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User:Tomruen/Noncrystallographic root systems

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Dynkin diagram foldings, showing related Coxeter ring diagram polytopes.

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

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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.

  1. A1A1 has 4 roots, from a square, {}×{}, , being the vertex figure of the apeirogon product: .
    • H1, as a folding of A1A1: , has 2 sets of 2 root, each seen as the vertices of a digon: .
  2. 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: .
    • H2, as a folding of A4: , has 2 sets of 10 roots, each seen as the vertices of a decagon: .
  3. 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: .
    • H3, as a folding of D6: , has 2 sets of 30 roots, each seen as the vertices of a icosidodecahedron: .
  4. 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: .
    • H4, as a folding of E8: , has 2 sets of 120 roots, each seen in the vertices of the 600-cell: .
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

Rank 2

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Rank 2 crystallographic root systems
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

References

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