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The 112 roots with integer entries form a D 8 root system. The E 8 root system also contains a copy of A 8 (which has 72 roots) as well as E 6 and E 7 (in fact, the latter two are usually defined as subsets of E 8). In the odd coordinate system, E 8 is given by taking the roots in the even coordinate system and changing the sign of any one ...
The root system E 7 is the set of vectors in E 8 that are perpendicular to a fixed root in E 8. The root system E 7 has 126 roots. The root system E 6 is not the set of vectors in E 7 that are perpendicular to a fixed root in E 7, indeed, one obtains D 6 that way. However, E 6 is the subsystem of E 8 perpendicular to two suitably chosen roots ...
The name derives from the fact that it is the root lattice of the E 8 root system. The norm [1] of the E 8 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.
For root systems, no root maps to zero, corresponding to the Coxeter element not fixing any root or rather axis (not having eigenvalue 1 or −1), so the projections of orbits under w form h-fold circular arrangements [9] and there is an empty center, as in the E 8 diagram at above right. For polytopes, a vertex may map to zero, as depicted below.
A binary clock might use LEDs to express binary values. In this clock, each column of LEDs shows a binary-coded decimal numeral of the traditional sexagesimal time.. The common names are derived somewhat arbitrarily from a mix of Latin and Greek, in some cases including roots from both languages within a single name. [27]
E 8, an exceptional simple Lie group with root lattice of rank 8; E 8 lattice, special lattice in R 8; E 8 manifold, mathematical object with no smooth structure or topological triangulation; E 8 polytope, alternate name for the 4 21 semiregular (uniform) polytope; Elementary abelian group of order 8
An alternative (6-dimensional) description of the root system, which is useful in considering E 6 × SU(3) as a subgroup of E 8, is the following: All () permutations of (,,,,,) preserving the zero at the last entry,
An alternative (7-dimensional) description of the root system, which is useful in considering E 7 × SU(2) as a subgroup of E 8, is the following: All () permutations of (±1,±1,0,0,0,0,0) preserving the zero at the last entry, all of the following roots with an even number of + 1 / 2