3 ms·

Having one point at the center does not feel intuitive. For example the N=19 feels like it would be more uniform as same as N=20 but without center point. Simil

by zokier 2y ago

Having one point at the center does not feel intuitive. For example the N=19 feels like it would be more uniform as same as N=20 but without center point. Similarly N=3 could be just the points on the outer circle, similar to what N=4 is now.

Sharlin 2y ago
N=19 (and likely N=18 etc) could also be more uniform if it already jumped to M=2, so that it would be like N=20 except with five points on the inner ring. Or maybe with six on the inner and twelve on the outer. Would probably be fairly easy to turn this into an optimization problem if one wanted a really uniform distribution (eg. distribute points on rings such that some sum of squared distances is minimized).
Jaxan 2y ago
I agree that N=3 (and also N=2 for that matter) could be distributed better. EDIT: but for large N these things probably don’t matter that much. And I guess that’s the application in the end.
mturmon 2y ago
It turns out there are differing interpretations of what to do about the edges. For an "interior" packing (there is a technical word for this that I don't recall rn), following the link by @mbauman below, the n=3 case is: http://hydra.nat.uni-magdeburg.de/packing/cci/d1.html http://hydra.nat.uni-magdeburg.de/packing/cci/d1.html and the other cases are nearby.
- zokier 2y ago
  circle packing feels slightly different problem, especially as it doesn't tend to have any obvious algorithmic solutions available; circle packing produces irregular packings which I'm not sure I'd call super uniform btw what are the colors indicating on that page? its missing a legend and its not immediately obvious to me.
  - mturmon 2y ago
    I get you. The purple circles are not kissing other circles, but the yellow ones -- not sure. In general it seems to have to do with how constrained they are. And there may be differently-arranged also-optimal packings - e.g. for N = 18 there are others besides what is shown.

by zokier 2y ago

Sharlin 2y ago

N=19 (and likely N=18 etc) could also be more uniform if it already jumped to M=2, so that it would be like N=20 except with five points on the inner ring. Or maybe with six on the inner and twelve on the outer. Would probably be fairly easy to turn this into an optimization problem if one wanted a really uniform distribution (eg. distribute points on rings such that some sum of squared distances is minimized).

Jaxan 2y ago

I agree that N=3 (and also N=2 for that matter) could be distributed better. EDIT: but for large N these things probably don’t matter that much. And I guess that’s the application in the end.

mturmon 2y ago

It turns out there are differing interpretations of what to do about the edges. For an "interior" packing (there is a technical word for this that I don't recall rn), following the link by @mbauman below, the n=3 case is: http://hydra.nat.uni-magdeburg.de/packing/cci/d1.html http://hydra.nat.uni-magdeburg.de/packing/cci/d1.html and the other cases are nearby.

zokier 2y ago
circle packing feels slightly different problem, especially as it doesn't tend to have any obvious algorithmic solutions available; circle packing produces irregular packings which I'm not sure I'd call super uniform btw what are the colors indicating on that page? its missing a legend and its not immediately obvious to me.
- mturmon 2y ago
  I get you. The purple circles are not kissing other circles, but the yellow ones -- not sure. In general it seems to have to do with how constrained they are. And there may be differently-arranged also-optimal packings - e.g. for N = 18 there are others besides what is shown.