[math-fun] LOOP: it's pool, but backwards.
I first read about elliptical pool tables in Martin Gardner. So I decided to go make one. http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/1... <http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/16/loop-new-cue-sport-pool-ellipse-elliptical>
It strikes me as unseemly for you to describe your own game as "thrilling" and "amazingly fun" (other than in commercial publicity materials). In any case, maybe the next time you describe the mathematical properties of an ellipse you might mention that the ellipse can be defined as the curve any point of whose sum of distances to two specific points is a constant. —Dan
On Jul 16, 2015, at 8:31 AM, Alex Bellos <alexbellos@gmail.com> wrote:
I first read about elliptical pool tables in Martin Gardner.
So I decided to go make one.
http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/1... <http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/16/loop-new-cue-sport-pool-ellipse-elliptical>
In my book i gave exactly that definition. We British are unseemly. It’s our most charming trait. ;)
On 16 Jul 2015, at 16:46, Dan Asimov <asimov@msri.org> wrote:
It strikes me as unseemly for you to describe your own game as "thrilling" and "amazingly fun" (other than in commercial publicity materials).
In any case, maybe the next time you describe the mathematical properties of an ellipse you might mention that the ellipse can be defined as the curve any point of whose sum of distances to two specific points is a constant.
—Dan
On Jul 16, 2015, at 8:31 AM, Alex Bellos <alexbellos@gmail.com> wrote:
I first read about elliptical pool tables in Martin Gardner.
So I decided to go make one.
http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/1... <http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/16/loop-new-cue-sport-pool-ellipse-elliptical>
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Maybe Alex's description of his game as "thrilling" is meant ironically, as an enactment of false immodesty (the seldom-seen flip side of false modesty). Subtext: "This is the sort of thing I'd say about the game I invented if I were a self-aggrandizing American rather than a self-deprecating Brit." :-) Jim Propp On Thu, Jul 16, 2015 at 12:08 PM, Alex Bellos <alexbellos@gmail.com> wrote:
In my book i gave exactly that definition.
We British are unseemly. It’s our most charming trait. ;)
On 16 Jul 2015, at 16:46, Dan Asimov <asimov@msri.org> wrote:
It strikes me as unseemly for you to describe your own game as "thrilling" and "amazingly fun" (other than in commercial publicity materials).
In any case, maybe the next time you describe the mathematical properties of an ellipse you might mention that the ellipse can be defined as the curve any point of whose sum of distances to two specific points is a constant.
—Dan
On Jul 16, 2015, at 8:31 AM, Alex Bellos <alexbellos@gmail.com> wrote:
I first read about elliptical pool tables in Martin Gardner.
So I decided to go make one.
http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/1... < http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/1...
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Let's focus on the important points here. The elliptical table is what's interesting. I'm sure Alex is eccentric and this is reflected by his choice of adjectives, but let us put away the axes because we all know he's affine fellow. On Thu, Jul 16, 2015 at 10:43 AM, James Propp <jamespropp@gmail.com> wrote:
Maybe Alex's description of his game as "thrilling" is meant ironically, as an enactment of false immodesty (the seldom-seen flip side of false modesty). Subtext: "This is the sort of thing I'd say about the game I invented if I were a self-aggrandizing American rather than a self-deprecating Brit."
:-)
Jim Propp
On Thu, Jul 16, 2015 at 12:08 PM, Alex Bellos <alexbellos@gmail.com> wrote:
In my book i gave exactly that definition.
We British are unseemly. It’s our most charming trait. ;)
On 16 Jul 2015, at 16:46, Dan Asimov <asimov@msri.org> wrote:
It strikes me as unseemly for you to describe your own game as "thrilling" and "amazingly fun" (other than in commercial publicity materials).
In any case, maybe the next time you describe the mathematical properties of an ellipse you might mention that the ellipse can be defined as the curve any point of whose sum of distances to two specific points is a constant.
—Dan
On Jul 16, 2015, at 8:31 AM, Alex Bellos <alexbellos@gmail.com> wrote:
I first read about elliptical pool tables in Martin Gardner.
So I decided to go make one.
http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/1... < http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/jul/1...
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Good point. If we define an ellipse by the locus of points whose sum-of-distances to two given points is a fixed constant, what is the shortest proof that these two segments make equal angles with the tangent line? (Assuming the ellipse surrounds a positive area.) —Dan
On Jul 16, 2015, at 10:52 AM, Tom Rokicki <rokicki@gmail.com> wrote:
Let's focus on the important points here. The elliptical table is what's interesting. I'm sure Alex is eccentric and this is reflected by his choice of adjectives, but let us put away the axes because we all know he's affine fellow.
My informal proof is that if the two angles were unequal, and you shifted the contact paint along the ellipse toward the smaller angle, you'd be making the distance shorter. On Thu, Jul 16, 2015 at 2:10 PM, Dan Asimov <asimov@msri.org> wrote:
Good point.
If we define an ellipse by the locus of points whose sum-of-distances to two given points is a fixed constant, what is the shortest proof that these two segments make equal angles with the tangent line?
(Assuming the ellipse surrounds a positive area.)
—Dan
On Jul 16, 2015, at 10:52 AM, Tom Rokicki <rokicki@gmail.com> wrote:
Let's focus on the important points here. The elliptical table is what's interesting. I'm sure Alex is eccentric and this is reflected by his choice of adjectives, but let us put away the axes because we all know he's affine fellow.
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Nice one! I think that can easily be made rigorous, say with a Taylor expansion to the first order. —Dan
On Jul 16, 2015, at 11:30 AM, Allan Wechsler <acwacw@gmail.com> wrote:
My informal proof is that if the two angles were unequal, and you shifted the contact paint along the ellipse toward the smaller angle, you'd be making the distance shorter.
On Thu, Jul 16, 2015 at 2:10 PM, Dan Asimov <asimov@msri.org <mailto:asimov@msri.org>> wrote:
If we define an ellipse by the locus of points whose sum-of-distances to two given points is a fixed constant, what is the shortest proof that these two segments make equal angles with the tangent line?
(Assuming the ellipse surrounds a positive area.)
Nice! Is there a proof this short and simple that the ellipse defined by these definitions is related to a circle by an affine transformation? The algebrist's proof of this is that both are degree-two curves that don't intersect the line at infinity, but I don't see an easy geometrical proof. On Thu, Jul 16, 2015 at 2:30 PM, Allan Wechsler <acwacw@gmail.com> wrote:
My informal proof is that if the two angles were unequal, and you shifted the contact paint along the ellipse toward the smaller angle, you'd be making the distance shorter.
On Thu, Jul 16, 2015 at 2:10 PM, Dan Asimov <asimov@msri.org> wrote:
Good point.
If we define an ellipse by the locus of points whose sum-of-distances to two given points is a fixed constant, what is the shortest proof that these two segments make equal angles with the tangent line?
(Assuming the ellipse surrounds a positive area.)
—Dan
On Jul 16, 2015, at 10:52 AM, Tom Rokicki <rokicki@gmail.com> wrote:
Let's focus on the important points here. The elliptical table is what's interesting. I'm sure Alex is eccentric and this is reflected by his choice of adjectives, but let us put away the axes because we all know he's affine fellow.
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-- Andy.Latto@pobox.com
Is there a proof this short and simple that the ellipse defined by these definitions is related to a circle by an affine transformation?
There is a perspex model of Dandelin’s spheres at the Mathematica exhibition (the one by Charles and Ray Eames). Just by the model of the Zeta function.
On 17 Jul 2015, at 20:05, Adam P. Goucher <apgoucher@gmx.com> wrote:
Is there a proof this short and simple that the ellipse defined by these definitions is related to a circle by an affine transformation?
Yes: https://en.wikipedia.org/wiki/Dandelin_spheres
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That's a great geometrical proof that "Set of points with a constant distant sum from two foci" is the same as "intersection of a circular cone and a plane". But what's the geometrical argument that either of those is the same as a stretched circle? Andy On Fri, Jul 17, 2015 at 3:05 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Is there a proof this short and simple that the ellipse defined by these definitions is related to a circle by an affine transformation?
Yes: https://en.wikipedia.org/wiki/Dandelin_spheres
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-- Andy.Latto@pobox.com
Ooh. There is a continuum of cones that have a given ellipse as cross-section. In the limit, with the apex of the cone at infinity, the cone becomes a cylinder. Intuitively, the Dandelin construction must still work with a cylinder. And proving that the cross-section of a cylinder is a stretched circle seems like it should be easy. I bet there is a good proof hiding here. On Fri, Jul 17, 2015 at 5:04 PM, Andy Latto <andy.latto@pobox.com> wrote:
That's a great geometrical proof that "Set of points with a constant distant sum from two foci" is the same as "intersection of a circular cone and a plane". But what's the geometrical argument that either of those is the same as a stretched circle?
Andy
On Fri, Jul 17, 2015 at 3:05 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Is there a proof this short and simple that the ellipse defined by these definitions is related to a circle by an affine transformation?
Yes: https://en.wikipedia.org/wiki/Dandelin_spheres
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-- Andy.Latto@pobox.com
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That limit works just fine geometrically, keeping say the ellipse, and the angle between the cutting-plane and the cone's axis, constant. Then the ellipse is the intersection of the cylinder x^2 + y^2 = R^2 and the cutting-plane. It's then clear that the ellipse becomes the circular cross-section of the cylinder by squeezing uniformly in the direction of its major axis, but not at all in the direction of its minor axis. —Dan
On Jul 17, 2015, at 4:43 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Ooh. There is a continuum of cones that have a given ellipse as cross-section. In the limit, with the apex of the cone at infinity, the cone becomes a cylinder. Intuitively, the Dandelin construction must still work with a cylinder. And proving that the cross-section of a cylinder is a stretched circle seems like it should be easy. I bet there is a good proof hiding here.
On Fri, Jul 17, 2015 at 5:04 PM, Andy Latto <andy.latto@pobox.com> wrote:
That's a great geometrical proof that "Set of points with a constant distant sum from two foci" is the same as "intersection of a circular cone and a plane". But what's the geometrical argument that either of those is the same as a stretched circle?
Andy
On Fri, Jul 17, 2015 at 3:05 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Is there a proof this short and simple that the ellipse defined by these definitions is related to a circle by an affine transformation?
Yes: https://en.wikipedia.org/wiki/Dandelin_spheres
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-- Andy.Latto@pobox.com
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participants (8)
-
Adam P. Goucher -
Alex Bellos -
Allan Wechsler -
Andy Latto -
Dan Asimov -
Dan Asimov -
James Propp -
Tom Rokicki