the moment map and non–periodic tilings -...

The moment map and non–periodic tilings

Elisa Prato

Universita degli Studi di Firenze

June 28th, 2012

joint work with Fiammetta Battaglia

Delzant theorem

Delzant theorem

Delzant theorem

Delzant theorem

Delzant theorem

Delzant polytopes ⇐⇒ symplectic toric manifolds

Delzant theorem

Delzant theorem


Delzant polytope

Delzant theorem

Delzant theorem


Delzant polytope

a Delzant polytope is a simple convex polytope ∆ ⊂ (Rn)∗ that isrational with respect to a lattice L ⊂ R

n and satisfies an additional”smoothness” condition

Delzant theorem

Delzant theorem


Delzant polytope



symplectic toric manifold

Delzant theorem

Delzant theorem


Delzant polytope




a symplectic toric manifold is a 2n–dimensional compact connectedsymplectic manifold M with an effective Hamiltonian action of thetorus T = R

n/L

Delzant theorem

Delzant theorem


Delzant polytope




a symplectic toric manifold is a 2n–dimensional compact connectedsymplectic manifold M with an effective Hamiltonian action of thetorus T = R

n/L

If Φ is the moment mapping of this action we have Φ(M) = ∆

Delzant construction



∆ =⇒ M

explicit construction using symplectic reduction



∆ =⇒ M


idea



∆ =⇒ M


idea

◮ ∆ =⇒ N, a subtorus of T d = Rd/Zd , d being the number of

facets of ∆



∆ =⇒ M


idea


facets of ∆

◮ M = Ψ−1(0)N

, where Ψ is a moment mapping for the inducedaction of N on C

d



∆ =⇒ M


idea


facets of ∆

◮ M = Ψ−1(0)N

, where Ψ is a moment mapping for the inducedaction of N on C

d

◮ M inherits an action of T d/N ≃ T = Rn/L from the standard

action of T d on Cd

generalized Delzant construction


natural question


natural question

what if ∆ is any (not necessarily rational) simple convex polytopein (Rn)∗?


natural question




natural question



∆ =⇒ M



natural question



∆ =⇒ M


formally, it works exactly like the Delzant construction but ...


◮ the lattice L is replaced by a quasilattice Q



◮ rationality is replaced by quasirationality




◮ N is a general subgroup of T d , not necessarily a subtorus





◮ M is a 2n–dimensional compact connected quasifold





◮ M is a 2n–dimensional compact connected quasifold

◮ the torus is replaced by a quasitorus T d/N ≃ Rn/Q

quasifold geometry

quasifold geometry

quasilattice

quasifold geometry

quasilattice

a generalization of a lattice L ⊂ Rn, a quasilattice Q is the Z–span

of a set of spanning vectors, Y1, . . . ,Yd , of Rn

quasifold geometry

quasilattice



quasifold

quasifold geometry

quasilattice



quasifold

a generalization of a manifold and a orbifold, a quasifold is locallymodeled by an open subset of a k–dimensional manifold modulothe smooth action of a discrete group

quasifold geometry

quasilattice



quasifold


quasitorus

quasifold geometry

quasilattice



quasifold


quasitorus

a generalization of a torus Rn/L, a quasitorus is the quotientRn/Q, Q being a quasilattice

rationality vs quasirationality

rationality vs quasirationalitygiven any convex polytope ∆ ⊂ (Rn)∗, then there exist vectorsX1, . . . ,Xd ∈ R

n and real numbers λ1, . . . , λd such that

∆ =d⋂

j=1

{ µ ∈ (Rn)∗ | 〈µ,Xj〉 ≥ λj }



∆ =d⋂

j=1

{ µ ∈ (Rn)∗ | 〈µ,Xj〉 ≥ λj }

rational polytope



∆ =d⋂

j=1

{ µ ∈ (Rn)∗ | 〈µ,Xj〉 ≥ λj }

rational polytope

we recall that ∆ ⊂ (Rn)∗ is rational if there exists a lattice L ⊂ Rn

such that the vectors X1, . . . ,Xd can be chosen in L



∆ =d⋂

j=1

{ µ ∈ (Rn)∗ | 〈µ,Xj〉 ≥ λj }

rational polytope



quasirational polytope



∆ =d⋂

j=1

{ µ ∈ (Rn)∗ | 〈µ,Xj〉 ≥ λj }

rational polytope




we say that ∆ ⊂ (Rn)∗ is quasirational with respect to aquasilattice Q ⊂ R

n if the vectors X1, . . . ,Xd can be chosen in Q



∆ =d⋂

j=1

{ µ ∈ (Rn)∗ | 〈µ,Xj〉 ≥ λj }

rational polytope






remark



∆ =d⋂

j=1

{ µ ∈ (Rn)∗ | 〈µ,Xj〉 ≥ λj }

rational polytope






remarkany given convex polytope is quasirational with respect to thequasilattice that is generated by the vectors X1, . . . ,Xd

Penrose tilings

Penrose tilings

2 examples non-periodic tilings of the plane

Penrose tilings

2 examples non-periodic tilings of the plane

Figure: a rhombus tiling Figure: a kite and dart tiling

figures by D. Austin, reprinted courtesy of the AMS

convex Penrose tiles


◮ the thin rhombus



◮ the thick rhombus



◮ the thick rhombus

◮ the kite

geometric properties of the tiles


◮ the tiles are obtained from a regular pentagon with a verysimple geometric construction



◮ all angles of the tiles are multiples of π5




◮ the following are all equal to the golden ratio

φ = 1+√5

2 = 2cos π5 :





φ = 1+√5

2 = 2cos π5 :

◮ the ratio of the edge of the thin rhombus to its short diagonal





φ = 1+√5

2 = 2cos π5 :

◮ the ratio of the edge of the thin rhombus to its short diagonal◮ the ratio of the long diagonal of the thick rhombus to its edge





φ = 1+√5

2 = 2cos π5 :

◮ the ratio of the edge of the thin rhombus to its short diagonal◮ the ratio of the long diagonal of the thick rhombus to its edge◮ the ratio of the long edge of the kite to its short edge

rationality issues

rationality issues

◮ the rhombuses of a given tiling are not simultaneously rationalwith respect to a same lattice

rationality issues


◮ there exists no lattice with respect to which any kite is rational

rationality issues



idea

rationality issues



ideafind an appropriate quasilattice

choice of quasilattice for the Penrose tilings


Let us consider the quasilattice Q ⊂ R2 generated by the vectors

Y0 = (1, 0)

Y1 = (cos 2π5 , sin 2π

5 ) = 12(

1φ ,

√2 + φ)


5 ) = 12(−φ, 1

φ

√2 + φ)


5 ) = 12(−φ,− 1

φ

√2 + φ)


5 ) = 12(

1φ ,−

√2 + φ)


facts


facts

◮ any rhombus, thick or thin, of a given rhombus tiling isquasirational with respect to Q


facts

◮ any rhombus, thick or thin, of a given rhombus tiling isquasirational with respect to Q

◮ any kite of a given kite and dart tiling is quasirational withrespect to Q

symplectic geometry of the rhombus tiling


we apply the generalized Delzant construction and we get



=⇒ M = S2r ×S2

r

Γ



=⇒ M = S2r ×S2

r

Γ

=⇒ M =S2R×S2

R

Γ

why? what are r , R and Γ?


thin rhombus


thin rhombus

◮ symplectic reduction yields M = S3r ×S3

r

N, where

N ={

exp (s, s + hφ, t, t + kφ) ∈ T 4 | s, t ∈ R, h, k ∈ Z}

and

r =(

12φ

√2 + φ

)1/2


thin rhombus


r

N, where

N ={


and

r =(

12φ

√2 + φ

)1/2

◮ consider S1 × S1 = { exp (s, s, t, t) ∈ T 4 | s, t ∈ R } ⊂ N


thin rhombus


r

N, where

N ={


and

r =(

12φ

√2 + φ

)1/2


◮ then M = S2r ×S2

r

Γ , with Γ = NS1×S1


thin rhombus


r

N, where

N ={


and

r =(

12φ

√2 + φ

)1/2



r


thick rhombus


thin rhombus


r

N, where

N ={


and

r =(

12φ

√2 + φ

)1/2



r


thick rhombus

◮ same, with R =(

12

√2 + φ

)1/2instead of r

symplectic geometry of the kite and dart tiling


=⇒ M =


=⇒ M =

{

(z1,z2,z3,z4)∈C4 | | z1|2+ 1φ|z2|2+|z3|2=

√

2+φ

2,−|z1|2+|z2|2+φ|z4|2=

√

2+φ

2φ

}

{ exp(−s+φt,s,t,−t+φs)∈T 4 | s,t∈R }


=⇒ M =

{

(z1,z2,z3,z4)∈C4 | | z1|2+ 1φ|z2|2+|z3|2=

√

2+φ

2,−|z1|2+|z2|2+φ|z4|2=

√

2+φ

2φ

}


remark


=⇒ M =

{

(z1,z2,z3,z4)∈C4 | | z1|2+ 1φ|z2|2+|z3|2=

√

2+φ

2,−|z1|2+|z2|2+φ|z4|2=

√

2+φ

2φ

}


remarkone can show that M is not the global quotient of a manifoldmodulo the action of a discrete group

an example of a chart

consider the open subset of C2 given by U ={

(z1, z2) ∈ C2 | |z1|2 + 1

φ |z2|2 <√2+φ2 , −|z1|2 + |z2|2 <

√2+φ2φ

}



(z1, z2) ∈ C2 | |z1|2 + 1

φ |z2|2 <√2+φ2 , −|z1|2 + |z2|2 <

√2+φ2φ

}

and the following slice of Ψ−1(0) that is transversal to the N–orbits

Uτ→ {(z1, z2, z3, z4) ∈ Ψ−1(0) | z3 6= 0, z4 6= 0}

(z1, z2) 7→(

z1, z2,√√

2+φ2 − |z1|2 − 1

φ |z2|2,√√

2+φ2φ2 + |z1|2−|z2|2

φ

)



(z1, z2) ∈ C2 | |z1|2 + 1

φ |z2|2 <√2+φ2 , −|z1|2 + |z2|2 <

√2+φ2φ

}


Uτ→ {(z1, z2, z3, z4) ∈ Ψ−1(0) | z3 6= 0, z4 6= 0}

(z1, z2) 7→(

z1, z2,√√

2+φ2 − |z1|2 − 1

φ |z2|2,√√

2+φ2φ2 + |z1|2−|z2|2

φ

)

it induces the homeomorphism

U/Γτ−→ U

[(z1, z2)] 7−→ [τ(z1, z2)]



(z1, z2) ∈ C2 | |z1|2 + 1

φ |z2|2 <√2+φ2 , −|z1|2 + |z2|2 <

√2+φ2φ

}


Uτ→ {(z1, z2, z3, z4) ∈ Ψ−1(0) | z3 6= 0, z4 6= 0}

(z1, z2) 7→(

z1, z2,√√

2+φ2 − |z1|2 − 1

φ |z2|2,√√

2+φ2φ2 + |z1|2−|z2|2

φ

)


U/Γτ−→ U

[(z1, z2)] 7−→ [τ(z1, z2)]

where Γ ={

(e−2πi 1

φh, e

2πi 1φ(h+k)

) ∈ T 2 | h, k ∈ Z

}



(z1, z2) ∈ C2 | |z1|2 + 1

φ |z2|2 <√2+φ2 , −|z1|2 + |z2|2 <

√2+φ2φ

}


Uτ→ {(z1, z2, z3, z4) ∈ Ψ−1(0) | z3 6= 0, z4 6= 0}

(z1, z2) 7→(

z1, z2,√√

2+φ2 − |z1|2 − 1

φ |z2|2,√√

2+φ2φ2 + |z1|2−|z2|2

φ

)


U/Γτ−→ U

[(z1, z2)] 7−→ [τ(z1, z2)]

where Γ ={

(e−2πi 1

φh, e

2πi 1φ(h+k)

) ∈ T 2 | h, k ∈ Z

}

and

U = {(z1, z2, z3, z4) ∈ Ψ−1(0) | z3 6= 0, z4 6= 0}/N ⊂ M

Ammann tilings

Ammann tilings

Ammann tilings

Ammann tilings

Ammann tilings

◮ they are three–dimensional generalization of Penrose rhombustilings

Ammann tilings

Ammann tilings


◮ they provide a geometrical model for the physics of certainquasicrystals

Ammann tilings

Ammann tilings


◮ they provide a geometrical model for the physics of certainquasicrystals

◮ their tiles are given by two types of rhombohedra

Ammann tiles

Ammann tiles

◮ the oblate rhombohedron

Ammann tiles

◮ the oblate rhombohedron

◮ the prolate rhombohedron

Ammann tiles

the facets of these rhombohedra are so–called golden rhombuses:the ratio of their diagonals is equal to φ

choice of quasilattice for the Ammann tilings


let us consider the quasilattice F ⊂ R3 generated by the vectors

U1 =1√2(1, φ − 1, φ)

U2 =1√2(φ, 1, φ − 1)

U3 =1√2(φ− 1, φ, 1)

U4 =1√2(−1, φ− 1, φ)

U5 =1√2(φ,−1, φ − 1)

U6 =1√2(φ− 1, φ,−1)



U1 =1√2(1, φ − 1, φ)

U2 =1√2(φ, 1, φ − 1)

U3 =1√2(φ− 1, φ, 1)

U4 =1√2(−1, φ− 1, φ)

U5 =1√2(φ,−1, φ − 1)

U6 =1√2(φ− 1, φ,−1)

the quasilattice F is known in the physics of quasicrystals as theface–centered lattice



U1 =1√2(1, φ − 1, φ)

U2 =1√2(φ, 1, φ − 1)

U3 =1√2(φ− 1, φ, 1)

U4 =1√2(−1, φ− 1, φ)

U5 =1√2(φ,−1, φ − 1)

U6 =1√2(φ− 1, φ,−1)


fact



U1 =1√2(1, φ − 1, φ)

U2 =1√2(φ, 1, φ − 1)

U3 =1√2(φ− 1, φ, 1)

U4 =1√2(−1, φ− 1, φ)

U5 =1√2(φ,−1, φ − 1)

U6 =1√2(φ− 1, φ,−1)


fact

◮ any rhombohedron, oblate or prolate, of a given Ammanntiling is quasirational with respect to F

the face–centered lattice


◮ the vectors Ui have norm equal to√2



◮ there are exactly 30 vectors in F having the same norm



◮ there are exactly 30 vectors in F having the same norm

◮ these 30 vectors point to the vertices of an icosidodecahedron

another important quasilattice in this setting


let us consider the quasilattice P ⊂ (R3)∗ generated by the vectors

α1 =1√2(φ− 1, 1, 0)

α2 =1√2(0, φ− 1, 1)

α3 =1√2(1, 0, φ − 1)

α4 =1√2(1− φ, 1, 0)

α5 =1√2(0, 1 − φ, 1)

α6 =1√2(1, 0, 1 − φ)



α1 =1√2(φ− 1, 1, 0)

α2 =1√2(0, φ− 1, 1)

α3 =1√2(1, 0, φ − 1)

α4 =1√2(1− φ, 1, 0)

α5 =1√2(0, 1 − φ, 1)

α6 =1√2(1, 0, 1 − φ)

the quasilattice P is known in the physics of quasicrystals as thesimple icosahedral lattice



α1 =1√2(φ− 1, 1, 0)

α2 =1√2(0, φ− 1, 1)

α3 =1√2(1, 0, φ − 1)

α4 =1√2(1− φ, 1, 0)

α5 =1√2(0, 1 − φ, 1)

α6 =1√2(1, 0, 1 − φ)


fact



α1 =1√2(φ− 1, 1, 0)

α2 =1√2(0, φ− 1, 1)

α3 =1√2(1, 0, φ − 1)

α4 =1√2(1− φ, 1, 0)

α5 =1√2(0, 1 − φ, 1)

α6 =1√2(1, 0, 1 − φ)


fact

◮ up to a suitable rescaling, P has the property of containing allof the vertices of the Ammann tiling

the simple icosahedral lattice


◮ the vectors αi have norm equal to√

3−φ2



3−φ2

◮ there are exactly 12 vectors in P having the same norm



3−φ2

◮ there are exactly 12 vectors in P having the same norm

◮ these 12 vectors point to the vertices of an icosahedron

symplectic geometry of Ammann tiles


=⇒ M = S2r ×S2

r ×S2r

Γ


=⇒ M = S2r ×S2

r ×S2r

Γ

=⇒ M =S2R×S2

R×S2

R

Γ


oblate rhombohedron


oblate rhombohedron


r ×S3r

N, where N ⊂ T 6 is

{ exp (p, p + φh, s, s + φk , t, t + φl , p, s, t) | p, s, t ∈ R, h, k , l ∈ Z }and r = 1√

φ 4√

2(3−φ)


oblate rhombohedron


r ×S3r



φ 4√

2(3−φ)

◮ considerS1 × S1 × S1 = { exp (p, p, s, s, t, t) ∈ T 6 | p, s, t ∈ R } ⊂ N


oblate rhombohedron


r ×S3r



φ 4√

2(3−φ)



r ×S2r

Γ , with Γ = NS1×S1×S1


oblate rhombohedron


r ×S3r



φ 4√

2(3−φ)



r ×S2r


prolate rhombohedron


oblate rhombohedron


r ×S3r



φ 4√

2(3−φ)



r ×S2r


prolate rhombohedron

◮ same, with R = 14√

2(3−φ)instead of r

visual aids

bibliography

◮ E. Prato,Simple Non–Rational Convex Polytopes via SymplecticGeometry,Topology 40 (2001), 961–975.

◮ F. Battaglia, E. Prato,The Symplectic Geometry of Penrose Rhombus Tilings,J. Symplectic Geom. 6 (2008), 139–158.

◮ F. Battaglia, E. Prato,The Symplectic Penrose Kite,Comm. Math. Phys. 299 (2010), 577–601.

◮ F. Battaglia, E. Prato,Ammann Tilings in Symplectic Geometry,arXiv:1004.2471 [math.SG].

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