Editing Leech lattice (section)

===Using the Lorentzian lattice II<sub>25,1</sub>===

The Leech lattice can also be constructed as <math>w^\perp/w</math> where ''w'' is the Weyl vector:

:<math>(0,1,2,3,\dots,22,23,24; 70)</math>

in the 26-dimensional even Lorentzian [[unimodular lattice]] [[II25,1|II<sub>25,1</sub>]]. The existence of such an integral vector of Lorentzian norm zero relies on the fact that 1<sup>2</sup> + 2<sup>2</sup> + ... + 24<sup>2</sup> is a [[square number|perfect square]] (in fact 70<sup>2</sup>); the [[24 (number)|number 24]] is the only integer bigger than 1 with this property (see [[cannonball problem]]). This was conjectured by [[Édouard Lucas]], but the proof came much later, based on [[elliptic functions]].

The vector
<math>(0,1,2,3,\dots,22,23,24)</math>
in this construction is really the [[Weyl vector]] of the even sublattice ''D''<sub>24</sub> of the odd unimodular lattice ''I''<sup>25</sup>. More generally, if ''L'' is any positive definite unimodular lattice of dimension 25 with at least 4 vectors of norm 1, then the Weyl vector of its norm 2 roots has integral length, and there is a similar construction of the Leech lattice using ''L'' and this Weyl vector.