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RE: [math-fun] You Can Balance a Wobbly Table by Rotating It
by Andy Latto 19 Feb '07

19 Feb '07

> -----Original Message----- > From: > math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission > .com] On Behalf Of Daniel Asimov > Sent: Monday, February 19, 2007 8:46 PM > To: math-fun > Subject: Re: [math-fun] You Can Balance a Wobbly Table by Rotating It > > > 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). > >> > > 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). > -------------------------------------------------------------- > ------------------------ I think this is a completely different conjecture than the "putting a table on the floor so it doesn't wobble" conjecture. In particular, most floors are surfaces, rather than curves, and most tables have a fixed distance between their legs. I think the "putting a table on the floor" theorem is a theorem of the form "Given a differentiable function of two variables, z = f(x,y), and a distance H, there exist four points in the plane, (x1,y1), (x2, y2), (x3, y3) (x4, y4) such that the four points in space, (x1, y1, f(x1, y1)), (x2, y2, f(x2, y2), (x3, y3, f(x3, y3)), (x4, y4, f(x4, y4)) form a square of side H. > > 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 (*). I'd be interested to see a proof of this. But I don't see any connection between this theorem and the "placing a table on the floor so it doesn't wobble" theorem. Andy Latto andy.latto(a)pobox.com

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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

<< 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). >>

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[math-fun] You Can Balance a Wobbly Table by Rotating It
by Robert Baillie 19 Feb '07

19 Feb '07

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). Links: Audio of NPR story: http://www.npr.org/templates/story/story.php?storyId=7466072 Keith Devlin's website with a description: http://www.maa.org/devlin/devlin_02_07.html Technical article: http://arxiv.org/PS_cache/math/pdf/0511/0511490.pdf Scientific American article: http://www.sciam.com/article.cfm?articleID=5B1CAEBA-E7F2-99DF-3CD6B81A81B51… Mathematica animated movie: http://www.maths.monash.edu.au/~bpolster/table.mov

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CORRECTION Re: [math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
by Daniel Asimov 19 Feb '07

19 Feb '07

I wrote: << 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 next few lines need correction. They should read: The resulting curve has the equation r = cos(theta)^c, |theta| <= pi/2. When c = p/q for integers p, q > 0 in lowest terms, then the curve is given by the polynomial equation (x^2 + y^2)^(p+q) = x^(2p), x >= 0. --Dan

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[math-fun] Simplest Ovals (WAS: sections of quadratic surfaces)
by Daniel Asimov 19 Feb '07

19 Feb '07

As per David Cantrell's suggestion, I've been trying do define, using the simplest possible equations, an "oval" that has one curvature maximum and one minimum, such as k(s) = 2 + sin(s) where s is arclength, and k must of course be periodic. But without more critical points, such a k is impossible for a simple closed planar curve, thanks to the Four-Vertex Theorem. (Cf. < http://en.wikipedia.org/wiki/Four-vertex_theorem >.) So, can anyone think of a real-analytic simple closed convex planar curve as a candidate for the Simplest Oval ? (Perhaps there is a polynomial P(z) such that alpha(t) := P(exp(it)), 0 <= t <= 2pi works? In fact any solution must be of the form alpha(t) = f(exp(it)) for some analytic f.) Here's a possibility (that I haven't checked, since I'm about to leave on a 3-day trip back to Philly from Vancouver Island): Take an ellipse with semiaxes = 1 and 2 in the plane, with the end of a major semiaxis at the origin, and then apply the polar-coordinate transformation (r,theta) |--> (r^2, theta) to the ellipse. (I.e., (x,y) |--> (x sqrt(x^2+y^2), y sqrt(x^2+y^2).) --Dan

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[math-fun] Lizard's sequence
by Christian Boyer 19 Feb '07

19 Feb '07

In the March 2007 issue of "Pour La Science" (French edition of Scientific American), very interesting paper by Jean-Paul Delahaye on the Eric Angelini's sequences. Paper titled "La suite du lézard et autres inventions". OEIS is of course mentioned. For example, after the "prime" numbers, we can define the "second" numbers http://www.research.att.com/~njas/sequences/A123929 the "third" numbers, and so on... http://www.cetteadressecomportecinquantesignes.com/ThousandZetas.htm Other Angelini's sequence discussed in the paper, called by Delahaye "la suite du lézard" = "the lizard's sequence": http://www.cetteadressecomportecinquantesignes.com/Decimation.htm Eric Angelini is photographed showing a pizza. On his pizza, we can read OEIS, written with anchovies! Christian.

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Re: [math-fun] Re: sections of quadratic surfaces
by Daniel Asimov 19 Feb '07

19 Feb '07

Fred Lunnon wrote: [attributions have been edited for accuracy]: <<<< On 2/17/07, Emma Cohen <emma(a)don-eve.dyndns.org> wrote: << Jim Propp wrote: <<< But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section? >>> Isn't this definition of a line segment just an ellipse whose minor axis is zero? >> However, if you attempt to construct a finite line segment as the intersection of a plane with a cone --- the origin of the term "conic section" --- you will find it impossible, WFL >>>> This is an interesting point. It seems that by rights, a line segment (= an ellipse with |minor axis| = 0) *should* be within the family of conic sections, as a degenerate case. So if a is fixed, the locus of (*) (x/a)^2 + (y/b)^2 = 1 seems as if it ought to approach the locus of some limiting version of the equation (*) as b decreases to 0. This seems possible, in a sense, if conic sections are thought of as occurring in projective geometry. 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. A cone like, say, x^2 + y^2 = z^2 is replaced by the addition of a circle at infinity to it, resulting in a cool conoid shape in P^3 whose name I forget, but which (projectively) resembles a torus of revolution with one meridional circle pinched to a point, and which gets fatter the farther away one is from that point. 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.) --Dan

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[math-fun] How NOT to draw an oval ...
by Fred lunnon 17 Feb '07

17 Feb '07

Over the years I have spent many happy but largely fruitless hours attempting to construct a genuinely independent interpolation algorithm. Here, for the delectation of math funners, is my innovatory technique for building Bezier-like closed-curve splines, recently disinterred for service in another ovipository thread. Suppose we are given n control points P_i for i = 0,...,n-1, [as Cartesian 2-vectors in the plane, homogeneous 4-vectors in space, or whatever]. We wish to construct a closed curve Q(u), passing near(?) to each control as 0 <= u <= 1, with Q(0) = Q(1) and similarly for all derivatives. Consider the weight functions f_i(u) = (2/n)*cos(Pi*(u - i/n))^2 : these are analytic, periodic over 0 <= u <= 1, and for all u sum to 1 over 0 <= i <= n-1. Furthermore f_i is nonnegative, and has its maximum at u = i/n. So a reasonable looking spline function might be Q(u) = \sum_i f_i(u) P_i. Exercise: prove that, whatever the controls, Q(u) is always an ellipse! Fred Lunnon

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[math-fun] Re: sections of quadratic surfaces
by James Propp 17 Feb '07

17 Feb '07

Bill Gosper writes: >Some find it counterintuitive that slicing a cone gives an ellipse and >not an oval. What is an oval? I thought it was just a word for a generic thing that looks kind of like an ellipse. >What about slicing a paraboloid of rotation? More generally >how can eliminating a quadratic mumbloid vs a linear (planar) equation >yield anything but a quadratic (conic)? Beats me! If you use the linear equation to eliminate one variable in a quadric, how can you get anything but a conic? But this reminds me of something that used to bother me in high school when I learned about conic sections: Was anyone else annoyed that a line segment (which can be defined as the set of all points X such that |AX| + |BX| = |AB| and hence seems to be a sort of ellipse under one common definition) is not in fact a conic section? Of course, what the line segment is trying to be is a line, which is a bona fide (albeit degenerate) conic section. (This might be a good example to use when introducing undergraduates to algebraic geometry: the intersection of a cone with a plane can't be a line segment because a line segment isn't Zariski-closed.) It'd be interesting to see an animation that shows, side by side, a plane cutting a cone and the associated quadratic plane curve (rotated in 3-space in some standard way) evolving as the latter becomes more and more eccentric, with fixed major axis AB; perhaps one could develop some intuition about the way in which the points on the line AB that don't lie between A and B "jump" into the locus when the cutting plane becomes tangent to the cone. Jim Propp

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[math-fun] RE: amazing photo
by Guy Haworth 16 Feb '07

16 Feb '07

Wonderful: I wonder what such a mapping of the view from inside a regular or semi-regular Platonic solid would look like - the viewpoint being at the centre, or off-centre down some specific or random axis. The faces of the polyhedron could be 'windows' onto 'other worlds', though the view might be a little confusing with the non-convex, stellated ones. g

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