Lattice in 8-dimensional space with special properties
In mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E8 root system.
The norm[1] of the E8 lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8.
The existence of such a form was first shown by H. J. S. Smith in 1867,[2] and the first explicit construction of this quadratic form was given by Korkin and Zolotarev in 1873.[3]
The E8 lattice is also called the Gosset lattice after Thorold Gosset who was one of the first to study the geometry of the lattice itself around 1900.[4]
^In this article, the norm of a vector refers to its length squared (the square of the ordinary norm).
^Smith, H. J. S. (1867). "On the orders and genera of quadratic forms containing more than three indeterminates". Proceedings of the Royal Society. 16: 197–208. doi:10.1098/rspl.1867.0036.
^Korkin, A.; Zolotarev, G. (1873). "Sur les formes quadratiques". Mathematische Annalen. 6: 366–389. doi:10.1007/BF01442795.
^Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.
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