User:Tomruen/Noncrystallographic root systems

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

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, t_0,3{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{3,3,3,3^1,1}, , 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, {3,3,3,3,3^2,1}, , 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

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	$\left[{\begin{smallmatrix}2&0\\0&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-1\\-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-2\\-1&2\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}2&-3\\-1&2\end{smallmatrix}}\right]$
Roots	4 roots	6 roots	12 roots 4×2 roots	24 roots 6×2 roots
Polytope

Here I2(p) as a folding of A_p-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

• Quaternionic Roots of E 8 Related Coxeter Graphs and Quasicrystals Mehmet KOCA, Nazife Ozdes KOCA, 1998 [1]
• Noncrystallographic Coxeter group H4 in E8 Mehmet Koca, Ramazan Koc and Muataz Al-Barwani, 2001
• Icosidodecahedron from D6 John Baez / 1 January, 2015