A funster who might or might not wish to remain anonymous has corrected me in private. A3 does not equal 2; it equals 1. The corrected sequence is: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... As before, I'm not entirely confident of the last couple of values. Thank you for the correction.
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one. I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton. For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272). Order 6: (273), (333). Order 7: (373). Order 8: (377), (2772). Order 9: (777), (2773). Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.) Order 11: (3777), (27f7), (67f6). I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type. I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.) On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
A funster who might or might not wish to remain anonymous has corrected me in private. A3 does not equal 2; it equals 1. The corrected sequence is:
1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ...
As before, I'm not entirely confident of the last couple of values. Thank you for the correction.
Here's a reference I can't consult right away, which seems at least partly germane: Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008. On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one.
I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton.
For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
Order 6: (273), (333).
Order 7: (373).
Order 8: (377), (2772).
Order 9: (777), (2773).
Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
Order 11: (3777), (27f7), (67f6).
I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type.
I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.)
On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
A funster who might or might not wish to remain anonymous has corrected me in private. A3 does not equal 2; it equals 1. The corrected sequence is:
1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ...
As before, I'm not entirely confident of the last couple of values. Thank you for the correction.
A related seqiuence to look at would be the following: For each d>0 look at sets of lattice points maximal with respect to the property that every 2 of them is distance <=d apart. For these type of polyominoes you don't have to specify a center. It's not clear that is identical with the circular polyominoes since ther are non-circular regions of constant diameter. Victor On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote:
Here's a reference I can't consult right away, which seems at least partly germane:
Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008.
On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one.
I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton.
For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
Order 6: (273), (333).
Order 7: (373).
Order 8: (377), (2772).
Order 9: (777), (2773).
Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
Order 11: (3777), (27f7), (67f6).
I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type.
I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.)
On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
A funster who might or might not wish to remain anonymous has corrected me in private. A3 does not equal 2; it equals 1. The corrected sequence is:
1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ...
As before, I'm not entirely confident of the last couple of values. Thank you for the correction.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I assume you would then define a[n] as the number of such maximal polyominoes containing n lattice points. For small n, starting at n=0, I get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly have to have d=sqrt(5), but so far the only maximal polyomino I have found with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] = 1, but I'm not sure yet. 2009/5/2 victor miller <victorsmiller@gmail.com>
A related seqiuence to look at would be the following:
For each d>0 look at sets of lattice points maximal with respect to the property that every 2 of them is distance <=d apart. For these type of polyominoes you don't have to specify a center. It's not clear that is identical with the circular polyominoes since ther are non-circular regions of constant diameter.
Victor
On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote:
Here's a reference I can't consult right away, which seems at least partly germane:
Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008.
On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one.
I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton.
For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
Order 6: (273), (333).
Order 7: (373).
Order 8: (377), (2772).
Order 9: (777), (2773).
Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
Order 11: (3777), (27f7), (67f6).
I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type.
I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.)
On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
A funster who might or might not wish to remain anonymous has corrected me in private. A3 does not equal 2; it equals 1. The corrected sequence is:
1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ...
As before, I'm not entirely confident of the last couple of values. Thank you for the correction.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
A long time ago I proposed a definition for "disk polyominoes", which are essentially rasterized disks. I was able to enumerate these polyominoes (as usual, considering congruent polyominoes to be identical) up to order 11, which was just enough to run off the end of OEIS. Neil was kind enough to key the new sequence in as A147680. Later I was able to add a[12], which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4. Still later, I managed to calculate a[13] = 4, but apparently I didn't inform anybody of this. At this point the clunkiness of my search methods caught up with me, and I didn't have the patience to work out a[14]. Finally, today, I can announce that a[14] = 4. I have some more powerful lemmas under my belt now, and in a few days I ought to be able to come up with a[15] as well. The actual polyominoes for order 13, in the same notation described earlier in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for that "v": it represents a decimal 31, binary 11111, a column of five lattice-points. The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and (4eve6). Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able to figure out how to enter this update for myself. What am I missing? If there is any interest, I will present my lemmas here, so other people can join the hunt for disk polyominoes. The sequence seems to grow quite slowly, and I have conflicting intuitions about whether the growth rate is polynomial. 2009/5/2 Allan Wechsler <acwacw@gmail.com>
I assume you would then define a[n] as the number of such maximal polyominoes containing n lattice points. For small n, starting at n=0, I get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly have to have d=sqrt(5), but so far the only maximal polyomino I have found with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] = 1, but I'm not sure yet.
2009/5/2 victor miller <victorsmiller@gmail.com>
A related seqiuence to look at would be the following:
For each d>0 look at sets of lattice points maximal with respect to the property that every 2 of them is distance <=d apart. For these type of polyominoes you don't have to specify a center. It's not clear that is identical with the circular polyominoes since ther are non-circular regions of constant diameter.
Victor
On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote:
Here's a reference I can't consult right away, which seems at least partly germane:
Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008.
On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one.
I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton.
For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
Order 6: (273), (333).
Order 7: (373).
Order 8: (377), (2772).
Order 9: (777), (2773).
Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
Order 11: (3777), (27f7), (67f6).
I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type.
I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.)
On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
A funster who might or might not wish to remain anonymous has corrected me in private. A3 does not equal 2; it equals 1. The corrected sequence is:
1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ...
As before, I'm not entirely confident of the last couple of values. Thank you for the correction.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Allan, I updated A147680. (But you certainly could have done it yourself. I don't know what the problem could have been. Perhaps take a look at the web page on the OEIS Wiki called - * Trouble registering, logging in, changing password<https://oeis.org/wiki/Trouble_registering,_logging_in,_changing_password> * Possibly you didn't log in, which you do at the little login button at the top right of any OEIS page. Your login name is Allan C. Wechsler, perhaps you left out the C? Possible your browser did an automatic fill that did not include the C?) Best regards Neil 2012/4/9 Allan Wechsler <acwacw@gmail.com>
A long time ago I proposed a definition for "disk polyominoes", which are essentially rasterized disks. I was able to enumerate these polyominoes (as usual, considering congruent polyominoes to be identical) up to order 11, which was just enough to run off the end of OEIS. Neil was kind enough to key the new sequence in as A147680. Later I was able to add a[12], which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4.
Still later, I managed to calculate a[13] = 4, but apparently I didn't inform anybody of this. At this point the clunkiness of my search methods caught up with me, and I didn't have the patience to work out a[14]. Finally, today, I can announce that a[14] = 4. I have some more powerful lemmas under my belt now, and in a few days I ought to be able to come up with a[15] as well.
The actual polyominoes for order 13, in the same notation described earlier in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for that "v": it represents a decimal 31, binary 11111, a column of five lattice-points.
The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and (4eve6).
Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able to figure out how to enter this update for myself. What am I missing?
If there is any interest, I will present my lemmas here, so other people can join the hunt for disk polyominoes. The sequence seems to grow quite slowly, and I have conflicting intuitions about whether the growth rate is polynomial.
2009/5/2 Allan Wechsler <acwacw@gmail.com>
I assume you would then define a[n] as the number of such maximal polyominoes containing n lattice points. For small n, starting at n=0, I get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly have to have d=sqrt(5), but so far the only maximal polyomino I have found with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] = 1, but I'm not sure yet.
2009/5/2 victor miller <victorsmiller@gmail.com>
A related seqiuence to look at would be the following:
For each d>0 look at sets of lattice points maximal with respect to the property that every 2 of them is distance <=d apart. For these type of polyominoes you don't have to specify a center. It's not clear that is identical with the circular polyominoes since ther are non-circular regions of constant diameter.
Victor
On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote:
Here's a reference I can't consult right away, which seems at least partly germane:
Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008.
On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one.
I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton.
For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
Order 6: (273), (333).
Order 7: (373).
Order 8: (377), (2772).
Order 9: (777), (2773).
Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
Order 11: (3777), (27f7), (67f6).
I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type.
I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.)
On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
A funster who might or might not wish to remain anonymous has corrected me in private. A3 does not equal 2; it equals 1. The corrected sequence is:
1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ...
As before, I'm not entirely confident of the last couple of values. Thank you for the correction.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
I was logged in successfully. I just couldn't figure out how to edit the sequence entry. I don't know what I was missing. 2012/4/9 Neil Sloane <njasloane@gmail.com>
Allan, I updated A147680.
(But you certainly could have done it yourself. I don't know what the problem could have been. Perhaps take a look at the web page on the OEIS Wiki called
- * Trouble registering, logging in, changing password< https://oeis.org/wiki/Trouble_registering,_logging_in,_changing_password> *
Possibly you didn't log in, which you do at the little login button at the top right of any OEIS page. Your login name is Allan C. Wechsler, perhaps you left out the C? Possible your browser did an automatic fill that did not include the C?)
Best regards
Neil
2012/4/9 Allan Wechsler <acwacw@gmail.com>
A long time ago I proposed a definition for "disk polyominoes", which are essentially rasterized disks. I was able to enumerate these polyominoes (as usual, considering congruent polyominoes to be identical) up to order 11, which was just enough to run off the end of OEIS. Neil was kind enough to key the new sequence in as A147680. Later I was able to add a[12], which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4.
Still later, I managed to calculate a[13] = 4, but apparently I didn't inform anybody of this. At this point the clunkiness of my search methods caught up with me, and I didn't have the patience to work out a[14]. Finally, today, I can announce that a[14] = 4. I have some more powerful lemmas under my belt now, and in a few days I ought to be able to come up with a[15] as well.
The actual polyominoes for order 13, in the same notation described earlier in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for that "v": it represents a decimal 31, binary 11111, a column of five lattice-points.
The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and (4eve6).
Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able to figure out how to enter this update for myself. What am I missing?
If there is any interest, I will present my lemmas here, so other people can join the hunt for disk polyominoes. The sequence seems to grow quite slowly, and I have conflicting intuitions about whether the growth rate is polynomial.
2009/5/2 Allan Wechsler <acwacw@gmail.com>
I assume you would then define a[n] as the number of such maximal polyominoes containing n lattice points. For small n, starting at n=0, I get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly have to have d=sqrt(5), but so far the only maximal polyomino I have found with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] = 1, but I'm not sure yet.
2009/5/2 victor miller <victorsmiller@gmail.com>
A related seqiuence to look at would be the following:
For each d>0 look at sets of lattice points maximal with respect to the property that every 2 of them is distance <=d apart. For these type of polyominoes you don't have to specify a center. It's not clear that is identical with the circular polyominoes since ther are non-circular regions of constant diameter.
Victor
On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote:
Here's a reference I can't consult right away, which seems at least partly germane:
Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008.
On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
Neil was kind enough to enter the sequence for me; the OEIS sequence number is A147680. He asks for some pictures, but I confess that what I've got doesn't actually show circles. Typically I've drawn a bunch of lattice points, highlighting the ones that lie on the circumference of the smallest containing circle. Other marked points need to lie outside that circle in order for the disk polyomino to be valid, but I'm not sure how illuminating my diagrams would be. Perhaps a picture of the lone 7-point example with its bounding circle and other points clearly lying outside would convey the idea, but I don't have a nice one.
I should probably list the polyominoes I've shown to be disks. I could use the notation we used to use for small Life patterns, where each row is represented by the value of a binary number whose ones show which points are part of the configuration. These numbers are usually small, and we write the different row-descriptors with no delimiter between them, going up to letters of the alphabet if we run out of digits. We usually pick a scan order that minimizes the maximum descripton.
For order 0, we of course have only (0), and for order 1 only (1). Order 2 gives (11), and order 3 gives the L-tromino (13). Order 4 has two examples, the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino (133) and the X-pentomino (272).
Order 6: (273), (333).
Order 7: (373).
Order 8: (377), (2772).
Order 9: (777), (2773).
Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent points in a row included in the polyomino.)
Order 11: (3777), (27f7), (67f6).
I only have 80% confidence that these lists are exhaustive. I'm 99% confident that all the polyominoes listed are in fact of the disk type.
I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? (I suspect RWG can generate much more fiendish conundra of this variety.)
On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com> wrote:
> A funster who might or might not wish to remain anonymous has corrected > me > in private. A3 does not equal 2; it equals 1. The corrected sequence is: > > 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... > > As before, I'm not entirely confident of the last couple of values. > Thank > you for the correction. >
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-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Dear Allan, When you are logged in and look at the sequence, https://oeis.org/A147680 you will see a small blue 'edit' link under the list of the first values of the sequence. It links to https://oeis.org/edit?seq=A147680 when you can propose modifications to the editor. Regards, Olivier GERARD Seqfan Mailing List 2012/4/10 Allan Wechsler <acwacw@gmail.com>
I was logged in successfully. I just couldn't figure out how to edit the sequence entry. I don't know what I was missing.
go to a sequence, say http://oeis.org/A000001 (assuming you are logged in) you will see a little "edit" button near the top of the entry! Neil 2012/4/10 Allan Wechsler <acwacw@gmail.com>
I was logged in successfully. I just couldn't figure out how to edit the sequence entry. I don't know what I was missing.
2012/4/9 Neil Sloane <njasloane@gmail.com>
Allan, I updated A147680.
(But you certainly could have done it yourself. I don't know what the problem could have been. Perhaps take a look at the web page on the OEIS Wiki called
- * Trouble registering, logging in, changing password< https://oeis.org/wiki/Trouble_registering,_logging_in,_changing_password
*
Possibly you didn't log in, which you do at the little login button at the top right of any OEIS page. Your login name is Allan C. Wechsler, perhaps you left out the C? Possible your browser did an automatic fill that did not include the C?)
Best regards
Neil
2012/4/9 Allan Wechsler <acwacw@gmail.com>
A long time ago I proposed a definition for "disk polyominoes", which are essentially rasterized disks. I was able to enumerate these polyominoes (as usual, considering congruent polyominoes to be identical) up to order 11, which was just enough to run off the end of OEIS. Neil was kind enough to key the new sequence in as A147680. Later I was able to add a[12], which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4.
Still later, I managed to calculate a[13] = 4, but apparently I didn't inform anybody of this. At this point the clunkiness of my search methods caught up with me, and I didn't have the patience to work out a[14]. Finally, today, I can announce that a[14] = 4. I have some more powerful lemmas under my belt now, and in a few days I ought to be able to come up with a[15] as well.
The actual polyominoes for order 13, in the same notation described earlier in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for that "v": it represents a decimal 31, binary 11111, a column of five lattice-points.
The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and (4eve6).
Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able to figure out how to enter this update for myself. What am I missing?
If there is any interest, I will present my lemmas here, so other people can join the hunt for disk polyominoes. The sequence seems to grow quite slowly, and I have conflicting intuitions about whether the growth rate is polynomial.
2009/5/2 Allan Wechsler <acwacw@gmail.com>
I assume you would then define a[n] as the number of such maximal polyominoes containing n lattice points. For small n, starting at n=0, I get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly have to have d=sqrt(5), but so far the only maximal polyomino I have found with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] = 1, but I'm not sure yet.
2009/5/2 victor miller <victorsmiller@gmail.com>
A related seqiuence to look at would be the following:
For each d>0 look at sets of lattice points maximal with respect to the property that every 2 of them is distance <=d apart. For these type of polyominoes you don't have to specify a center. It's not clear that is identical with the circular polyominoes since ther are non-circular regions of constant diameter.
Victor
On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote:
Here's a reference I can't consult right away, which seems at least partly germane:
Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. 299-309. Springer, Berlin, 2008.
On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler <acwacw@gmail.com> wrote:
> Neil was kind enough to enter the sequence for me; the OEIS sequence > number > is A147680. He asks for some pictures, but I confess that what I've got > doesn't actually show circles. Typically I've drawn a bunch of lattice > points, highlighting the ones that lie on the circumference of the > smallest > containing circle. Other marked points need to lie outside that circle in > order for the disk polyomino to be valid, but I'm not sure how > illuminating > my diagrams would be. Perhaps a picture of the lone 7-point example with > its bounding circle and other points clearly lying outside would convey > the > idea, but I don't have a nice one. > > I should probably list the polyominoes I've shown to be disks. I could > use > the notation we used to use for small Life patterns, where each row is > represented by the value of a binary number whose ones show which points > are > part of the configuration. These numbers are usually small, and we write > the different row-descriptors with no delimiter between them, going up to > letters of the alphabet if we run out of digits. We usually pick a scan > order that minimizes the maximum descripton. > > For order 0, we of course have only (0), and for order 1 only (1). Order > 2 > gives (11), and order 3 gives the L-tromino (13). Order 4 has two > examples, > the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino > (133) and the X-pentomino (272). > > Order 6: (273), (333). > > Order 7: (373). > > Order 8: (377), (2772). > > Order 9: (777), (2773). > > Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent > points in a row included in the polyomino.) > > Order 11: (3777), (27f7), (67f6). > > I only have 80% confidence that these lists are exhaustive. I'm 99% > confident that all the polyominoes listed are in fact of the disk type. > > I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? > (I > suspect RWG can generate much more fiendish conundra of this variety.) > > > On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler <acwacw@gmail.com
wrote:
> >> A funster who might or might not wish to remain anonymous has corrected >> me >> in private. A3 does not equal 2; it equals 1. The corrected sequence is: >> >> 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... >> >> As before, I'm not entirely confident of the last couple of values. >> Thank >> you for the correction. >> > > _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
OK, I _looked_ for an edit button, but I'll look again. I have another addition: a[15] = 4. (I expected 5.) The order-15 discoids are: (7fff), (2fff6), (4evee), (4evf6). 2012/4/10 Neil Sloane <njasloane@gmail.com>
go to a sequence, say http://oeis.org/A000001
(assuming you are logged in)
you will see a little "edit" button near the top of the entry!
Neil
2012/4/10 Allan Wechsler <acwacw@gmail.com>
I was logged in successfully. I just couldn't figure out how to edit the sequence entry. I don't know what I was missing.
2012/4/9 Neil Sloane <njasloane@gmail.com>
Allan, I updated A147680.
(But you certainly could have done it yourself. I don't know what the problem could have been. Perhaps take a look at the web page on the OEIS Wiki called
- * Trouble registering, logging in, changing password<
https://oeis.org/wiki/Trouble_registering,_logging_in,_changing_password
*
Possibly you didn't log in, which you do at the little login button at the top right of any OEIS page. Your login name is Allan C. Wechsler, perhaps you left out the C? Possible your browser did an automatic fill that did not include the C?)
Best regards
Neil
2012/4/9 Allan Wechsler <acwacw@gmail.com>
A long time ago I proposed a definition for "disk polyominoes", which
are
essentially rasterized disks. I was able to enumerate these polyominoes (as usual, considering congruent polyominoes to be identical) up to order 11, which was just enough to run off the end of OEIS. Neil was kind enough to key the new sequence in as A147680. Later I was able to add a[12], which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4.
Still later, I managed to calculate a[13] = 4, but apparently I didn't inform anybody of this. At this point the clunkiness of my search methods caught up with me, and I didn't have the patience to work out a[14]. Finally, today, I can announce that a[14] = 4. I have some more powerful lemmas under my belt now, and in a few days I ought to be able to come up with a[15] as well.
The actual polyominoes for order 13, in the same notation described earlier in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for that "v": it represents a decimal 31, binary 11111, a column of five lattice-points.
The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and (4eve6).
Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able to figure out how to enter this update for myself. What am I missing?
If there is any interest, I will present my lemmas here, so other people can join the hunt for disk polyominoes. The sequence seems to grow quite slowly, and I have conflicting intuitions about whether the growth rate is polynomial.
2009/5/2 Allan Wechsler <acwacw@gmail.com>
I assume you would then define a[n] as the number of such maximal polyominoes containing n lattice points. For small n, starting at n=0, I get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly have to have d=sqrt(5), but so far the only maximal polyomino I have found with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] = 1, but I'm not sure yet.
2009/5/2 victor miller <victorsmiller@gmail.com>
A related seqiuence to look at would be the following:
For each d>0 look at sets of lattice points maximal with respect to the property that every 2 of them is distance <=d apart. For these type of polyominoes you don't have to specify a center. It's not clear that is identical with the circular polyominoes since ther are non-circular regions of constant diameter.
Victor
On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote: > Here's a reference I can't consult right away, which seems at least partly > germane: > > Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of > Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. > 299-309. Springer, Berlin, 2008. > > On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler < acwacw@gmail.com> wrote: > >> Neil was kind enough to enter the sequence for me; the OEIS sequence >> number >> is A147680. He asks for some pictures, but I confess that what I've got >> doesn't actually show circles. Typically I've drawn a bunch of lattice >> points, highlighting the ones that lie on the circumference of the >> smallest >> containing circle. Other marked points need to lie outside that circle in >> order for the disk polyomino to be valid, but I'm not sure how >> illuminating >> my diagrams would be. Perhaps a picture of the lone 7-point example with >> its bounding circle and other points clearly lying outside would convey >> the >> idea, but I don't have a nice one. >> >> I should probably list the polyominoes I've shown to be disks. I could >> use >> the notation we used to use for small Life patterns, where each row is >> represented by the value of a binary number whose ones show which points >> are >> part of the configuration. These numbers are usually small, and we write >> the different row-descriptors with no delimiter between them, going up to >> letters of the alphabet if we run out of digits. We usually pick a scan >> order that minimizes the maximum descripton. >> >> For order 0, we of course have only (0), and for order 1 only (1). Order >> 2 >> gives (11), and order 3 gives the L-tromino (13). Order 4 has two >> examples, >> the block (33) and the T-tetromino (131). Order 5 gives the P-pentomino >> (133) and the X-pentomino (272). >> >> Order 6: (273), (333). >> >> Order 7: (373). >> >> Order 8: (377), (2772). >> >> Order 9: (777), (2773). >> >> Order 10: (2777), (3773), (27f6). (That "f" means 15, with four adjacent >> points in a row included in the polyomino.) >> >> Order 11: (3777), (27f7), (67f6). >> >> I only have 80% confidence that these lists are exhaustive. I'm 99% >> confident that all the polyominoes listed are in fact of the disk type. >> >> I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? >> (I >> suspect RWG can generate much more fiendish conundra of this variety.) >> >> >> On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler < acwacw@gmail.com
wrote: >> >>> A funster who might or might not wish to remain anonymous has corrected >>> me >>> in private. A3 does not equal 2; it equals 1. The corrected sequence is: >>> >>> 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... >>> >>> As before, I'm not entirely confident of the last couple of values. >>> Thank >>> you for the correction. >>> >> >> > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I made that change successfully, and also regularized the format in the examples section. 2012/4/10 Allan Wechsler <acwacw@gmail.com>
OK, I _looked_ for an edit button, but I'll look again. I have another addition: a[15] = 4. (I expected 5.)
The order-15 discoids are: (7fff), (2fff6), (4evee), (4evf6).
2012/4/10 Neil Sloane <njasloane@gmail.com>
go to a sequence, say http://oeis.org/A000001
(assuming you are logged in)
you will see a little "edit" button near the top of the entry!
Neil
2012/4/10 Allan Wechsler <acwacw@gmail.com>
I was logged in successfully. I just couldn't figure out how to edit the sequence entry. I don't know what I was missing.
2012/4/9 Neil Sloane <njasloane@gmail.com>
Allan, I updated A147680.
(But you certainly could have done it yourself. I don't know what the problem could have been. Perhaps take a look at the web page on the OEIS Wiki called
- * Trouble registering, logging in, changing password<
https://oeis.org/wiki/Trouble_registering,_logging_in,_changing_password
*
Possibly you didn't log in, which you do at the little login button at the top right of any OEIS page. Your login name is Allan C. Wechsler, perhaps you left out the C? Possible your browser did an automatic fill that did not include the C?)
Best regards
Neil
2012/4/9 Allan Wechsler <acwacw@gmail.com>
A long time ago I proposed a definition for "disk polyominoes",
which are
essentially rasterized disks. I was able to enumerate these polyominoes (as usual, considering congruent polyominoes to be identical) up to order 11, which was just enough to run off the end of OEIS. Neil was kind enough to key the new sequence in as A147680. Later I was able to add a[12], which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4.
Still later, I managed to calculate a[13] = 4, but apparently I didn't inform anybody of this. At this point the clunkiness of my search methods caught up with me, and I didn't have the patience to work out a[14]. Finally, today, I can announce that a[14] = 4. I have some more powerful lemmas under my belt now, and in a few days I ought to be able to come up with a[15] as well.
The actual polyominoes for order 13, in the same notation described earlier in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for that "v": it represents a decimal 31, binary 11111, a column of five lattice-points.
The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and (4eve6).
Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able to figure out how to enter this update for myself. What am I missing?
If there is any interest, I will present my lemmas here, so other people can join the hunt for disk polyominoes. The sequence seems to grow quite slowly, and I have conflicting intuitions about whether the growth rate is polynomial.
2009/5/2 Allan Wechsler <acwacw@gmail.com>
I assume you would then define a[n] as the number of such maximal polyominoes containing n lattice points. For small n, starting at n=0, I get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly have to have d=sqrt(5), but so far the only maximal polyomino I have found with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] = 1, but I'm not sure yet.
2009/5/2 victor miller <victorsmiller@gmail.com>
> A related seqiuence to look at would be the following: > > For each d>0 look at sets of lattice points maximal with respect to > the property that every 2 of them is distance <=d apart. For these > type of polyominoes you don't have to specify a center. It's not > clear that is identical with the circular polyominoes since ther are > non-circular regions of constant diameter. > > Victor > > On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote: > > Here's a reference I can't consult right away, which seems at least > partly > > germane: > > > > Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of > > Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. > > 299-309. Springer, Berlin, 2008. > > > > On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler < acwacw@gmail.com> > wrote: > > > >> Neil was kind enough to enter the sequence for me; the OEIS sequence > >> number > >> is A147680. He asks for some pictures, but I confess that what I've > got > >> doesn't actually show circles. Typically I've drawn a bunch of lattice > >> points, highlighting the ones that lie on the circumference of the > >> smallest > >> containing circle. Other marked points need to lie outside that > circle in > >> order for the disk polyomino to be valid, but I'm not sure how > >> illuminating > >> my diagrams would be. Perhaps a picture of the lone 7-point example > with > >> its bounding circle and other points clearly lying outside would convey > >> the > >> idea, but I don't have a nice one. > >> > >> I should probably list the polyominoes I've shown to be disks. I could > >> use > >> the notation we used to use for small Life patterns, where each row is > >> represented by the value of a binary number whose ones show which > points > >> are > >> part of the configuration. These numbers are usually small, and we > write > >> the different row-descriptors with no delimiter between them, going up > to > >> letters of the alphabet if we run out of digits. We usually pick a > scan > >> order that minimizes the maximum descripton. > >> > >> For order 0, we of course have only (0), and for order 1 only (1). > Order > >> 2 > >> gives (11), and order 3 gives the L-tromino (13). Order 4 has two > >> examples, > >> the block (33) and the T-tetromino (131). Order 5 gives the > P-pentomino > >> (133) and the X-pentomino (272). > >> > >> Order 6: (273), (333). > >> > >> Order 7: (373). > >> > >> Order 8: (377), (2772). > >> > >> Order 9: (777), (2773). > >> > >> Order 10: (2777), (3773), (27f6). (That "f" means 15, with four > adjacent > >> points in a row included in the polyomino.) > >> > >> Order 11: (3777), (27f7), (67f6). > >> > >> I only have 80% confidence that these lists are exhaustive. I'm 99% > >> confident that all the polyominoes listed are in fact of the disk type. > >> > >> I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? > >> (I > >> suspect RWG can generate much more fiendish conundra of this variety.) > >> > >> > >> On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler < acwacw@gmail.com
> wrote: > >> > >>> A funster who might or might not wish to remain anonymous has > corrected > >>> me > >>> in private. A3 does not equal 2; it equals 1. The corrected sequence > is: > >>> > >>> 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... > >>> > >>> As before, I'm not entirely confident of the last couple of values. > >>> Thank > >>> you for the correction. > >>> > >> > >> > > _______________________________________________ > > math-fun mailing list > > math-fun@mailman.xmission.com > > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Since my last message, I have found a[16] = 4, a[17] = 4, and, the big surprise, a[18] = 3. This last seems very unintuitive to me. "Provide a complete list of 18-pixel rasterized disks" might be a good Olympiad problem. This sequence is not growing anywhere near as fast as I thought it would. I was sure it would be at least linear (in fact, one intuitive line of thought, now lost, suggested low-coefficient quadratic), but it is looking sublinearer and sublinearer. Also, I'm a little frustrated that I don't have a real algorithm. I can generate an exhaustive list of candidates by algorithm, and I have ways of proving individual candidates bad or good. These proof techniques have sufficed for all the examples I have seen so far, but I have no confidence that they suffice to categorize every candidate. 2012/4/10 Allan Wechsler <acwacw@gmail.com>
I made that change successfully, and also regularized the format in the examples section.
2012/4/10 Allan Wechsler <acwacw@gmail.com>
OK, I _looked_ for an edit button, but I'll look again. I have another addition: a[15] = 4. (I expected 5.)
The order-15 discoids are: (7fff), (2fff6), (4evee), (4evf6).
2012/4/10 Neil Sloane <njasloane@gmail.com>
go to a sequence, say http://oeis.org/A000001
(assuming you are logged in)
you will see a little "edit" button near the top of the entry!
Neil
2012/4/10 Allan Wechsler <acwacw@gmail.com>
I was logged in successfully. I just couldn't figure out how to edit the sequence entry. I don't know what I was missing.
2012/4/9 Neil Sloane <njasloane@gmail.com>
Allan, I updated A147680.
(But you certainly could have done it yourself. I don't know what the problem could have been. Perhaps take a look at the web page on the OEIS Wiki called
- * Trouble registering, logging in, changing password<
https://oeis.org/wiki/Trouble_registering,_logging_in,_changing_password
*
Possibly you didn't log in, which you do at the little login button at the top right of any OEIS page. Your login name is Allan C. Wechsler, perhaps you left out the C? Possible your browser did an automatic fill that did not include the C?)
Best regards
Neil
2012/4/9 Allan Wechsler <acwacw@gmail.com>
A long time ago I proposed a definition for "disk polyominoes",
which are
essentially rasterized disks. I was able to enumerate these polyominoes (as usual, considering congruent polyominoes to be identical) up to order 11, which was just enough to run off the end of OEIS. Neil was kind enough to key the new sequence in as A147680. Later I was able to add a[12], which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3, 4.
Still later, I managed to calculate a[13] = 4, but apparently I didn't inform anybody of this. At this point the clunkiness of my search methods caught up with me, and I didn't have the patience to work out a[14]. Finally, today, I can announce that a[14] = 4. I have some more powerful lemmas under my belt now, and in a few days I ought to be able to come up with a[15] as well.
The actual polyominoes for order 13, in the same notation described earlier in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for that "v": it represents a decimal 31, binary 11111, a column of five lattice-points.
The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and (4eve6).
Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able to figure out how to enter this update for myself. What am I missing?
If there is any interest, I will present my lemmas here, so other people can join the hunt for disk polyominoes. The sequence seems to grow quite slowly, and I have conflicting intuitions about whether the growth rate is polynomial.
2009/5/2 Allan Wechsler <acwacw@gmail.com>
> I assume you would then define a[n] as the number of such maximal > polyominoes containing n lattice points. For small n, starting at n=0, I > get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly > have to have d=sqrt(5), but so far the only maximal polyomino I have found > with that diameter is (373), which has 7 points. I suspect a[6] = 0, a[7] > = 1, but I'm not sure yet. > > > 2009/5/2 victor miller <victorsmiller@gmail.com> > >> A related seqiuence to look at would be the following: >> >> For each d>0 look at sets of lattice points maximal with respect to >> the property that every 2 of them is distance <=d apart. For these >> type of polyominoes you don't have to specify a center. It's not >> clear that is identical with the circular polyominoes since ther are >> non-circular regions of constant diameter. >> >> Victor >> >> On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote: >> > Here's a reference I can't consult right away, which seems at least >> partly >> > germane: >> > >> > Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of >> > Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. >> > 299-309. Springer, Berlin, 2008. >> > >> > On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler < acwacw@gmail.com> >> wrote: >> > >> >> Neil was kind enough to enter the sequence for me; the OEIS sequence >> >> number >> >> is A147680. He asks for some pictures, but I confess that what I've >> got >> >> doesn't actually show circles. Typically I've drawn a bunch of lattice >> >> points, highlighting the ones that lie on the circumference of the >> >> smallest >> >> containing circle. Other marked points need to lie outside that >> circle in >> >> order for the disk polyomino to be valid, but I'm not sure how >> >> illuminating >> >> my diagrams would be. Perhaps a picture of the lone 7-point example >> with >> >> its bounding circle and other points clearly lying outside would convey >> >> the >> >> idea, but I don't have a nice one. >> >> >> >> I should probably list the polyominoes I've shown to be disks. I could >> >> use >> >> the notation we used to use for small Life patterns, where each row is >> >> represented by the value of a binary number whose ones show which >> points >> >> are >> >> part of the configuration. These numbers are usually small, and we >> write >> >> the different row-descriptors with no delimiter between them, going up >> to >> >> letters of the alphabet if we run out of digits. We usually pick a >> scan >> >> order that minimizes the maximum descripton. >> >> >> >> For order 0, we of course have only (0), and for order 1 only (1). >> Order >> >> 2 >> >> gives (11), and order 3 gives the L-tromino (13). Order 4 has two >> >> examples, >> >> the block (33) and the T-tetromino (131). Order 5 gives the >> P-pentomino >> >> (133) and the X-pentomino (272). >> >> >> >> Order 6: (273), (333). >> >> >> >> Order 7: (373). >> >> >> >> Order 8: (377), (2772). >> >> >> >> Order 9: (777), (2773). >> >> >> >> Order 10: (2777), (3773), (27f6). (That "f" means 15, with four >> adjacent >> >> points in a row included in the polyomino.) >> >> >> >> Order 11: (3777), (27f7), (67f6). >> >> >> >> I only have 80% confidence that these lists are exhaustive. I'm 99% >> >> confident that all the polyominoes listed are in fact of the disk type. >> >> >> >> I leave you with a puzzle: Is the duodecomino (67f7) a disk polyomino? >> >> (I >> >> suspect RWG can generate much more fiendish conundra of this variety.) >> >> >> >> >> >> On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler < acwacw@gmail.com
>> wrote: >> >> >> >>> A funster who might or might not wish to remain anonymous has >> corrected >> >>> me >> >>> in private. A3 does not equal 2; it equals 1. The corrected sequence >> is: >> >>> >> >>> 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... >> >>> >> >>> As before, I'm not entirely confident of the last couple of values. >> >>> Thank >> >>> you for the correction. >> >>> >> >> >> >> >> > _______________________________________________ >> > math-fun mailing list >> > math-fun@mailman.xmission.com >> > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > >> >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
And since *then*, I have found a[19]=a[20]=3, and then a[21]=5. This is all very much not what I expected, and even more interesting than I thought it would be. I'm going to update OEIS, but then I think I'm going to stop for a while, because this is all being done by hand, and the farther I go the more chance there is of getting something wrong. Still no well-defined algorithm. 2012/4/10 Allan Wechsler <acwacw@gmail.com>
Since my last message, I have found a[16] = 4, a[17] = 4, and, the big surprise, a[18] = 3. This last seems very unintuitive to me. "Provide a complete list of 18-pixel rasterized disks" might be a good Olympiad problem.
This sequence is not growing anywhere near as fast as I thought it would. I was sure it would be at least linear (in fact, one intuitive line of thought, now lost, suggested low-coefficient quadratic), but it is looking sublinearer and sublinearer.
Also, I'm a little frustrated that I don't have a real algorithm. I can generate an exhaustive list of candidates by algorithm, and I have ways of proving individual candidates bad or good. These proof techniques have sufficed for all the examples I have seen so far, but I have no confidence that they suffice to categorize every candidate.
2012/4/10 Allan Wechsler <acwacw@gmail.com>
I made that change successfully, and also regularized the format in the examples section.
2012/4/10 Allan Wechsler <acwacw@gmail.com>
OK, I _looked_ for an edit button, but I'll look again. I have another addition: a[15] = 4. (I expected 5.)
The order-15 discoids are: (7fff), (2fff6), (4evee), (4evf6).
2012/4/10 Neil Sloane <njasloane@gmail.com>
go to a sequence, say http://oeis.org/A000001
(assuming you are logged in)
you will see a little "edit" button near the top of the entry!
Neil
2012/4/10 Allan Wechsler <acwacw@gmail.com>
I was logged in successfully. I just couldn't figure out how to edit the sequence entry. I don't know what I was missing.
2012/4/9 Neil Sloane <njasloane@gmail.com>
Allan, I updated A147680.
(But you certainly could have done it yourself. I don't know what the problem could have been. Perhaps take a look at the web page on the OEIS Wiki called
- * Trouble registering, logging in, changing password<
https://oeis.org/wiki/Trouble_registering,_logging_in,_changing_password
*
Possibly you didn't log in, which you do at the little login button at the top right of any OEIS page. Your login name is Allan C. Wechsler, perhaps you left out the C? Possible your browser did an automatic fill that did not include the C?)
Best regards
Neil
2012/4/9 Allan Wechsler <acwacw@gmail.com>
> A long time ago I proposed a definition for "disk polyominoes",
which are
> essentially rasterized disks. I was able to enumerate these polyominoes > (as usual, considering congruent polyominoes to be identical) up to order > 11, which was just enough to run off the end of OEIS. Neil was kind enough > to key the new sequence in as A147680. Later I was able to add a[12], > which Neil added for me. These 13 elements were: 1, 1, 1, 1, 2, 2, 2, 1, > 2, 2, 3, 3, 4. > > Still later, I managed to calculate a[13] = 4, but apparently I didn't > inform anybody of this. At this point the clunkiness of my search methods > caught up with me, and I didn't have the patience to work out a[14]. > Finally, today, I can announce that a[14] = 4. I have some more powerful > lemmas under my belt now, and in a few days I ought to be able to come up > with a[15] as well. > > The actual polyominoes for order 13, in the same notation described earlier > in this thread, are (77f7), (6ff7), (27ff2), and (4eve4). I apologize for > that "v": it represents a decimal 31, binary 11111, a column of five > lattice-points. > > The four disk polyominoes of order 14 are (7ff7), (2fff2), (27ff6), and > (4eve6). > > Now, a meta-question: I have an account at the OEIS wiki, but I wasn't able > to figure out how to enter this update for myself. What am I missing? > > If there is any interest, I will present my lemmas here, so other people > can join the hunt for disk polyominoes. The sequence seems to grow quite > slowly, and I have conflicting intuitions about whether the growth rate is > polynomial. > > 2009/5/2 Allan Wechsler <acwacw@gmail.com> > > > I assume you would then define a[n] as the number of such maximal > > polyominoes containing n lattice points. For small n, starting at n=0, I > > get 0, 1, 1, 0, 1, 1. But a[6] has me temporarily stalled. We clearly > > have to have d=sqrt(5), but so far the only maximal polyomino I have > found > > with that diameter is (373), which has 7 points. I suspect a[6] = 0, > a[7] > > = 1, but I'm not sure yet. > > > > > > 2009/5/2 victor miller <victorsmiller@gmail.com> > > > >> A related seqiuence to look at would be the following: > >> > >> For each d>0 look at sets of lattice points maximal with respect to > >> the property that every 2 of them is distance <=d apart. For these > >> type of polyominoes you don't have to specify a center. It's not > >> clear that is identical with the circular polyominoes since ther are > >> non-circular regions of constant diameter. > >> > >> Victor > >> > >> On 5/2/09, Allan Wechsler <acwacw@gmail.com> wrote: > >> > Here's a reference I can't consult right away, which seems at least > >> partly > >> > germane: > >> > > >> > Srečko Brlek, Gilbert Labelle, and Annie Lacasse; On Minimal Moment of > >> > Inertia Polyominoes; in Discrete Geometry For Computer Imagery, pp. > >> > 299-309. Springer, Berlin, 2008. > >> > > >> > On Sat, May 2, 2009 at 6:24 PM, Allan Wechsler < acwacw@gmail.com> > >> wrote: > >> > > >> >> Neil was kind enough to enter the sequence for me; the OEIS sequence > >> >> number > >> >> is A147680. He asks for some pictures, but I confess that what I've > >> got > >> >> doesn't actually show circles. Typically I've drawn a bunch of > lattice > >> >> points, highlighting the ones that lie on the circumference of the > >> >> smallest > >> >> containing circle. Other marked points need to lie outside that > >> circle in > >> >> order for the disk polyomino to be valid, but I'm not sure how > >> >> illuminating > >> >> my diagrams would be. Perhaps a picture of the lone 7-point example > >> with > >> >> its bounding circle and other points clearly lying outside would > convey > >> >> the > >> >> idea, but I don't have a nice one. > >> >> > >> >> I should probably list the polyominoes I've shown to be disks. I > could > >> >> use > >> >> the notation we used to use for small Life patterns, where each row > is > >> >> represented by the value of a binary number whose ones show which > >> points > >> >> are > >> >> part of the configuration. These numbers are usually small, and we > >> write > >> >> the different row-descriptors with no delimiter between them, going > up > >> to > >> >> letters of the alphabet if we run out of digits. We usually pick a > >> scan > >> >> order that minimizes the maximum descripton. > >> >> > >> >> For order 0, we of course have only (0), and for order 1 only (1). > >> Order > >> >> 2 > >> >> gives (11), and order 3 gives the L-tromino (13). Order 4 has two > >> >> examples, > >> >> the block (33) and the T-tetromino (131). Order 5 gives the > >> P-pentomino > >> >> (133) and the X-pentomino (272). > >> >> > >> >> Order 6: (273), (333). > >> >> > >> >> Order 7: (373). > >> >> > >> >> Order 8: (377), (2772). > >> >> > >> >> Order 9: (777), (2773). > >> >> > >> >> Order 10: (2777), (3773), (27f6). (That "f" means 15, with four > >> adjacent > >> >> points in a row included in the polyomino.) > >> >> > >> >> Order 11: (3777), (27f7), (67f6). > >> >> > >> >> I only have 80% confidence that these lists are exhaustive. I'm 99% > >> >> confident that all the polyominoes listed are in fact of the disk > type. > >> >> > >> >> I leave you with a puzzle: Is the duodecomino (67f7) a disk > polyomino? > >> >> (I > >> >> suspect RWG can generate much more fiendish conundra of this > variety.) > >> >> > >> >> > >> >> On Fri, May 1, 2009 at 2:12 PM, Allan Wechsler < acwacw@gmail.com
> >> wrote: > >> >> > >> >>> A funster who might or might not wish to remain anonymous has > >> corrected > >> >>> me > >> >>> in private. A3 does not equal 2; it equals 1. The corrected > sequence > >> is: > >> >>> > >> >>> 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 3 ... > >> >>> > >> >>> As before, I'm not entirely confident of the last couple of values. > >> >>> Thank > >> >>> you for the correction. > >> >>> > >> >> > >> >> > >> > _______________________________________________ > >> > math-fun mailing list > >> > math-fun@mailman.xmission.com > >> > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >> > > >> > >> _______________________________________________ > >> math-fun mailing list > >> math-fun@mailman.xmission.com > >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >> > > > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Dear Friends, I will soon be retiring from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I have kind of lazily been following this thread. By "disk polyomino", do we mean a polyomino consisting of the grid cells lying entirely within some circular disk? On 4/12/2012 4:43 PM, Allan Wechsler wrote:
And since *then*, I have found a[19]=a[20]=3, and then a[21]=5. This is all very much not what I expected, and even more interesting than I thought it would be. I'm going to update OEIS, but then I think I'm going to stop for a while, because this is all being done by hand, and the farther I go the more chance there is of getting something wrong.
Still no well-defined algorithm.
2012/4/10 Allan Wechsler<acwacw@gmail.com>
That's mostly correct. My definition is that it is a set of lattice points, not cells, of the form D intersect Z^2, for some closed disk D. (Open disks give the same polyominoes, but require slightly more finicky reasoning.) On Thu, Apr 12, 2012 at 10:05 PM, David Wilson <davidwwilson@comcast.net>wrote:
I have kind of lazily been following this thread.
By "disk polyomino", do we mean a polyomino consisting of the grid cells lying entirely within some circular disk?
On 4/12/2012 4:43 PM, Allan Wechsler wrote:
And since *then*, I have found a[19]=a[20]=3, and then a[21]=5. This is
all very much not what I expected, and even more interesting than I thought it would be. I'm going to update OEIS, but then I think I'm going to stop for a while, because this is all being done by hand, and the farther I go the more chance there is of getting something wrong.
Still no well-defined algorithm.
2012/4/10 Allan Wechsler<acwacw@gmail.com>
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participants (5)
-
Allan Wechsler -
David Wilson -
Neil Sloane -
Olivier Gerard -
victor miller