[math-fun] Voronoi tilings from grids with periodic deleted points
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle. This is the picture you get (use VoronoiMesh[] in Mathematica) https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/ Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm) https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/ Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap. (1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way? -- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
I would be very surprised if all the wallpaper groups could be realized. Surely the symmetry group of the Voronoi tiling is the same as that of the underlying collection of points -- work with that instead. I'll bet almost anything that, say, *236 can't be realized. On Sat, Jan 9, 2016 at 12:03 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle.
This is the picture you get (use VoronoiMesh[] in Mathematica)
https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/
Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm)
https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/
Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap.
(1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way?
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Allan> I'll bet almost anything that, say, *236 can't be realized. yes that sounds right....but maybe if a person starts with a hexagonal array of points, instead? On Sat, Jan 9, 2016 at 10:18 AM, Allan Wechsler <acwacw@gmail.com> wrote:
I would be very surprised if all the wallpaper groups could be realized. Surely the symmetry group of the Voronoi tiling is the same as that of the underlying collection of points -- work with that instead. I'll bet almost anything that, say, *236 can't be realized.
On Sat, Jan 9, 2016 at 12:03 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle.
This is the picture you get (use VoronoiMesh[] in Mathematica)
https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/
Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm)
https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/
Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap.
(1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way?
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
Note the following slightly unintuitive fact: "There exist full-rank sublattices L of the cubic lattice Z^d such that the point-fixing automorphism group of L is greater than that of Z^d." For instance, in Z^4 we have that: L := {(a, b, c, d) : a + b + c + d is even} has an origin-fixing automorphism group of order 1152 (compared with 384 for the lattice Z^4). Similarly, in Z^8 we can take a scaled E_8 lattice: L := {(a_0, a_1, ..., a_7) : all coordinates have the same parity and a_0 + a_1 + ... + a_7 is divisible by 4} which has an automorphism group of 696729600, compared with 10321920 for the cubic lattice Z^8. The reason this phenomenon is slightly unintuitive is that it doesn't occur in dimensions d <= 3 so defies our geometric intuition. Best wishes, Adam P. Goucher
Sent: Saturday, January 09, 2016 at 6:18 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Voronoi tilings from grids with periodic deleted points
I would be very surprised if all the wallpaper groups could be realized. Surely the symmetry group of the Voronoi tiling is the same as that of the underlying collection of points -- work with that instead. I'll bet almost anything that, say, *236 can't be realized.
On Sat, Jan 9, 2016 at 12:03 PM, Thane Plambeck <tplambeck@gmail.com> wrote:
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle.
This is the picture you get (use VoronoiMesh[] in Mathematica)
https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/
Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm)
https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/
Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap.
(1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way?
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Another thing that only happens in d > 3 is for the automorphism group to have subgroups whose invariant subspaces are completely irrational. Penrose tilings are non-intuitive mostly for this reason. -Veit
On Jan 14, 2016, at 8:01 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Note the following slightly unintuitive fact:
"There exist full-rank sublattices L of the cubic lattice Z^d such that the point-fixing automorphism group of L is greater than that of Z^d."
For instance, in Z^4 we have that:
L := {(a, b, c, d) : a + b + c + d is even}
has an origin-fixing automorphism group of order 1152 (compared with 384 for the lattice Z^4).
Similarly, in Z^8 we can take a scaled E_8 lattice:
L := {(a_0, a_1, ..., a_7) : all coordinates have the same parity and a_0 + a_1 + ... + a_7 is divisible by 4}
which has an automorphism group of 696729600, compared with 10321920 for the cubic lattice Z^8.
The reason this phenomenon is slightly unintuitive is that it doesn't occur in dimensions d <= 3 so defies our geometric intuition.
Best wishes,
Adam P. Goucher
Adam or Veit or anyone, do you know if somewhere there is a list of (some or all) maximally symmetric lattices* in R^n for various dimensions n ? For n=2 I guess these are the square and triangular lattices, and for n=3 they're the cubic, fcc, and bcc lattices. But I don't know the list in any higher dimensions, just a few of the lattices. —Dan ______________________________ * in his book "Three-Dimensional Geometry and Topology, Vol. 1", Bill Thurston considers subgroups F of GL(n,Z), and all embeddings rho: Z^n —> R^n such that F acts by isometries on the image rho(Z^n). He writes: "If some lattice with symmetry F has no other symmetries, we say that F is an *exact lattice group*. An example of an F that is not exact is the trivial group — every lattice has this symmetry, but also the symmetry -Id. If every lattice with symmetry F has no other symmetries, F is a maximal finite subgroup of GL(n,Z) and the corresponding lattices are *maximally symmetric*."
On Jan 14, 2016, at 5:01 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Note the following slightly unintuitive fact:
"There exist full-rank sublattices L of the cubic lattice Z^d such that the point-fixing automorphism group of L is greater than that of Z^d."
For instance, in Z^4 we have that:
L := {(a, b, c, d) : a + b + c + d is even}
has an origin-fixing automorphism group of order 1152 (compared with 384 for the lattice Z^4).
Similarly, in Z^8 we can take a scaled E_8 lattice:
L := {(a_0, a_1, ..., a_7) : all coordinates have the same parity and a_0 + a_1 + ... + a_7 is divisible by 4}
which has an automorphism group of 696729600, compared with 10321920 for the cubic lattice Z^8.
The reason this phenomenon is slightly unintuitive is that it doesn't occur in dimensions d <= 3 so defies our geometric intuition.
To answer Dan's question, see here: http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/ Gabriele Nebe (following work by W. Plesken and many others) has extensively studied this kind of question 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 Thu, Jan 14, 2016 at 10:45 AM, Dan Asimov <asimov@msri.org> wrote:
Adam or Veit or anyone, do you know if somewhere there is a list of (some or all) maximally symmetric lattices* in R^n for various dimensions n ?
For n=2 I guess these are the square and triangular lattices, and for n=3 they're the cubic, fcc, and bcc lattices.
But I don't know the list in any higher dimensions, just a few of the lattices.
—Dan ______________________________ * in his book "Three-Dimensional Geometry and Topology, Vol. 1", Bill Thurston considers subgroups F of GL(n,Z), and all embeddings rho: Z^n —> R^n such that F acts by isometries on the image rho(Z^n).
He writes: "If some lattice with symmetry F has no other symmetries, we say that F is an *exact lattice group*. An example of an F that is not exact is the trivial group — every lattice has this symmetry, but also the symmetry -Id. If every lattice with symmetry F has no other symmetries, F is a maximal finite subgroup of GL(n,Z) and the corresponding lattices are *maximally symmetric*."
On Jan 14, 2016, at 5:01 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Note the following slightly unintuitive fact:
"There exist full-rank sublattices L of the cubic lattice Z^d such that the point-fixing automorphism group of L is greater than that of Z^d."
For instance, in Z^4 we have that:
L := {(a, b, c, d) : a + b + c + d is even}
has an origin-fixing automorphism group of order 1152 (compared with 384 for the lattice Z^4).
Similarly, in Z^8 we can take a scaled E_8 lattice:
L := {(a_0, a_1, ..., a_7) : all coordinates have the same parity and a_0 + a_1 + ... + a_7 is divisible by 4}
which has an automorphism group of 696729600, compared with 10321920 for the cubic lattice Z^8.
The reason this phenomenon is slightly unintuitive is that it doesn't occur in dimensions d <= 3 so defies our geometric intuition.
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Thanks, Neil! That is quite a database, which I was not aware of. But, do you know if it uses a particular term for "maximally symmetric" as defined below? —Dan
On Jan 14, 2016, at 9:32 AM, Neil Sloane <njasloane@gmail.com> wrote:
To answer Dan's question, see here: http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/
Gabriele Nebe (following work by W. Plesken and many others) has extensively studied this kind of question
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 Thu, Jan 14, 2016 at 10:45 AM, Dan Asimov <asimov@msri.org> wrote:
Adam or Veit or anyone, do you know if somewhere there is a list of (some or all) maximally symmetric lattices* in R^n for various dimensions n ?
For n=2 I guess these are the square and triangular lattices, and for n=3 they're the cubic, fcc, and bcc lattices.
But I don't know the list in any higher dimensions, just a few of the lattices.
—Dan ______________________________ * in his book "Three-Dimensional Geometry and Topology, Vol. 1", Bill Thurston considers subgroups F of GL(n,Z), and all embeddings rho: Z^n —> R^n such that F acts by isometries on the image rho(Z^n).
He writes: "If some lattice with symmetry F has no other symmetries, we say that F is an *exact lattice group*. An example of an F that is not exact is the trivial group — every lattice has this symmetry, but also the symmetry -Id. If every lattice with symmetry F has no other symmetries, F is a maximal finite subgroup of GL(n,Z) and the corresponding lattices are *maximally symmetric*."
On Jan 14, 2016, at 5:01 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Note the following slightly unintuitive fact:
"There exist full-rank sublattices L of the cubic lattice Z^d such that the point-fixing automorphism group of L is greater than that of Z^d."
For instance, in Z^4 we have that:
L := {(a, b, c, d) : a + b + c + d is even}
has an origin-fixing automorphism group of order 1152 (compared with 384 for the lattice Z^4).
Similarly, in Z^8 we can take a scaled E_8 lattice:
L := {(a_0, a_1, ..., a_7) : all coordinates have the same parity and a_0 + a_1 + ... + a_7 is divisible by 4}
which has an automorphism group of 696729600, compared with 10321920 for the cubic lattice Z^8.
The reason this phenomenon is slightly unintuitive is that it doesn't occur in dimensions d <= 3 so defies our geometric intuition.
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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I should have mentioned that in that terrific database, Nebe lists "Lattices from the maximal finite subgroups of GL(n,Q)", which I suspect are the same ones I'm seeking. But, these are listed under that phrase only for various n >= 12. Ideally I'd like to know about dimensions 4 through 11 as well. —Dan
On Jan 14, 2016, at 9:35 AM, Dan Asimov <asimov@msri.org> wrote:
Thanks, Neil! That is quite a database, which I was not aware of.
But, do you know if it uses a particular term for "maximally symmetric" as defined below?
—Dan
On Jan 14, 2016, at 9:32 AM, Neil Sloane <njasloane@gmail.com> wrote:
To answer Dan's question, see here: http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/
Gabriele Nebe (following work by W. Plesken and many others) has extensively studied this kind of question.
On Thu, Jan 14, 2016 at 10:45 AM, Dan Asimov <asimov@msri.org> wrote:
Adam or Veit or anyone, do you know if somewhere there is a list of (some or all) maximally symmetric lattices* in R^n for various dimensions n ?
For n=2 I guess these are the square and triangular lattices, and for n=3 they're the cubic, fcc, and bcc lattices.
But I don't know the list in any higher dimensions, just a few of the lattices.
—Dan ______________________________ * in his book "Three-Dimensional Geometry and Topology, Vol. 1", Bill Thurston considers subgroups F of GL(n,Z), and all embeddings rho: Z^n —> R^n such that F acts by isometries on the image rho(Z^n).
He writes: "If some lattice with symmetry F has no other symmetries, we say that F is an *exact lattice group*. An example of an F that is not exact is the trivial group — every lattice has this symmetry, but also the symmetry -Id. If every lattice with symmetry F has no other symmetries, F is a maximal finite subgroup of GL(n,Z) and the corresponding lattices are *maximally symmetric*."
In our database we give lists in lower dimensions that include all those irreducible lattices whose groups are locally maximal. You can go through them and take your pick. There are also various sequences in the OEIS that give the max order 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 Thu, Jan 14, 2016 at 12:42 PM, Dan Asimov <asimov@msri.org> wrote:
I should have mentioned that in that terrific database, Nebe lists "Lattices from the maximal finite subgroups of GL(n,Q)", which I suspect are the same ones I'm seeking.
But, these are listed under that phrase only for various n >= 12.
Ideally I'd like to know about dimensions 4 through 11 as well.
—Dan
On Jan 14, 2016, at 9:35 AM, Dan Asimov <asimov@msri.org> wrote:
Thanks, Neil! That is quite a database, which I was not aware of.
But, do you know if it uses a particular term for "maximally symmetric" as defined below?
—Dan
On Jan 14, 2016, at 9:32 AM, Neil Sloane <njasloane@gmail.com> wrote:
To answer Dan's question, see here: http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/
Gabriele Nebe (following work by W. Plesken and many others) has extensively studied this kind of question.
On Thu, Jan 14, 2016 at 10:45 AM, Dan Asimov <asimov@msri.org> wrote:
Adam or Veit or anyone, do you know if somewhere there is a list of (some or all) maximally symmetric lattices* in R^n for various dimensions n ?
For n=2 I guess these are the square and triangular lattices, and for n=3 they're the cubic, fcc, and bcc lattices.
But I don't know the list in any higher dimensions, just a few of the lattices.
—Dan ______________________________ * in his book "Three-Dimensional Geometry and Topology, Vol. 1", Bill Thurston considers subgroups F of GL(n,Z), and all embeddings rho: Z^n —> R^n such that F acts by isometries on the image rho(Z^n).
He writes: "If some lattice with symmetry F has no other symmetries, we say that F is an *exact lattice group*. An example of an F that is not exact is the trivial group — every lattice has this symmetry, but also the symmetry -Id. If every lattice with symmetry F has no other symmetries, F is a maximal finite subgroup of GL(n,Z) and the corresponding lattices are *maximally symmetric*."
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Very interesting tilings, but — I'm not sure I understand: If the points to be removed depend only on their row numbers and nothing else, then the Voronoi regions should show a constancy in the vertical direction, should they not? —Dan
On Jan 9, 2016, at 9:03 AM, Thane Plambeck <tplambeck@gmail.com> wrote:
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle.
This is the picture you get (use VoronoiMesh[] in Mathematica)
https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/
Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm)
https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/
Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap.
(1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way?
Never mind! I get it now. My brain was in neutral, sorry. —Dan
On Jan 9, 2016, at 10:55 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Very interesting tilings, but — I'm not sure I understand: If the points to be removed depend only on their row numbers and nothing else, then the Voronoi regions should show a constancy in the vertical direction, should they not?
—Dan
On Jan 9, 2016, at 9:03 AM, Thane Plambeck <tplambeck@gmail.com> wrote:
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle.
This is the picture you get (use VoronoiMesh[] in Mathematica)
https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/
Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm)
https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/
Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap.
(1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way?
Suggest to show the points. This will make the images less mysterious but more helpful for visual analysis. Best regards, jj * Thane Plambeck <tplambeck@gmail.com> [Jan 09. 2016 18:48]:
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle.
This is the picture you get (use VoronoiMesh[] in Mathematica)
https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/
Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm)
https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/
Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap.
(1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way?
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Joerg, here's a patch of that 23x23 Voronoi tiling with the points. If you look down a particular column of dots, they repeat mod 5 in the pattern dot, dot, dot, nodot, nodot with a phase shift each column since the grid dimension is 23x23 rather than a multiple of 5 https://www.flickr.com/photos/thane/23932185269/in/dateposted-public/ On Sat, Jan 9, 2016 at 11:39 PM, Joerg Arndt <arndt@jjj.de> wrote:
Suggest to show the points. This will make the images less mysterious but more helpful for visual analysis.
Best regards, jj
* Thane Plambeck <tplambeck@gmail.com> [Jan 09. 2016 18:48]:
Make a 23x23 square array of points, numbered from 1 to 529 row-wise. Remove all the points whose number is congruent to 0, 1, or 2 modulo 5. Then compute the Voronoi cells of that arrangement of points inside a bounding rectangle.
This is the picture you get (use VoronoiMesh[] in Mathematica)
https://www.flickr.com/photos/thane/24191215291/in/dateposted-public/
Here's another one with different values of 23, 0, 1, 2, 5, so to speak (unfortunately I forgot to write them down in my enthusiasm)
https://www.flickr.com/photos/thane/24191270151/in/dateposted-public/
Anyway, this seems to be a simple way to make lots of interesting looking tilings on the cheap.
(1) I can't possibly be the first person to do this...I'd welcome references (2) Can all the wallpaper groups be realized this way?
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
Thanks. My impression is that imposing conditions mod something on x^2 + y^2 would give results with structures that are easier seen (similarly, conditions on x^2 + x*y + y^2 for the Eisenstein integers). I especially suspect that some of the k-uniform grids are obtained in that manner. Best regards, jj * Thane Plambeck <tplambeck@gmail.com> [Jan 11. 2016 11:31]:
Joerg, here's a patch of that 23x23 Voronoi tiling with the points. If you look down a particular column of dots, they repeat mod 5 in the pattern
dot, dot, dot, nodot, nodot
with a phase shift each column since the grid dimension is 23x23 rather than a multiple of 5
https://www.flickr.com/photos/thane/23932185269/in/dateposted-public/
On Sat, Jan 9, 2016 at 11:39 PM, Joerg Arndt <arndt@jjj.de> wrote:
Suggest to show the points. This will make the images less mysterious but more helpful for visual analysis.
Best regards, jj
[...]
Joerg's last e-mail reminded me of my first computer graphics experiment (1981), which I since learned that many other people have also tried: ----- Make a color table of N colors (labeled 0 through N-1) representing a discrete circle of hues. (One easy way to do this is to use the circle inscribed in the hexagon cross-section of the RGB cube [0,1]^3 that is halfway between black = (0,0,0) and white = (1,1,1). At that time the maximum I could use was N = 256 colors (equally spaced around the circle of hues). Then for each pixel on the screen, with coordinates given by the integer point (x,y), calculate C(x,y) := x^2 + y^2 (mod N) and then color pixel (x,y) with the color having number C(x,y) in the color table. ----- The effect is an extremely complicated Moire-type pattern. I later used 1024 colors and the effect was even more interesting. I have not seen any good explanation of the very complicated image that this produces. I also tried many other polynomials P(x,y) in lieu of x^2 + y^2, and got some weird results, but nothing was as interesting and I daresay beautiful as x^2 + y^2. —Dan
On Jan 11, 2016, at 2:49 AM, Joerg Arndt <arndt@jjj.de> wrote:
Thanks. My impression is that imposing conditions mod something on x^2 + y^2 would . . . . . . . . .
Does anybody have a link to a relevant pic? WFL On 1/11/16, Dan Asimov <asimov@msri.org> wrote:
Joerg's last e-mail reminded me of my first computer graphics experiment (1981), which I since learned that many other people have also tried:
----- Make a color table of N colors (labeled 0 through N-1) representing a discrete circle of hues. (One easy way to do this is to use the circle inscribed in the hexagon cross-section of the RGB cube [0,1]^3 that is halfway between black = (0,0,0) and white = (1,1,1). At that time the maximum I could use was N = 256 colors (equally spaced around the circle of hues).
Then for each pixel on the screen, with coordinates given by the integer point (x,y), calculate
C(x,y) := x^2 + y^2 (mod N)
and then color pixel (x,y) with the color having number C(x,y) in the color table. -----
The effect is an extremely complicated Moire-type pattern. I later used 1024 colors and the effect was even more interesting.
I have not seen any good explanation of the very complicated image that this produces. I also tried many other polynomials P(x,y) in lieu of x^2 + y^2, and got some weird results, but nothing was as interesting and I daresay beautiful as x^2 + y^2.
—Dan
On Jan 11, 2016, at 2:49 AM, Joerg Arndt <arndt@jjj.de> wrote:
Thanks. My impression is that imposing conditions mod something on x^2 + y^2 would . . . . . . . . .
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Here you go: http://checkmyworking.com/misc/dan-asimov-pattern.png I coded it up earlier today - http://codepen.io/christianp/pen/PZjGmV On Mon, 11 Jan 2016 at 14:29 Fred Lunnon <fred.lunnon@gmail.com> wrote:
Does anybody have a link to a relevant pic?
WFL
On 1/11/16, Dan Asimov <asimov@msri.org> wrote:
Joerg's last e-mail reminded me of my first computer graphics experiment (1981), which I since learned that many other people have also tried:
----- Make a color table of N colors (labeled 0 through N-1) representing a discrete circle of hues. (One easy way to do this is to use the circle inscribed in the hexagon cross-section of the RGB cube [0,1]^3 that is halfway between black = (0,0,0) and white = (1,1,1). At that time the maximum I could use was N = 256 colors (equally spaced around the circle of hues).
Then for each pixel on the screen, with coordinates given by the integer point (x,y), calculate
C(x,y) := x^2 + y^2 (mod N)
and then color pixel (x,y) with the color having number C(x,y) in the color table. -----
The effect is an extremely complicated Moire-type pattern. I later used 1024 colors and the effect was even more interesting.
I have not seen any good explanation of the very complicated image that this produces. I also tried many other polynomials P(x,y) in lieu of x^2 + y^2, and got some weird results, but nothing was as interesting and I daresay beautiful as x^2 + y^2.
—Dan
On Jan 11, 2016, at 2:49 AM, Joerg Arndt <arndt@jjj.de> wrote:
Thanks. My impression is that imposing conditions mod something on x^2 + y^2 would . . . . . . . . .
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Nice! Christian, I don't know if you take requests, but: Would you be willing to do the same thing but with 1024 or more colors, on a larger square? That will show the fine features of the Moire pattern more clearly. (Also: The (x,y) coordinates should ideally refer to physical pixels of the computer, rather than virtual pixels; I'm not sure whether you already did it this way or not.) —Dan
On Jan 11, 2016, at 9:55 AM, Christian Lawson-Perfect <christianperfect@gmail.com> wrote:
Here you go: http://checkmyworking.com/misc/dan-asimov-pattern.png I coded it up earlier today - http://codepen.io/christianp/pen/PZjGmV
On Mon, 11 Jan 2016 at 14:29 Fred Lunnon <fred.lunnon@gmail.com> wrote:
Does anybody have a link to a relevant pic?
WFL
On 1/11/16, Dan Asimov <asimov@msri.org> wrote:
Joerg's last e-mail reminded me of my first computer graphics experiment (1981), which I since learned that many other people have also tried:
----- Make a color table of N colors (labeled 0 through N-1) representing a discrete circle of hues. (One easy way to do this is to use the circle inscribed in the hexagon cross-section of the RGB cube [0,1]^3 that is halfway between black = (0,0,0) and white = (1,1,1). At that time the maximum I could use was N = 256 colors (equally spaced around the circle of hues).
Then for each pixel on the screen, with coordinates given by the integer point (x,y), calculate
C(x,y) := x^2 + y^2 (mod N)
and then color pixel (x,y) with the color having number C(x,y) in the color table. -----
The effect is an extremely complicated Moire-type pattern. I later used 1024 colors and the effect was even more interesting.
I have not seen any good explanation of the very complicated image that this produces. I also tried many other polynomials P(x,y) in lieu of x^2 + y^2, and got some weird results, but nothing was as interesting and I daresay beautiful as x^2 + y^2.
—Dan
On Jan 11, 2016, at 2:49 AM, Joerg Arndt <arndt@jjj.de> wrote:
Thanks. My impression is that imposing conditions mod something on x^2 + y^2 would . . . . . . . . .
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Here's one on a 2048 x 2048 grid, mod 1024: http://www.kerrymitchellart.com/temp/moire.png Kerry On Mon, Jan 11, 2016 at 4:02 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
That will show the fine features of the Moire pattern more clearly.
(Also: The (x,y) coordinates should ideally refer to physical pixels of the computer, rather than virtual pixels; I'm not sure whether you already did it this way or not.)
—Dan
On Jan 11, 2016, at 9:55 AM, Christian Lawson-Perfect < christianperfect@gmail.com> wrote:
Here you go: http://checkmyworking.com/misc/dan-asimov-pattern.png I coded it up earlier today - http://codepen.io/christianp/pen/PZjGmV
On Mon, 11 Jan 2016 at 14:29 Fred Lunnon <fred.lunnon@gmail.com> wrote:
Does anybody have a link to a relevant pic?
WFL
On 1/11/16, Dan Asimov <asimov@msri.org> wrote:
Joerg's last e-mail reminded me of my first computer graphics experiment (1981), which I since learned that many other people have also tried:
----- Make a color table of N colors (labeled 0 through N-1) representing a discrete circle of hues. (One easy way to do this is to use the circle inscribed in the hexagon cross-section of the RGB cube [0,1]^3 that is halfway between black = (0,0,0) and white = (1,1,1). At that time the maximum I could use was N = 256 colors (equally spaced around the circle of hues).
Then for each pixel on the screen, with coordinates given by the integer point (x,y), calculate
C(x,y) := x^2 + y^2 (mod N)
and then color pixel (x,y) with the color having number C(x,y) in the color table. -----
The effect is an extremely complicated Moire-type pattern. I later used 1024 colors and the effect was even more interesting.
I have not seen any good explanation of the very complicated image that this produces. I also tried many other polynomials P(x,y) in lieu of x^2 + y^2, and got some weird results, but nothing was as interesting and I daresay beautiful as x^2 + y^2.
—Dan
On Jan 11, 2016, at 2:49 AM, Joerg Arndt <arndt@jjj.de> wrote:
Thanks. My impression is that imposing conditions mod something on x^2 + y^2 would . . . . . . . . .
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On 2016-01-11 15:57, Kerry Mitchell wrote:
Here's one on a 2048 x 2048 grid, mod 1024:
http://www.kerrymitchellart.com/temp/moire.png
Kerry
Inverse motion blur!?! Fullscreen. Then the contrast increases dramatically, but only while scrolling. WtF? --rwg (MacBook Pro)
On Mon, Jan 11, 2016 at 4:02 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
That will show the fine features of the Moire pattern more clearly.
(Also: The (x,y) coordinates should ideally refer to physical pixels of the computer, rather than virtual pixels; I'm not sure whether you already did it this way or not.)
—Dan
On Jan 11, 2016, at 9:55 AM, Christian Lawson-Perfect < christianperfect@gmail.com> wrote:
Here you go: http://checkmyworking.com/misc/dan-asimov-pattern.png I coded it up earlier today - http://codepen.io/christianp/pen/PZjGmV
On Mon, 11 Jan 2016 at 14:29 Fred Lunnon <fred.lunnon@gmail.com> wrote:
Does anybody have a link to a relevant pic?
WFL
On 1/11/16, Dan Asimov <asimov@msri.org> wrote:
Joerg's last e-mail reminded me of my first computer graphics experiment (1981), which I since learned that many other people have also tried:
----- Make a color table of N colors (labeled 0 through N-1) representing a discrete circle of hues. (One easy way to do this is to use the circle inscribed in the hexagon cross-section of the RGB cube [0,1]^3 that is halfway between black = (0,0,0) and white = (1,1,1). At that time the maximum I could use was N = 256 colors (equally spaced around the circle of hues).
Then for each pixel on the screen, with coordinates given by the integer point (x,y), calculate
C(x,y) := x^2 + y^2 (mod N)
and then color pixel (x,y) with the color having number C(x,y) in the color table. -----
The effect is an extremely complicated Moire-type pattern. I later used 1024 colors and the effect was even more interesting.
I have not seen any good explanation of the very complicated image that this produces. I also tried many other polynomials P(x,y) in lieu of x^2 + y^2, and got some weird results, but nothing was as interesting and I daresay beautiful as x^2 + y^2.
—Dan
On Jan 11, 2016, at 2:49 AM, Joerg Arndt <arndt@jjj.de> wrote:
Thanks. My impression is that imposing conditions mod something on x^2 + y^2 would . . . . . . . . .
On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
Try this: http://codepen.io/anon/pen/ZQyaYm which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel. (For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.) The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste. If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed. With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of. Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure. -- g
Oh, yeah! That's exactly the picture I had in mind. Thanks so much for making it! Now maybe some number theorist can explain the patterns. —Dan P.S. Though for a hexagonal pixel structure with pixels labeled via x * 1 + y * exp(2pi*i/3) I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
Try this: http://codepen.io/anon/pen/ZQyaYm
which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel.
(For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.)
The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste.
If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed.
With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of.
Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure.
-- g
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On 2016-01-11 17:03, Dan Asimov wrote:
Oh, yeah! That's exactly the picture I had in mind.
You did? Including the part where you can only see two levels of bullseyes unless you're scrolling? WtF? At least on my MacBook Pro. --rwg
Thanks so much for making it!
Now maybe some number theorist can explain the patterns.
—Dan
P.S. Though for a hexagonal pixel structure with pixels labeled via
x * 1 + y * exp(2pi*i/3)
I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
Try this: http://codepen.io/anon/pen/ZQyaYm
which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel.
(For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.)
The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste.
If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed.
With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of.
Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure.
-- g
It's kinda standard that you'll see spatiotemporal moire patterns when you move fine lines on a pixelated screen. Ever see someone on TV wearing a pinstriped tie? —Dan
On Jan 12, 2016, at 11:41 AM, rwg <rwg@sdf.org> wrote:
On 2016-01-11 17:03, Dan Asimov wrote:
Oh, yeah! That's exactly the picture I had in mind.
You did? Including the part where you can only see two levels of bullseyes unless you're scrolling? WtF? At least on my MacBook Pro. --rwg
Thanks so much for making it! Now maybe some number theorist can explain the patterns. —Dan P.S. Though for a hexagonal pixel structure with pixels labeled via x * 1 + y * exp(2pi*i/3) I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote: On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but: Would you be willing to do the same thing but with 1024 or more colors, on a larger square? Try this: http://codepen.io/anon/pen/ZQyaYm which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel. (For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.) The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste. If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed. With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of. Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure. -- g
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I don't think this is related to the Moire patterns you see due to sampling on a grid. I suspect what rwg may be seeing is more along the lines of scrolling without any antialiasing/Retina correction but when you stop scrolling the browser applies "prettiness" filters to make things look "better". Alternatively, it might just be nonlinearities when turning the screen pixels on or off. I definitely see what he's referring to when I scroll vs look at the static image at 100% zoom. On Tue, Jan 12, 2016 at 4:24 PM, Dan Asimov <dasimov@earthlink.net> wrote:
It's kinda standard that you'll see spatiotemporal moire patterns when you move fine lines on a pixelated screen. Ever see someone on TV wearing a pinstriped tie?
—Dan
On Jan 12, 2016, at 11:41 AM, rwg <rwg@sdf.org> wrote:
On 2016-01-11 17:03, Dan Asimov wrote:
Oh, yeah! That's exactly the picture I had in mind.
You did? Including the part where you can only see two levels of bullseyes unless you're scrolling? WtF? At least on my MacBook Pro. --rwg
Thanks so much for making it! Now maybe some number theorist can explain the patterns. —Dan P.S. Though for a hexagonal pixel structure with pixels labeled via x * 1 + y * exp(2pi*i/3) I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan < gareth.mccaughan@pobox.com> wrote: On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but: Would you be willing to do the same thing but with 1024 or more colors, on a larger square? Try this: http://codepen.io/anon/pen/ZQyaYm which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel. (For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.) The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste. If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed. With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of. Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure. -- g
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On 2016-01-12 16:24, Dan Asimov wrote:
It's kinda standard that you'll see spatiotemporal moire patterns when you move fine lines on a pixelated screen. Ever see someone on TV wearing a pinstriped tie?
—Dan
That's a shimmering artifact. I don't think this is a Moire effect. It's more like being able to see what's in your microwave when it's revolving. --rwg
On Jan 12, 2016, at 11:41 AM, rwg <rwg@sdf.org> wrote:
On 2016-01-11 17:03, Dan Asimov wrote:
Oh, yeah! That's exactly the picture I had in mind.
You did? Including the part where you can only see two levels of bullseyes unless you're scrolling? WtF? At least on my MacBook Pro. --rwg
Thanks so much for making it! Now maybe some number theorist can explain the patterns. —Dan P.S. Though for a hexagonal pixel structure with pixels labeled via x * 1 + y * exp(2pi*i/3) I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote: On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but: Would you be willing to do the same thing but with 1024 or more colors, on a larger square? Try this: http://codepen.io/anon/pen/ZQyaYm which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel. (For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.) The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste. If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed. With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of. Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure. -- g
The number theory has been treated at length for the one-dimensional case and where the modulus is the product of two primes, since finding x^2 = y^2 mod N often lets you factor N. On Mon, Jan 11, 2016 at 5:03 PM, Dan Asimov <asimov@msri.org> wrote:
Oh, yeah! That's exactly the picture I had in mind. Thanks so much for making it!
Now maybe some number theorist can explain the patterns.
—Dan
P.S. Though for a hexagonal pixel structure with pixels labeled via
x * 1 + y * exp(2pi*i/3)
I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
Try this: http://codepen.io/anon/pen/ZQyaYm
which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel.
(For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.)
The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste.
If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed.
With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of.
Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure.
-- g
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Why do Kerry and Gareth's pictures look so different? I gather that the latter is closer to what Dan had in mind, but the former seems to me to have a richer Moire structure. Jim Propp On Monday, January 11, 2016, Dan Asimov <asimov@msri.org <javascript:_e(%7B%7D,'cvml','asimov@msri.org');>> wrote:
Oh, yeah! That's exactly the picture I had in mind. Thanks so much for making it!
Now maybe some number theorist can explain the patterns.
—Dan
P.S. Though for a hexagonal pixel structure with pixels labeled via
x * 1 + y * exp(2pi*i/3)
I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan < gareth.mccaughan@pobox.com> wrote:
On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
Try this: http://codepen.io/anon/pen/ZQyaYm
which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel.
(For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.)
The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste.
If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed.
With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of.
Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure.
-- g
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I've updated my code so it renders in realtime, like Gareth's, but I've found that replacing half of the colour spectrum with black makes it much easier to see the pattern. You can see that the centres of circles lie on the lines y = x*a and y = x/a, for integers a. There are also symmetries obtained by the fact that x^2=(-x)^2, y^2=(-y)^2, and (x+N)^2 is congruent to x^2 mod N, where N is the number of colours. Here's the link again: http://codepen.io/christianp/full/PZjGmV/ On Wed, 13 Jan 2016 at 05:18 James Propp <jamespropp@gmail.com> wrote:
Why do Kerry and Gareth's pictures look so different?
I gather that the latter is closer to what Dan had in mind, but the former seems to me to have a richer Moire structure.
Jim Propp
On Monday, January 11, 2016, Dan Asimov <asimov@msri.org <javascript:_e(%7B%7D,'cvml','asimov@msri.org');>> wrote:
Oh, yeah! That's exactly the picture I had in mind. Thanks so much for making it!
Now maybe some number theorist can explain the patterns.
—Dan
P.S. Though for a hexagonal pixel structure with pixels labeled via
x * 1 + y * exp(2pi*i/3)
I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan < gareth.mccaughan@pobox.com> wrote:
On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
Try this: http://codepen.io/anon/pen/ZQyaYm
which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel.
(For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.)
The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste.
If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed.
With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of.
Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure.
-- g
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On 13/01/2016 08:44, Christian Lawson-Perfect wrote:
I've updated my code so it renders in realtime, like Gareth's, but I've found that replacing half of the colour spectrum with black makes it much easier to see the pattern.
That's the effect you get in my code by reducing the value of "colours" while leaving "modulus" unaltered. I agree that it makes it easier to see what's going on. -- g
Extemely cool!!! Especially with all those parameters to play with, even the polynomial!!! But I am wondering if the white diagonal of slope = -1 that I am seeing (so far unchanged with all parameter combinations that I've tried) is intentional. Or alternatively: How can I get rid of it? (iMac 27"). —Dan
On Jan 13, 2016, at 12:44 AM, Christian Lawson-Perfect <christianperfect@gmail.com> wrote:
I've updated my code so it renders in realtime, like Gareth's, but I've found that replacing half of the colour spectrum with black makes it much easier to see the pattern. You can see that the centres of circles lie on the lines y = x*a and y = x/a, for integers a. There are also symmetries obtained by the fact that x^2=(-x)^2, y^2=(-y)^2, and (x+N)^2 is congruent to x^2 mod N, where N is the number of colours. Here's the link again: http://codepen.io/christianp/full/PZjGmV/
On Wed, 13 Jan 2016 at 05:18 James Propp <jamespropp@gmail.com> wrote:
Why do Kerry and Gareth's pictures look so different?
I gather that the latter is closer to what Dan had in mind, but the former seems to me to have a richer Moire structure.
Jim Propp
On Monday, January 11, 2016, Dan Asimov <asimov@msri.org <javascript:_e(%7B%7D,'cvml','asimov@msri.org');>> wrote:
Oh, yeah! That's exactly the picture I had in mind. Thanks so much for making it!
Now maybe some number theorist can explain the patterns.
—Dan
P.S. Though for a hexagonal pixel structure with pixels labeled via
x * 1 + y * exp(2pi*i/3)
I would want to try x^2 + xy + y^2 (with all coefficients = 1, no 2's.)
On Jan 11, 2016, at 4:30 PM, Gareth McCaughan < gareth.mccaughan@pobox.com> wrote:
On 11/01/2016 23:02, Dan Asimov wrote:
Nice! Christian, I don't know if you take requests, but:
Would you be willing to do the same thing but with 1024 or more colors, on a larger square?
Try this: http://codepen.io/anon/pen/ZQyaYm
which uses more colours, on a larger square, at one pixel per pixel, and does it using an HTML5 canvas object rather than by constructing an enormous table with one cell per pixel.
(For reasons I don't at all understand, this has stopped actually working for me on the computer where I created it -- it sits there saying "Loading" for ever and never actually draws anything -- but on another one it works just fine. My apologies if whatever I've screwed up causes it not to work for some of you too.)
The "skeleton" of the code is copy-and-pasted from some tutorial thing that obviously envisages turning this into an animation. I've experimented with making it animate but what it's doing is a bit too expensive for my taste.
If on line 10 you set colours to a smaller value (try, say, 100) more of the finer structure of the Moire fringes will be exposed.
With the default values (1024 for both), on some monitors you may see flickering in some parts of the image. I think this is not an optical illusion but the result of temporal dithering by a monitor that uses that technique to represent a larger number of brightness levels for each channel than the display is actually physically capable of.
Try replacing x*x+y*y with 2*(x*x+y*y)+x*y for a hexagonal structure.
-- g
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Never mind — I found that by slightly perturbing the "line gradient" parameter, the diagonal line disappeared. —Dan
On Jan 13, 2016, at 3:21 PM, Dan Asimov <asimov@msri.org> wrote:
But I am wondering if the white diagonal of slope = -1 that I am seeing (so far unchanged with all parameter combinations that I've tried) is intentional. Or alternatively: How can I get rid of it?
Possible headline on January 14: 529 STUDENTS HOSPITALIZED AFTER FALLING INTO TRANCE FROM STARING AT COMPUTER GRAPHICS APP In other news, homeless woman becomes billionaire after winning Powerball jackpot —Dan
On Jan 13, 2016, at 3:21 PM, Dan Asimov <asimov@msri.org> wrote:
On Jan 13, 2016, at 12:44 AM, Christian Lawson-Perfect <christianperfect@gmail.com> wrote:
I've updated my code so it renders in realtime, like Gareth's, but I've found that replacing half of the colour spectrum with black makes it much easier to see the pattern. You can see that the centres of circles lie on the lines y = x*a and y = x/a, for integers a. There are also symmetries obtained by the fact that x^2=(-x)^2, y^2=(-y)^2, and (x+N)^2 is congruent to x^2 mod N, where N is the number of colours. Here's the link again: http://codepen.io/christianp/full/PZjGmV/
participants (16)
-
Adam P. Goucher -
Allan Wechsler -
Christian Lawson-Perfect -
Dan Asimov -
Dan Asimov -
Fred Lunnon -
Gareth McCaughan -
James Propp -
Joerg Arndt -
Kerry Mitchell -
Mike Stay -
Neil Sloane -
rwg -
Thane Plambeck -
Tom Rokicki -
Veit Elser