[math-fun] French Hens: open problem on space-filling curves / IFS
Since Joerg mentioned Ventrella's book _Brain-Filling Curves: A Fractal Bestiary_, let me take this opportunity to re-mention an open problem I'm fond of. (At least, I think it's still open.) Recall that 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+...+n)^2. That means that one 1x1 square, two 2x2 squares, ..., n nxn squares together all have the same total area as one square of side length 1+2+...+n. Robert Wainright's "Partridge problem" asks whether you can realize this numerical identity as a dissection of a big square into the requisite set of smaller squares. It turns out to be possible for all n >= 8; you can see some pictures of solutions on pages by Funsters at http://www2.stetson.edu/~efriedma/mathmagic/0802.html or http://www.mathpuzzle.com/partridge.html. (The "all n >=" part is very not obvious even once you have an n=8 solution.) The name "partridge problem" comes from "four calling birds, three french hens, two turtledoves, and a partridge in a square tree", of course. This identity can manifest itself as a tiling problem not just for squares, but for an arbitrary planar figure P. If kP means a copy of P scaled linearly by a factor of k, then the question is whether you can use one P, two 2P's, ..., n nP's to cover a (1+2+...+n)P. If so then we say P has "partridge number" n (or maybe only for the minimal such n, depending on who you ask). For example, the square has partridge number 8, the equilateral triangle has partridge number 7, the 30-60-90 right triangle has a partridge number of 4, and so on. The 30-60-90 find is by Patrick Hamlyn, and is a truly beautiful puzzle; I've had a physical copy sitting on my desk for years. Anyway, the open problem part of all this! Patrick Hamlyn's 30-60-90 is the polygon with the smallest known partridge number. As far as I know, it is open whether there are polygons OR fractals with partridge numbers 2 or 3. The fact that 2 is open, even for polygons, really bugs me. I really want to know whether there is a polygon P such that a P and two 2P's fit together to make a 3P. (For a while I thought I had a proof of impossibility, and I was all set to write a paper called "On the Necessity of French Hens", and that my proof fell apart is a great disappointment, though I have to admit more for the title of the non-paper than for its theorem.) Anyway, the fractal version of the partridge-2 or partridge-3 problem is very much along the lines of Ventrella's book. In particular, I think that if a solution exists then then must be a space-filling curve that realizes it. --Michael On Sun, Nov 3, 2013 at 9:19 AM, Joerg Arndt <arndt@jjj.de> wrote:
A pointer to a neat picture book featuring very many "space filling" curves:
Jeffrey J. Ventrella: Brain-Filling Curves: A Fractal Bestiary, LuLu.com, (2012).
The book can be downloaded for free at
http://archive.org/download/BrainfillingCurves-AFractalBestiary/BrainFilling... Warning: it is a hefty 235 Megabytes!
Two pertinent URLs are http://www.fractalcurves.com/HorrorVacui.html (book as html) http://www.brainfillingcurves.com/ (needs scripting activated)
On page 107 (top) there is a curve corresponding to the L-system (and turns of 120 degrees) Start: F Rules: F --> F-F+F+FF-FF + --> + - --> -
Here is a rendering http://jjj.de/3frac/R7-x1-curve.pdf
What Ventrella does not show is how neatly 3 of these combine into either http://jjj.de/3frac/R7-x1-closed-trihook.pdf or http://jjj.de/3frac/R7-x1-closed-gosper-island.pdf
The better known splittings of Gosper's island are http://jjj.de/3frac/gosper-split7.pdf and possibly (3 renderings) http://jjj.de/3frac/R7-2-gosper.pdf http://jjj.de/3frac/R7-2-sty1-gosper.pdf http://jjj.de/3frac/R7-alt-hex-gosper.pdf Here is a bubbly one: http://jjj.de/3frac/R7-bubble-gosper.pdf
(cf. http://www.fractalcurves.com/Root7.html )
For the truly bored there are more images under http://jjj.de/3frac/
Best, jj
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-- Forewarned is worth an octopus in the bush.
Is there a P and a number n such that P is n-partridgeable but not (n+1)-partridgeable? Is there a polygonal P that is not n-partridgeable for any n? I make bold to venture that one way to figure out whether you are a recreational mathematician who isn't afraid to do some hard work, or a serious mathematician who likes to have fun, is to ask yourself whether you prefer concrete problems like "What's the partridge number of the 30-60-90 triangle?" or more structural questions like the two I pose. Jim Propp On Sunday, November 3, 2013, Michael Kleber <michael.kleber@gmail.com> wrote:
Since Joerg mentioned Ventrella's book _Brain-Filling Curves: A Fractal Bestiary_, let me take this opportunity to re-mention an open problem I'm fond of. (At least, I think it's still open.)
Recall that 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+...+n)^2. That means that one 1x1 square, two 2x2 squares, ..., n nxn squares together all have the same total area as one square of side length 1+2+...+n. Robert Wainright's "Partridge problem" asks whether you can realize this numerical identity as a dissection of a big square into the requisite set of smaller squares. It turns out to be possible for all n >= 8; you can see some pictures of solutions on pages by Funsters at http://www2.stetson.edu/~efriedma/mathmagic/0802.html or http://www.mathpuzzle.com/partridge.html. (The "all n >=" part is very not obvious even once you have an n=8 solution.)
The name "partridge problem" comes from "four calling birds, three french hens, two turtledoves, and a partridge in a square tree", of course.
This identity can manifest itself as a tiling problem not just for squares, but for an arbitrary planar figure P. If kP means a copy of P scaled linearly by a factor of k, then the question is whether you can use one P, two 2P's, ..., n nP's to cover a (1+2+...+n)P. If so then we say P has "partridge number" n (or maybe only for the minimal such n, depending on who you ask). For example, the square has partridge number 8, the equilateral triangle has partridge number 7, the 30-60-90 right triangle has a partridge number of 4, and so on. The 30-60-90 find is by Patrick Hamlyn, and is a truly beautiful puzzle; I've had a physical copy sitting on my desk for years.
Anyway, the open problem part of all this!
Patrick Hamlyn's 30-60-90 is the polygon with the smallest known partridge number. As far as I know, it is open whether there are polygons OR fractals with partridge numbers 2 or 3.
The fact that 2 is open, even for polygons, really bugs me. I really want to know whether there is a polygon P such that a P and two 2P's fit together to make a 3P.
(For a while I thought I had a proof of impossibility, and I was all set to write a paper called "On the Necessity of French Hens", and that my proof fell apart is a great disappointment, though I have to admit more for the title of the non-paper than for its theorem.)
Anyway, the fractal version of the partridge-2 or partridge-3 problem is very much along the lines of Ventrella's book. In particular, I think that if a solution exists then then must be a space-filling curve that realizes it.
--Michael
On Sun, Nov 3, 2013 at 9:19 AM, Joerg Arndt <arndt@jjj.de> wrote:
A pointer to a neat picture book featuring very many "space filling" curves:
Jeffrey J. Ventrella: Brain-Filling Curves: A Fractal Bestiary, LuLu.com, (2012).
The book can be downloaded for free at
http://archive.org/download/BrainfillingCurves-AFractalBestiary/BrainFilling...
Warning: it is a hefty 235 Megabytes!
Two pertinent URLs are http://www.fractalcurves.com/HorrorVacui.html (book as html) http://www.brainfillingcurves.com/ (needs scripting activated)
On page 107 (top) there is a curve corresponding to the L-system (and turns of 120 degrees) Start: F Rules: F --> F-F+F+FF-FF + --> + - --> -
Here is a rendering http://jjj.de/3frac/R7-x1-curve.pdf
What Ventrella does not show is how neatly 3 of these combine into either http://jjj.de/3frac/R7-x1-closed-trihook.pdf or http://jjj.de/3frac/R7-x1-closed-gosper-island.pdf
The better known splittings of Gosper's island are http://jjj.de/3frac/gosper-split7.pdf and possibly (3 renderings) http://jjj.de/3frac/R7-2-gosper.pdf http://jjj.de/3frac/R7-2-sty1-gosper.pdf http://jjj.de/3frac/R7-alt-hex-gosper.pdf Here is a bubbly one: http://jjj.de/3frac/R7-bubble-gosper.pdf
(cf. http://www.fractalcurves.com/Root7.html )
For the truly bored there are more images under http://jjj.de/3frac/
Best, jj
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It's easy to make non-partridgeable. If you're allowed concavities it's trivial to make a smallest P's indent unfillable by any larger copy of P. Even with convex P, constraining the angles to make it hard for copies of P to fit around a vertex isn't challenging. I'm sure Patrick Hamlyn's computer could tell us whether there's a 30-60-90 partridge=5 solution in a matter of minutes, but I'd put money on the answer being no. --Michael On Sun, Nov 3, 2013 at 11:19 AM, James Propp <jamespropp@gmail.com> wrote:
Is there a P and a number n such that P is n-partridgeable but not (n+1)-partridgeable?
Is there a polygonal P that is not n-partridgeable for any n?
I make bold to venture that one way to figure out whether you are a recreational mathematician who isn't afraid to do some hard work, or a serious mathematician who likes to have fun, is to ask yourself whether you prefer concrete problems like "What's the partridge number of the 30-60-90 triangle?" or more structural questions like the two I pose.
Jim Propp
On Sunday, November 3, 2013, Michael Kleber <michael.kleber@gmail.com> wrote:
Since Joerg mentioned Ventrella's book _Brain-Filling Curves: A Fractal Bestiary_, let me take this opportunity to re-mention an open problem I'm fond of. (At least, I think it's still open.)
Recall that 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+...+n)^2. That means that one 1x1 square, two 2x2 squares, ..., n nxn squares together all have the same total area as one square of side length 1+2+...+n. Robert Wainright's "Partridge problem" asks whether you can realize this numerical identity as a dissection of a big square into the requisite set of smaller squares. It turns out to be possible for all n >= 8; you can see some pictures of solutions on pages by Funsters at http://www2.stetson.edu/~efriedma/mathmagic/0802.html or http://www.mathpuzzle.com/partridge.html. (The "all n >=" part is very not obvious even once you have an n=8 solution.)
The name "partridge problem" comes from "four calling birds, three french hens, two turtledoves, and a partridge in a square tree", of course.
This identity can manifest itself as a tiling problem not just for squares, but for an arbitrary planar figure P. If kP means a copy of P scaled linearly by a factor of k, then the question is whether you can use one P, two 2P's, ..., n nP's to cover a (1+2+...+n)P. If so then we say P has "partridge number" n (or maybe only for the minimal such n, depending on who you ask). For example, the square has partridge number 8, the equilateral triangle has partridge number 7, the 30-60-90 right triangle has a partridge number of 4, and so on. The 30-60-90 find is by Patrick Hamlyn, and is a truly beautiful puzzle; I've had a physical copy sitting on my desk for years.
Anyway, the open problem part of all this!
Patrick Hamlyn's 30-60-90 is the polygon with the smallest known partridge number. As far as I know, it is open whether there are polygons OR fractals with partridge numbers 2 or 3.
The fact that 2 is open, even for polygons, really bugs me. I really want to know whether there is a polygon P such that a P and two 2P's fit together to make a 3P.
(For a while I thought I had a proof of impossibility, and I was all set to write a paper called "On the Necessity of French Hens", and that my proof fell apart is a great disappointment, though I have to admit more for the title of the non-paper than for its theorem.)
Anyway, the fractal version of the partridge-2 or partridge-3 problem is very much along the lines of Ventrella's book. In particular, I think that if a solution exists then then must be a space-filling curve that realizes it.
--Michael
On Sun, Nov 3, 2013 at 9:19 AM, Joerg Arndt <arndt@jjj.de> wrote:
A pointer to a neat picture book featuring very many "space filling" curves:
Jeffrey J. Ventrella: Brain-Filling Curves: A Fractal Bestiary, LuLu.com, (2012).
The book can be downloaded for free at
http://archive.org/download/BrainfillingCurves-AFractalBestiary/BrainFilling...
Warning: it is a hefty 235 Megabytes!
Two pertinent URLs are http://www.fractalcurves.com/HorrorVacui.html (book as html) http://www.brainfillingcurves.com/ (needs scripting activated)
On page 107 (top) there is a curve corresponding to the L-system (and turns of 120 degrees) Start: F Rules: F --> F-F+F+FF-FF + --> + - --> -
Here is a rendering http://jjj.de/3frac/R7-x1-curve.pdf
What Ventrella does not show is how neatly 3 of these combine into either http://jjj.de/3frac/R7-x1-closed-trihook.pdf or http://jjj.de/3frac/R7-x1-closed-gosper-island.pdf
The better known splittings of Gosper's island are http://jjj.de/3frac/gosper-split7.pdf and possibly (3 renderings) http://jjj.de/3frac/R7-2-gosper.pdf http://jjj.de/3frac/R7-2-sty1-gosper.pdf http://jjj.de/3frac/R7-alt-hex-gosper.pdf Here is a bubbly one: http://jjj.de/3frac/R7-bubble-gosper.pdf
(cf. http://www.fractalcurves.com/Root7.html )
For the truly bored there are more images under http://jjj.de/3frac/
Best, jj
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush.
participants (2)
-
James Propp -
Michael Kleber