There are just a few irreducible noncrystallographic root systems: H4, H3, H2, and I2(p) for p=5,7,8....
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, φ.
A1A1 → H1+φH1 | A4 → H2+φH2 | D6 → H3+φH3 | E8 → H4+φH4 |
---|---|---|---|
4 roots 2×2 roots |
20 roots 10×2 roots |
60 roots 30×2 roots |
240 roots 120×2 roots |
Simply-laced group folding |
D2 A1+A1 |
A2 | A3 to B2 D2+√2 D2 |
D4 to G2 A2+√3 A2 |
---|---|---|---|---|
Dynkin | ||||
Coxeter | ||||
Cartan | ||||
Roots |
4 roots |
6 roots |
12 roots 4×2 roots |
24 roots 6×2 roots |
Polytope |
Here I2(p) as a folding of Ap-1. I2(p) is considered the undirected group, while this article references the directed ones.
Crystallographic | Non-crystallographic | ||||||||
---|---|---|---|---|---|---|---|---|---|
Simply-laced group folding |
A2 → I2(3)×2 → I2(6) |
A3 → I2(3)×2 → G2 |
A4 → I2(5)×2 → I2(10) |
A5 → I2(5)×2 → I2(10) |
A6 → I2(7)×2 → I2(14) |
A7 → I2(7)×2 → I2(14) |
A8 → I2(9)×2 → I2(18) | ||
Dynkin | |||||||||
Coxeter | → | → → | → → | → → | → → | → → | → → | ||
Plane | A2 | A4 | A6 | A8 | |||||
Roots Polytope |
6 roots 6×1 roots |
12 roots 6×2 roots |
20 roots 10×2 roots |
30 roots 10×3 roots |
42 roots 14×3 roots |
56 roots 14×4 roots |
72 roots 18×4 roots | ||
Ring Radius ratios |
1 | √3:1 | φ:1 | φ:1:? | ?:?:1 | ?:?:?:1 | ?:?:?:1 |