February 2007 - mathfun@mailman.xmission.com

Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
by Daniel Asimov 24 Feb '07

24 Feb '07

There ought to be a simple criterion under which there exists a unique "best" simple closed curve in the plane. E.g.g., suppose 1) The curve has bilateral symmetry; 2) the curvature attains two local maxima with values k = M and k = m; 3) the interior of the curve has a fixed area A. Then I'd think there might be a unique "optimal" such curve having minimum length (or equally, fix the length and maximize the area), and that the unique optimal curve would automatically be analytic. (If so, then quite possibly condition 1) is redundant.) --Dan

2 1

Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
by Daniel Asimov 23 Feb '07

23 Feb '07

I plotted this with equal scale using the Mac utility "Grapher", and from the plot it's clear that this closed curve is strictly convex everywhere. It does have arcs (near the maximum of curvature) where the curvature comes very close to 0. NOTE: There is a converse to the four-vertex theorem (which says that the curvature function k(s) (s=arclength) of a C^2 simple closed planar curve must have a least two local maxima and two local minima). The converse says that for any continuous real-valued function on the circle with at least two local maxima and two local minima is the curvature function of some simple closed curve in the plane. (Cf. < http://www.ams.org/notices/200702/fea-gluck.pdf >.) We want our egg to be strictly convex, bilaterally symmetric, and as simple as possible. Hence we'd like the curvature function to be everywhere positive, satisfy k(t) = k(-t) (0 <= t <= 2pi), and have the fewest possible critical points: two local maxes and two local mins. One simple way to obtain such a function is via k(t) = (cos(t) + 1/2)^2 + 1/2 (or more generally, k(t) = (cos(t) + a)^2 + b^2 where 0 < a < 1 and b > 0). I'd like to see simple closed curves with curvature functions like these. --Dan -------------------------------- RWG wrote: << >> Cheap and smooth: r=(5+cos(t))^2+(3+cos(2*t))^2 . WFL>Looks more like a squash or pear to me Whoa, nonconvex even? It looks quite Mossy when I plot it with Macsyma, equalscale:true, but I don't know how to get equalscale in Mma, which gives me various avocados. (Are there any birds with nonconvex eggs?) >--- must be some funny-shaped birds in your part of the world, Bill! Amen. I recently rounded a corner on which calmly browsed an East African crested crane, perhaps in town to see "The Last King of Scotland." Or maybe it just wanted some action with an American whoopee crane. >> P.S. Fred, they're called *whooping* cranes, not whoopee cranes.

3 4

[math-fun] Interesting ovoids
by franktaw＠netscape.net 23 Feb '07

23 Feb '07

This discussion of ovals reminds me of a problem I was working on a while ago, which I gave up since I was not getting anywhere. The problem starts with the remarkable observation that if we take a(n+1) = (a(n)+1)/a(n-1), the resulting sequence is periodic with period five. (I saw this in one of Martin Gardner's columns in Scientific American; I don't know what issue at this point. He puts it into the mouth of the fictional Dr. Matrix, but I think he gave a reference for it later. It probably also appeared in one of his books.) (This excludes cases where a zero appears in the sequence. It is sufficient to take a(1) and a(2) both positive.) I asked, what happens if the 1 in that equation is replaced by some other constant, c: a(n+1) = (a(n)+c)/a(n-1). The case c=0 gives us period 6: starting with x, y, the sequence continues y/x, 1/x, 1/y, x/y, and then loops. In general, the sequence exhibits quasi-periodic behavior. It goes up and down regularly, with the average distance between peaks being some real number. This quasi-period is between 4 and 5 for c>1, and between 5 and 6 for 0 < c < 1; however in general the quasi-period depends on the initial values as well as on c. (Of course, if the quasi-period is rational, the sequence is actually periodic, with period equal to the numerator of the quasi-period.) Note that such a sequence has a fixed point u_c where u_c = (1+sqrt(4c+1))/2. (So in particular, u_1 is the golden ratio, phi, aka tau.) When you graph a(n) vs. a(n+1), with suitable parameters the result is an ovoid. If a(1), a(2) is close to u_c,u_c you get an oval, approaching a circle as you approach u_c,u_c. When the starting values are far from this point, the graph approximates an isoceles right triangle, with the right angle near the origin. The obvious next step is to find a non-trivial function f_c which satisfies the functional equation f_c(x,y) = f_c(y,(y+c)/x). The function must obviously have a minimum or maximum at u_c,u_c. Then the ovals should be the curves defined by f_c(x,y) = k. This is where I got stuck - I wasn't able to find such an f_c. Maybe somebody with better analytic skills than I have will be able to. Regardless, have a look at some of these curves. Franklin T. Adams-Watters ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.

2 2

[math-fun] Re: sections of quadratic surfaces
by James Propp 22 Feb '07

22 Feb '07

Dan Asimov wrote: >If we replace R^3 by its "projective compactification" by adding the P^2 of >the "lines at infinity", the result is P^3, projective 3-space. >... >Then the conic sections become the intersection of one of these conoids with >an "affine projective plane" in P^3. All the nondegenerate intersections are >equivalent to ellipses, but the segmentlike degenerate ellipse at issue becomes >a circle. (At least it's compact.) This is a nice point of view. Thanks, Dan! Jim Propp

1 0

[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

1 0

[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?

1 0

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

4 6

Re: [math-fun] Ovaltine Dreams
by Fred lunnon 22 Feb '07

22 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

1 0

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

1 0

[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.

3 2