[math-fun] Anti-kissing number puzzle
The kissing number in n dimensions is the maximum number K(n) of non-overlapping unit spheres that can be placed tangent to the one centered at the origin. Known kissing numbers: dimension kissing number —————————————————————————— 1 2 2 6 3 12 4 24 8 240 24 196560 No other kissing numbers are known. A related problem is to find the "anti-kissing number" in each dimension n: the smallest number A(n) of non-overlapping unit spheres in n-space, all tangent to the unit sphere centered at the origin, so that there's no room for any additional non-overlapping unit spheres tangent to the central one. It's obvious that A(1) = 2 and easy to show that A(2) = 4. Puzzle: What is A(3)? —Dan
I'm going to say 6, octahedral arrangement. If one more sphere could fit, then more could fit, but only 12 unit spheres can fit around a unit sphere. --Ed Pegg Jr On Monday, June 1, 2020, 06:08:05 PM CDT, Dan Asimov <dasimov@earthlink.net> wrote: The kissing number in n dimensions is the maximum number K(n) of non-overlapping unit spheres that can be placed tangent to the one centered at the origin. Known kissing numbers: dimension kissing number —————————————————————————— 1 2 2 6 3 12 4 24 8 240 24 196560 No other kissing numbers are known. A related problem is to find the "anti-kissing number" in each dimension n: the smallest number A(n) of non-overlapping unit spheres in n-space, all tangent to the unit sphere centered at the origin, so that there's no room for any additional non-overlapping unit spheres tangent to the central one. It's obvious that A(1) = 2 and easy to show that A(2) = 4. Puzzle: What is A(3)? —Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Conway's Sphere Packing, Lattices and Groups has an interesting (and surprising!) discussion of this particular case. On Mon, Jun 1, 2020 at 6:37 PM ed pegg <ed@mathpuzzle.com> wrote:
I'm going to say 6, octahedral arrangement. If one more sphere could fit, then more could fit, but only 12 unit spheres can fit around a unit sphere. --Ed Pegg Jr On Monday, June 1, 2020, 06:08:05 PM CDT, Dan Asimov < dasimov@earthlink.net> wrote:
The kissing number in n dimensions is the maximum number K(n) of non-overlapping unit spheres that can be placed tangent to the one centered at the origin.
Known kissing numbers:
dimension kissing number —————————————————————————— 1 2
2 6
3 12
4 24
8 240
24 196560
No other kissing numbers are known.
A related problem is to find the "anti-kissing number" in each dimension n: the smallest number A(n) of non-overlapping unit spheres in n-space, all tangent to the unit sphere centered at the origin, so that there's no room for any additional non-overlapping unit spheres tangent to the central one.
It's obvious that A(1) = 2 and easy to show that A(2) = 4.
Puzzle: What is A(3)?
—Dan
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I have to apologize on two counts. First, SPLAG is by both Conway and Sloane, not just Conway. Secondly, I misinterpreted the second question; as far as I recall (and can see through a quick examination), SPLAG only addresses the first question. For those interested, discussion of the first question starts at page 21 in the third edition, but there's a very interesting discussion starting on page 29 that is also related. -tom On Mon, Jun 1, 2020 at 7:07 PM Tomas Rokicki <rokicki@gmail.com> wrote:
Conway's Sphere Packing, Lattices and Groups has an interesting (and surprising!) discussion of this particular case.
On Mon, Jun 1, 2020 at 6:37 PM ed pegg <ed@mathpuzzle.com> wrote:
I'm going to say 6, octahedral arrangement. If one more sphere could fit, then more could fit, but only 12 unit spheres can fit around a unit sphere. --Ed Pegg Jr On Monday, June 1, 2020, 06:08:05 PM CDT, Dan Asimov < dasimov@earthlink.net> wrote:
The kissing number in n dimensions is the maximum number K(n) of non-overlapping unit spheres that can be placed tangent to the one centered at the origin.
Known kissing numbers:
dimension kissing number —————————————————————————— 1 2
2 6
3 12
4 24
8 240
24 196560
No other kissing numbers are known.
A related problem is to find the "anti-kissing number" in each dimension n: the smallest number A(n) of non-overlapping unit spheres in n-space, all tangent to the unit sphere centered at the origin, so that there's no room for any additional non-overlapping unit spheres tangent to the central one.
It's obvious that A(1) = 2 and easy to show that A(2) = 4.
Puzzle: What is A(3)?
—Dan
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Dan, A configuration of unit spheres, either in Euclidean space or kissing a given unit sphere, is called “saturated” when it is not possible to add another unit sphere without intersecting one of the spheres. Torquato and co-workers have speculated that “random” saturated packings (sequentially select positions from the uniform distribution on the remaining volume) achieve the *densest* possible packing in the limit of high dimension. I do not know if anyone has looked into your variant of saturated packing, where one is interested in the *lowest* density (fewest number, in the kissing case). I doubt “anti-kissing number” will catch on. Maybe “blocking number”? The corresponding property of a graph is the “minimum size of a maximal independent set”. A(3) <= 6. When you place 6 spheres at the vertices of the regular octahedron no additional spheres can be placed since the positions closest to the central sphere are at the 8 octahedron faces. But these cannot be kissing since we know a 14 kissing configuration does not exist. I’ll guess A(3) = 6, because the most symmetrical 5-sphere configuration — the triangular bi-pyramid — does not work (3 spheres can be added at the equator). -Veit
On Jun 1, 2020, at 7:07 PM, Dan Asimov <dasimov@earthlink.net> wrote:
The kissing number in n dimensions is the maximum number K(n) of non-overlapping unit spheres that can be placed tangent to the one centered at the origin.
Known kissing numbers:
dimension kissing number —————————————————————————— 1 2
2 6
3 12
4 24
8 240
24 196560
No other kissing numbers are known.
A related problem is to find the "anti-kissing number" in each dimension n: the smallest number A(n) of non-overlapping unit spheres in n-space, all tangent to the unit sphere centered at the origin, so that there's no room for any additional non-overlapping unit spheres tangent to the central one.
It's obvious that A(1) = 2 and easy to show that A(2) = 4.
Puzzle: What is A(3)?
—Dan
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participants (4)
-
Dan Asimov -
ed pegg -
Tomas Rokicki -
Veit Elser