Writing series
A series on the 120 unit icosians, and how far one set of quaternions goes.
There are exactly 120 unit icosians. They are quaternions, they are the rotational symmetry of the icosahedron, and their arithmetic runs on the golden ratio. That is already three descriptions of one object, and this series is an attempt to follow it as far as it goes.
Part 1 fixes the arithmetic. Part 2 turns the icosians into the E8 lattice twice over, by two routes that look nothing alike. Part 3 asks what shape they make and finds the 600-cell, the four-dimensional solid with no analogue above four. Part 4 follows them up to sixteen dimensions, where the heterotic string is waiting. Part 5 steps down instead of up, to the older and easier arithmetic of the Gaussian and Eisenstein integers, and asks why five-fold symmetry is the difficult one.
from a pentagon's proportion to the symmetry of the icosahedron
The golden ratio is the arithmetic backbone of the icosians - a ring of quaternions that is the rotational symmetry of the icosahedron, and a doorway to the E8 lattice.
the same 120 quaternions seed the E8 lattice twice, by two routes that look nothing alike
The 120 icosians seed the E8 lattice twice over - once arithmetically, once through the McKay correspondence.
not just a group and a lattice, but a shape
The 120 unit icosians are the vertices of the 600-cell, the regular four-dimensional solid with no analogue in any dimension above four.
sixteen dimensions, two even unimodular lattices, and which one the icosians pick
Past E8 are the sixteen-dimensional even unimodular lattices the heterotic string was built on. The icosian ring has something to say about which is which.
step down from five-fold symmetry to four and three, and the arithmetic gets older and easier
The golden ring is the exotic case. Step down to four-fold and three-fold symmetry and you land on the Gaussian and Eisenstein integers.