From: mcintosh@redvax1.dgsca.unam.mx (McIntosh Harold V.-UAP) Date: Mon, 16 May 94 21:44:40 CST To: math-fun@cs.arizona.edu Subject: Re: spaceball > What, if anything, is minimized by the closed space curve > approximated by a baseball stitch? Is there an equation for it? Perhaps I've missed just what is being asked for here. The STITCHING works back and forth in a sort of X-pattern, and that's obviously to keep the thread from pulling toward the seam and ripping the leather. Also why the holes are a ways back from the edge. As to why the leather has that yin-yang shape, that is evidently because leather is planar but you want a spherical baseball. Less forcing. But why baseballs, but neither soccer balls, basket balls, beach balls, nor footballs? (Students here say basketballs yes; a book on (intellectual) games shows a tennis ball knotted out of a bunch of parallel stems (doesn't say why that's intellectual). Why is the resulting boundary curve the shape of the conductor you use when you want to make a magnetic quadrupole for plasma confinement? The figure is similar to a Hamiltonian edge path on a cube, only rounded out. Maybe we're maximizing the quadrupole moment of the induced magnetic field for a given volume enclosed. From: sjk@netcom.com (Shel Kaphan) Date: Tue, 17 May 1994 21:54:51 -0700 To: math-fun@cs.arizona.edu Subject: spaceball season One place to start in thinking about the shape of the two "fruit peels" used to form a baseball shape is to consider a cube of edge 1, and two 1x3 pieces which can wrap the cube, the edges where they join forming a hamiltonian circuit of the cube. So I started thinking about how to cover the surface of a cube with just two pieces that were more baseball-like, i.e. more like two interlocking dumbbells... The puzzle is to describe the possible shapes of two planar pieces than can be folded to cover a cube (no overlapping). From: John Conway Date: Fri, 3 Jun 94 17:41:32 EDT Subject: Re: Baseball Curves: Striking Out in a New Direction Cc: math-fun@cs.arizona.edu What IS all this stuff about baseball curves? Haven't we all done enough to suggest that the correct answer to Peter Doyle's original question "What is `the correct' baseball curve?" is "What correct baseball curve?" Is there much point in one person's conjuring up something defined by a clever conformal map, while another one considers an electrically charged strip of phosphor-bronze, and so on? I announce my conjecture: No two definitions of "the correct baseball curve" will give the same answer unless their equivalence is obvious from the start. John Conway From: Mike Beeler Date: Mon, 20 Jun 1994 17:32:46 -0400 To: math-fun@cs.arizona.edu Subject: baseball seams Well, I'm more of an engineer, and not much of a mathematician, so I took one apart. I was surprised that a few circular arcs seem to be a pretty good approximation for the shape. ------ -------------------- / \ ^ / \ | | | | A | | B | | | | L \ / C length K \ / D | \ / | | | | ---> | | <--- waist | | | | / \ | J / \ E | I / \ F | | | | H | | G | | | | \ / | \ / v ------ -------------------- A to B (21 stitches) ~= semicircle of radius 9/8 inches B to C ( 5 stitches) ~= arc of radius 9/4 inches, length 5/8 inch C to D ( 2 stitches) ~= straight, length 5/16 inch D to E (19 stitches) ~= arc of radius 33/8 inches total boundary (108 stitches) ~= 65/4 inches, but I don't much trust this measurement because of the leather's stretchiness circumference of ball = 9 inches waist ~= 5/4 inches (or maybe 39/32) length must = circumference - waist = 31/4 inches, but it was hard to get it to measure that long. The leather is slippery. Removed and flattened it measured only 57/8. The leather is stretchy enough that it is hard to get precise, repeatable measurements. And patches that didn't quite fit mathematically might easily be stretched into fitting by the stitching. The surface of the body threads underneath the leather is sticky, so any slightly loose part of the leather is likely to stick in place and not be obviously aspherical. specs: Wilson A1054, 9-in. 5-oz., official baseball, "official size and weight". Also, although I don't expect much from it, I wrote to Wilson and asked them. -- Mike Date: Thu, 30 Jun 1994 17:02-0700 From: Bill Gosper Subject: pump, grind, and swing To: math-fun@cs.arizona.edu Delete an opposing pair of the cyclically tangent disks. The resulting circle-plus-two-interior-quarter-circles coincides with the suprapolar (epi-isthmic) view of the baseball closer than the manufacturing asymmetries. I applaud Mike's pragmatism in writing to the baseball manufacturer. (But they are probably preoccupied with arranging next year's production, which was heretofore stitched by Haitian women.) From: Date: Tue, 6 Sep 1994 18:01-0700 Subject: [rucker@SJSUMCS.SJSU.EDU: kappa tau algorithm] To: math-fun@cs.arizona.edu Rudy sez it's OK to forward this to math-fun, and expressed interest in joining. --------- Date: Thu, 1 Sep 1994 09:35 PDT From: Rudy Rucker To: rwg@macsyma.com Subject: kappa tau algorithm KAPPA TAU CURVES by Rudy Rucker Excerpts from a Mathematica Notebook Copryright Rudy Rucker (C) 1994 Mathematical Background One can associate a trihedron of unit vectors with each point of a space curve. The tangent vector lies on the striaght line that best matches the curve and can be thought of as determined by two "consecutive" points of the curve. The normal vector points in the direction in which the tip of the tangent vector is moving, If one takes three "consecutive" points of the curve one gets a plane, known as the osculating or "kissing" plane which best fits the curve at that point. The tangent and the normal vectors lie in the osculating plane and the binormal vector is perpendicular to the osculating plane. For each point of a space curve one can also define the quantities known as curvature and torsion. The curve's curvature measures its tendency to bendaway from its tangent line, and the curve's torsion measures the tendency of the curve to twist out of the osculating plane. A theorem of differential geometry states that a space curve's shape is uniquely characterized by the functions kappa and tau which give the curve's curvature and torsion as functions of the arclength along the curve. In this notebook, I state and use an algorithm for "kappa tau curves," which are space curves defined in terms of functions for curvature and torsion. The algorithm is based on a set of three formulas known as "Frenet's formlas for the moving trihedron of a space curve." These formulas were published by the French mathematicians Serret and Frenet in 1852. Suppose that s parametrizes the arclength along a space curve, and that kappa and tau are functions of s which give the curvature and the torsion of the curve. Suppose also that T, N,and B are vector functions giving the tangent, normal, and binormal as functions of s. Then the Frenet formulas are as follows: dT/ds = kappa * N dN/ds = -kappa * T + tau * B dB/ds = -tau * N You can get an intuitive feel for the truth of the Frenet formulas by trying to "fly" your hand around as if it were the moving trihedron of a space curve. Make a fist with your right hand and then extend your index finger, middle finger, and thumb so that they are at right angles to each other. These fingers play the role of tangent, normal, and binormal. There seem to be three intrinsic ways your hand can rotate: around your thumb, your middle finger, or your index finger. To cystallize what is meant, try grabbing these with your left hand and performing an "unscrewing" motion on each. These produce motions that are also called yaw, pitch, and roll. If your hand moves along an imaginary space curve so that your index is always pointing in the direction of motion and your middle finger is always pointing in the direction in which the index finger is turning, then there will not actually be any "pitch" or "middle finger unscrewing" motion. The reason is that all the turning of the index finger or tangent vector is in the direction of the middle finger or normal. This is all "thumb unscrewing" motion. The only other motion allowed is motion which does not move the index finger: this is the "index finger unscrewing" motion. The "thumb unscrewing" motion is summarized in the first of the Frenet formulae: the change of the tangent is in the direction of the normal with a magnitude corresponding to the curvature. The "index finger unscrewing" motion appears in the third Frenet formula: the change of the binormal is in the reverse direction of the normal and with a magnitude corresponding to the torsion. The second of the Frenet formulas captures the "back effects" of the first and third formulas: As the tangent turns towards the normal, the normal turns away from the tangent; and as the binormal turns away from the normal, the normal turns towards the binormal. Our algorithm performs a simple "Euler-style" numerical integration to compute the successive points and trihedra of a space curve given by a kappa and a tau function. The curve is displayed as a ribbon by letting rungs stick out along the normal vectors and by connecting these rungs to make paved tiles. Usage and Limitations First select and evaluate the cell containing the Algorithm. Then create and evaluate a cell of this form: clear; top = ; kappa[s_] = ; tau[s_] = ; width = ; ds = ; pribbonfrenet The most significant limitation is that when Mathematica runs out of memory it does not issue a warning, instead it crashes by locking up the machine and commonly destroying the active notebook. Values of top over 300 reliably crash a machine with 12 Meg of RAM. It is good practice to save a backup copy of this notebook under a different name from the working copy. A second limitation is that we are using the crude "Euler style" method of numerical integration instead of a more accurate "Runge Kutte" style method. This can in priciple be changed if there is sufficient need. Algorithm to Display a Kappa-Tau Space Curve as a Ribbon clear:=Clear[ds,width,kappa,tau,top,point,tangent,normal, binormal,curve,ncurve,tiles]; normalize[vector_]:=(1/Sqrt[vector.vector])vector; cross[u_,v_]:={u[[2]]v[[3]]-u[[3]]v[[2]], u[[3]]v[[1]]-u[[1]]v[[3]], u[[1]]v[[2]]-u[[2]]v[[1]]}; pribbonfrenet:= ( point = {0.,0.,0.}; edgepoint = {0.,width,0.}; tangent= If[kappa[0] != 0. || tau[0] != 0, normalize[{kappa[0],0.,tau[0]}], {1.,0.,0.}]; normal={0.,1.,0.}; binormal=cross[tangent,normal]; curve=List[point]; edgecurve=List[edgepoint]; tiles=List[]; rungs=List[Line[{point,edgepoint}]]; Do [ newpoint=point+ds tangent; newtangent= normalize[tangent+ kappa[n ds]normal ds]; newnormal= normalize[normal+ (tau[n ds]binormal- kappa[n ds]tangent)ds]; newedgepoint=newpoint+width newnormal; AppendTo[curve,newpoint]; AppendTo[edgecurve,newedgepoint]; AppendTo[tiles,Polygon[{point,edgepoint, newedgepoint,newpoint}]]; AppendTo[rungs,Line[{newpoint,newedgepoint}]]; tangent=newtangent; normal=newnormal; binormal=cross[tangent,normal]; point=newpoint; edgepoint=newedgepoint; , {n,1,top,1} ]; Show [ Graphics3D[{Thickness[.006],Line[curve]}], Graphics3D[{Thickness[.003],RGBColor[1,0,0], Line[edgecurve]}], Graphics3D[{RGBColor[1,1,0],EdgeForm[],tiles}], Graphics3D[{GrayLevel[0.25],rungs}], BoxRatios->{1,1,1} ] ) Examples Helix: Kappa constant, Tau constant. clear;top=60;kappa[s_]=1.;tau[s_]=1.;width=.3;ds=.1; pribbonfrenet Baseball Stitch: Kappa constant, Tau periodic. You get a perfect baseball stitch if tau is Sin and kappa is 1. clear;top=125;kappa[s_]=1;tau[s_]:=Sin[s];width=.3;ds=.1; pribbonfrenet Vortex: Kappa linear, Tau inverse linear. clear;top=200;kappa[s_]=s;tau[s_]=50./(1+s);width=.001; ds=.02;pribbonfrenet Reversing Helix: Kappa periodic, Tau constant. Stretch this vertically so you can see it better. clear;top=160;kappa[s_]:=10 Sin[s];tau[s_]:=3.0;ds:=0.1; width=.2;pribbonfrenet Ruffle: Kappa periodic, Tau periodic; out of phase. Multiply Cos and Sin by 0.6 as otherwise it looks too bunchy. Bigger width than 1.5 makes nice conical overlaps. Swapping Cos and Sin gives a curve that looks the same. clear;top=100;kappa[s_]:=0.6Cos[s];tau[s_]:=0.6Sin[s]; ds:=N[12Pi/top];width=3.5;pribbonfrenet ----- Let me know if you get this to work! From: Rudy Rucker Date: Tue, 27 Sep 1994 21:02:35 -0700 To: math-fun@cs.arizona.edu Subject: kappa tau flies I raised the subject of how flies fly because I have been studying kappa tau curves, and think of flies as good virtual animators of these curves. To recap, one way to define a 3D space curve is to give the curve's curvature (kappa) and torsion (tau) as functions of the arclength along the curve. The attractiveness of this specification is that it is wholly coordinate independent. A few years ago, I did an initial exploration of kappa tau curves in Mathematica. In the last few months I've been developing a Windows based kappa tau curve program with my grad students here at SJSU. One interesting kind of kappa tau curve --- my "discovery" in this field --- is that the stich on a baseball is gotten by setting kappa constantly equal to one, and setting tau to the sine of the arclength. Now I've started looking at random kappa tau curves; a nice ribbon evolves if you do a random walk, bumping kappa and tau up or down randomly, say, 0.1 each at each rung of the ladder. The ladder being the visual representation of them, a ribbon computed by the trace of the points and the trace of the tips of the normal vectors extruded out to whatever width you want to set. The next step is flies who flie about, with as much vision as you care to endow them with, capable of changing their curvature and torsion at each step. The kappa and tau functions are now, in other words, under the control of an agent that can be designed, randomized, evolved, or whatever. I want to keep the flies in a room so they don't fly offscreen. How do I keep them from running into the walls? I've been wrestling with the complexities of it, whether and when to increase or decrease curvature and/or torsion. The example of the airplane was for quite some time (years) very misleading. One tends to think of an airplane as a moving trihedron with fuselage = tangent, wing = normal, the whaddyacallit-tail? thing that sticks up in back = binormal. But the whaddyacallit isn't the binormal and the wing isn't the normal. Counterexample: A plane banks to perform a horizontal curve. Now I've decided that instead of thinking, I'll program. I make an OOP object of the room and it returns a realnumer Danger value when the fly is within some Tooclose of the walls. Instead of thinking about the walls location (and why should it, as it is a coordinate free agent) the fly simply simulates ahead and asks what the value of Danger will be if it tries one of several obvious strategies. The four most obvious strategies are to (a) increment kappa and tau (b) increment kappa and decrement tau (c) decrement kappa and increment tau (d) decrement both. Some fixed amount of inc and dec being a parameter of the agent. Choose the best of (a)-(d) and if all of them make things worse, try doubling the incremnt/decrement. Might also consider the option of changing them by different amounts... Buzzzzzz. I don't see why after this has been achieved I couldn't make the flies goal to be in fact to change their wall distance to 0 relative to one wall, while also rotating their binormal to be normal to that wall. They will, I predict, land on the walls much like real flies do, with not fixed method, but with a heartening liveliness and chaoticity. The only problem is that my students are getting rebellious... From: Rudy Rucker Date: Tue, 27 Sep 1994 21:51:23 -0700 To: math-fun@cs.arizona.edu Subject: kappa tau flies addendum I forgot to mention the reason why I think flies do in fact fly along kappa tau curves. Flies don't (in my vision of them) speed up and slow down. That's why they have to leap up discontinuously to start flying. They always fly at the same speed. They just use up time with helical behaviour if need be. And the main thing about the kappa tau parametrization besides coordinate invariancy is that, as parametrized by arclength, the point on the kappa tau curve move always with velocity one. The patient buzzing fly. From: "John Conway" Date: Sun, 2 Oct 94 19:06:19 EDT To: math-fun@cs.arizona.edu, rucker@jupiter.SJSU.EDU Subject: Re: kappa tau flies [Mostly to Rudy Rucker:] You mention your "discovery" that the seam on a baseball curve is obtained by setting kappa constant and tau equal to the sine of the arclength. What does this "discovery" mean? Does it have anything to do with real baseballs? Or (as I suspect) does it merely mean that this is your favorite definition of a baseball-like curve? Anyway, I'd be happy if someone were to disprove my thesis that any two definitions of "the" baseball curve define distinct curves unless their equivalence was obvious from the start. One way to do this might be to show that Rucker's curve coincides (DOES IT?) with the one you get from an annular strip of phosphor-bronze with total angle bigger than 360 degrees. About 720 degrees approximates the baseball seam - about 630 degrees the tennisball one. John Conway From: Rudy Rucker Date: Thu, 6 Oct 1994 17:35:41 -0700 To: math-fun@cs.arizona.edu Subject: baseball curves Thanks for the insights from John Conway. It hadn't occured to me that were many possible baseball-like curves. The one given by curvature = 1 and torsion = sin arclength seems to correspond to the one you'd get from an annular strip of total angle 4 Pi, i.e. two full circles. "Seems" in that my program lets me determine the arclength of the curve and it is close to 4 Pi. (It's off a bit because I'm using Euler integration instead of Runge-Kutte, but it's within a few hundredths). But is this the REAL baseball curve? I don't know, how is a baseball curve defined by baseball makers? Is it in fact a curve of constant curvature? Given that the curve lies on a unit sphere one is tempted to think that the osculating circle is everywhere a great circle of the sphere, provided that the curvature is everywher one, is this true? Date: Fri, 7 Oct 1994 01:12-0700 From: Subject: JHC's phosphor bronze thingy To: math-fun@cs.arizona.edu In-Reply-To: <9409012344.AA01114@broccoli.princeton.edu> Message-Id: <"19941007081205.2.rwg@TSUNAMI"@SWEATHOUSE.macsyma.com> Status: RO Date: Thu, 1 Sep 1994 16:44 PDT From: "John Conway" [. . .] I was given in Zurich a phosphor-bronze strip in the form of a sector of an annulus, with angle 720 degrees (so it goes twice round) When you "open this up" by sliding it against itself, you get one "baseball curve" for each angle up to 720 degrees, and if there were more strip, you could go further. If the angle is less than one revolution, the strip just lies along a circular cone, as you'd expect, which flattens out into the plane as the angle reaches 360. Then as you increase the angle past 360, it sinuizes itself (so to speak) into a really baseball-like curve. It's a fascinating object, because you can hold it and wiggle so that the curve "passes along itself", assuming all of its possible positions that have a given line as tangent. I have tried to recall enough Mechanics to work out what analytic form it's giving, but can't for the life of me see why it isn't just the one made of 4 circular arcs, which careful examination shows me it really isn't. Perhaps your model assumes flexibility in the thin direction but none in the thick? Still trying to picture it. I assume it's not Moebius-twisted, and when not "opened up", the "two strips" are atop each other on the left and beside each other on the right, say?: ***** ******* * * ****** . * ****** **** ***** . **** **** *** *********************** **** *** *** ******** . ****** *** *** *** ******* . *** ** *** ** ****** . *** *** ** *** ***** . *** * *** ** **** . ** ** * ** **** . ** * ** * *** . ** ** ** * **** . ** ** * * *** . ** ** ** * *** . * * ** * *** . ** ** * * ** . ** ** * * ** . * * * * ** . * ** * * ** . * * * * * . ** * * * * . * ** * *....*..........................................................................*...**....* * * . * ** * * * . ** * ** ** ** . * * * ** * . * ** * ** ** . ** * * ** * . ** ** * ** * . * ** * *** * . ** * ** *** * . ** ** ** *** ** . *** ** ** **** ** . ** ** * **** ** . *** ** ** **** ** . *** ** * **** ** . *** ** ** ***** *** . **** *** *** ***** **** . ******** ** ****** ***** . ********* *** ****** ********* . ************* ** * ******* *********** **** ******* ** . **** ** *********** **** **** * (Hope that wasn't too wide. To increase the density, I needed to bump a dedocumented variable named CALCOMPNUM! Amazing I didn't need to set HOLLERITH to TRUE.) And when you open it up, the outer edge and inner edge make baseball seams on concentric spheres, with the strip's breadth nearly normal to them? From: Rudy Rucker Date: Fri, 7 Oct 1994 10:54:06 -0700 To: math-fun@cs.arizona.edu Subject: baseball bronze Aha. Gosper's post of JHC's bronze annulus experience is most enlightening. The key phrase is that the curve "sineusizes itself" so as to dawdle enough to include more arclength. Say that the bronze curve BZa for angle a is the curve assumed when the total annulus angle is a. Sat that the kappa tau baseball curve KTb for amplitude b is the curve given by integrating the Frenet formulas for an arc-length parametrized curve with curvature constantly 1 and torsion given as b*sine(arclength). Conjecture: BZa = KTf(a), for f a simple function. We know that f(2 Pi) = 0 and my simulations indicate that f(4 Pi) = 1, so Second conjecture: f(a) = (a - 2 Pi) / 2 Pi. Note that the first conjecture contradicts Conway's thesis that "all baseball curves defined in superficially different ways are analytically different." From: "John Conway" Date: Tue, 11 Oct 94 19:19:45 EDT To: math-fun@cs.arizona.edu, rucker@jupiter.SJSU.EDU Subject: Re: baseball curves [To Rudy Rucker:] I'm glad to see that you, like everyone else, don't know what the "real" baseball curve is. I'm not at all convinced of the truth of my conjecture that any two definitions of "the real baseball curve" will disagree unless theie equivalence was obvious from the start. You might like to disprove it, perhaps by showing that your curve is the same as that assumed by a phosphor-bronze strip subject to [...]. John Conway ., From: "John Conway" Date: Tue, 11 Oct 94 19:27:01 EDT To: math-fun@cs.arizona.edu, rwg@TSUNAMI.macsyma.com Subject: Re: JHC's phosphor bronze thingy Thank you for your big diagram, which I didn't understand one little bit. My phosphor-bronze thingy is just an annular strip of ph-br whose total angle exceeds 360 degrees. If it were less than 360 degrees, then, as you can surely imagine, it would assume a conical form (and it does!). But at more than 360 degrees, it really does take up a baseball-seam-like curve, whose analytic form I have not been able to understand. John Conway From: asimov@nas.nasa.gov (Daniel A. Asimov) Date: Tue, 11 Oct 1994 16:59:10 -0700 To: "John Conway" , math-fun@cs.arizona.edu Subject: Re: JHC's phosphor bronze thingy Back around 1970, another grad student at Berkeley -- Dave Bleecker -- took two copies of a planar annulus (the space between two concentric circles) made of rubber, and attached them together, thereby creating a 720-degree annulus embedded in R^3. It tended to assume the baseball-curve shape, and generally behaved strangely when one pushed it around. Dave dubbed his invention the "Intergalactic Nurdle." --Dan Asimov From: Rudy Rucker Date: Tue, 11 Oct 1994 19:43:01 -0700 To: math-fun@cs.arizona.edu Subject: phosphor bronze I tried simulating kt curves with curvature 1 and torsion equal to B*sine(arclength) for various B, and the damn thing seems only to smoothly meet up with itself when B is 1. In other words, while Ihad conjectured that if B is, e.g. 1/2, I would get a "baseball curve" more like an annulus, (flatter), --- this is not the case. We tried a bunch of variants (square roots, etc) and nothing so far seems to work. Gosper: speaking of Mobius strips, do you have a nifty way of generating the closed curve which is the edge of one? So JHC's "baseball curves are weird" thesis looks stronger than before... From: Rudy Rucker Date: Wed, 12 Oct 1994 11:31:08 -0700 To: math-fun@cs.arizona.edu Subject: phosphor bronze re. asimov's mobius with a circular edge, ok, I'll bite, describe it. re. my conjecture that a curve with constant curvature and sinusoidal torsion might match the phosphor bronze curve --- I was completely wrong. Duh level wrong. How so? The phosophr bronze curves arise as a physical solution, a minimal energy configuration, so it seems (phosphor bronze constancy conjecture) that the curvature and torsion ought to be the same at each point of one. Conway's observation that he could slowly turn the curve keeping its appreance the same while he did a virtual walk along it --- this suggests the curve is everywhere similar to itself which again indicates constant kappa and tau. But the kappa = 1 tau = sin(s) curve has torsion 0 at one spot. Moving out to the right, the torsion gets positive, moving hte other way the torsion gets negative. Right and left curling helices. They amp up to max torsion of 1 and -1, then damp back down to 0 and by a kind of *miracle* happen to smoothly meet back up overhead. If tau = a sin(s) for any a other than 1, they seem not to meet smoothly. This curve probably doesn't even fit on a sphere like I thought. it's not a baseball stitch ---- it's a CHAIR! There's still flies in the ointment, tho. I always thought the only curves with constant kappa and tau were helices. But it seems like the phosphor bronze curves ought to have constant kappa and tau, as mentioned above. How to resolve this discrepancy? From: hoey@AIC.NRL.Navy.Mil Date: Thu, 2 May 96 19:17:03 EDT To: math-fun@cs.arizona.edu Subject: Baseball season In case anyone's still interested in baseball curves, Richard B. Thompson's paper "Designing a Baseball Cover" is on http://www.mathsoft.com/asolve/baseball/baseball.html. He gives a little of the history. Apparently a trial-and-error drawing by C. H. Jackson in the 1860's has become enshrined in practice. Thompson also gives a family of solutions that I don't particularly like, but I'll describe. By symmetry, the northern hemisphere is a rotated inversion of the southern hemisphere, and the four equator crossings are at equal intervals. He takes the width of the neck as a given parameter, which determines the northernmost and southernmost points of the curve. Between two equator crossings and the intervening latitude maximum, he interpolates by cutting the sphere with a plane. I think it's surprising how good the result looks. Dan Hoey From: John Conway Date: Fri, 3 May 1996 14:08:15 -0400 (EDT) Subject: Re: Baseball season To: doyle@euclid.ucsd.edu Cc: math-fun@cs.arizona.edu On Thu, 2 May 1996 hoey@AIC.NRL.Navy.Mil wrote: > In case anyone's still interested in baseball curves, Richard > B. Thompson's paper "Designing a Baseball Cover" is on > http://www.mathsoft.com/asolve/baseball/baseball.html. He gives a > little of the history. Apparently a trial-and-error drawing by > C. H. Jackson in the 1860's has become enshrined in practice. > > Thompson also gives a family of solutions that I don't particularly > like, but I'll describe. By symmetry, the northern hemisphere is a > rotated inversion of the southern hemisphere, and the four equator > crossings are at equal intervals. He takes the width of the neck as a > given parameter, which determines the northernmost and southernmost > points of the curve. Between two equator crossings and the > intervening latitude maximum, he interpolates by cutting the sphere > with a plane. > > I think it's surprising how good the result looks. > > Dan > Do you have a reference for this paper, or at least a date, Dan? Peter Doyle, to whom I'm sending a copy of this message, has been raising the mathematical question of what "is" "the" baseball curve for some time. All sorts of people have produced answers that are mathematically "nicer" than the ones described above. However, I think that I myself have made the most valuable contribution to this problem (apart perhaps from Thompson's) by propounding my own BASEBALL CURVE CONJECTURE : No two definitions of "the" baseball curve will give the same answer unless their equivalence was obvious from the start. Certainly nobody seems to have disproved this conjecture in the 4 or 5 years it's been circulating. JHC From: hoey@AIC.NRL.Navy.Mil Date: Fri, 3 May 96 15:13:17 EDT Cc: math-fun@cs.arizona.edu Subject: Re: Baseball season conway@math.Princeton.EDU writes: > On Thu, 2 May 1996 hoey@AIC.NRL.Navy.Mil wrote: > > Richard > > B. Thompson's paper "Designing a Baseball Cover" is on > > http://www.mathsoft.com/asolve/baseball/baseball.html. > Do you have a reference for this paper, or at least a date, Dan? Well, it says "This page was updated March 05, 1996" but who's to say when most of it was written. Richard B. Thompson is at the U of Arizona math department: rbt@math.arizona.edu seems to be his e-mail address if you want to check if it's been published, but he doesn't list it on his c.v. (http://www.math.arizona.edu/profiles/thompson.html). > All sorts of people have produced answers that are mathematically > "nicer" than the ones described above. Eventually I figured out that the main reason it looks even as nice as it does is that symmetry forces the figure to be differentiable at the equator. Seems not to be second differentiable, though. It is nice, though, that this family of curves, parameterized by neck-width, goes from Pi, where it is the equator, down to -Pi, where it is a triple cover of the equator. Then it comes back. It's equivalent to taking arcs from four like circles with points of tangency at the equator. > BASEBALL CURVE CONJECTURE : No two definitions of "the" > baseball curve will give the same answer unless their equivalence > was obvious from the start. Well, Thompson's definition was sufficiently mired in symbology that it took me a while for me to figure out the geometric description. His definition essentially was that the projection of each loop in a plane perpendicular to the equator was linear. (Except he doesn't call it an equator and he draws it horizontally....) Dan Hoey From: Bill Gosper Date: Wed, 8 May 1996 06:57-0700 Subject: Re: Baseball season To: hoey@AIC.NRL.Navy.Mil Cc: doyle@euclid.ucsd.edu, math-fun@cs.arizona.edu Date: Mon, 6 May 1996 07:50 PDT From: hoey@AIC.NRL.Navy.Mil Mostly to RWG > It is nice, though, that this family of curves, parameterized by > neck-width, goes from Pi, where it is the equator, down to -Pi, where > it is a triple cover of the equator. Then it comes back. It's > What is? The family? Every member? I was talking about the family of baseball curves that consist of arcs from four circles on the sphere. These will be four equal circles, tangent each to the next in a cycle. The four points of tangency occur on the equator at intervals of Pi/2. The family is parameterized by one real number, be it diameter of the circle, the width of the baseball cover's neck, the latitude of the circle's center, etc. When I said "it comes back" I meant that the family is periodic in the parameter, or that the parameter might be better taken from a circle than from the reals. But now I'm not sure--it seems to me that while the family of curves is periodic, the family of baseball covers may become multiple. I find it confusing because the neck width becomes negative, and I'm not too sure what that means physically. > Does this family include the suggestion I make below? [...] > During the last recrudescence of this topic, I somewhat unclearly > remarked that, within manufacturing tolerances, the stitches I've > seen project into four quarter-circles in the equatorial plane. No the family I'm talking about, would project to ellipses, except when the circles were (degenerately) centered at the poles. The quarter-circles have the advantage of being "vertical" at the equator, whereat the lowest discontinuous derivative (wrt arclength) is the fourth. (The specific curve is four copies of the z>=0 half of the intersection of the unit sphere with the cylinder (x-1)^2 + (y-1)^2 = 1, i.e., z = sqrt(2 (1-x-y)).) From: Bill Gosper Date: Sat, 29 Jun 1996 03:54-0700 Subject: yet more seamliness To: math-fun@cs.arizona.edu Let a baseball have unit radius. I proposed that four equal arcs of the seam lay on parallel, pairwise tangent cylinders of unit radius. R. Weyhrauch proposes to do it with two perpendicular cylinders of radius r, with axes displaced by b and -b from the ball's center. If r+b<1, each cylinder intersects the sphere in two ovoids. If r^2-b^2 = 1/2, each ovoid is tangent to two others. Discard the short arcs joining points of tangency, and what's left is a seam. Somewhere between (r=1/sqrt 2, b=0) (i.e., four semicircles) and (r=3/4, b=1/4) (where the isthmi (2 arccos r+b) narrow to zero) is the (r,b) pair for which 2 arccos r+b matches the actual isthmus. (.707 < r < .75, a rather short interval.) Eyeballing a real baseball is tricky due to the wide stitching, which exaggerates the narrowness of the isthmus. If you mentally vaporize the stitching, the seam seems less dramatic. Re hat-boxes, I think most of us have at one time read that the cylindrically projected map of the globe is area-preserving.