# Wheels within wheels

October 19, 2011 7:08 PM Subscribe

The best known packings of equal circles within a circle. Best packings with 1-12 circles. Best packings with 49-60 circles. Best packings with 1093-1104 circles. Also, circles whose areas form a harmonic series. Circles in an isosceles right triangle. Or generate your own circle packings. (Background for beginners: circle packing. Background for experts: circle packing.)

During my horrible career as a busser this summer, I often wondered while stacking glasses on a countertop how many glasses wide by glasses wide a surface would have to be before it became a more efficient use of space to stagger the rows instead of laying them out in a straight grid.

posted by dunkadunc at 7:20 PM on October 19, 2011

posted by dunkadunc at 7:20 PM on October 19, 2011

Awesome. I love packing problems, and I love that people who can actually solve them cleverly do so so thoroughly that I don't waste time trying to somehow work them out by hand. The ghost of Paul Erdős is popping bennies and nodding in approval.

posted by cortex at 7:27 PM on October 19, 2011 [1 favorite]

posted by cortex at 7:27 PM on October 19, 2011 [1 favorite]

In my much much earlier career of pre-teen model rocket designer, I thought it would be cool to have tubes for fins, it seemed obvious that tubes would line up around a central tube. That lack of success is likely when I deep down realized I would not have a career as a mathematician.

posted by sammyo at 7:29 PM on October 19, 2011 [1 favorite]

posted by sammyo at 7:29 PM on October 19, 2011 [1 favorite]

Sammyo are you me? I also thought this would work. I also remember the tumbling D engine fire tube racing toward my head like it was yesterday.

posted by mrgroweler at 7:50 PM on October 19, 2011 [1 favorite]

posted by mrgroweler at 7:50 PM on October 19, 2011 [1 favorite]

Per some discussion on google+, I think the color coding is by number of neighbors -- the red guys aren't touching any other circles, the orange ones touch 5 or 6.

posted by escabeche at 8:01 PM on October 19, 2011 [1 favorite]

posted by escabeche at 8:01 PM on October 19, 2011 [1 favorite]

So is there any kind of periodicity or something else meaningful to the symmetrical arrangements or that have an equal number of touches?

posted by cgk at 8:02 PM on October 19, 2011

posted by cgk at 8:02 PM on October 19, 2011

Circles make hexagons. We derive pi from a circle.

A hexagon is - I believe - the simplest polygon that does

There's something there, there, but I'll leave it to the math nerds to figure that one out. I was just musing about this (and circle packing) the other day as I was trying to segment a symmetrical hexagon into three equal parallelograms for the purposes of creating an illustration of an obliquely viewed cube.

It's harder than you'd think at first glance in a GUI design program that's not math or CAD software. You can't just tell most vector illustration programs to make a polygon with X sides and Z angles on each corner. The only way I figured out how to do it was to fudge it at a sub-sub-pixel resolution by eye in wireframe mode. Even the mathematically derived skews and transforms I was trying weren't matching up just right.

Granted I could do in it seconds with a t-square, a compass and a set of triangles, but desktop publishing software doesn't really work like that, when sometimes it totally should.

(Though, now that I think about it I could have used grids and guidelines like traingles and a t-square, but it was fast enough and good enough to just use the polygon tool for a hexagon and then eyeball the three segments.)

posted by loquacious at 8:43 PM on October 19, 2011

A hexagon is - I believe - the simplest polygon that does

*not*tessellate (fold) into a Platonic solid.There's something there, there, but I'll leave it to the math nerds to figure that one out. I was just musing about this (and circle packing) the other day as I was trying to segment a symmetrical hexagon into three equal parallelograms for the purposes of creating an illustration of an obliquely viewed cube.

It's harder than you'd think at first glance in a GUI design program that's not math or CAD software. You can't just tell most vector illustration programs to make a polygon with X sides and Z angles on each corner. The only way I figured out how to do it was to fudge it at a sub-sub-pixel resolution by eye in wireframe mode. Even the mathematically derived skews and transforms I was trying weren't matching up just right.

Granted I could do in it seconds with a t-square, a compass and a set of triangles, but desktop publishing software doesn't really work like that, when sometimes it totally should.

(Though, now that I think about it I could have used grids and guidelines like traingles and a t-square, but it was fast enough and good enough to just use the polygon tool for a hexagon and then eyeball the three segments.)

posted by loquacious at 8:43 PM on October 19, 2011

N=31 FTW

posted by alex_skazat at 9:17 PM on October 19, 2011 [1 favorite]

posted by alex_skazat at 9:17 PM on October 19, 2011 [1 favorite]

Is it just me, or do the packings where

Also, the packings start to look quite random once

posted by clorox at 10:43 PM on October 19, 2011

*n*is prime tend to be more symmetric than the rest? Take a look at 1027 and compare it to 1028.Also, the packings start to look quite random once

*n*gets over 150 or so.posted by clorox at 10:43 PM on October 19, 2011

I'll pay anyone $55 who finds one with five-fold rotational symmetry

posted by claudius at 10:45 PM on October 19, 2011

posted by claudius at 10:45 PM on October 19, 2011

*I'll pay anyone $55 who finds one with five-fold rotational symmetry*

I found two: 5 and 15.

posted by clorox at 10:53 PM on October 19, 2011 [3 favorites]

Holy shit, I suppose I'll have to deliver on that...

posted by claudius at 12:10 AM on October 20, 2011

posted by claudius at 12:10 AM on October 20, 2011

That, or get stuffed in your locker by guys with pocket protectors.

posted by obiwanwasabi at 12:52 AM on October 20, 2011

posted by obiwanwasabi at 12:52 AM on October 20, 2011

*That, or get stuffed in your locker by guys with pocket protectors.*

or this guy...

posted by ennui.bz at 1:12 AM on October 20, 2011

*Holy shit, I suppose I'll have to deliver on that...*

I found 5 and 15, too; memail me and I'll give you my paypal address.

posted by Philosopher Dirtbike at 1:38 AM on October 20, 2011 [1 favorite]

My description of the colors was wrong. I have no idea what they mean now.

posted by escabeche at 4:58 AM on October 20, 2011

posted by escabeche at 4:58 AM on October 20, 2011

*Is it just me, or do the packings where n is prime tend to be more symmetric than the rest?*

It wouldn't surprise me to find that odd packings will more likely have nice symmetry than even ones, just based on a tendency of a snowflake-like packing to radiate evenly from a central point with one circle at the center and even-numbered multiple-of-six and multiple-of-three rings proceeding out from there.

In other words, nice symmetry seems like it'd might be correlated with 6n+1 and 6n+4 packings, the former of which at least would be prime in some cases. Whether primality comes directly into that is another question and I guess would require sorting out all of those cases into prime vs. not prime and comparing relative rates of symmetry.

posted by cortex at 6:24 AM on October 20, 2011

But then 6n+3 packings could also look nice if you start with a core of three in a triangle; something like n=27 shows that off well. I'm not sure if that structure would translate to much larger circles, though.

posted by cortex at 6:34 AM on October 20, 2011

posted by cortex at 6:34 AM on October 20, 2011

Also, Ken Stephenson (author of the pdf) has a lot of info on circle packing and some software (Java) to implement packings at his website...

posted by ennui.bz at 7:11 AM on October 20, 2011

posted by ennui.bz at 7:11 AM on October 20, 2011

I kind of want a giant print out of all of these from 1-1000 to hang up on my wall. That would be awesome right?

posted by cirrostratus at 8:21 AM on October 20, 2011

posted by cirrostratus at 8:21 AM on October 20, 2011

Anything with hexagonal symmetry would have to have either 6n or 6n+1 circles, and 6n only if there's a hole in the middle, so with the exception of N=6 there's probably more efficient packings without hexagonal symmetry, and the only other ones you're likely to find with hexagonal symmetry are 6n+1.

Which, as cortex pointed out, are more likely to be primes (except for 2 and 3, all primes are congruent to either 1 or 5 mod 6), but I don't think the primality is directly related to symmetry.

For example, non-primes of form 6n+1 with hexagonal symmetry include 55 and 235, and primes of form 6n+1 which have no symmetry include 43, 67, and 73. (Not to mention many primes of the form 6n+5.)

posted by DevilsAdvocate at 8:33 AM on October 20, 2011

Which, as cortex pointed out, are more likely to be primes (except for 2 and 3, all primes are congruent to either 1 or 5 mod 6), but I don't think the primality is directly related to symmetry.

For example, non-primes of form 6n+1 with hexagonal symmetry include 55 and 235, and primes of form 6n+1 which have no symmetry include 43, 67, and 73. (Not to mention many primes of the form 6n+5.)

posted by DevilsAdvocate at 8:33 AM on October 20, 2011

*Holy shit, I suppose I'll have to deliver on that...*

posted by claudius at 12:10 AM on October 20 [+] [!]

posted by claudius at 12:10 AM on October 20 [+] [!]

Wow, that seemed shockingly coherent! Oh, I see - a case of mistaken identity.

posted by FatherDagon at 10:23 AM on October 20, 2011

Great post!

Amazingly counterintuitive to me.

I wouldn't have guessed, for example, that the upper limit of the number of circles that don't have to touch any other circle would continue to grow (without bound?), and I wouldn't have said the triangular/hexagonal close packing that covers the plane would appear as the entire circle packing in fully convex form only in n=3 and n=7 (and n=2, I guess), though I suppose it is the limit as n goes to infinity.

It would be nice to see a graph of the fraction of the containing circle that the packing circles cover as n increases-- some surprises there too, it looks like.

By the way, escabeche, congratulations on your audiocameo on NPR yesterday; Pesca didn't quite talk to you long enough for me to hear your voice as I read this post, but you were very interesting-- and it's too bad he didn't ask you to review the complete segment before it aired, as I imagine you would have scotched (not an ethnic slur, I hope!) what seemed to me a blatant example of a Monte Carlo fallacy in "regression to the mean" clothing there at the end.

posted by jamjam at 4:55 PM on October 20, 2011

Amazingly counterintuitive to me.

I wouldn't have guessed, for example, that the upper limit of the number of circles that don't have to touch any other circle would continue to grow (without bound?), and I wouldn't have said the triangular/hexagonal close packing that covers the plane would appear as the entire circle packing in fully convex form only in n=3 and n=7 (and n=2, I guess), though I suppose it is the limit as n goes to infinity.

It would be nice to see a graph of the fraction of the containing circle that the packing circles cover as n increases-- some surprises there too, it looks like.

By the way, escabeche, congratulations on your audiocameo on NPR yesterday; Pesca didn't quite talk to you long enough for me to hear your voice as I read this post, but you were very interesting-- and it's too bad he didn't ask you to review the complete segment before it aired, as I imagine you would have scotched (not an ethnic slur, I hope!) what seemed to me a blatant example of a Monte Carlo fallacy in "regression to the mean" clothing there at the end.

posted by jamjam at 4:55 PM on October 20, 2011

Holy crow! I listen to Pesca on the superb Slate "Hang Up And Listen" podcast and I assumed that's what he was interviewing me for -- thanks for the tip, I'd have totally missed it!

posted by escabeche at 6:46 PM on October 20, 2011

posted by escabeche at 6:46 PM on October 20, 2011

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posted by Scientist at 7:13 PM on October 19, 2011 [1 favorite]