19 of them, yoursGallery

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Two hundred and forty dots in eight rings, every dot joined to fifty-six others by faint lines, the whole thing turning about a dark centre with thirty-fold symmetry. The dots come in three colours: amber, teal and rose.

E8, all of it
240 roots, 6,720 lines, coloured by the three-way split The E8 shadow with each root coloured by which of three eigenspaces it falls in: 84 amber, 78 teal, 78 rose.

The E8 shadow with each root coloured by which of three eigenspaces it falls in: 84 amber, 78 teal, 78 rose.

E8 cut three ways
84 fixed, 78 and 78 turning The E8 shadow with 112 roots in teal and 128 in rose.

E8 cut two ways
120 + 128 The E8 root system in the Coxeter plane: 240 dots on eight rings of thirty.

E8
eight rings of thirty The E7 root system: 126 dots on seven rings of eighteen.

E7
126 roots The E6 root system: 72 dots, twelve-fold.

E6
72 roots The F4 root system: 48 dots on four rings of twelve.

F4
48 roots The D4 root system: 24 dots, six-fold.

D4
24 roots — the one with triality The G2 root system: 12 dots, six long and six short.

G2
12 roots — the octonions' own symmetry A table of five algebras against four laws. The reals and the complex numbers keep every law; the quaternions lose commutativity; the octonions lose associativity; the sedenions lose the norm law too.

A table of five algebras against four laws. The reals and the complex numbers keep every law; the quaternions lose commutativity; the octonions lose associativity; the sedenions lose the norm law too.

The ladder
what each doubling costs, measured Seven labelled circles at the corners, edge midpoints and centre of a triangle, joined by three sides, three medians and one inner circle, with arrows around each line.

Seven labelled circles at the corners, edge midpoints and centre of a triangle, joined by three sides, three medians and one inner circle, with arrows around each line.

The wheel
the octonion table as seven lines Four coloured grids, two by two up to eight by eight, one for each algebra's multiplication table.

The tables
ℂ, ℍ, 𝕆 and split 𝕆, coloured Twenty-four dots in three colours with three-fold symmetry: a third of a turn carries each colour onto the next.

8v, 8s, 8c
the three faces of so(8) A three-pointed star with a rotation arrow, beside the three exceptional diagrams drawn as lines of nodes with one node branching off.

The diagrams
D4's three-fold turn, and E6, E7, E8 A circle with three spokes 120 degrees apart, labelled I, II and III, beside the rotation matrix that carries one spoke to the next.

A circle with three spokes 120 degrees apart, labelled I, II and III, beside the rotation matrix that carries one spoke to the next.

A third of a turn
what triality does to a generator Four boxes: su(2), su(3), sp(3) and f4, each showing its triality algebra plus three copies of a division algebra.

Four boxes: su(2), su(3), sp(3) and f4, each showing its triality algebra plus three copies of a division algebra.

su(3, 𝔻)
four sizes of one shape A seven by seven grid of Lie algebras, each box carrying a name, a dimension and the arithmetic behind it. E8 at 248 sits in the corner.

A seven by seven grid of Lie algebras, each box carrying a name, a dimension and the arithmetic behind it. E8 at 248 sits in the corner.

The magic square
forty-nine boxes, one formula Six stacked bars, each split into a fixed part and two equal moving parts.

The three-way splits
fixed part and two halves

The share card