Triacontagonal coordinates for the E(8) root system

TL;DR

This paper presents an explicit, symmetric formula for the E(8) root system elements, highlighting the triacontagonal symmetry derived from the Coxeter number 30, which clarifies the structure of E(8).

Contribution

It provides the first explicit triacontagonally symmetric formula for E(8) roots, revealing new geometric insights into the root system.

Findings

Explicit triacontagonal symmetric formula for E(8) roots

Demonstrates cyclic group action of order 30 on the root system

Connects Coxeter number to geometric symmetry in E(8)

Abstract

This note gives an explicit formula for the elements of the E(8) root system. The formula is triacontagonally symmetric in that one may clearly see an action by the cyclic group with 30 elements. The existence of such a formula is due to the fact that the Coxeter number of E(8) is 30.