[math-fun] Steiner's porism
Using the result of that 51.2 hr Simplification, homogc[xp_, u_] := xp /. (CD : Circle | Disk)[{x_, y_}, r_] :> CD[{-((x + u*(1 - r^2 + x*(u + x) + y^2))/(r^2*u^2 - (1 + u*x)^2 - u^2*y^2)), (y - u^2*y)/(1 + u*(2*x + u*(-r^2 + x^2 + y^2)))}, Abs[ (r*(-1 + u^2))/(r^2*u^2 - (1 + u*x)^2 - u^2*y^2)]] will subject all the Circles and Disks in xp to (z+u)/(u z+1), which, modulo pre and post rotation, is, I think the most general homographic transformation preserving the unit circle. This facilitates gosper.org/stein.gif , an animation (of the 3.5 circles case) unsurprising to both the average Joe and the highly astute, but making the rest of us say, "Hey, wait a minute.". (Can there be only one decent Youtube of this?) In the 60s, I animated this on the PDP6 using the obvious algorithm of drawing (as fast as I could) the circles for a given "frame", updating the sizes and positions, and then drawing the next "frame". You control the rotational speed ω, # of circles, and u parameter variation with the 36 bit test word switches on the front panel. Freezing u and simply varying ω revealed a puzzling gap between two of the orbiting circles, said gap widening with increasing ω, and disappearing when ω → 0. This seemed paradoxical. The phosphor persistence made it clear that none of the circles changed size. How could there be room for more simply because they were moving? It's a relativistic foreshortening pun! Video in those days had no "frames"--you just threw dots on the screen. So the procedure Draw circles, Draw updated circles,... when scrutinized, contains the sequence Draw last old circle, Draw first updated circle, so your eye sees the gap, instead connecting first old circle to last old circle. However, a movie shot right off the screen sees no gap! Elegant fix. Don't rotate at all. Just draw 6.999 circles instead of 7 to rotate slowly, 6.99 to rotate rapidly, and 7.01 to spin the other way. No gap. But now, the persistence reveals them changing size when you change speed! And now filming the screen shows the mismatch. --rwg
While not claiming to have precisely grasped the purpose of that heroic computation, I should like to remind everybody that there are much less resource-intensive methods available for circle geometry, using the geometric algebra Cl(3, 2) based on pentacyclic coordinates for plane contact geometry. These incorporate the potential for generalisation to higher dimensions; also for investigation of the Laguerre "equilong" group alongside the familiar Moebius conformal group, besides the full Lie-sphere behemoth (which cosmologists contrarily denote "conformal" instead). It's a great shame these beautiful ideas have not gained better traction among the general mathematical community. WFL On 8/18/14, Bill Gosper <billgosper@gmail.com> wrote:
Using the result of that 51.2 hr Simplification, homogc[xp_, u_] := xp /. (CD : Circle | Disk)[{x_, y_}, r_] :> CD[{-((x + u*(1 - r^2 + x*(u + x) + y^2))/(r^2*u^2 - (1 + u*x)^2 - u^2*y^2)), (y - u^2*y)/(1 + u*(2*x + u*(-r^2 + x^2 + y^2)))}, Abs[ (r*(-1 + u^2))/(r^2*u^2 - (1 + u*x)^2 - u^2*y^2)]]
will subject all the Circles and Disks in xp to (z+u)/(u z+1), which, modulo pre and post rotation, is, I think the most general homographic transformation preserving the unit circle. This facilitates gosper.org/stein.gif , an animation (of the 3.5 circles case) unsurprising to both the average Joe and the highly astute, but making the rest of us say, "Hey, wait a minute.". (Can there be only one decent Youtube of this?)
In the 60s, I animated this on the PDP6 using the obvious algorithm of drawing (as fast as I could) the circles for a given "frame", updating the sizes and positions, and then drawing the next "frame". You control the rotational speed ω, # of circles, and u parameter variation with the 36 bit test word switches on the front panel. Freezing u and simply varying ω revealed a puzzling gap between two of the orbiting circles, said gap widening with increasing ω, and disappearing when ω → 0. This seemed paradoxical. The phosphor persistence made it clear that none of the circles changed size. How could there be room for more simply because they were moving? It's a relativistic foreshortening pun! Video in those days had no "frames"--you just threw dots on the screen. So the procedure Draw circles, Draw updated circles,... when scrutinized, contains the sequence Draw last old circle, Draw first updated circle, so your eye sees the gap, instead connecting first old circle to last old circle. However, a movie shot right off the screen sees no gap!
Elegant fix. Don't rotate at all. Just draw 6.999 circles instead of 7 to rotate slowly, 6.99 to rotate rapidly, and 7.01 to spin the other way. No gap. But now, the persistence reveals them changing size when you change speed! And now filming the screen shows the mismatch. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Bill's animation is extremely beautiful. Just in case this hasn't been mentioned already: It occurs to me that for any p/q circles case (say GCD(p,q) = 1), if one takes the q-fold cover of the annulus this occurs in, one gets a (topological) annulus -- call it A' -- with a well-defined geometry, and which contains a circular chain of p circles. Now, every geometrical annulus is conformally equivalent to a standard one: { z in C | 1 <= |z| <= R} for a unique R > 1. Let the standard annulus conformally equivalent to A' be called A. Since (locally isometric) covering maps, and conformal maps, preserve circles, we can apply Steiner's porism to annulus A, and this implies that it works in the "fractional circles" case in the first place. --Dan P.S. Steiner's porism is very similar to Poncelet's porism. What the hell is a porism, anyway? Aha -- Webster's New World dictionary defines it as follows. Now it's clear why Steiner's and Poncelet's discoveries are called porisms: ----- porism • a proposition that uncovers the possibility of finding such conditions as to make a specific problem capable of innumerable solutions ----- PORISM / PRIMOS On Aug 21, 2014, at 4:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
gosper.org/stein.gif , an animation (of the 3.5 circles case)
The proof of Steiner's porism, whether for n integer or rational, is anyway trivial via inversive / conformal geometry / Moebius group: invert the chain to a concentric pair of frontier circles, rotate that, then invert back to the new chain. But Poncelet's is rather harder. At first sight it looks as though a similar trick might work using the Laguerre group: but though this conserves both circles and (separately) lines, it fails to conserve points, whence polygonal vertices bloat up into circles. The proof offered in Wikipedia involves elliptic functions, and is too terse for me to get a handle. Anybody know of a good elementary demonstration? WFL On 8/22/14, Dan Asimov <dasimov@earthlink.net> wrote:
Bill's animation is extremely beautiful.
Just in case this hasn't been mentioned already:
It occurs to me that for any p/q circles case (say GCD(p,q) = 1), if one takes the q-fold cover of the annulus this occurs in, one gets a (topological) annulus -- call it A' -- with a well-defined geometry, and which contains a circular chain of p circles.
Now, every geometrical annulus is conformally equivalent to a standard one: { z in C | 1 <= |z| <= R} for a unique R > 1.
Let the standard annulus conformally equivalent to A' be called A.
Since (locally isometric) covering maps, and conformal maps, preserve circles, we can apply Steiner's porism to annulus A, and this implies that it works in the "fractional circles" case in the first place.
--Dan
P.S. Steiner's porism is very similar to Poncelet's porism.
What the hell is a porism, anyway? Aha -- Webster's New World dictionary defines it as follows. Now it's clear why Steiner's and Poncelet's discoveries are called porisms:
----- porism
• a proposition that uncovers the possibility of finding such conditions as to make a specific problem capable of innumerable solutions -----
PORISM / PRIMOS
On Aug 21, 2014, at 4:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
gosper.org/stein.gif , an animation (of the 3.5 circles case)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Detailed exposition at http://mathworld.wolfram.com/PonceletsPorism.html gives reference to King (1994). The fact that this porism generalises to ellipses suggests that it is a theorem of projective geometry, rather than inversive or Laguerre, etc. Elliptic functions apparently take up the role played by trig (or hyperbolic) functions in the Steiner porism. Noteworthy that the harder Poncelet is worked out more thoroughly than the Steiner! WFL On 8/22/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The proof of Steiner's porism, whether for n integer or rational, is anyway trivial via inversive / conformal geometry / Moebius group: invert the chain to a concentric pair of frontier circles, rotate that, then invert back to the new chain.
But Poncelet's is rather harder. At first sight it looks as though a similar trick might work using the Laguerre group: but though this conserves both circles and (separately) lines, it fails to conserve points, whence polygonal vertices bloat up into circles.
The proof offered in Wikipedia involves elliptic functions, and is too terse for me to get a handle. Anybody know of a good elementary demonstration?
WFL
On 8/22/14, Dan Asimov <dasimov@earthlink.net> wrote:
Bill's animation is extremely beautiful.
Just in case this hasn't been mentioned already:
It occurs to me that for any p/q circles case (say GCD(p,q) = 1), if one takes the q-fold cover of the annulus this occurs in, one gets a (topological) annulus -- call it A' -- with a well-defined geometry, and which contains a circular chain of p circles.
Now, every geometrical annulus is conformally equivalent to a standard one: { z in C | 1 <= |z| <= R} for a unique R > 1.
Let the standard annulus conformally equivalent to A' be called A.
Since (locally isometric) covering maps, and conformal maps, preserve circles, we can apply Steiner's porism to annulus A, and this implies that it works in the "fractional circles" case in the first place.
--Dan
P.S. Steiner's porism is very similar to Poncelet's porism.
What the hell is a porism, anyway? Aha -- Webster's New World dictionary defines it as follows. Now it's clear why Steiner's and Poncelet's discoveries are called porisms:
----- porism
• a proposition that uncovers the possibility of finding such conditions as to make a specific problem capable of innumerable solutions -----
PORISM / PRIMOS
On Aug 21, 2014, at 4:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
gosper.org/stein.gif , an animation (of the 3.5 circles case)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The Poncelet porism is elementary for fixed number of vertices n and pair of circles frontier. Eg. for triangles n = 3 we have "Euler's theorem" --- apparently due to Chapple, see http://en.wikipedia.org/wiki/Euler%27s_theorem_in_geometry --- relating circumradius R , inradius r , offset d by the elegant 1/(R - r + d) + 1/(R - r - d) = 1/r . Since every triangle with given values of the parameters satisfies this, they all fit (continuously) around the corresponding pair of circum- and in-circles. Now factoring out similarities (freedom 4) from the plane projective group (freedom 8), the pair of circles (freedom 2) is transformed into a pair of conics (freedom 6). Since 8-4+2 = 6 , all pairs of conics are generated, so the theorem holds for general disjoint conic frontier. The comprehensive http://mathworld.wolfram.com/PonceletsPorism.html mentions that For n even, the diagonals are concurrent at the limiting point of the two circles, whereas for n odd, the lines connecting the vertices to the opposite points of tangency are concurrent at the limiting point. --- the "limiting point" is presumably one of two limit points in the coaxal system generated by the circles. Does this point have a name when associated with a triangle? The projective argument shows that the same fixed point of concurrency exists for any given pair of disjoint conics --- again, does it have a name, and other interesting properties? Fred Lunnon On 8/22/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Detailed exposition at http://mathworld.wolfram.com/PonceletsPorism.html gives reference to King (1994).
The fact that this porism generalises to ellipses suggests that it is a theorem of projective geometry, rather than inversive or Laguerre, etc. Elliptic functions apparently take up the role played by trig (or hyperbolic) functions in the Steiner porism.
Noteworthy that the harder Poncelet is worked out more thoroughly than the Steiner!
WFL
On 8/22/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
The proof of Steiner's porism, whether for n integer or rational, is anyway trivial via inversive / conformal geometry / Moebius group: invert the chain to a concentric pair of frontier circles, rotate that, then invert back to the new chain.
But Poncelet's is rather harder. At first sight it looks as though a similar trick might work using the Laguerre group: but though this conserves both circles and (separately) lines, it fails to conserve points, whence polygonal vertices bloat up into circles.
The proof offered in Wikipedia involves elliptic functions, and is too terse for me to get a handle. Anybody know of a good elementary demonstration?
WFL
On 8/22/14, Dan Asimov <dasimov@earthlink.net> wrote:
Bill's animation is extremely beautiful.
Just in case this hasn't been mentioned already:
It occurs to me that for any p/q circles case (say GCD(p,q) = 1), if one takes the q-fold cover of the annulus this occurs in, one gets a (topological) annulus -- call it A' -- with a well-defined geometry, and which contains a circular chain of p circles.
Now, every geometrical annulus is conformally equivalent to a standard one: { z in C | 1 <= |z| <= R} for a unique R > 1.
Let the standard annulus conformally equivalent to A' be called A.
Since (locally isometric) covering maps, and conformal maps, preserve circles, we can apply Steiner's porism to annulus A, and this implies that it works in the "fractional circles" case in the first place.
--Dan
P.S. Steiner's porism is very similar to Poncelet's porism.
What the hell is a porism, anyway? Aha -- Webster's New World dictionary defines it as follows. Now it's clear why Steiner's and Poncelet's discoveries are called porisms:
----- porism
• a proposition that uncovers the possibility of finding such conditions as to make a specific problem capable of innumerable solutions -----
PORISM / PRIMOS
On Aug 21, 2014, at 4:15 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
gosper.org/stein.gif , an animation (of the 3.5 circles case)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Bill Gosper -
Dan Asimov -
Fred Lunnon