steve ovaltine levation olive-tan velation violante not-alive vital-one elan-vito toenail-V a-violent intel-ova naive-lot ovate-nil ion-valet ten-viola ain't-love evil-NATO

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[math-fun] employing asinh for false perspective
by Steve Witham 20 Feb '07

20 Feb '07

I was plotting a fractal and wanted a false-perspective view, where the magnification is really high at one point, and decreases in a log-like way further out from there. Like that cartoon map of the U.S. from the point of view of Manhattan. The mapping needs to be like a log until you get close to zero, then flatten out. I don't know whether it was Henry Baker's comment from last August that triggered it, but I thought of the asinh function. Here is the nice result: http://www.tiac.net/~sw/2007/02/asinh_perspective In the process of programming this, I needed an asinh for Postscript. The one I found is in a package called "pst-math.pro". (.pro is apparently a suffix for Postscript libraries.) The definition basically means asinh( x ) = ln( x + sqrt( x^2 + 1 ) ) which is the standard definition, but there is a precision-slash- overflow problem when x < 0 if you actually calculate it that way in floating point. The link above has a trippy graph of the problem. There is Postscript source code there for both pictures. --Steve >From: Henry Baker <hbaker1(a)pipeline.com> >Date: Sun, 06 Aug 2006 21:39:29 -0700 >BTW, "asinh" now has a real paying job. I searched using Google and >I found that astronomers are now using asinh instead of log to >compute brightness of stars in a star catalog. > >I have advocated the use of asinh in audio & video processing to do >"dynamic range compression" in a way which is somewhat smoother than >current ad hoc methods and floating point methods. > >Another job is its use in genetic engineering & statistics to smooth >data better than log when very small magnitudes are encountered, which >would otherwise drive log nuts (i.e., very substantially negative).

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Re: [math-fun] You Can Balance a Wobbly Table by Rotating It
by Daniel Asimov 19 Feb '07

19 Feb '07

Robert Baillie writes: << On NPR's Saturday Weekend Edition, they did a story with "Math Guy" Keith Devlin. He described a recent (last year) proof that, if the slope of the ground is at most about 35 degrees (actually, ArcTan[1/Sqrt[2]]), then you can balance a wobbly table by rotating it. (This assumes a mathematically perfect table, with the ground being a continuous function of x and y). >> A closely related old problem that seems to be a "folk theorem" (but *only* a folk theorem, apparently without a written proof), is this: Claim: Any simple closed curve in the plane contains the four corners of some square. ------------------------------------------------------------------------------------- I guess the Claim that Robert refers to can be stated rigorously as follows: Any continuous space curve of the form z = f(exp(i*theta)) (i.e., {cos(theta), sin(theta), z(theta), 0 <= theta <= 2pi}, where z has period theta) contains 4 points in space that form the corners of a square . . . as long as z = f(x+iy) is differentiable and satisfies (dz/dy)/(dz/dy) <= sqrt(1/2). -------------------------------------------------------------------------------------- A[n apparently as yet unproved] theorem that would imply both of these is this: (*) Claim: Any simple closed curve in R^3 contains the 4 corners of a (planar) square. --Dan P.S. I will say, smugly, that I believe I know how to prove (*).

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