Not being well-versed in the area, I looked up "Kissing Number" at Mathworld.

In the table, I notice that the "NL" (non lattice kissing number) values are missing

for dimensions 1 thru 8. Don't we know that NL(1,2,3) = (2,6,12)? And certainly

at worst NL(d) >= L(d) for d = 4 thru. I gather from our current discussion that we

know 24 = L(4) <= NL(4) <= 25, and are discussing means to establish or eliminate

NL(4) = 25 (though I've been following the thread only cursorily). Do we know of

any lower bounds better than NL(d) for d <= 8 that we might put in the table?

Also, bear with me here, I'm going to rehash some elementary stuff. In 2

dimensions, we easily see that a unit circle kissing unit circle S "uses up" pi/3

radians of the circumference of S, which cannot be used by any other kissing

circle. This bounds NL(2) <= (2pi)/(pi/3) = 6. Similarly, a unit sphere kissing

unit sphere S "uses up" an area A of the surface of C subsumed by a circle

with a 60-degree diameter, so that NL(3) <= (4pi)/A. Does this observation

generalize to higher dimensions? Does it result in an upper bound formula

for NL(d)? Is there a nice bounding asymptotic? Does the formula grow

relatively worse or better for increasing d? I'm sure this is textbook, but I've

never seen it developed. If there is an accessible online development, that

would be great.