[math-fun] Gosper island double surprise
Two renderings of Gosper's island, one traverses all points of the (3.4.6.4)-tiling once, the other (was awfully hard to obtain and) traverses all edges once: http://jjj.de/tmp-xmas/ The edge version is to my best knowledge the first such curve for (3.4.6.4), versions for (3^6) and (4^4) are known for quite some time. Ventrella gave the first for (3.6.3.6) in 2012 in his book "Brain-Filling Curves". All other uniform tilings have odd incidences on some points, hence no such curve exists on them. The curve I use is not Gosper's (I need a curve that turns by 120 degrees after every edge). Enjoy! Now that is my chance to ask about terminology (again). Do the following appear OK? I call... ... the arrangement of points and edges of some tiling a "grid", as in the "square grid" for what is (4^4), the (uniform) tiling into unit squares in Gruenbaum and Shephard. ... specifically, the grids for (3^6) the "triangular grid", for (6^3) the "hexagonal grid", for (4^4) the "square grid" (as said), and for (3.6.3.6) the "tri-hexagonal grid". ... curves that traverse all points once "point-covering" (and could in analogy call those that traverse all edges once "edge-covering", but indeed call them "grid-filling", should I prefer "edge-covering"?). The (pdf) images above are examples of each. These are two corner cases of "plane-filling" on a grid. Also I define Eisenstein integers as numbers of the form x + \omega_6 * y while the rest of the world seems to use x + \omega_3 * y (my norm is x^2 + x*y + y^2, the other is x^2 - x*y + y^2). Does anybody want to kill me for that? Yes, I am writing something up and would like to avoid annoying the readers with bad terminology (and nobody in my personal reach could possibly answer the questions above). Best regards, jj
JJ, Concerning the ring of Eisenstein integers: There is no room for argument! They are the complex numbers of the form a + b omega, where omega = e^(2 Pi i / 3) = -1/2 + i sqrt(3)/2 and a and b are ordinary integers. I'll use w for omega from here on The norm of a+b w is (a + b w)(a + b w^2) = a^2 -ab + b^2. w satisfies w^3 = 1. You should not use a sixth root of unity. (Yes, -w is in the ring, but so what) Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Dec 21, 2015 at 10:00 AM, Joerg Arndt <arndt@jjj.de> wrote:
Two renderings of Gosper's island, one traverses all points of the (3.4.6.4)-tiling once, the other (was awfully hard to obtain and) traverses all edges once: http://jjj.de/tmp-xmas/
The edge version is to my best knowledge the first such curve for (3.4.6.4), versions for (3^6) and (4^4) are known for quite some time. Ventrella gave the first for (3.6.3.6) in 2012 in his book "Brain-Filling Curves". All other uniform tilings have odd incidences on some points, hence no such curve exists on them.
The curve I use is not Gosper's (I need a curve that turns by 120 degrees after every edge).
Enjoy!
Now that is my chance to ask about terminology (again). Do the following appear OK? I call...
... the arrangement of points and edges of some tiling a "grid", as in the "square grid" for what is (4^4), the (uniform) tiling into unit squares in Gruenbaum and Shephard. ... specifically, the grids for (3^6) the "triangular grid", for (6^3) the "hexagonal grid", for (4^4) the "square grid" (as said), and for (3.6.3.6) the "tri-hexagonal grid".
... curves that traverse all points once "point-covering" (and could in analogy call those that traverse all edges once "edge-covering", but indeed call them "grid-filling", should I prefer "edge-covering"?). The (pdf) images above are examples of each. These are two corner cases of "plane-filling" on a grid.
Also I define Eisenstein integers as numbers of the form x + \omega_6 * y while the rest of the world seems to use x + \omega_3 * y (my norm is x^2 + x*y + y^2, the other is x^2 - x*y + y^2). Does anybody want to kill me for that?
Yes, I am writing something up and would like to avoid annoying the readers with bad terminology (and nobody in my personal reach could possibly answer the questions above).
Best regards, jj
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On Monday, December 21, 2015, Neil Sloane <njasloane@gmail.com> wrote: JJ,
Concerning the ring of Eisenstein integers: There is no room for argument! They are the complex numbers of the form a + b omega, where omega = e^(2 Pi i / 3) = -1/2 + i sqrt(3)/2 and a and b are ordinary integers.
That is one way to describe them. Joerg's description is equally valid. It's a matter of esthetics and convenience. Joerg, can you say why you prefer your way of describing the ring? You should not use a sixth root of unity. Why shouldn't he?
(Yes, -w is in the ring, but so what)
Maybe because for some applications it's nice to have a norm-form in which all coefficients are positive? (I'm just guessing.) Best regards
Neil
Jim Propp
* James Propp <jamespropp@gmail.com> [Dec 22. 2015 08:18]:
On Monday, December 21, 2015, Neil Sloane <njasloane@gmail.com> wrote:
JJ,
Concerning the ring of Eisenstein integers: There is no room for argument! They are the complex numbers of the form a + b omega, where omega = e^(2 Pi i / 3) = -1/2 + i sqrt(3)/2 and a and b are ordinary integers.
That is one way to describe them. Joerg's description is equally valid. It's a matter of esthetics and convenience. Joerg, can you say why you prefer your way of describing the ring?
I did all my drawings (lots!), calculations, and programming with the basis { 1, \omega_6 }, so it is a bit of a practical thing. It will be painless for my write-up to switch to the third primitive root, so I will do that (unless I'll find compelling reason not to).
You should not use a sixth root of unity.
Why shouldn't he?
(Yes, -w is in the ring, but so what)
Maybe because for some applications it's nice to have a norm-form in which all coefficients are positive? (I'm just guessing.)
Best regards
Neil
Jim Propp
Thanks for the answers! If anyone wants to murder some of the remaining terminology, kindly speak up. Best regards, jj
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Jörg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice. Your island/3 looks to be self-similarly dissectible. Could you render a multicolored one to show how many pieces? --rwg On 2015-12-21 07:00, Joerg Arndt wrote:
Two renderings of Gosper's island, one traverses all points of the (3.4.6.4)-tiling once, the other (was awfully hard to obtain and) traverses all edges once: http://jjj.de/tmp-xmas/
The edge version is to my best knowledge the first such curve for (3.4.6.4), versions for (3^6) and (4^4) are known for quite some time. Ventrella gave the first for (3.6.3.6) in 2012 in his book "Brain-Filling Curves". All other uniform tilings have odd incidences on some points, hence no such curve exists on them.
The curve I use is not Gosper's (I need a curve that turns by 120 degrees after every edge).
Enjoy!
Now that is my chance to ask about terminology (again). Do the following appear OK? I call...
... the arrangement of points and edges of some tiling a "grid", as in the "square grid" for what is (4^4), the (uniform) tiling into unit squares in Gruenbaum and Shephard. ... specifically, the grids for (3^6) the "triangular grid", for (6^3) the "hexagonal grid", for (4^4) the "square grid" (as said), and for (3.6.3.6) the "tri-hexagonal grid".
... curves that traverse all points once "point-covering" (and could in analogy call those that traverse all edges once "edge-covering", but indeed call them "grid-filling", should I prefer "edge-covering"?). The (pdf) images above are examples of each. These are two corner cases of "plane-filling" on a grid.
Also I define Eisenstein integers as numbers of the form x + \omega_6 * y while the rest of the world seems to use x + \omega_3 * y (my norm is x^2 + x*y + y^2, the other is x^2 - x*y + y^2). Does anybody want to kill me for that?
Yes, I am writing something up and would like to avoid annoying the readers with bad terminology (and nobody in my personal reach could possibly answer the questions above).
Best regards, jj
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* rwg <rwg@sdf.org> [Dec 23. 2015 20:26]:
Jörg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice.
Your island/3 looks to be self-similarly dissectible. Could you render a multicolored one to show how many pieces? --rwg
There are many ways to do this, one is http://jjj.de/tmp-xmas/gosper-tiling-by-border.pdf But you do know _that_ one (more than on page in the file!). Another one would give 21 parts, just by splitting each of the thirds into 7 similar copies. As it is past beer o'clock here I just simply uploaded http://jjj.de/tmp-xmas/arndt-curve-search.pdf I hope there is more than one instance "oh!" in there, though the text (note the \marginpars) still needs serious polishing (notably the part about complex numeration systems, I sadly may have to drop it finally). The big question about a lot of what I found is: why hasn't that been done 25 years ago? Best regards, j-ö-j <--= Umlaut!
[...]
Added the file r07-t-5-island-split.pdf showing the splitting of the three order-7 curves (without any (3.4.6.4)-trickery) into 7 parts each. One can spot the decomposition into 7 smaller islands and instances of what Davis/Knuth and Dekking call "carousel" (the arrangements of 6 curves with 6-fold symmetry). Is this pertinent? Best regards, jj * rwg <rwg@sdf.org> [Dec 23. 2015 20:26]:
Jörg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice.
Your island/3 looks to be self-similarly dissectible. Could you render a multicolored one to show how many pieces? --rwg
On 2015-12-21 07:00, Joerg Arndt wrote: [...]
On 2015-12-25 09:51, Joerg Arndt wrote:
Added the file r07-t-5-island-split.pdf showing the splitting of the three order-7 curves (without any (3.4.6.4)-trickery) into 7 parts each.
One can spot the decomposition into 7 smaller islands and instances of what Davis/Knuth and Dekking call "carousel" (the arrangements of 6 curves with 6-fold symmetry).
Is this pertinent?
Best regards, jj
http://jjj.de/tmp-xmas/r07-t-5-island-split.pdf OMG!! You can heptasect the island in the usual way, trisect the sub- islands, and rotate them independently by π/3, as well as three overlapping sub-islands! Then Fig 6.1-C, page 52, http://jjj.de/tmp-xmas/arndt-curve-search-2015.12.25.pdf it *looks* like the figure is similar to 7/12 of itself, which is bloody impossible. Holy something, I'm confused! --Bill PS: Thanks!
* rwg <rwg@sdf.org> [Dec 23. 2015 20:26]:
Jörg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice.
Your island/3 looks to be self-similarly dissectible. Could you render a multicolored one to show how many pieces? --rwg
On 2015-12-21 07:00, Joerg Arndt wrote: [...]
OK, NeilB straightened me out. The frac-12-tile, Fig 6.1-C, page 52 is *not* 1/3 of the usual Franceoid island, but rather 1/3 of a *different* Franceoid. (And not the Floppy Franceoid you get by mirror imaging alternate generations.) Island/3 is a frac-7-tile. It probably needs a better name than our "pepperoncino". --rwg On 2015-12-25 10:40, rwg wrote:
On 2015-12-25 09:51, Joerg Arndt wrote:
Added the file r07-t-5-island-split.pdf showing the splitting of the three order-7 curves (without any (3.4.6.4)-trickery) into 7 parts each.
One can spot the decomposition into 7 smaller islands and instances of what Davis/Knuth and Dekking call "carousel" (the arrangements of 6 curves with 6-fold symmetry).
Is this pertinent?
Best regards, jj
http://jjj.de/tmp-xmas/r07-t-5-island-split.pdf OMG!! You can heptasect the island in the usual way, trisect the sub- islands, and rotate them independently by π/3, as well as three overlapping sub-islands!
Then Fig 6.1-C, page 52, http://jjj.de/tmp-xmas/arndt-curve-search-2015.12.25.pdf it *looks* like the figure is similar to 7/12 of itself, which is bloody impossible. Holy something, I'm confused! --Bill PS: Thanks!
* rwg <rwg@sdf.org> [Dec 23. 2015 20:26]:
Jörg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice.
Your island/3 looks to be self-similarly dissectible. Could you render a multicolored one to show how many pieces? --rwg
On 2015-12-21 07:00, Joerg Arndt wrote: [...]
* rwg <rwg@sdf.org> [Dec 26. 2015 07:19]:
OK, NeilB straightened me out. The frac-12-tile, Fig 6.1-C, page 52 is *not* 1/3 of the usual Franceoid island, but rather 1/3 of a *different* Franceoid. (And not the Floppy Franceoid you get by mirror imaging alternate generations.) Island/3 is a frac-7-tile.
Yes, it is a curve of order 12 (hence cannot give the Gosper/France island): F F0F+F+F-F-F0F+F+F-F-F0F # R12-4 # symm-dr (here 0 is "no turn", aka "do nothing). The L-systems for this one and the other 943344 curves are online at: http://jjj.de/3frac/ As one tar ball: http://jjj.de/3frac/short-lsys.tar.xz (size 4.3 MByte, unpacks into a directory ./short-lsys/ of size about 140 MByte).
It probably needs a better name than our "pepperoncino".
I gave up naming those things already at the stage of finding them with pencil and paper, when I hit the letter 'z' within one grid and order. Now "names" are non-negative numbers, the triple (number, order, grid) uniquely specifies a curve. Bonus track for eyeballing (not mentioned in the draft): The following images are of families of curves that were constructed. In http://jjj.de/tmp-xmas/ see the files thin-*.pdf Best regards, jj
--rwg [...]
On 2015-12-25 23:02, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Dec 26. 2015 07:19]:
OK, NeilB straightened me out. The frac-12-tile, Fig 6.1-C, page 52 is *not* 1/3 of the usual Franceoid island, but rather 1/3 of a *different* Franceoid. (And not the Floppy Franceoid you get by mirror imaging alternate generations.) Island/3 is a frac-7-tile.
Yes, it is a curve of order 12 (hence cannot give the Gosper/France island): F F0F+F+F-F-F0F+F+F-F-F0F # R12-4 # symm-dr (here 0 is "no turn", aka "do nothing). The L-systems for this one and the other 943344 curves are online at: http://jjj.de/3frac/ As one tar ball: http://jjj.de/3frac/short-lsys.tar.xz (size 4.3 MByte, unpacks into a directory ./short-lsys/ of size about 140 MByte).
It probably needs a better name than our "pepperoncino".
I gave up naming those things already at the stage of finding them with pencil and paper, when I hit the letter 'z' within one grid and order. Now "names" are non-negative numbers, the triple (number, order, grid) uniquely specifies a curve.
This one seems noteworthy.
Bonus track for eyeballing (not mentioned in the draft): The following images are of families of curves that were constructed. In http://jjj.de/tmp-xmas/ see the files thin-*.pdf
Best regards, jj
--rwg [...]
_____________
http://jjj.de/tmp-xmas/thin-3-tiles-sty1.pdf switches from L-system: 2 iterations axiom = _F_+F_+F sty = 3 1 0 F |--> F+F0F+F-F+F0F0F+F-F+F-F+F0F0F0F+F-F+F-F+F-F+F0F0F0F-F+F-F+F-F+F-F0F0F0F-F+F- F+F-F0F0F-F+F-F0F-F # mk-thin-3-dragon.gp: R48-1 # dragon # symm-dr to L-system: 2 iterations axiom = _F_+F_+F sty = 3 1 0 F |--> F+F0F+F-F+F0F0F+F-F+F-F+F0F0F0F+F-F+F-F+F-F+F0F0F0F0F-F+F-F+F-F+F-F0F0F0F-F+F- F+F-F0F0F-F+F-F0F-F # mk-thin-3-dragon.gp: R49-1 # dragon # symm-dr about 1/3 of the way in. It starts out identically(?) to a base 2+omega system where the omega^1, omega^3, and omega^5 digits of "France" are replaced with larger ones, producing a feathery tile. --rwg
* rwg <rwg@sdf.org> [Dec 26. 2015 14:40]:
[...]
_____________
http://jjj.de/tmp-xmas/thin-3-tiles-sty1.pdf switches from L-system: 2 iterations axiom = _F_+F_+F sty = 3 1 0 F |--> F+F0F+F-F+F0F0F+F-F+F-F+F0F0F0F+F-F+F-F+F-F+F0F0F0F-F+F-F+F-F+F-F0F0F0F-F+F- F+F-F0F0F-F+F-F0F-F # mk-thin-3-dragon.gp: R48-1 # dragon # symm-dr to L-system: 2 iterations axiom = _F_+F_+F sty = 3 1 0 F |--> F+F0F+F-F+F0F0F+F-F+F-F+F0F0F0F+F-F+F-F+F-F+F0F0F0F0F-F+F-F+F-F+F-F0F0F0F-F+F- F+F-F0F0F-F+F-F0F-F # mk-thin-3-dragon.gp: R49-1 # dragon # symm-dr about 1/3 of the way in. It starts out identically(?) to a base 2+omega system where the omega^1, omega^3, and omega^5 digits of "France" are replaced with larger ones, producing a feathery tile. --rwg
Not sure I understand. One can take _any_ tile for the smallest surrounded sets (the very many tiny triangles, which one can render as hexagons). Choosing one tile for those "atoms" corresponds to the tile of a product curve (two iterates of the curve we are looking at, followed by one iterate of "whatever", as in my Section 5). Every curve has a tile, we can multiply curves, hence tiles. The tiles shown in thin-3-tiles-sty1.pdf correspond to two families of curves, each with again two sub-families.
From my file: Family 1: orders 3,4, 12,13, 27,28, 48,49, 75,76, 108,109, ... Family 2: orders 7,9, 19,21, 37,39, 61,63, 91,93, 127,129, ...
And yes, switching back and fourth between orders written next to each other is somewhat lovely (stare at the center of the image to get the difference beyond rotation). Btw. the "manta" curves show that curves exists that move without any turn as long as theoretically possible (_one_ more straight move and they'd be at the end point!). Best regards, jj
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On 2015-12-26 06:16, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Dec 26. 2015 14:40]:
[...]
_____________
http://jjj.de/tmp-xmas/thin-3-tiles-sty1.pdf switches from L-system: 2 iterations axiom = _F_+F_+F sty = 3 1 0 F |--> F+F0F+F-F+F0F0F+F-F+F-F+F0F0F0F+F-F+F-F+F-F+F0F0F0F-F+F-F+F-F+F-F0F0F0F-F+F- F+F-F0F0F-F+F-F0F-F # mk-thin-3-dragon.gp: R48-1 # dragon # symm-dr to L-system: 2 iterations axiom = _F_+F_+F sty = 3 1 0 F |--> F+F0F+F-F+F0F0F+F-F+F-F+F0F0F0F+F-F+F-F+F-F+F0F0F0F0F-F+F-F+F-F+F-F0F0F0F-F+F- F+F-F0F0F-F+F-F0F-F # mk-thin-3-dragon.gp: R49-1 # dragon # symm-dr about 1/3 of the way in. It starts out identically(?) to a base 2+omega system where the omega^1, omega^3, and omega^5 digits of "France" are replaced with larger ones, producing a feathery tile. --rwg
Dawk, I forgot I had a picture! http://www.tweedledum.com/rwg/tril7.htm . The pictured tile of order 2 is of 7 tiles of order 1, each of which is 7 tiles of order 0 (hexagons), where the grouping at every level is 3 around 3 around 1, vs 6 around 1 for the Island. So the base is still 2+(-1)^(1/3), but the digits are (-1)^(0),(-1)^0+(-1)^(1/3), (-1)^(2,3), (-1)^(2/3)+(-1)^(3/3), (-1)^(4/3), (-1)^(4/3)+(-1)^(5/3) instead of (-1)^(0..5/3).
And a (xerographically printed) spacefill: http://gosper.org/IMG_0245.JPG .
Not sure I understand. One can take _any_ tile for the smallest surrounded sets (the very many tiny triangles, which one can render as hexagons). Choosing one tile for those "atoms" corresponds to the tile of a product curve (two iterates of the curve we are looking at, followed by one iterate of "whatever", as in my Section 5).
Every curve has a tile, we can multiply curves, hence tiles.
The tiles shown in thin-3-tiles-sty1.pdf correspond to two families of curves, each with again two sub-families.
From my file: Family 1: orders 3,4, 12,13, 27,28, 48,49, 75,76, 108,109, ... Family 2: orders 7,9, 19,21, 37,39, 61,63, 91,93, 127,129, ...
And yes, switching back and fourth between orders written next to each other is somewhat lovely (stare at the center of the image to get the difference beyond rotation).
Ah, so the presence or absence in an image of a central hexagon containing a tricolor spiral, which in successive images goes ... YES YES NO NO YES YES NO NO ... is you switching among rules rather than a bizarre consequence of a single rule. I thought the mixing was some kind of editing accident! --rwg
Btw. the "manta" curves show that curves exists that move without any turn as long as theoretically possible (_one_ more straight move and they'd be at the end point!).
Best regards, jj
Btw, that tricolor spiral can make a nasty "physical illusion". (http://gosper.org/esch2.PNG) Just by brightening and darkening the three colors, you can permute which surfaces appear horizontal, and which vertical. I want to see a life-sized contradictory pair of these in a (well-insured and well-carpeted) math museum. --rwg
_______________________________________________
* rwg <rwg@sdf.org> [Dec 26. 2015 19:33]:
[...] Dawk, I forgot I had a picture! http://www.tweedledum.com/rwg/tril7.htm . The pictured tile of order 2 is of 7 tiles of order 1, each of which is 7 tiles of order 0 (hexagons), where the grouping at every level is 3 around 3 around 1, vs 6 around 1 for the Island. So the base is still 2+(-1)^(1/3), but the digits are (-1)^(0),(-1)^0+(-1)^(1/3), (-1)^(2,3), (-1)^(2/3)+(-1)^(3/3), (-1)^(4/3), (-1)^(4/3)+(-1)^(5/3) instead of (-1)^(0..5/3).
Could you check against my numeration system? See http://jjj.de/tmp-xmas/arndt-curve-search-2015.12.26.pdf (new, better errors!) in section 3.3 pp.21ff, especially Figure 3.3-C. (btw. negating the base gives a tile that has a region around zero covered, and that I cannot identify with any curve I know).
And a (xerographically printed) spacefill: http://gosper.org/IMG_0245.JPG .
Just the minute I was staring at this image... I tried to re-create it, but my rendering methods give superficially similar but different images. It would also be nice to learn how you got the rendering in the image IMG_0246.JPG (looks like morphed from IMG_0247.JPG to me, but I cannot quite see how).
[...] And yes, switching back and fourth between orders written next to each other is somewhat lovely (stare at the center of the image to get the difference beyond rotation).
Ah, so the presence or absence in an image of a central hexagon containing a tricolor spiral, which in successive images goes ... YES YES NO NO YES YES NO NO ... is you switching among rules rather than a bizarre consequence of a single rule. I thought the mixing was some kind of editing accident! --rwg
Btw. the "manta" curves show that curves exists that move without any turn as long as theoretically possible (_one_ more straight move and they'd be at the end point!).
Best regards, jj
Btw, that tricolor spiral can make a nasty "physical illusion". (http://gosper.org/esch2.PNG) Just by brightening and darkening the three colors, you can permute which surfaces appear horizontal, and which vertical. I want to see a life-sized contradictory pair of these in a (well-insured and well-carpeted) math museum. --rwg
I have quite a few prints where I pencilled (is that a word?) in similar things. But then the triangular grid is sort-of-ish a projection of the simple cubic lattice, looked upon from direction (say) (1, 1, 1).
[...]
Best regards, jj
On 2015-12-26 11:03, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Dec 26. 2015 19:33]:
[...] Dawk, I forgot I had a picture! http://www.tweedledum.com/rwg/tril7.htm . The pictured tile of order 2 is of 7 tiles of order 1, each of which is 7 tiles of order 0 (hexagons), where the grouping at every level is 3 around 3 around 1, vs 6 around 1 for the Island. So the base is still 2+(-1)^(1/3), but the digits are (-1)^(0),(-1)^0+(-1)^(1/3), (-1)^(2,3), (-1)^(2/3)+(-1)^(3/3), (-1)^(4/3), (-1)^(4/3)+(-1)^(5/3) instead of (-1)^(0..5/3).
Could you check against my numeration system? See http://jjj.de/tmp-xmas/arndt-curve-search-2015.12.26.pdf (new, better errors!) in section 3.3 pp.21ff, especially Figure 3.3-C. Yes, I'm pretty sure my IMG_0245 is your R7.1 . (btw. negating the base gives a tile that has a region around zero covered, and that I cannot identify with any curve I know). No picture? Anyway, 0.* failing to represent 0 is annoying but nonfatal. E.g., balanced negabinary doesn't even have a 0 digit, but can represent 0 as an infinite string.
And a (xerographically printed) spacefill: http://gosper.org/IMG_0245.JPG .
Just the minute I was staring at this image... I tried to re-create it, but my rendering methods give superficially similar but different images.
It would also be nice to learn how you got the rendering in the image IMG_0246.JPG (looks like morphed from IMG_0247.JPG to me, but I cannot quite see how).
They actually form a trio, starting with http://www.tweedledum.com/rwg/7posies.bmp . They're topologically different--not morphs. Recursively join together clusters of seven to form a tree. Then "ensausage" it. In the limit, inside and outside are replaced by boundary, and all points are hit at least twice.
[...] And yes, switching back and fourth between orders written next to each other is somewhat lovely (stare at the center of the image to get the difference beyond rotation).
Ah, so the presence or absence in an image of a central hexagon containing a tricolor spiral, which in successive images goes ... YES YES NO NO YES YES NO NO ... is you switching among rules rather than a bizarre consequence of a single rule. I thought the mixing was some kind of editing accident! --rwg
Btw. the "manta" curves show that curves exists that move without any turn as long as theoretically possible (_one_ more straight move and they'd be at the end point!).
Best regards, jj
Btw, that tricolor spiral can make a nasty "physical illusion". (http://gosper.org/esch2.PNG) Just by brightening and darkening the three colors, you can permute which surfaces appear horizontal, and which vertical. I want to see a life-sized contradictory pair of these in a (well-insured and well-carpeted)
(but poorly lighted)
math museum. --rwg
I have quite a few prints where I pencilled (is that a word?)
In the UK. Stateside, it's penciled.
in similar things. But then the triangular grid is sort-of-ish a projection of the simple cubic lattice, looked upon from direction (say) (1, 1, 1).
Somebody thus treated the flowsnake.
Point of grammar: On p22 you say As the set contains a neighborhood of zero no signs are needed neither. Nobody says this because it sounds like a double negative. A chatbot might say it and blow Turing's test. --rwg
[...]
Best regards, jj
* rwg <rwg@sdf.org> [Dec 27. 2015 07:48]:
On 2015-12-26 11:03, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Dec 26. 2015 19:33]:
[...] Dawk, I forgot I had a picture! http://www.tweedledum.com/rwg/tril7.htm . [...]
Could you check against my numeration system? See http://jjj.de/tmp-xmas/arndt-curve-search-2015.12.26.pdf (new, better errors!) in section 3.3 pp.21ff, especially Figure 3.3-C.
Yes, I'm pretty sure my IMG_0245 is your R7.1 .
(btw. negating the base gives a tile that has a region around zero covered, and that I cannot identify with any curve I know). No picture?
OK: http://jjj.de/tmp-xmas/r7-numsys-tile.png (the one above) http://jjj.de/tmp-xmas/r7-neg-numsys-tile.png (negated base)
[...]
They actually form a trio, starting with http://www.tweedledum.com/rwg/7posies.bmp . They're topologically different--not morphs. Recursively join together clusters of seven to form a tree. Then "ensausage" it. In the limit, inside and outside are replaced by boundary, and all points are hit at least twice.
Ah! Gotta revisit those scripts peeling out boundaries (and generate L-systems for boundaries) .
[...]
I have quite a few prints where I pencilled (is that a word?)
In the UK. Stateside, it's penciled.
in similar things. But then the triangular grid is sort-of-ish a projection of the simple cubic lattice, looked upon from direction (say) (1, 1, 1).
Somebody thus treated the flowsnake.
Point of grammar: On p22 you say As the set contains a neighborhood of zero no signs are needed neither. Nobody says this because it sounds like a double negative. A chatbot might say it and blow Turing's test. --rwg
OK, thanks. I just dropped the word "neither". Still flagged as "to be reworded", will have to wait for the edit from my editor Edith. Best regards, jj
[...]
On Sat, Dec 26, 2015 at 5:15 AM, rwg <rwg@sdf.org> wrote:
On 2015-12-25 23:02, Joerg Arndt wrote:
* rwg <rwg@sdf.org> [Dec 26. 2015 07:19]:
OK, NeilB straightened me out. The frac-12-tile, Fig 6.1-C, page 52 is *not* 1/3 of the usual Franceoid island, but rather 1/3 of a *different* Franceoid. (And not the Floppy Franceoid you get by mirror imaging alternate generations.) Island/3 is a frac-7-tile.
Yes, it is a curve of order 12 (hence cannot give the Gosper/France island): F F0F+F+F-F-F0F+F+F-F-F0F # R12-4 # symm-dr (here 0 is "no turn", aka "do nothing). The L-systems for this one and the other 943344 curves are online at: http://jjj.de/3frac/ As one tar ball: http://jjj.de/3frac/short-lsys.tar.xz (size 4.3 MByte, unpacks into a directory ./short-lsys/ of size about 140 MByte).
It probably needs a better name
than our "pepperoncino".
I gave up naming those things already at the stage of finding them with pencil and paper, when I hit the letter 'z' within one grid and order. Now "names" are non-negative numbers, the triple (number, order, grid) uniquely specifies a curve.
This one seems noteworthy.
Bonus track for eyeballing (not mentioned in the draft):
The following images are of families of curves that were constructed. In http://jjj.de/tmp-xmas/ see the files thin-*.pdf
Best regards, jj
--rwg
[...]
Well 900K curves is pretty impressive, but how about a continuum of them? When building a frac-tile, you have, at every level of recursion, the option of mirror imaging. Hence, e.g., uncountably many slight variations of "France". --rwg. "France. We're from France." --Beldar Conehead
Yes, but unless the mirroring is periodic with respect to the levels of recursion, can the end result be self-similar? —Dan
On Dec 26, 2015, at 10:54 AM, Bill Gosper <billgosper@gmail.com> wrote:
When building a frac-tile, you have, at every level of recursion, the option of mirror imaging. Hence, e.g., uncountably many slight variations of"France".
On 2015-12-26 12:01, Dan Asimov wrote:
Yes, but unless the mirroring is periodic with respect to the levels of recursion, can the end result be self-similar?
—Dan
No. Interesting point. Only countably many self-similar ones. But more than 900K !-) --rwg
On Dec 26, 2015, at 10:54 AM, Bill Gosper <billgosper@gmail.com> wrote:
When building a frac-tile, you have, at every level of recursion, the option of mirror imaging. Hence, e.g., uncountably many slight variations of "France".
Aha. By the way, just in case I haven't mentioned this: For some time I wondered about the possibility of an analogue of the map-of-France in higher dimensions. (Not the space-filling curve per se, but the Franceoid shape that it fills.) For example, can you start by taking a truncated octahedron and surrounding it with 14 copies of itself, then iterate, so that in the limit we have a bounded shape which when surrounded by 14 copies of itself results in the identical shape only larger? Apparently not, so an analogue of the map-of-France in 3D seems not to exist, at least not with this particular strategy. But with Joe Gerver (Rutgers Camden), a couple of years ago we were able to find such things in 4 and 8 dimensions (based on the quaternions and octonions, respectively). Since there are no real division algebras except in dimensions 1, 2, 4, 8, is it possible that such shapes don't exist in any other dimensions? —Dan
On Dec 26, 2015, at 3:34 PM, rwg <rwg@sdf.org> wrote:
On 2015-12-26 12:01, Dan Asimov wrote:
Yes, but unless the mirroring is periodic with respect to the levels of recursion, can the end result be self-similar? —Dan
No. Interesting point. Only countably many self-similar ones. But more than 900K !-) --rwg
On Dec 26, 2015, at 10:54 AM, Bill Gosper <billgosper@gmail.com> wrote: When building a frac-tile, you have, at every level of recursion, the option of mirror imaging. Hence, e.g., uncountably many slight variations of "France".
* Dan Asimov <asimov@msri.org> [Dec 27. 2015 07:48]:
Aha.
By the way, just in case I haven't mentioned this:
For some time I wondered about the possibility of an analogue of the map-of-France in higher dimensions. (Not the space-filling curve per se, but the Franceoid shape that it fills.)
For example, can you start by taking a truncated octahedron and surrounding it with 14 copies of itself, then iterate, so that in the limit we have a bounded shape which when surrounded by 14 copies of itself results in the identical shape only larger?
Apparently not, so an analogue of the map-of-France in 3D seems not to exist, at least not with this particular strategy.
But with Joe Gerver (Rutgers Camden), a couple of years ago we were able to find such things in 4 and 8 dimensions (based on the quaternions and octonions, respectively).
Since there are no real division algebras except in dimensions 1, 2, 4, 8, is it possible that such shapes don't exist in any other dimensions?
For dim=3, see, http://jjj.de/tmp-cubic/ (all found March 2015, and posted here). There are STL files so you can whirl these objects around (DO THIS). These are self-similar (not only self-affine, which is rather trivial to obtain)! It appears these critters exist only for third powers for the number of similar subsets that the object can be decomposed into. these are built from either cubes, truncated octahedra, or rhombic dodecahedra. No proper success with curves giving these as "tiles" as in my draft (except for the well known 3D-Peano which gives a cube). I can build self-affine objects that give some of the "islands" ("tiles" for me) that we are talking about. Best regards, jj
—Dan
On Dec 26, 2015, at 3:34 PM, rwg <rwg@sdf.org> wrote:
On 2015-12-26 12:01, Dan Asimov wrote:
Yes, but unless the mirroring is periodic with respect to the levels of recursion, can the end result be self-similar? —Dan
No. Interesting point. Only countably many self-similar ones. But more than 900K !-) --rwg
On Dec 26, 2015, at 10:54 AM, Bill Gosper <billgosper@gmail.com> wrote: When building a frac-tile, you have, at every level of recursion, the option of mirror imaging. Hence, e.g., uncountably many slight variations of "France".
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Very interesting, jjj! I didn't list all the conditions we had hoped that the higher dimensional versions would satisfy, but one of them is that the limiting shapes would have at least the same rotational symmetry as the starting tile. (Just as the map-of-France has 6-fold rotational symmetry based as it is on the hexagon — though it lacks the reflectional symmetry.) I won't have a chance to look at your files carefully until tomorrow. But at least when Joe and I tried to make this work with the truncated octahedron, it seemed to always lose some of the rotational symmetry of the truncated octahedron, i.e., the rotational symmetry of a cube. —Dan
On Dec 26, 2015, at 11:05 PM, Joerg Arndt <arndt@jjj.de> wrote:
* Dan Asimov <asimov@msri.org> [Dec 27. 2015 07:48]:
Aha.
By the way, just in case I haven't mentioned this:
For some time I wondered about the possibility of an analogue of the map-of-France in higher dimensions. (Not the space-filling curve per se, but the Franceoid shape that it fills.)
For example, can you start by taking a truncated octahedron and surrounding it with 14 copies of itself, then iterate, so that in the limit we have a bounded shape which when surrounded by 14 copies of itself results in the identical shape only larger?
Apparently not, so an analogue of the map-of-France in 3D seems not to exist, at least not with this particular strategy.
But with Joe Gerver (Rutgers Camden), a couple of years ago we were able to find such things in 4 and 8 dimensions (based on the quaternions and octonions, respectively).
Since there are no real division algebras except in dimensions 1, 2, 4, 8, is it possible that such shapes don't exist in any other dimensions?
For dim=3, see, http://jjj.de/tmp-cubic/ (all found March 2015, and posted here). There are STL files so you can whirl these objects around (DO THIS). These are self-similar (not only self-affine, which is rather trivial to obtain)!
It appears these critters exist only for third powers for the number of similar subsets that the object can be decomposed into.
these are built from either cubes, truncated octahedra, or rhombic dodecahedra.
No proper success with curves giving these as "tiles" as in my draft (except for the well known 3D-Peano which gives a cube).
I can build self-affine objects that give some of the "islands" ("tiles" for me) that we are talking about.
Best regards, jj
—Dan
On Dec 26, 2015, at 3:34 PM, rwg <rwg@sdf.org> wrote:
On 2015-12-26 12:01, Dan Asimov wrote:
Yes, but unless the mirroring is periodic with respect to the levels of recursion, can the end result be self-similar? —Dan
No. Interesting point. Only countably many self-similar ones. But more than 900K !-) --rwg
On Dec 26, 2015, at 10:54 AM, Bill Gosper <billgosper@gmail.com> wrote: When building a frac-tile, you have, at every level of recursion, the option of mirror imaging. Hence, e.g., uncountably many slight variations of "France".
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* Bill Gosper <billgosper@gmail.com> [Dec 27. 2015 07:48]:
[...]
Well 900K curves is pretty impressive,
However, these give only about 20000 shapes of curves, as more and more curves for high orders are of the same shape. I'll stop at two Giga-shapes(*). For that a completely new search method will be employed, hopefully started soon-ish by a brilliant student I talked to a week ago. The method should be faster by at least a factor 1000, and progressively more for high orders. (*) one shape for every second of a human life.
but how about a continuum of them?
Sorry, only discretuums on offer. A few from the constructions and a lot by the product mechanism. Could you look at my _division_ method in section 5.2? It appears that some people use something like it, but I have never seen it spelled out in any form. Many of the more curious curves by various authors are just special cases of this. Examples: http://www.tweedledum.com/rwg/semizerp.htm http://www.tweedledum.com/rwg/semizerp1.htm and (when done on tiles) http://www.tweedledum.com/rwg/frac5.htm Section 5.1 of course gives just the "folding product" as in Davis/Knuth and Dekking. Best regards, jj
[...]
participants (8)
-
Bill Gosper -
Dan Asimov -
Dan Asimov -
James Propp -
Joerg Arndt -
Marc LeBrun -
Neil Sloane -
rwg