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.