According to en.wikipedia.org, https://en.wikipedia.org/wiki/Kissing_number#Some_known_bounds

It says the kissing numbers $K$ have lower bound $K_L$ and upper bound $K_S$: $$ K_L < K < K_U. $$

I am slightly puzzled by what these lower bounds and upper bounds really constrain.

enter image description here

Here are my puzzles:

  1. Isn't that the lower bound can be just 0? The ball does not kiss with its nearest neighbors?

  2. Suppose the goal is to achieve the densest sphere packing - then shouldn't the solution always be the highest kissing number $K_U$? What are the exceptions?

  3. Are there examples of the lower bound kissing numbers $K_L$ that can give rise to the densest sphere packing?

  4. Only when the solutions are found, then the $K_L=K_U$ always? What exactly is the logic here? Why not $K_L \neq K_U$ and $K_L < K < K_U$ at dimensions = 1,2,3,8,24?

  5. How about dimension = 4? Why $K_L = K = K_U = 24$?


  • 1
    $\begingroup$ Maybe you'd be interested in the paper "A survey on the kissing numbers" by Peter Boyvalenkov, Stefan Dodunekov, and Oleg R. Musin: arxiv.org/abs/1507.03631 $\endgroup$ May 29 at 20:38

1 Answer 1


Lots of questions here, I'll see how many I can address. To be clear, $K_L$ and $K_U$ are summaries of our current knowledge. There is some true kissing number in each dimension, and hopefully we'll learn what it is someday; this is just what we know now.

(1) The $n$-dimensional kissing number, $K$, is defined to be the maximum number of non-overlapping spheres capable of touching an $S^{n-1}$. So saying that $K_L \leq K$ means that we have found a way to place $K_L$ many non-overlapping spheres so that they touch an $S^{n-1}$. Of course, we could place fewer spheres in contact with a sphere (as you say, we could place $0$), but $K_L$ is the largest number of spheres we have managed to place.

(2) Probably not. The conjectured best packings in $9$ and $10$ dimensions are the $9$ and $10$ dimensional laminated lattices, with kissing numbers $272$ and $336$ (see here) and the conjectured best lattice in $12$ dimensions is the Coxeter-Todd lattice with kissing number $756$; these are less than the numbers in your table. I see no reason that the densest packing should also have the highest kissing number, but since we have very little understanding of either problem, I don't know if we can rule it out.

(3), (4) and (5): The dimensions $1$, $2$, $8$ and $24$ are very special, because we know exactly what the densest lattice is and it also achieves the maximum kissing number. In dimension $3$, the Kepler conjecture, now proved by Hales, establishes that the optimal density is achieved by the face centered cubic lattice, which indeed has kissing number $12$. I don't know what is going on in dimension $4$.

Again, what is going on here is that there is special structure which allows us to know $K$ exactly in these cases, and so $K_L = K_U$. The other cases where $K_L < K_U$ in the table, reflect our current ignorance, not eternal mathematical truth.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.