Editing Leech lattice (section)

==Geometry==

{{harvtxt|Conway|Parker|Sloane|1982}} showed that the covering radius of the Leech lattice is <math>\sqrt 2</math>; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space. The points at distance at least <math>\sqrt 2</math> from all lattice points are called the '''''deep holes''''' of the Leech lattice. There are 23 orbits of them under the automorphism group of the Leech lattice, and these orbits correspond to the 23 [[Niemeier lattices]] other than the Leech lattice: the set of vertices of  deep hole is isometric to the affine Dynkin diagram of the corresponding Niemeier lattice.

The Leech lattice has a density of <math>\tfrac{\pi^{12}}{12!}\approx 0.001930</math>. {{harvtxt|Cohn|Kumar|2009}} showed that it gives the densest lattice [[sphere packing|packing of balls]] in 24-dimensional space. {{harvs|txt|title=The sphere packing problem in dimension 24|
first1=Henry|last1= Cohn|first2= Abhinav|last2= Kumar|first3= Stephen D. |last3=Miller|first4= Danylo |last4=Radchenko|first5= Maryna |last5=Viazovska|year=2017|arxiv=1603.06518}} improved this by showing that it is the densest sphere packing, even among non-lattice packings.

The 196560 minimal vectors are of three different varieties, known as ''shapes'':

* <math>1104 = \binom {24}{2} \cdot 2^2</math> vectors of shape (4<sup>2</sup>,0<sup>22</sup>), for all permutations and sign choices;
* <math>97152 = 759 \cdot 2^8 \cdot \frac {1}{2}</math> vectors of shape (2<sup>8</sup>,0<sup>16</sup>), where the '2's correspond to an octad in the Golay code, and there are any even number of minus signs;
* <math>98304 = 2^{12} \cdot 24</math> vectors of shape (∓3,±1<sup>23</sup>), where the lower sign is used for the '1's of any codeword of the Golay code, and the '∓3' can appear in any position.

The [[ternary Golay code]], [[binary Golay code]] and Leech lattice give very efficient 24-dimensional [[spherical code]]s of 729, 4096 and 196560 points, respectively. Spherical codes are higher-dimensional analogues of [[Tammes problem]], which arose as an attempt to explain the distribution of pores on pollen grains. These are distributed as to maximise the minimal angle between them. In two dimensions, the problem is trivial, but in three dimensions and higher it is not. An example of a spherical code in three dimensions is the set of the 12 vertices of the regular icosahedron.