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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