Re: [math-fun] On a clear day at Battersea power station
Fred: Pls email me these pix and I'll host them. --rwg I cannot refrain from adding: Screw DropBox. << Scale diagrams of these solutions might be appreciated here ...? >> Quite so. In a pleasant inversion of the scholastic dictum that a diagram does not constitute a proof, a clutch of accurate diagrams immediately discloses that: The obvious interpretation of a signed sightline distance --- that the correct angle is made only with a line segment directed _away_ from the tower --- was after all perfectly correct. Hence --- assuming an observer lacking eyes in the back of the head --- my second and fourth solutions were duds, and only the other two remain standing (outside the rectangle). Corrected summary & diagrams, and program & results, are posted at https://www.dropbox.com/s/gva6w9ni9y6svp1/battersea.pdf https://www.dropbox.com/s/175plix3vkvxqoj/battersea_run.txt This has turned out to be an unexpectedly involved exercise: in particular, a neat elementary application of Gröbner bases, as well as an excellent topic for an undergraduate project; perhaps in a situation calling for an especially --- ahem --- challenging one ... Fred Lunnon
gosper.org/battersea_run.txt gosper.org/battersea.pdf should be up now. --rwg On Tue, Jul 17, 2018 at 7:29 PM Bill Gosper <billgosper@gmail.com> wrote:
Fred: Pls email me these pix and I'll host them. --rwg I cannot refrain from adding: Screw DropBox.
<< Scale diagrams of these solutions might be appreciated here ...? >>
Quite so. In a pleasant inversion of the scholastic dictum that a diagram does not constitute a proof, a clutch of accurate diagrams immediately discloses that:
The obvious interpretation of a signed sightline distance --- that the correct angle is made only with a line segment directed _away_ from the tower --- was after all perfectly correct.
Hence --- assuming an observer lacking eyes in the back of the head --- my second and fourth solutions were duds, and only the other two remain standing (outside the rectangle).
Corrected summary & diagrams, and program & results, are posted at https://www.dropbox.com/s/gva6w9ni9y6svp1/battersea.pdf https://www.dropbox.com/s/175plix3vkvxqoj/battersea_run.txt
This has turned out to be an unexpectedly involved exercise: in particular, a neat elementary application of Gröbner bases, as well as an excellent topic for an undergraduate project; perhaps in a situation calling for an especially --- ahem --- challenging one ...
Fred Lunnon
If we leave out the constraint about the distance to the closest tower, then there is a continuum of solutions. Each solution requires a different aspect ratio for the rectangular power station. If you fix one edge of the station (letting the other telescope in and out as needed), what does the locus of viewpoints look like? Is this a classic curve, or is it something new ("Splictrix of Hess")? On Tue, Jul 17, 2018 at 11:50 PM, Bill Gosper <billgosper@gmail.com> wrote:
gosper.org/battersea_run.txt gosper.org/battersea.pdf should be up now. --rwg
On Tue, Jul 17, 2018 at 7:29 PM Bill Gosper <billgosper@gmail.com> wrote:
Fred: Pls email me these pix and I'll host them. --rwg I cannot refrain from adding: Screw DropBox.
<< Scale diagrams of these solutions might be appreciated here ...? >>
Quite so. In a pleasant inversion of the scholastic dictum that a diagram does not constitute a proof, a clutch of accurate diagrams immediately discloses that:
The obvious interpretation of a signed sightline distance --- that the correct angle is made only with a line segment directed _away_ from the tower --- was after all perfectly correct.
Hence --- assuming an observer lacking eyes in the back of the head --- my second and fourth solutions were duds, and only the other two remain standing (outside the rectangle).
Corrected summary & diagrams, and program & results, are posted at https://www.dropbox.com/s/gva6w9ni9y6svp1/battersea.pdf https://www.dropbox.com/s/175plix3vkvxqoj/battersea_run.txt
This has turned out to be an unexpectedly involved exercise: in particular, a neat elementary application of Gröbner bases, as well as an excellent topic for an undergraduate project; perhaps in a situation calling for an especially --- ahem --- challenging one ...
Fred Lunnon
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gosper.org/battersea_run.txt gosper.org/battersea.pdf
For reference, I have a slightly edited version of David Singmaster's original posing of the problem (of unknown date) wherein I've added a photo at the bottom of page 3 and an addendum page 4, both taken from an alternate (but presumably subsequent) version of the pdf. I've left out the photo of Elton John. http://chesswanks.com/txt/BatterseaPowerStationPuzzle.pdf
Nice find, HH. http://chesswanks.com/txt/BatterseaPowerStationPuzzle.pdf It is hard to imagine how Messrs Salomon, Armstrong & Sylvester could have managed to convince themselves about the conic shown in the final figure of Singmaster's document: for instance, their point o = 0 coincides with the origin tower, from where the other angles cannot possibly be equal. [Unless of course BPS has transmogrified into a square floorplan since last I set eyes on it, which would require shifting an awful lot of bricks in addition to the roof.] But it's comforting to know how many other people this wicked problem has tied into knots. Incidentally in Hess' case where angle v = pi/14 , I find the (nontrivial) locus of viewpoint V = (x,y) for variable distance o to be the cubic curve 4*x^3 - 4*x*y^2 - 3*x^2 + y^2 = 0 , which (ummm) also meets the origin! Wait, wait --- at the origin c_0 c_2 - c_1^2 = 3 > 0 indicating an acnode (isolated point), see https://en.wikipedia.org/wiki/Singular_point_of_a_curve Phew --- so everything is alright? Probably; though ACW's telescope may undergo extensive structural modification at this point ... I suppose some of this is going to have be included in my screed. WFL On 7/18/18, Hans Havermann <gladhobo@bell.net> wrote:
gosper.org/battersea_run.txt gosper.org/battersea.pdf
For reference, I have a slightly edited version of David Singmaster's original posing of the problem (of unknown date) wherein I've added a photo at the bottom of page 3 and an addendum page 4, both taken from an alternate (but presumably subsequent) version of the pdf. I've left out the photo of Elton John.
http://chesswanks.com/txt/BatterseaPowerStationPuzzle.pdf
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There were errors and misunderstandings in my previous post. The Salamon et al curve is unrelated to Wechsler's problem. The cubic equation for the curve through Hess's viewpoint in sector <r, o, q, p> at the fixed corner origin is correct, but the curve has a crunode (double point) at the origin --- all these curves pass continuously through the origin, when side s = 0 . The congruent curve reflected in centre line x = 1/2 meets my viewpoint in sector <q, p, r, o> . [There is more of this stuff, my friends; much more!] I went searching and turned up the Singmaster article at Gathering for Gardner 11 (2016), as well as an article and slides by Salamon (2014) at https://s3-eu-west-1.amazonaws.com/content.gresham.ac.uk/data/binary/2365/20... https://nms.kcl.ac.uk/simon.salamon/X/dulwich.pdf These sketch a neat-looking treatment which appears to avoid heavy computation, but considers curves generated by viewpoints with two equal angles of a fixed rectangle. In the meantime it dawned on me that the Wechsler curve problem as it stood was incompletely posed: the answer would depend both on choice of fixed origin (at rectangle centre, rather than corner?), and on some relation between rectangle sides ( s + t = 1 , rather than t = 1 ?). At the moment it's unclear what refinement of these choices might best simplify the outcome. That telescope is sure going to have its work cut out! WFL On 7/18/18, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Nice find, HH. http://chesswanks.com/txt/BatterseaPowerStationPuzzle.pdf
It is hard to imagine how Messrs Salomon, Armstrong & Sylvester could have managed to convince themselves about the conic shown in the final figure of Singmaster's document: for instance, their point o = 0 coincides with the origin tower, from where the other angles cannot possibly be equal. [Unless of course BPS has transmogrified into a square floorplan since last I set eyes on it, which would require shifting an awful lot of bricks in addition to the roof.]
But it's comforting to know how many other people this wicked problem has tied into knots. Incidentally in Hess' case where angle v = pi/14 , I find the (nontrivial) locus of viewpoint V = (x,y) for variable distance o to be the cubic curve
4*x^3 - 4*x*y^2 - 3*x^2 + y^2 = 0 ,
which (ummm) also meets the origin!
Wait, wait --- at the origin c_0 c_2 - c_1^2 = 3 > 0 indicating an acnode (isolated point), see https://en.wikipedia.org/wiki/Singular_point_of_a_curve Phew --- so everything is alright? Probably; though ACW's telescope may undergo extensive structural modification at this point ...
I suppose some of this is going to have be included in my screed.
WFL
On 7/18/18, Hans Havermann <gladhobo@bell.net> wrote:
gosper.org/battersea_run.txt gosper.org/battersea.pdf
For reference, I have a slightly edited version of David Singmaster's original posing of the problem (of unknown date) wherein I've added a photo at the bottom of page 3 and an addendum page 4, both taken from an alternate (but presumably subsequent) version of the pdf. I've left out the photo of Elton John.
http://chesswanks.com/txt/BatterseaPowerStationPuzzle.pdf
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An updated version of my investigations into the mysteries of the Battersea power station towers, boasting new sections on Wechsler's and Salamon's problems, has been posted at https://www.dropbox.com/s/gva6w9ni9y6svp1/battersea.pdf https://www.dropbox.com/s/175plix3vkvxqoj/battersea_run.txt --- and for the benefit of those disdaining commercial repositories, may shortly make an appearance on gosper.org , courtesy of RWG ? Copyright claims, defamation suits and cease-and-desist notices should not be sent to the author, who is considering into hiding on the pretext of getting something marginally useful done for a change. WFL On 7/19/18, Fred Lunnon <fred.lunnon@gmail.com> wrote:
There were errors and misunderstandings in my previous post. The Salamon et al curve is unrelated to Wechsler's problem.
The cubic equation for the curve through Hess's viewpoint in sector <r, o, q, p> at the fixed corner origin is correct, but the curve has a crunode (double point) at the origin --- all these curves pass continuously through the origin, when side s = 0 . The congruent curve reflected in centre line x = 1/2 meets my viewpoint in sector <q, p, r, o> . [There is more of this stuff, my friends; much more!]
I went searching and turned up the Singmaster article at Gathering for Gardner 11 (2016), as well as an article and slides by Salamon (2014) at
https://s3-eu-west-1.amazonaws.com/content.gresham.ac.uk/data/binary/2365/20... https://nms.kcl.ac.uk/simon.salamon/X/dulwich.pdf
These sketch a neat-looking treatment which appears to avoid heavy computation, but considers curves generated by viewpoints with two equal angles of a fixed rectangle.
In the meantime it dawned on me that the Wechsler curve problem as it stood was incompletely posed: the answer would depend both on choice of fixed origin (at rectangle centre, rather than corner?), and on some relation between rectangle sides ( s + t = 1 , rather than t = 1 ?).
At the moment it's unclear what refinement of these choices might best simplify the outcome. That telescope is sure going to have its work cut out!
WFL
On 7/18/18, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Nice find, HH. http://chesswanks.com/txt/BatterseaPowerStationPuzzle.pdf
It is hard to imagine how Messrs Salomon, Armstrong & Sylvester could have managed to convince themselves about the conic shown in the final figure of Singmaster's document: for instance, their point o = 0 coincides with the origin tower, from where the other angles cannot possibly be equal. [Unless of course BPS has transmogrified into a square floorplan since last I set eyes on it, which would require shifting an awful lot of bricks in addition to the roof.]
But it's comforting to know how many other people this wicked problem has tied into knots. Incidentally in Hess' case where angle v = pi/14 , I find the (nontrivial) locus of viewpoint V = (x,y) for variable distance o to be the cubic curve
4*x^3 - 4*x*y^2 - 3*x^2 + y^2 = 0 ,
which (ummm) also meets the origin!
Wait, wait --- at the origin c_0 c_2 - c_1^2 = 3 > 0 indicating an acnode (isolated point), see https://en.wikipedia.org/wiki/Singular_point_of_a_curve Phew --- so everything is alright? Probably; though ACW's telescope may undergo extensive structural modification at this point ...
I suppose some of this is going to have be included in my screed.
WFL
On 7/18/18, Hans Havermann <gladhobo@bell.net> wrote:
gosper.org/battersea_run.txt gosper.org/battersea.pdf
For reference, I have a slightly edited version of David Singmaster's original posing of the problem (of unknown date) wherein I've added a photo at the bottom of page 3 and an addendum page 4, both taken from an alternate (but presumably subsequent) version of the pdf. I've left out the photo of Elton John.
http://chesswanks.com/txt/BatterseaPowerStationPuzzle.pdf
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participants (4)
-
Allan Wechsler -
Bill Gosper -
Fred Lunnon -
Hans Havermann