User:Tomruen/Noncrystallographic root systems

There are just a few irreducible noncrystallographic root systems: H4, H3, H2, and I2(p) for p=5,7,8....

Folding A4, D6, and E8
These can be constructed from simply laced root systems. H4 is a folding of E8, H3 is a folding of D6, and H2 is a folding of A4. A final H1 can be seen as a folding of A1A1. The ratio in length of the long to short roots is the Golden ratio, &phi;.
 * 1) A1A1 has 4 roots, from a square, {}×{},, being the vertex figure of the apeirogon product:.
 * 2) * H1, as a folding of A1A1:, has 2 sets of 2 root, each seen as the vertices of a digon:.
 * 3) A4 has 20 roots, from runcinated 5-cell polytope, t0,3{3,3,3},, being the vertex figure of the 5-cell honeycomb and A4 lattice:.
 * 4) * H2, as a folding of A4:, has 2 sets of 10 roots, each seen as the vertices of a decagon:.
 * 5) D6 has 60 roots, from the rectified 6-orthoplex polytope, t1{3,3,3,31,1},, being the vertex figure of the 222 honeycomb and D6 lattice:.
 * 6) * H3, as a folding of D6:, has 2 sets of 30 roots, each seen as the vertices of a icosidodecahedron:.
 * 7) E8 has 240 roots, from the 421 polytope, {3,3,3,3,32,1},, being the vertex figure of the 521 honeycomb and E8 lattice:.
 * 8) * H4, as a folding of E8:, has 2 sets of 120 roots, each seen in the vertices of the 600-cell:.

Rank 2
Here I2(p) as a folding of Ap-1. I2(p) is considered the undirected group, while this article references the directed ones.