## 1887 – E8

In 1887, Willhelm Killing came up with a classification scheme for Lie algebras. The designation E-8 was eventually applied to an exceptional, simple Lie algegra and the associated Lie group of rank 8 and dimension 248. This is one of the most complex objects known to mankind.

**Mathematicians Map E8** – [aimath.org]

Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E8. This achievement is significant both as an advance in basic knowledge and because of the many connections between E8 and other areas, including string theory and geometry. The magnitude of the calculation is staggering: the answer, if written out in tiny print, would cover an area the size of Manhattan. Mathematicians are known for their solitary work style, but the assault on E8 is part of a large project bringing together 18 mathematicians from the U.S. and Europe for an intensive four-year collaboration.

**‘Most beautiful’ math structure appears in lab for first time** – [newscientist.com]

07 January 2010

A complex form of mathematical symmetry linked to string theory has been glimpsed in the real world for the first time, in laboratory experiments on exotic crystals.

Mathematicians discovered a complex 248-dimensional symmetry called E8 in the late 1800s. The dimensions in the structure are not necessarily spatial, like the three dimensions we live in, but they correspond to mathematical degrees of freedom, where each dimension represents a different variable.

**A graphical representation of the root system of E-8** – [mit.edu]

In eight dimensions each root is a vector of the same length (often chosen to be the square root of 2). In this projection, some of the vectors are shorter. In fact there are 30 vectors of the greatest possible length (the small dots on the outer circle), 30 of the next greatest length, and so on. There may appear to be lots more than eight circles of dots; many of the smaller ones are visual artifacts of intersecting lines in the drawing.

Each root is connected by a line to its 56 nearest neighbors, each of which is at distance square root of two; so two adjacent roots and the origin make an equilateral triangle in eight dimensions. After projection to two dimensions, pairs of these 56 lines coincide, so you see only 28 lines coming out of each root image. The picture does not show lines from the origin to each root.

PRECURSOR:

1799 – Algebraic solution to quintic equation doesn’t exist (**Ruffini**)

1824 – No algebraic solution to quintic and higher equations (**Abel**)

**1829 – Non-Euclidean geometry**

1831 – Solvability question (**Galois**)

1873 – **Lie Groups**

CONCURRENT:

1894 – Classification extended (Cartan)

SUBSEQUENT:

SEE ALSO:

**Timeline of Group Theory**

[…] 1887 – discovery of exceptional group E-8 […]

[…] CONCURRENT: 1838 – Jordan (Camille) 1849 – Klein 1887 – E8 […]