The E8 lattice for Beginners: Understand the E8 structure behind physics using an easy game

Описание: We describe an easy to think about board game that describes the exceptional
Lie lattice E8 exactly without needing to use any mathematical terminology or algebra.
This is remarkable because E8 has been proposed by physicists as a theory of everything


Some scientists believe that the the laws of physics are all described by the E8 lattice. In a sense E8 is a structure in 8 dimensions, but since it corresponds to the largest exceptional Lie group
it very mathematically central. Recently Wildberger showed a dramatically
simpler way to study ADE graphs in these videos


There is a way to understand the exact structure of the E8 lattice that underlyies this theory of fundamental physics

An Exceptionally Simple Theory of Everything A. Garrett Lisi
https://arxiv.org/abs/0711.0770



In terms of a simple game called the "mutation game", which is played on a graph. This changes the populations, which we obtain descriptions of from Wildberger

The Ubiquity of ADE Graphs, and the Mutation and Numbers Games | Math Seminars | NJ Wildberger
   • The Ubiquity of ADE Graphs, and the Mutati...  

   • Exceptional Structures in Mathematics and ...  

Wildberger's mutation game gives an elementary way to understand the E8 lattice without requiring any prior math skills. I'm very new to these ideas, and so try try and test my understand I've tried to make this video to see how far I could get towards describing E8 without using any math notation. But I how experts in E8/Lie theory etc. will understand that I'm very much a newcomer in this subject, and I hope they will point out and forgive mistakes I make. Like I say, this is how I see Wildberger's way of looking at ADE graphs like E8 using his population game.
No math skills needed

So we have n dimensional space, with an unusual dot product (non-euclidean geometry). Here
n is the number of cities/vertices in our graph.
A population allocation is hence a vector/arrow in n dimensional "population space"
A unit vector associated with blue city is population allocated to blue city
and other cities/vertices has zero population.
Dotting a unit vector with self yields 2.
Dotting unit vectors of adjacent cities yields -1
Dotting unit vectors of distinct non-adjacent vertices yields 0.
Every population can be written as a linear combination of unit vectors,
and so we can understand how the dot product works on general
population vectors by algebraically as usual. Given the blue city there
will be a corresponding blue population unit vector, and there will be a "blue hyperplane"
spanned by the vectors that are perpendicular to our blue unit vector (they are perpendicular in
the sense that when we dot them with the blue unit vector we get zero), and
doing a mutation of the blue city corresponds to a linear map from the population space to itself.
This linear map / square matrix corresponds to reflecting everything in
the hyperplane perpendicular to the blue unit vector. Population mutation
moves correspond to reflections in hyperplanes, that gives a Kaledescope
like described by Coxter,'s book on regular polytopes and this can give the E8 lattice.

When we
play this "mutation game" of Wildberger on the E8 graphs (which is one of
the ADE graphs known to explain lots of Lie theory).