Timeline for Random polycube shapes
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Mar 11, 2017 at 14:09 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Image links broken; now fixed.
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| May 22, 2011 at 18:58 | vote | accept | Joseph O'Rourke | ||
| May 21, 2011 at 20:23 | comment | added | Joseph O'Rourke | I stand corrected! A cavity can form by plugging the top of a well, just as described. | |
| May 21, 2011 at 20:14 | comment | added | j.c. | @ARupinski: Your construction looks good to me. Apologies, I was too sloppy in my previous remarks. | |
| May 21, 2011 at 18:05 | comment | added | Gerhard Paseman | Isn't it possible to have the middle cube removed on the second iteration, and then have that space completely surrounded on the third? Or are you adding cubes in a special way that prevents such from happening? Of course forming a cavity that way is much less likely, but I do not see it being precluded. Gerhard "Ask Me About System Design" Paseman, 2011.05.21 | |
| May 21, 2011 at 18:05 | comment | added | ARupinski | On the 3x3x3 setup: Add a cube on a corner, then delete a central face cube. Add another cube to a corner cube, then delete the interior cube. Finally, fill in the central face cube deleted in step 1 and delete one of the cubes you added to the corners. This leaves a hole in the interior and an extra cube sticking out on the exterior. Where does this sequence violate the rules you set out in the setup? | |
| May 21, 2011 at 17:52 | answer | added | Omer | timeline score: 13 | |
| May 19, 2011 at 9:59 | comment | added | Joseph O'Rourke | @ARupinski & jc: jc is correct. I add before I delete, so no interior cavity can form. It would be equally interesting to delete before adding, in which case cavities could be created. @jc: Thanks for the CHomP and Guttmann references! | |
| May 19, 2011 at 4:32 | comment | added | j.c. | One last comment: A good place to read about what's known about "polycubes chosen uniformly from the set of all connected face-to-face gluing of $n^3$ unit cubes" is the recent book "Polygons, Polyominoes and Polycubes", edited by A.J. Guttmann. | |
| May 19, 2011 at 4:27 | comment | added | j.c. | If someone is inclined to try to compute "experimentally" the genus, I suggest trying CHomP chomp.rutgers.edu/software by the group of Konstantin Mischaikow at Rutgers. I've played around with it before to compute the homology of other randomly created shapes made out of cubes. | |
| May 19, 2011 at 4:23 | comment | added | j.c. | @ARupinski Don't rules (1) and (3) require that cube additions / removals occur only for those with exposed faces, so that "interior holes" can't be formed? This seems to be the main difference between the "$n\times n\times n$ random polycubes shapes" as defined in the question and polycubes chosen uniformly from the set of all connected face-to-face gluing of $n^3$ unit cubes. | |
| May 19, 2011 at 1:08 | comment | added | ARupinski | Very interesting and awesome animation. Because you are working with polycubes, you might even need to expand from just asking about the genus to asking about the full homology since even in the $3\times 3\times 3$ case one can get an interior hole as soon as $t=3$, and the number of potential interior holes grows with $n$. | |
| May 18, 2011 at 21:47 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
edited body
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| May 18, 2011 at 21:40 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
URL correction
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| May 18, 2011 at 21:34 | history | asked | Joseph O'Rourke | CC BY-SA 3.0 |