Re: [math-fun] bizarre algebraic trivium
Message: 7 Date: Sat, 2 Nov 2013 16:41:51 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] bizarre algebraic trivium
I am using Maple. Denote the polynomial in that Root by p, the given root by x0. Using Sturm sequences one sees, that p has exactly 2 real roots, both are positive and 1 < x0 is confirmed by the system (the other one is smaller than 1). Now substitute x = 1/X^8 (ok, a bit strange) and assuming 0 < X that becomes 0 = -X^(31/4)*sqrt(sqrt(X+1)+sqrt(X))+1 Maple finds those roots, 2 of them are Reals, one is -1 and the other one is X0:= RootOf(_Z^30-_Z^29+_Z^28-_Z^27+_Z^26-_Z^25+_Z^24-_Z^23+_Z^22-_Z^21+_Z^20 -_Z^19+_Z^18-_Z^17+_Z^16+_Z^15-_Z^14+_Z^13-_Z^12+_Z^11-_Z^10 +_Z^9-_Z^8+_Z^7-_Z^6+_Z^5-_Z^4+_Z^3-_Z^2+_Z-1, index = 1); Feeding ln(x)/ln(x^2-2) with x = (1/X0)^8 is simplified to -8/15 by Maple.
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces. Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
This article shows how to construct a tennis ball curve from four circles (scroll down a little to see it): http://www.qedcat.com/archive/165.html I think the four points where the circles meet are required, but that the arcs that connect them can be altered, either exaggerating or reducing their size. Tom rkg writes:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
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On 11/3/2013 12:09 PM, rkg wrote:
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
There is a very revealing video on YouTube called "How Tennis Balls are made" (http://www.youtube.com/watch?v=yuZI9zgY4wM) that shows the processes inside a Wilson factory. The cutting and application of the two pieces of the felt cover are shown from 1:29 to 1:50. Here is a transcript of the narration: The felt cover of a tennis ball is made of two dumbbell-shaped pieces of material. These pass through a glue bath and will shortly be applied to the rubber ball. Covering is either done by hand or by machine. Hand covering is more accurate, and this can be seen by the very consistent width in the glue line that runs around the ball. Despite saying "dumbbell-shaped" the pieces shown in the video (especially clear around 1:42) seem to be convex; the boundary looks like two semicircles joined by straight lines. Perhaps this varies among manufacturers; the Penn balls I have at hand seem to be formed from nonconvex pieces. In fact, a brief perusal of images of Wilson tennis balls suggests that they make balls with varied curves. After watching this, I'd guess that the shapes are the result of refinements made in practice without any formal mathematics intervening. -- Fred W. Helenius fredh@ix.netcom.com
Is a tennis ball seam the same shape as a baseball seam? http://math.arizona.edu/~rbt/baseball.PDF "Designing a Baseball Cover" Richard B. Thompson College Mathematics Journal, Jan. 1998. At 09:09 AM 11/3/2013, rkg wrote:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
I'd always casually assumed that the tennis-ball / baseball curve was probably some simple space quartic that everybody but me knew about. But since it seems this is not the case, I'll put in my two-penn'orth. The classic minimal Enneper's surface intersects concentric spheres in a family of such curves. With the parameterisation in the Wikipedia page, a sphere of small radius meets it in an approximate circle; a sphere of radius (approx?) 1/4 meets it in a typical tennis-ball curve, with expected symmetry and arcs parallel at the extremities; a sphere of radius (approx?) 1 meets it in a curve with touching arcs; for larger radius the curve has 4 self-intersections. See http://www.indiana.edu/~minimal/maze/enneper.html http://en.wikipedia.org/wiki/Enneper_surface The degree of Enneper's surface equals 9, so presumably these curves have degree 18. Fred Lunnon On 11/3/13, Henry Baker <hbaker1@pipeline.com> wrote:
Is a tennis ball seam the same shape as a baseball seam?
http://math.arizona.edu/~rbt/baseball.PDF
"Designing a Baseball Cover"
Richard B. Thompson
College Mathematics Journal, Jan. 1998.
At 09:09 AM 11/3/2013, rkg wrote:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Update on Enneper-sphere intersections: For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +/- 0.1849108100, +/- 0.1380711874] , so the cuboid boxing the curve is somewhat flattened. For the extreme waisted curve with tacnodes at its tips, dividing the sphere into four teardrop regions, radius 1 turns out to be exact. Note that the usual parameterisation for Enneper's surface scales all coordinates involved up by a factor 3 . Magma script and Maple graphic are available on request. Fred Lunnon On 11/4/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I'd always casually assumed that the tennis-ball / baseball curve was probably some simple space quartic that everybody but me knew about. But since it seems this is not the case, I'll put in my two-penn'orth.
The classic minimal Enneper's surface intersects concentric spheres in a family of such curves. With the parameterisation in the Wikipedia page, a sphere of small radius meets it in an approximate circle; a sphere of radius (approx?) 1/4 meets it in a typical tennis-ball curve, with expected symmetry and arcs parallel at the extremities; a sphere of radius (approx?) 1 meets it in a curve with touching arcs; for larger radius the curve has 4 self-intersections.
See http://www.indiana.edu/~minimal/maze/enneper.html http://en.wikipedia.org/wiki/Enneper_surface
The degree of Enneper's surface equals 9, so presumably these curves have degree 18.
Fred Lunnon
On 11/3/13, Henry Baker <hbaker1@pipeline.com> wrote:
Is a tennis ball seam the same shape as a baseball seam?
http://math.arizona.edu/~rbt/baseball.PDF
"Designing a Baseball Cover"
Richard B. Thompson
College Mathematics Journal, Jan. 1998.
At 09:09 AM 11/3/2013, rkg wrote:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Customary typo correction --- should have read For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +0.1849108100, -0.1380711874] , [0, -0.1849108100, -0.1380711874] , [+0.1849108100, 0, +0.1380711874] , [-0.1849108100, 0, +0.1380711874] ; so the cuboid boxing the curve is somewhat flattened. On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Update on Enneper-sphere intersections:
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +/- 0.1849108100, +/- 0.1380711874] , so the cuboid boxing the curve is somewhat flattened.
For the extreme waisted curve with tacnodes at its tips, dividing the sphere into four teardrop regions, radius 1 turns out to be exact.
Note that the usual parameterisation for Enneper's surface scales all coordinates involved up by a factor 3 .
Magma script and Maple graphic are available on request.
Fred Lunnon
On 11/4/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I'd always casually assumed that the tennis-ball / baseball curve was probably some simple space quartic that everybody but me knew about. But since it seems this is not the case, I'll put in my two-penn'orth.
The classic minimal Enneper's surface intersects concentric spheres in a family of such curves. With the parameterisation in the Wikipedia page, a sphere of small radius meets it in an approximate circle; a sphere of radius (approx?) 1/4 meets it in a typical tennis-ball curve, with expected symmetry and arcs parallel at the extremities; a sphere of radius (approx?) 1 meets it in a curve with touching arcs; for larger radius the curve has 4 self-intersections.
See http://www.indiana.edu/~minimal/maze/enneper.html http://en.wikipedia.org/wiki/Enneper_surface
The degree of Enneper's surface equals 9, so presumably these curves have degree 18.
Fred Lunnon
On 11/3/13, Henry Baker <hbaker1@pipeline.com> wrote:
Is a tennis ball seam the same shape as a baseball seam?
http://math.arizona.edu/~rbt/baseball.PDF
"Designing a Baseball Cover"
Richard B. Thompson
College Mathematics Journal, Jan. 1998.
At 09:09 AM 11/3/2013, rkg wrote:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The Enneper-sphere tennis-ball curve is smooth, being the intersection of two algebraic surfaces. However it lacks a rational parameterisation, since (according to Maple) the plane curve |[x, y, z]|^2 = r^2 , qua function of parameters u,v , has genus = 8 rather than 0 . In practice such considerations are irrelevant, since a template needs to be computed only once, and to working tolerance. To "unroll" a strip of the corresponding spherical region onto a plane template requires a decision to be made about the appropriate projection. There doesn't appear to be a canonical answer to the latter question: it depends upon how much the cover material can be expected to stretch across its central line of symmetry, as opposed to wrinkling up along its boundary. Fred Lunnon On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Customary typo correction --- should have read
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +0.1849108100, -0.1380711874] , [0, -0.1849108100, -0.1380711874] , [+0.1849108100, 0, +0.1380711874] , [-0.1849108100, 0, +0.1380711874] ; so the cuboid boxing the curve is somewhat flattened.
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Update on Enneper-sphere intersections:
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +/- 0.1849108100, +/- 0.1380711874] , so the cuboid boxing the curve is somewhat flattened.
For the extreme waisted curve with tacnodes at its tips, dividing the sphere into four teardrop regions, radius 1 turns out to be exact.
Note that the usual parameterisation for Enneper's surface scales all coordinates involved up by a factor 3 .
Magma script and Maple graphic are available on request.
Fred Lunnon
On 11/4/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I'd always casually assumed that the tennis-ball / baseball curve was probably some simple space quartic that everybody but me knew about. But since it seems this is not the case, I'll put in my two-penn'orth.
The classic minimal Enneper's surface intersects concentric spheres in a family of such curves. With the parameterisation in the Wikipedia page, a sphere of small radius meets it in an approximate circle; a sphere of radius (approx?) 1/4 meets it in a typical tennis-ball curve, with expected symmetry and arcs parallel at the extremities; a sphere of radius (approx?) 1 meets it in a curve with touching arcs; for larger radius the curve has 4 self-intersections.
See http://www.indiana.edu/~minimal/maze/enneper.html http://en.wikipedia.org/wiki/Enneper_surface
The degree of Enneper's surface equals 9, so presumably these curves have degree 18.
Fred Lunnon
On 11/3/13, Henry Baker <hbaker1@pipeline.com> wrote:
Is a tennis ball seam the same shape as a baseball seam?
http://math.arizona.edu/~rbt/baseball.PDF
"Designing a Baseball Cover"
Richard B. Thompson
College Mathematics Journal, Jan. 1998.
At 09:09 AM 11/3/2013, rkg wrote:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
For arbitrary sphere radius r , the curve has four nodal singularities where x = 0 or y = 0 : these are real and obvious where the curve intersects itself when r > 1 , but safely complex and out of harm's way when r < 1 . Rather unexpectedly, there are four more real singularities at the inflections where z = 0 and x = y : these correspond to a pair of lines on the Enneper surface which superficially appear innocuous. There is a theorem that any TB curve must have (at least) four inflections, see http://www.qedcat.com/archive/165.html --- it intrigues me that these points turn out to be so special here. Despite getting down and dirty with Magma's algebraic geometry feature, I haven't yet managed to decide whether there are further singularities. [Don't even think about trying to understand what schemes are --- just scope the examples, then hack them!] Having gone this far, I couldn't resist putting the r ~ 0.2307718797455473 curve onto a sphere. OK, I know it looks boring --- that's the whole point! https://www.dropbox.com/s/ypsqc07tisw2gf2/tennis_ball.jpg Fred Lunnon On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The Enneper-sphere tennis-ball curve is smooth, being the intersection of two algebraic surfaces. However it lacks a rational parameterisation, since (according to Maple) the plane curve |[x, y, z]|^2 = r^2 , qua function of parameters u,v , has genus = 8 rather than 0 .
In practice such considerations are irrelevant, since a template needs to be computed only once, and to working tolerance.
To "unroll" a strip of the corresponding spherical region onto a plane template requires a decision to be made about the appropriate projection. There doesn't appear to be a canonical answer to the latter question: it depends upon how much the cover material can be expected to stretch across its central line of symmetry, as opposed to wrinkling up along its boundary.
Fred Lunnon
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Customary typo correction --- should have read
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +0.1849108100, -0.1380711874] , [0, -0.1849108100, -0.1380711874] , [+0.1849108100, 0, +0.1380711874] , [-0.1849108100, 0, +0.1380711874] ; so the cuboid boxing the curve is somewhat flattened.
On 11/6/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Update on Enneper-sphere intersections:
For the nice tennis-ball curve with parallel osculating planes at its tips, the exact sphere radius is the root of (9r)^4 + 14(9r)^2 - 79 , ie. r ~ 0.2307718797455473 --- rather less than 1/4 . At the tips [x,y,z] ~ [0, +/- 0.1849108100, +/- 0.1380711874] , so the cuboid boxing the curve is somewhat flattened.
For the extreme waisted curve with tacnodes at its tips, dividing the sphere into four teardrop regions, radius 1 turns out to be exact.
Note that the usual parameterisation for Enneper's surface scales all coordinates involved up by a factor 3 .
Magma script and Maple graphic are available on request.
Fred Lunnon
On 11/4/13, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I'd always casually assumed that the tennis-ball / baseball curve was probably some simple space quartic that everybody but me knew about. But since it seems this is not the case, I'll put in my two-penn'orth.
The classic minimal Enneper's surface intersects concentric spheres in a family of such curves. With the parameterisation in the Wikipedia page, a sphere of small radius meets it in an approximate circle; a sphere of radius (approx?) 1/4 meets it in a typical tennis-ball curve, with expected symmetry and arcs parallel at the extremities; a sphere of radius (approx?) 1 meets it in a curve with touching arcs; for larger radius the curve has 4 self-intersections.
See http://www.indiana.edu/~minimal/maze/enneper.html http://en.wikipedia.org/wiki/Enneper_surface
The degree of Enneper's surface equals 9, so presumably these curves have degree 18.
Fred Lunnon
On 11/3/13, Henry Baker <hbaker1@pipeline.com> wrote:
Is a tennis ball seam the same shape as a baseball seam?
http://math.arizona.edu/~rbt/baseball.PDF
"Designing a Baseball Cover"
Richard B. Thompson
College Mathematics Journal, Jan. 1998.
At 09:09 AM 11/3/2013, rkg wrote:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Dear Richard, Just between you and me: Here's my method for constructing a nice baseball curve: First divide a sphere S into 2 hemispheres by drawing an equator Q. Now draw 2 great semicircles C_1, C_2 on S, with their endpoints on Q -- at -pi/2, 0, pi/2, pi -- with their midpoints at the poles. Conformal mapping theory (e.g., in Ahlfors, 3rd ed.) tells us that S minus C_1 u C_2 is conformally equivalent to a right cylinder Cyl of fixed ratio (of diameter to height). And the equivalence is unique up to a rotation. Call the conformal equivalence F: F: S - (C_1 u C_2) -> Cyl The cylinder has a well-defined "waist" circle W (halfway between its boundary circles). Finally, take the inverse image B of W by F: B := Finv(W) and then B is the baseball curve. (I mentioned this on math-fun to Conway maybe 15 years ago, and he objected because C_1 and C_2 are part of a continuum of curves that we could also use instead: curves with the same symmetry with respect to S, but longer -- or maybe even shorter -- portions of a great circle.) Every so often I work on figuring out the exact formula for B -- which I expect to be simple -- but I haven't done that yet. Regards, Dan On 2013-11-03, at 9:09 AM, rkg wrote:
Dear funsters, A tennis-ball appears to be made from 2 congruent pieces of material, seamed together in a curve. Are possible equations to the curve known? I'd like a smooth algebraic equation, probably of degree 4, and preferably with a maximum number of rational points on it. A first approximation might be to take a sphere of radius root(3) and centre at (0,0,0) and take the 8 great circle arcs (1,1,1) to (-1,1,1) to (-1,-1,1), (1,-1,1), (1,-1,-1), (-1,-1,-1), (-1,1,-1), (1,1,-1) and back to (1,1,1). However, this isn't smooth at the 8 corners of the cube, and I think that it doesn't even partition the sphere into two congruent pieces.
Is this well-known to those who well know it? What do the tennis-ball manufacturers do? R.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Sorry about that. The only reason I didn't intend to post that to math-fun is that I already have, and didn't mean to bother anyone with a rerun. But as long as I screwed up, I welcome anyone's comment on the idea for a baseball curve that I explained in that inadvertent post. --Dan On 2013-11-03, at 2:59 PM, James Propp wrote:
On Sunday, November 3, 2013, Dan Asimov <dasimov@earthlink.net> wrote:
Dear Richard,
Just between you and me:
And the rest of us. :-)
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (8)
-
Axel Vogt -
Dan Asimov -
Fred Lunnon -
Fred W. Helenius -
Henry Baker -
James Propp -
rkg -
Tom Karzes