[math-fun] a circle packing theorem
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name? Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is: a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.) For a given D, the collection is either empty or infinite. erich
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain. Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold. And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking! I believe there is at least one app for sale that displays the Steiner Porism. — Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >> within the outer circle, maybe? Though that's still a tad vague ... Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds. WFL On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Unless you thought about it, that display hack was "Ho-hum, the circles go around and get bigger and smaller. So what?" So you had to be somewhat astute to be surprised by it. Ah, the days of asynchronous, random point plotting non raster display hacking! Completely apart from the Steinerism, consider just portraying circulating ball bearings of fixed size confined between concentric circles. Obviously, you just draw the confining circles, draw n ball bearings rotated by omega t, increment t, and repeat. Wrong. What this draws is the n balls rotating, but with a gap proportional to omega between two of them! Because you see the last circle of the previous "frame" immediately followed by the first circle of the new "frame". It was amazing-- you could continuously vary omega and the gap would grow and shrink, yet the circles remained of constant size. Where did the gap space come from? Where did it go when omega -> 0? The fix: To go clockwise, display n + omega*epsilon circulators; anticlockwise, n - omega*epsilon. The inner confining circle radius is necessarily a function of n, so it grew or shrank slightly as a function of omega! But then you get a gap or overlap when you try to film it, so I had to reintroduce the original bug! Do I smell relativity? There were other amazing effects just from varying the timing and ordering of points displayed in a rectangular grid. While developing his original integer linear relation finder, Rich asked to see an x-y grid time-ordered by x pi + y e mod 1. 30 usec/point would take half a minute to paint the whole 1024 by 1024 screen, so we may have derezzed. Anyway, you saw a slowly drifting non-rectangular grid. But if you looked away and then back, you might see a different grid with a different velocity! You could train yourself to see several different grids. You could conjecture a random large grid velocity, swipe your finger across the screen, and see it if your guess was close. You could thus hide an image on the screen by slightly dislocating the time-ordering of its points. Only someone who knew the approximate velocity could see it. To maximize this effect, I wrote a trivial-looking hack that marched a small parallelogram-shaped grid repeatedly across the screen at a fixed rate. You could control which way the parallelogram leaned, and how much, by adjusting the console "test word". But if you looked at the yellow persistent phosphor, you saw that it was constantly painting square boxes. The only thing the test word changed was time delays between points vs between lines vs between boxes. So you could skew the moving box left and right while leaving perfectly square afterimages. --rwg On 2015-12-18 09:21, Mike Beeler wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich
These sound cool. Is there a way to simulate them on modern devices? Jim On Friday, December 18, 2015, rwg <rwg@sdf.org> wrote:
Unless you thought about it, that display hack was "Ho-hum, the circles go around and get bigger and smaller. So what?" So you had to be somewhat astute to be surprised by it.
Ah, the days of asynchronous, random point plotting non raster display hacking! Completely apart from the Steinerism, consider just portraying circulating ball bearings of fixed size confined between concentric circles.
Obviously, you just draw the confining circles, draw n ball bearings rotated by omega t, increment t, and repeat.
Wrong.
What this draws is the n balls rotating, but with a gap proportional to omega between two of them! Because you see the last circle of the previous "frame" immediately followed by the first circle of the new "frame". It was amazing-- you could continuously vary omega and the gap would grow and shrink, yet the circles remained of constant size. Where did the gap space come from? Where did it go when omega -> 0?
The fix: To go clockwise, display n + omega*epsilon circulators; anticlockwise, n - omega*epsilon. The inner confining circle radius is necessarily a function of n, so it grew or shrank slightly as a function of omega!
But then you get a gap or overlap when you try to film it, so I had to reintroduce the original bug! Do I smell relativity?
There were other amazing effects just from varying the timing and ordering of points displayed in a rectangular grid.
While developing his original integer linear relation finder, Rich asked to see an x-y grid time-ordered by x pi + y e mod 1. 30 usec/point would take half a minute to paint the whole 1024 by 1024 screen, so we may have derezzed. Anyway, you saw a slowly drifting non-rectangular grid. But if you looked away and then back, you might see a different grid with a different velocity! You could train yourself to see several different grids. You could conjecture a random large grid velocity, swipe your finger across the screen, and see it if your guess was close. You could thus hide an image on the screen by slightly dislocating the time-ordering of its points. Only someone who knew the approximate velocity could see it.
To maximize this effect, I wrote a trivial-looking hack that marched a small parallelogram-shaped grid repeatedly across the screen at a fixed rate. You could control which way the parallelogram leaned, and how much, by adjusting the console "test word". But if you looked at the yellow persistent phosphor, you saw that it was constantly painting square boxes. The only thing the test word changed was time delays between points vs between lines vs between boxes. So you could skew the moving box left and right while leaving perfectly square afterimages. --rwg
On 2015-12-18 09:21, Mike Beeler wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net>
wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com
> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Sure. Just fade out the previous frame. https://www.khanacademy.org/computer-programming/phosphor/5497756172550144 On Fri, Dec 18, 2015 at 1:11 PM, James Propp <jamespropp@gmail.com> wrote:
These sound cool. Is there a way to simulate them on modern devices?
Jim
On Friday, December 18, 2015, rwg <rwg@sdf.org> wrote:
Unless you thought about it, that display hack was "Ho-hum, the circles go around and get bigger and smaller. So what?" So you had to be somewhat astute to be surprised by it.
Ah, the days of asynchronous, random point plotting non raster display hacking! Completely apart from the Steinerism, consider just portraying circulating ball bearings of fixed size confined between concentric circles.
Obviously, you just draw the confining circles, draw n ball bearings rotated by omega t, increment t, and repeat.
Wrong.
What this draws is the n balls rotating, but with a gap proportional to omega between two of them! Because you see the last circle of the previous "frame" immediately followed by the first circle of the new "frame". It was amazing-- you could continuously vary omega and the gap would grow and shrink, yet the circles remained of constant size. Where did the gap space come from? Where did it go when omega -> 0?
The fix: To go clockwise, display n + omega*epsilon circulators; anticlockwise, n - omega*epsilon. The inner confining circle radius is necessarily a function of n, so it grew or shrank slightly as a function of omega!
But then you get a gap or overlap when you try to film it, so I had to reintroduce the original bug! Do I smell relativity?
There were other amazing effects just from varying the timing and ordering of points displayed in a rectangular grid.
While developing his original integer linear relation finder, Rich asked to see an x-y grid time-ordered by x pi + y e mod 1. 30 usec/point would take half a minute to paint the whole 1024 by 1024 screen, so we may have derezzed. Anyway, you saw a slowly drifting non-rectangular grid. But if you looked away and then back, you might see a different grid with a different velocity! You could train yourself to see several different grids. You could conjecture a random large grid velocity, swipe your finger across the screen, and see it if your guess was close. You could thus hide an image on the screen by slightly dislocating the time-ordering of its points. Only someone who knew the approximate velocity could see it.
To maximize this effect, I wrote a trivial-looking hack that marched a small parallelogram-shaped grid repeatedly across the screen at a fixed rate. You could control which way the parallelogram leaned, and how much, by adjusting the console "test word". But if you looked at the yellow persistent phosphor, you saw that it was constantly painting square boxes. The only thing the test word changed was time delays between points vs between lines vs between boxes. So you could skew the moving box left and right while leaving perfectly square afterimages. --rwg
On 2015-12-18 09:21, Mike Beeler wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net>
wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com > > > wrote: > > back in undergraduate complex analysis, i learned the following > theorem, > whose proof is easy enough using conformal mappings. does this > theorem > have a name? > > Let D be a disk entirely inside the unit circle. Consider all > collections of non-overlapping disks C so that each member of C is: > > a) inside the unit circle and tangent to the unit circle, > b) doesn't overlap D but is tangent to D, and > c) tangent to exactly two other members of C. > (Thus the collection C forms a tangent ring around D.) > > For a given D, the collection is either empty or infinite. > > erich >
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). WFL On 12/18/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate >>
The number of discs on the chain must be infinite at tangency, of course. Otherwise, there is no theoretical reason why one boundary circle (strictly, directed cycle in the sense of contact / Lie-sphere geometry) should be internal to the other. Steiner's porism applies whether their relation is internal, external, or intersecting; also whether one, both or neither is external to the chain. A "rotating" system of finite chains may exist even when one boundary is a straight line.
WFL
On 12/18/15, James Buddenhagen <jbuddenh@gmail.com> wrote:
So an example would be https://flic.kr/p/em4XbX
On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Steiner chain interactive demo --- free, slightly tricky to control --- at http://codepen.io/yukulele/pen/OVOEdX/ This might be a good time to remind people (especially grumpy Yuletide refuseniks) about the excellent problem posed on this list by Gosper in August 2014, concerning polynomial relations between members of a chain: for example, where k1, ..., k5 denote curvatures of successive discs, (k1 - k4)(k4 - k3) = (k2 - k5)(k3 - k2) . This material may well have been previously unexplored: no reference to it appears under https://en.wikipedia.org/wiki/Steiner_chain http://mathworld.wolfram.com/SteinerChain.html . Among my notes I discover a 3-page summary of the theory, together with 2-page to-do list of amendments (such as "write down proof of the main theorem"). Plus ça change ... Fred Lunnon On 12/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). WFL
On 12/18/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate >>
The number of discs on the chain must be infinite at tangency, of course. Otherwise, there is no theoretical reason why one boundary circle (strictly, directed cycle in the sense of contact / Lie-sphere geometry) should be internal to the other. Steiner's porism applies whether their relation is internal, external, or intersecting; also whether one, both or neither is external to the chain. A "rotating" system of finite chains may exist even when one boundary is a straight line.
WFL
On 12/18/15, James Buddenhagen <jbuddenh@gmail.com> wrote:
So an example would be https://flic.kr/p/em4XbX
On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
On Dec 18, 2015, at 9:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Very nice graphical demo!!! —Dan
On Dec 19, 2015, at 3:30 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Steiner chain interactive demo --- free, slightly tricky to control --- at http://codepen.io/yukulele/pen/OVOEdX/
This might be a good time to remind people (especially grumpy Yuletide refuseniks) about the excellent problem posed on this list by Gosper in August 2014, concerning polynomial relations between members of a chain: for example, where k1, ..., k5 denote curvatures of successive discs,
(k1 - k4)(k4 - k3) = (k2 - k5)(k3 - k2) .
This material may well have been previously unexplored: no reference to it appears under https://en.wikipedia.org/wiki/Steiner_chain http://mathworld.wolfram.com/SteinerChain.html . Among my notes I discover a 3-page summary of the theory, together with 2-page to-do list of amendments (such as "write down proof of the main theorem"). Plus ça change ...
Fred Lunnon
On 12/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). WFL
On 12/18/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate >>
The number of discs on the chain must be infinite at tangency, of course. Otherwise, there is no theoretical reason why one boundary circle (strictly, directed cycle in the sense of contact / Lie-sphere geometry) should be internal to the other. Steiner's porism applies whether their relation is internal, external, or intersecting; also whether one, both or neither is external to the chain. A "rotating" system of finite chains may exist even when one boundary is a straight line.
WFL
On 12/18/15, James Buddenhagen <jbuddenh@gmail.com> wrote:
So an example would be https://flic.kr/p/em4XbX
On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> wrote:
Aha, a real math question that I think I can answer! The theorem is Steiner’s Porism, and the collection of circles in the ring is a Steiner Chain.
Back around 1980, Gosper programmed a demonstration of this on the PDP-6. It took as input parameters the number of circles in the chain, and the “wobble rate” at which the inner circle moved from side to side within the inner circle, and the “phase” at which the chain was positioned within the ring. From these, it calculated the size of the inner circle so that the chain existed. The resulting pictures on the (type 340) display were amazing to behold.
And this trivia: David Silver perturbed Bill’s program in some small way, such as changing one instruction or exchanging a pair of instructions. The result was, instead of circles, “teapot” or “cat face” outlines. Setting the parameters to exotic values caused the teapots to not percolate in a ring, but rather whoosh into existence as tiny, grow and swoop through an orbit, then shrink and vanish — rather comet like. Ah, the days of assembly language display hacking!
I believe there is at least one app for sale that displays the Steiner Porism.
— Mike
> On Dec 18, 2015, at 9:56 AM, Erich Friedman > <erichfriedman68@gmail.com> > wrote: > > back in undergraduate complex analysis, i learned the following > theorem, > whose proof is easy enough using conformal mappings. does this > theorem > have a name? > > Let D be a disk entirely inside the unit circle. Consider all > collections of non-overlapping disks C so that each member of C is: > > a) inside the unit circle and tangent to the unit circle, > b) doesn't overlap D but is tangent to D, and > c) tangent to exactly two other members of C. > (Thus the collection C forms a tangent ring around D.) > > For a given D, the collection is either empty or infinite. > > erich > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I said << Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). >> Exercises: invoke the demo http://codepen.io/yukulele/pen/OVOEdX/ then manoeuvre the "inner" (centre) to the outer circumference; discuss the ensuing behaviour. Next, with inner outside outer (!), review the points of tangency at the instant some disc degenerates to a straight line: what goes pear-shaped, and how might it be patched up? Finally, figure out how to change the number of discs (I couldn't). WFL On 12/20/15, Dan Asimov <asimov@msri.org> wrote:
Very nice graphical demo!!!
—Dan
On Dec 19, 2015, at 3:30 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Steiner chain interactive demo --- free, slightly tricky to control --- at http://codepen.io/yukulele/pen/OVOEdX/
This might be a good time to remind people (especially grumpy Yuletide refuseniks) about the excellent problem posed on this list by Gosper in August 2014, concerning polynomial relations between members of a chain: for example, where k1, ..., k5 denote curvatures of successive discs,
(k1 - k4)(k4 - k3) = (k2 - k5)(k3 - k2) .
This material may well have been previously unexplored: no reference to it appears under https://en.wikipedia.org/wiki/Steiner_chain http://mathworld.wolfram.com/SteinerChain.html . Among my notes I discover a 3-page summary of the theory, together with 2-page to-do list of amendments (such as "write down proof of the main theorem"). Plus ça change ...
Fred Lunnon
On 12/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). WFL
On 12/18/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate >>
The number of discs on the chain must be infinite at tangency, of course. Otherwise, there is no theoretical reason why one boundary circle (strictly, directed cycle in the sense of contact / Lie-sphere geometry) should be internal to the other. Steiner's porism applies whether their relation is internal, external, or intersecting; also whether one, both or neither is external to the chain. A "rotating" system of finite chains may exist even when one boundary is a straight line.
WFL
On 12/18/15, James Buddenhagen <jbuddenh@gmail.com> wrote:
So an example would be https://flic.kr/p/em4XbX
On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
erich _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Correction: the “phase” parameter was actually the RATE OF CHANGE of phase. For example, with number of circles fixed and wobble rate zero, the phase change rate specified how fast the “ball bearings” (circles in the chain) moved around the annulus. If the inner circle was not concentric with the outer, the “ball bearings” of course had to grow and shrink as they rolled around.
> On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> > wrote: > > Aha, a real math question that I think I can answer! The theorem is > Steiner’s Porism, and the collection of circles in the ring is a > Steiner > Chain. > > Back around 1980, Gosper programmed a demonstration of this on the > PDP-6. > It took as input parameters the number of circles in the chain, and > the > “wobble rate” at which the inner circle moved from side to side > within > the > inner circle, and the “phase” at which the chain was positioned > within > the > ring. From these, it calculated the size of the inner circle so that > the > chain existed. The resulting pictures on the (type 340) display were > amazing to behold. > > And this trivia: David Silver perturbed Bill’s program in some small > way, > such as changing one instruction or exchanging a pair of > instructions. > The result was, instead of circles, “teapot” or “cat face” outlines. > Setting the parameters to exotic values caused the teapots to not > percolate in a ring, but rather whoosh into existence as tiny, grow > and > swoop through an orbit, then shrink and vanish — rather comet like. > Ah, > the days of assembly language display hacking! > > I believe there is at least one app for sale that displays the > Steiner > Porism. > > — Mike > >> On Dec 18, 2015, at 9:56 AM, Erich Friedman >> <erichfriedman68@gmail.com> >> wrote: >> >> back in undergraduate complex analysis, i learned the following >> theorem, >> whose proof is easy enough using conformal mappings. does this >> theorem >> have a name? >> >> Let D be a disk entirely inside the unit circle. Consider all >> collections of non-overlapping disks C so that each member of C is: >> >> a) inside the unit circle and tangent to the unit circle, >> b) doesn't overlap D but is tangent to D, and >> c) tangent to exactly two other members of C. >> (Thus the collection C forms a tangent ring around D.) >> >> For a given D, the collection is either empty or infinite. >> >> erich >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
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If you rotate your middle-button wheel, you change the number of circles. Seb On 20 December 2015 at 22:33, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I said << Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). >>
Exercises: invoke the demo http://codepen.io/yukulele/pen/OVOEdX/ then manoeuvre the "inner" (centre) to the outer circumference; discuss the ensuing behaviour.
Next, with inner outside outer (!), review the points of tangency at the instant some disc degenerates to a straight line: what goes pear-shaped, and how might it be patched up?
Finally, figure out how to change the number of discs (I couldn't).
WFL
On 12/20/15, Dan Asimov <asimov@msri.org> wrote:
Very nice graphical demo!!!
—Dan
On Dec 19, 2015, at 3:30 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Steiner chain interactive demo --- free, slightly tricky to control --- at http://codepen.io/yukulele/pen/OVOEdX/
This might be a good time to remind people (especially grumpy Yuletide refuseniks) about the excellent problem posed on this list by Gosper in August 2014, concerning polynomial relations between members of a chain: for example, where k1, ..., k5 denote curvatures of successive discs,
(k1 - k4)(k4 - k3) = (k2 - k5)(k3 - k2) .
This material may well have been previously unexplored: no reference to it appears under https://en.wikipedia.org/wiki/Steiner_chain http://mathworld.wolfram.com/SteinerChain.html . Among my notes I discover a 3-page summary of the theory, together with 2-page to-do list of amendments (such as "write down proof of the main theorem"). Plus ça change ...
Fred Lunnon
On 12/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). WFL
On 12/18/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate >>
The number of discs on the chain must be infinite at tangency, of course. Otherwise, there is no theoretical reason why one boundary circle (strictly, directed cycle in the sense of contact / Lie-sphere geometry) should be internal to the other. Steiner's porism applies whether their relation is internal, external, or intersecting; also whether one, both or neither is external to the chain. A "rotating" system of finite chains may exist even when one boundary is a straight line.
WFL
On 12/18/15, James Buddenhagen <jbuddenh@gmail.com> wrote:
So an example would be https://flic.kr/p/em4XbX
On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
> back in undergraduate complex analysis, i learned the following > theorem, > whose proof is easy enough using conformal mappings. does this > theorem > have a name? > > Let D be a disk entirely inside the unit circle. Consider all > collections > of non-overlapping disks C so that each member of C is: > > a) inside the unit circle and tangent to the unit circle, > b) doesn't overlap D but is tangent to D, and > c) tangent to exactly two other members of C. > (Thus the collection C forms a tangent ring around D.) > > For a given D, the collection is either empty or infinite. > > erich > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< the “wobble rate” at which the inner circle moved from side to side within the inner circle >>
within the outer circle, maybe? Though that's still a tad vague ...
Modulo similarity, the static configuration has 3 parameters: radius of inner disc, centre displacement, and phase of chain; dynamically, displacement and phase might be replaced by wobble and rotation speeds.
WFL
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote: > Correction: the “phase” parameter was actually the RATE OF CHANGE of > phase. > For example, with number of circles fixed and wobble rate zero, the > phase > change rate specified how fast the “ball bearings” (circles in the > chain) > moved around the annulus. If the inner circle was not concentric with > the > outer, the “ball bearings” of course had to grow and shrink as they > rolled > around. > >> On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net> >> wrote: >> >> Aha, a real math question that I think I can answer! The theorem is >> Steiner’s Porism, and the collection of circles in the ring is a >> Steiner >> Chain. >> >> Back around 1980, Gosper programmed a demonstration of this on the >> PDP-6. >> It took as input parameters the number of circles in the chain, and >> the >> “wobble rate” at which the inner circle moved from side to side >> within >> the >> inner circle, and the “phase” at which the chain was positioned >> within >> the >> ring. From these, it calculated the size of the inner circle so that >> the >> chain existed. The resulting pictures on the (type 340) display were >> amazing to behold. >> >> And this trivia: David Silver perturbed Bill’s program in some small >> way, >> such as changing one instruction or exchanging a pair of >> instructions. >> The result was, instead of circles, “teapot” or “cat face” outlines. >> Setting the parameters to exotic values caused the teapots to not >> percolate in a ring, but rather whoosh into existence as tiny, grow >> and >> swoop through an orbit, then shrink and vanish — rather comet like. >> Ah, >> the days of assembly language display hacking! >> >> I believe there is at least one app for sale that displays the >> Steiner >> Porism. >> >> — Mike >> >>> On Dec 18, 2015, at 9:56 AM, Erich Friedman >>> <erichfriedman68@gmail.com> >>> wrote: >>> >>> back in undergraduate complex analysis, i learned the following >>> theorem, >>> whose proof is easy enough using conformal mappings. does this >>> theorem >>> have a name? >>> >>> Let D be a disk entirely inside the unit circle. Consider all >>> collections of non-overlapping disks C so that each member of C is: >>> >>> a) inside the unit circle and tangent to the unit circle, >>> b) doesn't overlap D but is tangent to D, and >>> c) tangent to exactly two other members of C. >>> (Thus the collection C forms a tangent ring around D.) >>> >>> For a given D, the collection is either empty or infinite. >>> >>> erich >>> _______________________________________________ >>> math-fun mailing list >>> math-fun@mailman.xmission.com >>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
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And holding the Control key while rotating your mouse wheel allows some overlap between circles! On 20 December 2015 at 22:46, Seb Perez-D <sbprzd+mathfun@gmail.com> wrote:
If you rotate your middle-button wheel, you change the number of circles.
Seb
On 20 December 2015 at 22:33, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I said << Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). >>
Exercises: invoke the demo http://codepen.io/yukulele/pen/OVOEdX/ then manoeuvre the "inner" (centre) to the outer circumference; discuss the ensuing behaviour.
Next, with inner outside outer (!), review the points of tangency at the instant some disc degenerates to a straight line: what goes pear-shaped, and how might it be patched up?
Finally, figure out how to change the number of discs (I couldn't).
WFL
On 12/20/15, Dan Asimov <asimov@msri.org> wrote:
Very nice graphical demo!!!
—Dan
On Dec 19, 2015, at 3:30 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Steiner chain interactive demo --- free, slightly tricky to control
at http://codepen.io/yukulele/pen/OVOEdX/
This might be a good time to remind people (especially grumpy Yuletide refuseniks) about the excellent problem posed on this list by Gosper in August 2014, concerning polynomial relations between members of a chain: for example, where k1, ..., k5 denote curvatures of successive discs,
(k1 - k4)(k4 - k3) = (k2 - k5)(k3 - k2) .
This material may well have been previously unexplored: no reference to it appears under https://en.wikipedia.org/wiki/Steiner_chain http://mathworld.wolfram.com/SteinerChain.html . Among my notes I discover a 3-page summary of the theory, together with 2-page to-do list of amendments (such as "write down proof of the main theorem"). Plus ça change ...
Fred Lunnon
On 12/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Minor correction: for a finite number of discs in the chain, the boundary cycles must be disjoint (this also excludes parallel lines). WFL
On 12/18/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate >>
The number of discs on the chain must be infinite at tangency, of course. Otherwise, there is no theoretical reason why one boundary circle (strictly, directed cycle in the sense of contact / Lie-sphere geometry) should be internal to the other. Steiner's porism applies whether their relation is internal, external, or intersecting; also whether one, both or neither is external to the chain. A "rotating" system of finite chains may exist even when one boundary is a straight line.
WFL
On 12/18/15, James Buddenhagen <jbuddenh@gmail.com> wrote: > So an example would be https://flic.kr/p/em4XbX > > On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman > <erichfriedman68@gmail.com> > wrote: > >> back in undergraduate complex analysis, i learned the following >> theorem, >> whose proof is easy enough using conformal mappings. does this >> theorem >> have a name? >> >> Let D be a disk entirely inside the unit circle. Consider all >> collections >> of non-overlapping disks C so that each member of C is: >> >> a) inside the unit circle and tangent to the unit circle, >> b) doesn't overlap D but is tangent to D, and >> c) tangent to exactly two other members of C. >> (Thus the collection C forms a tangent ring around D.) >> >> For a given D, the collection is either empty or infinite. >> >> erich >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote:
Yes, within the outer circle. Thanks, Fred! I think the center of the inner circle moved at a constant rate along the horizontal diameter of the outer circle. When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate, but to Bill’s credit, the animation looked smooth and convincing. We made a movie of it; some day I should get it scanned and put on YouTube. — Mike
> On Dec 18, 2015, at 11:23 AM, Fred Lunnon <fred.lunnon@gmail.com> > wrote: > > << the “wobble rate” at which the inner circle moved from side to side > within the inner circle >> > > within the outer circle, maybe? Though that's still a tad vague ... > > Modulo similarity, the static configuration has 3 parameters: radius > of inner disc, > centre displacement, and phase of chain; dynamically, displacement and > phase > might be replaced by wobble and rotation speeds. > > WFL > > > > > On 12/18/15, Mike Beeler <mikebeeler@verizon.net> wrote: >> Correction: the “phase” parameter was actually the RATE OF CHANGE of >> phase. >> For example, with number of circles fixed and wobble rate zero, the >> phase >> change rate specified how fast the “ball bearings” (circles in the >> chain) >> moved around the annulus. If the inner circle was not concentric with >> the >> outer, the “ball bearings” of course had to grow and shrink as they >> rolled >> around. >> >>> On Dec 18, 2015, at 10:38 AM, Mike Beeler <mikebeeler@verizon.net
>>> wrote: >>> >>> Aha, a real math question that I think I can answer! The theorem is >>> Steiner’s Porism, and the collection of circles in the ring is a >>> Steiner >>> Chain. >>> >>> Back around 1980, Gosper programmed a demonstration of this on the >>> PDP-6. >>> It took as input parameters the number of circles in the chain, and >>> the >>> “wobble rate” at which the inner circle moved from side to side >>> within >>> the >>> inner circle, and the “phase” at which the chain was positioned >>> within >>> the >>> ring. From these, it calculated the size of the inner circle so that >>> the >>> chain existed. The resulting pictures on the (type 340) display were >>> amazing to behold. >>> >>> And this trivia: David Silver perturbed Bill’s program in some small >>> way, >>> such as changing one instruction or exchanging a pair of >>> instructions. >>> The result was, instead of circles, “teapot” or “cat face” outlines. >>> Setting the parameters to exotic values caused the teapots to not >>> percolate in a ring, but rather whoosh into existence as tiny, grow >>> and >>> swoop through an orbit, then shrink and vanish — rather comet like. >>> Ah, >>> the days of assembly language display hacking! >>> >>> I believe there is at least one app for sale that displays the >>> Steiner >>> Porism. >>> >>> — Mike >>> >>>> On Dec 18, 2015, at 9:56 AM, Erich Friedman >>>> <erichfriedman68@gmail.com> >>>> wrote: >>>> >>>> back in undergraduate complex analysis, i learned the following >>>> theorem, >>>> whose proof is easy enough using conformal mappings. does this >>>> theorem >>>> have a name? >>>> >>>> Let D be a disk entirely inside the unit circle. Consider all >>>> collections of non-overlapping disks C so that each member of C is: >>>> >>>> a) inside the unit circle and tangent to the unit circle, >>>> b) doesn't overlap D but is tangent to D, and >>>> c) tangent to exactly two other members of C. >>>> (Thus the collection C forms a tangent ring around D.) >>>> >>>> For a given D, the collection is either empty or infinite. >>>> >>>> erich >>>> _______________________________________________ >>>> math-fun mailing list >>>> math-fun@mailman.xmission.com >>>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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On 20/12/2015 21:33, Fred Lunnon wrote:
Finally, figure out how to change the number of discs (I couldn't).
I don't think there's any UI provided for doing so, but if you change "let n = 7;" on line 62 of the code to, say, "let n = 12;" then you will get a different number of circles. -- g
I wrote:
On 20/12/2015 21:33, Fred Lunnon wrote:
Finally, figure out how to change the number of discs (I couldn't).
I don't think there's any UI provided for doing so, but if you change "let n = 7;" on line 62 of the code to, say, "let n = 12;" then you will get a different number of circles.
As Seb Perez-D has found, there is in fact UI for doing it, namely the scroll wheel. My claim was based only on having played with the thing for a minute and not found anything that did that. I will be more thorough next time. -- g
So an example would be https://flic.kr/p/em4XbX On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
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<< When it reaches the outer circle and reverses direction, I think the math becomes a bit indeterminate >> The number of discs on the chain must be infinite at tangency, of course. Otherwise, there is no theoretical reason why one boundary circle (strictly, directed cycle in the sense of contact / Lie-sphere geometry) should be internal to the other. Steiner's porism applies whether their relation is internal, external, or intersecting; also whether one, both or neither is external to the chain. A "rotating" system of finite chains may exist even when one boundary is a straight line. WFL On 12/18/15, James Buddenhagen <jbuddenh@gmail.com> wrote:
So an example would be https://flic.kr/p/em4XbX
On Fri, Dec 18, 2015 at 8:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
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This is not hard to prove with the group of conformal automorphisms of the open unit disk int(D^2) about the origin. (Which can be seen to be isomorphic to PSL(2,R) — or equivalently, this is the group of conformal automorphisms of the upper half-plane.) The group Aut(int(D^2)) is isomorphic to SL(2,R) / {I,-I}, since the elements of G = Aut(int(D^2)) = {z |—> (az+b)/(cz+d) | a, b, c, d in R with ad-bc = 1} multiply just as do the 2x2 matrices of SL(2,R), and any element (az+b)/(cz+d) is the same mapping when (a,b,c,d) is changed to (-a,-b,-c,-d), and only then. Now given the setup with D below: Since elements of G carries circles to circles, we may find an element of G that takes D to a disk centered at the origin. This takes the other relevant circles to circles, but these new circles all have the same radius. Without loss of generality we assume the center of D is at the origin. In this conformally equivalent picture to the original one, either a) Copies of the first additional disk C below (i.e., having the same radius as C) either fit consecutively around D and return exactly to C after finitely many new disks are placed as below, or b) not. And a) will be true or not independent of where the first C is placed, by circular symmetry. –Dan
On Dec 18, 2015, at 6:56 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:
back in undergraduate complex analysis, i learned the following theorem, whose proof is easy enough using conformal mappings. does this theorem have a name?
Let D be a disk entirely inside the unit circle. Consider all collections of non-overlapping disks C so that each member of C is:
a) inside the unit circle and tangent to the unit circle, b) doesn't overlap D but is tangent to D, and c) tangent to exactly two other members of C. (Thus the collection C forms a tangent ring around D.)
For a given D, the collection is either empty or infinite.
participants (11)
-
Dan Asimov -
Dan Asimov -
Erich Friedman -
Fred Lunnon -
Gareth McCaughan -
James Buddenhagen -
James Propp -
Mike Beeler -
Mike Stay -
rwg -
Seb Perez-D