[math-fun] Octocog
This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible. Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges. Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun? Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry? Fred Lunnon
Hi Mathfun friends, See here a 3D-printed version of the Seven-Planet Somsky Gears. https://www.youtube.com/watch?v=3iyO-OvIS_k https://www.shapeways.com/product/SPGDQSRLE/seven-planet-somsky-gears Someone at the Twisty Puzzles Forum asked the question "Is there a link to the arguments for the maximum number of planets?" http://twistypuzzles.com/forum/viewtopic.php?p=342755#p342755 Enjoy! Oskar
It is not quite accurate to say (as the video does) that there is only one 8-planet configuration: my conjecture is rather that all others are related to this one via rescaling and concentric increment. I have several times sat down to try to write all this stuff up, but somehow seem to finish up concluding that I don't quite understand it sufficiently ... Fred Lunnon On 10/19/15, M. Oskar van Deventer <m.o.vandeventer@planet.nl> wrote:
Hi Mathfun friends,
See here a 3D-printed version of the Seven-Planet Somsky Gears. https://www.youtube.com/watch?v=3iyO-OvIS_k https://www.shapeways.com/product/SPGDQSRLE/seven-planet-somsky-gears
Someone at the Twisty Puzzles Forum asked the question "Is there a link to the arguments for the maximum number of planets?" http://twistypuzzles.com/forum/viewtopic.php?p=342755#p342755
Enjoy!
Oskar
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About all I can say is: COOL! Seeing it as a computer generated graphic is one thing, but to see it in real, physical form? WOW! :-) I'm kinda amazed that the three-tooth gear worked! Probably wouldn't have if you hadn't made them all double-helical. Well done! - Bill Somsky On 2015-10-19 06:39, M. Oskar van Deventer wrote:
Hi Mathfun friends,
See here a 3D-printed version of the Seven-Planet Somsky Gears. https://www.youtube.com/watch?v=3iyO-OvIS_k https://www.shapeways.com/product/SPGDQSRLE/seven-planet-somsky-gears
Someone at the Twisty Puzzles Forum asked the question "Is there a link to the arguments for the maximum number of planets?" http://twistypuzzles.com/forum/viewtopic.php?p=342755#p342755
Enjoy!
Oskar
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The site is not responding at the moment, so I can't check my facts: www.torsen.com/files/Traction_Control_Article.pdf But I think Gleasman commented there that his "torsen" differential employs 3-tooth gears, the design of which involved tearing up conventional textbook wisdom that 6 teeth was the practical minimum to avoid "overlapping" (maybe re-entrant?) teeth. WFL On 10/20/15, William R Somsky <wrsomsky@gmail.com> wrote:
About all I can say is: COOL!
Seeing it as a computer generated graphic is one thing, but to see it in real, physical form? WOW! :-)
I'm kinda amazed that the three-tooth gear worked! Probably wouldn't have if you hadn't made them all double-helical.
Well done!
- Bill Somsky
On 2015-10-19 06:39, M. Oskar van Deventer wrote:
Hi Mathfun friends,
See here a 3D-printed version of the Seven-Planet Somsky Gears. https://www.youtube.com/watch?v=3iyO-OvIS_k https://www.shapeways.com/product/SPGDQSRLE/seven-planet-somsky-gears
Someone at the Twisty Puzzles Forum asked the question "Is there a link to the arguments for the maximum number of planets?" http://twistypuzzles.com/forum/viewtopic.php?p=342755#p342755
Enjoy!
Oskar
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(Excepting the crank), was it 3D-printed pre-assembled?? How did he put it together? --rwg On 2015-10-19 17:10, William R Somsky wrote:
About all I can say is: COOL!
Seeing it as a computer generated graphic is one thing, but to see it in real, physical form? WOW! :-)
I'm kinda amazed that the three-tooth gear worked! Probably wouldn't have if you hadn't made them all double-helical.
Well done!
- Bill Somsky
On 2015-10-19 06:39, M. Oskar van Deventer wrote:
Hi Mathfun friends,
See here a 3D-printed version of the Seven-Planet Somsky Gears. https://www.youtube.com/watch?v=3iyO-OvIS_k https://www.shapeways.com/product/SPGDQSRLE/seven-planet-somsky-gears
Someone at the Twisty Puzzles Forum asked the question "Is there a link to the arguments for the maximum number of planets?" http://twistypuzzles.com/forum/viewtopic.php?p=342755#p342755
Enjoy!
Oskar
We do!
I *think* (to the extent that my insomniated brain functions at this hour) that a necessary (though not necessarily sufficient) condition for an N-fold rotationally symmetric solution is that cos[pi/N] be rational, which (Niven's theorem?) only occurs for N=1,2,3. N=1 is essentially no symmetry (which is a whole 'nother ball-of-wax), so it'll be two "orbits" of two or of three for symmetric cases. On Oct 19, 2015 05:58, "Fred Lunnon" <fred.lunnon@gmail.com> wrote:
This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible.
Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges.
Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun?
Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry?
Fred Lunnon
The planar graph associated with the octahedron --- see eg. first diagram at http://mathworld.wolfram.com/OctahedralGraph.html has symmetry group of size 6,3,2,1 according to the lengths assigned to its edges. Replacing edge-adjacent triangular faces by tangent circles excludes 3-cyclic symmetry; hence the numbers in my rider. WFL On 10/19/15, William R. Somsky <wrsomsky@gmail.com> wrote:
I *think* (to the extent that my insomniated brain functions at this hour) that a necessary (though not necessarily sufficient) condition for an N-fold rotationally symmetric solution is that cos[pi/N] be rational, which (Niven's theorem?) only occurs for N=1,2,3. N=1 is essentially no symmetry (which is a whole 'nother ball-of-wax), so it'll be two "orbits" of two or of three for symmetric cases. On Oct 19, 2015 05:58, "Fred Lunnon" <fred.lunnon@gmail.com> wrote:
This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible.
Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges.
Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun?
Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry?
Fred Lunnon
I hunted the "octocog" with 6-fold symmetry: like the snark, it remains elusive. Such a gear train would comprise a sun gear centred at the origin, of radius s ; which meshes with three counter-rotating inner planets of radius q ; which each mesh also with two of three outer planets of radius p ; which all mesh with an external ring of radius r . The constraint that corresponding discs should touch reduces to 4*X^2 - 3*Y^2 = W^2 (*) where X == p-q , Y == s-q , Z == r-p , W = 2 Z - Y . The meshing constraints involve belts around discs s,q,p,q and r,p,q,p resp. Essentially the only solutions found for X < 1024 are scalar dilations of [X, Y, Z, W] = 12 [1, 1, 1, 1] corresponding to concentric increments of the degenerate [r, s, p, q] = [18, 6, 6, -6] --- six equal planets touching both a sun and a ring of triple radius. Any increment of this has overlapping discs. The diophantine tangency equation (*) has some independent interest. Apparently a complete set of primitive, non-negative solutions is generated (after reducing by GCD and eliminating repetitions) by the set of three parametric forms [X, Y, Z, W] , [X, Z, Y, X+Y-W] , [X, |Y-Z|, Y, X+Y] (**) where X = u^2 + 3*v^2 , Y = 4*u*v , W = 2*|u^2 - 3*v^2| , Z = (Y+W)/2 . The first few primitive solutions are 1, 0, 1, 2 1, 1, 1, 1 7, 3, 8, 13 7, 5, 8, 11 7, 8, 5, 2 13, 7, 15, 23 13, 8, 15, 22 13, 15, 8, 1 19, 5, 21, 37 19, 16, 21, 26 19, 21, 16, 11 31, 11, 35, 59 31, 24, 35, 46 31, 35, 24, 13 This is a special case of the natural extension of Pythagorean triples to solutions of (say) a X^2 = b Y^2 + c Z^2 ; however I could not locate any relevant discussion --- references, anyone? Fred Lunnon On 10/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible.
Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges.
Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun?
Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry?
Fred Lunnon
Dickson's "Introduction to the Theory of Numbers" is mostly about Diophantine Equations. There's a chapter on ax2+by2+cz2=0. I think it's called Legendre's Equation. I have the Dover reprint, but can't put my hands on it at the moment. Hardy & Wright have a tiny DEq chapter. They cover the vanilla x2+y2=z2, and might stretch to include one coefficient. The typical situation is a small number of parametric solutions similar to the Pythagorean case (x2+y2=z2). I think the rational case can be written with just one parameter, not sure. The Pyth case can arrange all primitive solutions in a nice ternary tree, starting from 345 (or 011). The children are x' = 2x+y+2z, y' = x+2y+2z, z' = 2x+2y+3z. Varying the signs of x & y takes you in four directions: 3 children, one parent. Something similar probably exists for your modification (*). This arrangement of the variables gives a nice ordering of the tree by atan(x/y). If you swap the defns of x' & y', then you can preserve even-odd parity in the solutions, useful for your 4x2 term. Rich -------- Quoting Fred Lunnon <fred.lunnon@gmail.com>:
I hunted the "octocog" with 6-fold symmetry: like the snark, it remains elusive.
Such a gear train would comprise a sun gear centred at the origin, of radius s ; which meshes with three counter-rotating inner planets of radius q ; which each mesh also with two of three outer planets of radius p ; which all mesh with an external ring of radius r .
The constraint that corresponding discs should touch reduces to 4*X^2 - 3*Y^2 = W^2 (*) where X == p-q , Y == s-q , Z == r-p , W = 2 Z - Y . The meshing constraints involve belts around discs s,q,p,q and r,p,q,p resp.
Essentially the only solutions found for X < 1024 are scalar dilations of [X, Y, Z, W] = 12 [1, 1, 1, 1] corresponding to concentric increments of the degenerate [r, s, p, q] = [18, 6, 6, -6] --- six equal planets touching both a sun and a ring of triple radius. Any increment of this has overlapping discs.
The diophantine tangency equation (*) has some independent interest. Apparently a complete set of primitive, non-negative solutions is generated (after reducing by GCD and eliminating repetitions) by the set of three parametric forms [X, Y, Z, W] , [X, Z, Y, X+Y-W] , [X, |Y-Z|, Y, X+Y] (**) where X = u^2 + 3*v^2 , Y = 4*u*v , W = 2*|u^2 - 3*v^2| , Z = (Y+W)/2 .
The first few primitive solutions are 1, 0, 1, 2 1, 1, 1, 1 7, 3, 8, 13 7, 5, 8, 11 7, 8, 5, 2 13, 7, 15, 23 13, 8, 15, 22 13, 15, 8, 1 19, 5, 21, 37 19, 16, 21, 26 19, 21, 16, 11 31, 11, 35, 59 31, 24, 35, 46 31, 35, 24, 13
This is a special case of the natural extension of Pythagorean triples to solutions of (say) a X^2 = b Y^2 + c Z^2 ; however I could not locate any relevant discussion --- references, anyone?
Fred Lunnon
On 10/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible.
Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges.
Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun?
Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry?
Fred Lunnon
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* rcs@xmission.com <rcs@xmission.com> [Oct 24. 2015 10:03]:
Dickson's "Introduction to the Theory of Numbers" is mostly about Diophantine Equations. There's a chapter on ax2+by2+cz2=0. I think it's called Legendre's Equation. I have the Dover reprint, but can't put my hands on it at the moment.
arxiv.org has scans: http://www.archive.org/search.php?query=creator%3A%22Leonard+Eugene+Dickson%... http://ia700200.us.archive.org/19/items/HistoryOfTheTheoryOfNumbersI/ http://ia331335.us.archive.org/0/items/HistoryOfTheTheoryOfNumbersIii/ http://ia331304.us.archive.org/2/items/HistoryOfTheTheoryOfNumbersVolII/ More links at http://en.wikipedia.org/wiki/L._E._Dickson Best regards, jj
[...]
It finally occurred to me to compute the dimension of the solution set. Consider an arbitrary "octocog" as the union of a pair of disc subsets comprising given radii p,p',p",r and q,q',q",s respectively. Modulo isometry, each subset has freedom 2 (eg. discs p', p" move freely relative to disc p while touching disc r ); fitting them together imposes none for q to touch p',p" ; 2 for q' to touch p,p" , 2 for q" . Hence suitable disc configurations have dimension 2*2 - 2*2 = 0 --- so finitely (though not in this case denumerably) many for given radii. Now a gear train must satisfy 6 additional belt constraints (around discs s, q', q", p etc). Only 5 of these are linearly independent (any belt is effectively the path sum of the remaining 5 ); there might conceivably be further nonlinear dependencies between the constraints. But even a single independent constraint is sufficient to reduce the freedom of the solution set to less than zero --- hence any octocog is inevitably going to be special or degenerate in some way, eg. the symmetric touching specimen discussed earlier. Similar considerations apply to analogous "(2 n)-cogs" for n = 3,4,... , irrespective of imposed symmetry: the solution set has negative dimension, so any solution must degenerate in some way. Seems the snark really was a boojum after all ... Fred Lunnon On 10/23/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I hunted the "octocog" with 6-fold symmetry: like the snark, it remains elusive.
Such a gear train would comprise a sun gear centred at the origin, of radius s ; which meshes with three counter-rotating inner planets of radius q ; which each mesh also with two of three outer planets of radius p ; which all mesh with an external ring of radius r .
The constraint that corresponding discs should touch reduces to 4*X^2 - 3*Y^2 = W^2 (*) where X == p-q , Y == s-q , Z == r-p , W = 2 Z - Y . The meshing constraints involve belts around discs s,q,p,q and r,p,q,p resp.
Essentially the only solutions found for X < 1024 are scalar dilations of [X, Y, Z, W] = 12 [1, 1, 1, 1] corresponding to concentric increments of the degenerate [r, s, p, q] = [18, 6, 6, -6] --- six equal planets touching both a sun and a ring of triple radius. Any increment of this has overlapping discs.
The diophantine tangency equation (*) has some independent interest. Apparently a complete set of primitive, non-negative solutions is generated (after reducing by GCD and eliminating repetitions) by the set of three parametric forms [X, Y, Z, W] , [X, Z, Y, X+Y-W] , [X, |Y-Z|, Y, X+Y] (**) where X = u^2 + 3*v^2 , Y = 4*u*v , W = 2*|u^2 - 3*v^2| , Z = (Y+W)/2 .
The first few primitive solutions are 1, 0, 1, 2 1, 1, 1, 1 7, 3, 8, 13 7, 5, 8, 11 7, 8, 5, 2 13, 7, 15, 23 13, 8, 15, 22 13, 15, 8, 1 19, 5, 21, 37 19, 16, 21, 26 19, 21, 16, 11 31, 11, 35, 59 31, 24, 35, 46 31, 35, 24, 13
This is a special case of the natural extension of Pythagorean triples to solutions of (say) a X^2 = b Y^2 + c Z^2 ; however I could not locate any relevant discussion --- references, anyone?
Fred Lunnon
On 10/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible.
Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges.
Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun?
Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry?
Fred Lunnon
participants (7)
-
Fred Lunnon -
Joerg Arndt -
M. Oskar van Deventer -
rcs＠xmission.com -
rwg -
William R Somsky -
William R. Somsky