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 polytopes ⇐⇒ symplectic toric manifolds
Delzant polytope
Delzant theorem
Delzant theorem
Delzant polytopes ⇐⇒ symplectic toric manifolds
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 polytopes ⇐⇒ symplectic toric manifolds
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
symplectic toric manifold
Delzant theorem
Delzant theorem
Delzant polytopes ⇐⇒ symplectic toric manifolds
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
symplectic toric manifold
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 polytopes ⇐⇒ symplectic toric manifolds
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
symplectic toric manifold
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
Delzant construction
Delzant construction
Delzant construction
Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
Delzant construction
Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
idea
Delzant construction
Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
idea
◮ ∆ =⇒ N, a subtorus of T d = Rd/Zd , d being the number of
facets of ∆
Delzant construction
Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
idea
◮ ∆ =⇒ N, a subtorus of T d = Rd/Zd , d being the number of
facets of ∆
◮ M = Ψ−1(0)N
, where Ψ is a moment mapping for the inducedaction of N on C
d
Delzant construction
Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
idea
◮ ∆ =⇒ N, a subtorus of T d = Rd/Zd , d being the number of
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
generalized Delzant construction
natural question
generalized Delzant construction
natural question
what if ∆ is any (not necessarily rational) simple convex polytopein (Rn)∗?
generalized Delzant construction
natural question
what if ∆ is any (not necessarily rational) simple convex polytopein (Rn)∗?
generalized Delzant construction
generalized Delzant construction
natural question
what if ∆ is any (not necessarily rational) simple convex polytopein (Rn)∗?
generalized Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
generalized Delzant construction
natural question
what if ∆ is any (not necessarily rational) simple convex polytopein (Rn)∗?
generalized Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
formally, it works exactly like the Delzant construction but ...
generalized Delzant construction
generalized Delzant construction
◮ the lattice L is replaced by a quasilattice Q
generalized Delzant construction
◮ the lattice L is replaced by a quasilattice Q
◮ rationality is replaced by quasirationality
generalized Delzant construction
◮ 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
generalized Delzant construction
◮ 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
generalized Delzant construction
◮ 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
◮ 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
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
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
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
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
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
quasitorus
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
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
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 }
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 }
rational polytope
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 }
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
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 }
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
quasirational polytope
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 }
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
quasirational polytope
we say that ∆ ⊂ (Rn)∗ is quasirational with respect to aquasilattice Q ⊂ R
n if the vectors X1, . . . ,Xd can be chosen in Q
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 }
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
quasirational polytope
we say that ∆ ⊂ (Rn)∗ is quasirational with respect to aquasilattice Q ⊂ R
n if the vectors X1, . . . ,Xd can be chosen in Q
remark
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 }
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
quasirational polytope
we say that ∆ ⊂ (Rn)∗ is quasirational with respect to aquasilattice Q ⊂ R
n if the vectors X1, . . . ,Xd can be chosen in Q
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
convex Penrose tiles
◮ the thin rhombus
convex Penrose tiles
◮ the thin rhombus
◮ the thick rhombus
convex Penrose tiles
◮ the thin rhombus
◮ the thick rhombus
convex Penrose tiles
◮ the thin rhombus
◮ the thick rhombus
◮ the kite
convex Penrose tiles
◮ the thin rhombus
◮ the thick rhombus
◮ the kite
geometric properties of the tiles
geometric properties of the tiles
◮ the tiles are obtained from a regular pentagon with a verysimple geometric construction
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
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 :
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 :
◮ the ratio of the edge of the thin rhombus to its short diagonal
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 :
◮ 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
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 :
◮ 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
◮ the rhombuses of a given tiling are not simultaneously rationalwith respect to a same lattice
◮ there exists no lattice with respect to which any kite is rational
rationality issues
◮ the rhombuses of a given tiling are not simultaneously rationalwith respect to a same lattice
◮ there exists no lattice with respect to which any kite is rational
idea
rationality issues
◮ the rhombuses of a given tiling are not simultaneously rationalwith respect to a same lattice
◮ there exists no lattice with respect to which any kite is rational
ideafind an appropriate quasilattice
choice of quasilattice for the Penrose tilings
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)
Y2 = (cos 4π5 , sin 4π
5 ) = 12(−φ, 1
φ
√2 + φ)
Y3 = (cos 6π5 , sin 6π
5 ) = 12(−φ,− 1
φ
√2 + φ)
Y4 = (cos 8π5 , sin 8π
5 ) = 12(
1φ ,−
√2 + φ)
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)
Y2 = (cos 4π5 , sin 4π
5 ) = 12(−φ, 1
φ
√2 + φ)
Y3 = (cos 6π5 , sin 6π
5 ) = 12(−φ,− 1
φ
√2 + φ)
Y4 = (cos 8π5 , sin 8π
5 ) = 12(
1φ ,−
√2 + φ)
choice of quasilattice for the Penrose tilings
choice of quasilattice for the Penrose tilings
facts
choice of quasilattice for the Penrose tilings
facts
◮ any rhombus, thick or thin, of a given rhombus tiling isquasirational with respect to Q
choice of quasilattice for the Penrose tilings
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
symplectic geometry of the rhombus tiling
we apply the generalized Delzant construction and we get
symplectic geometry of the rhombus tiling
we apply the generalized Delzant construction and we get
=⇒ M = S2r ×S2
r
Γ
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
why? what are r , R and Γ?
why? what are r , R and Γ?
thin rhombus
why? what are r , R and Γ?
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
why? what are r , R and Γ?
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
◮ consider S1 × S1 = { exp (s, s, t, t) ∈ T 4 | s, t ∈ R } ⊂ N
why? what are r , R and Γ?
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
◮ consider S1 × S1 = { exp (s, s, t, t) ∈ T 4 | s, t ∈ R } ⊂ N
◮ then M = S2r ×S2
r
Γ , with Γ = NS1×S1
why? what are r , R and Γ?
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
◮ consider S1 × S1 = { exp (s, s, t, t) ∈ T 4 | s, t ∈ R } ⊂ N
◮ then M = S2r ×S2
r
Γ , with Γ = NS1×S1
thick rhombus
why? what are r , R and Γ?
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
◮ consider S1 × S1 = { exp (s, s, t, t) ∈ T 4 | s, t ∈ R } ⊂ N
◮ then M = S2r ×S2
r
Γ , with Γ = NS1×S1
thick rhombus
◮ same, with R =(
12
√2 + φ
)1/2instead of r
symplectic geometry of the kite and dart tiling
symplectic geometry of the kite and dart tiling
=⇒ M =
symplectic geometry of the kite and dart tiling
=⇒ 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 }
symplectic geometry of the kite and dart tiling
=⇒ 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 }
remark
symplectic geometry of the kite and dart tiling
=⇒ 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 }
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
}
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
}
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
)
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
}
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
)
it induces the homeomorphism
U/Γτ−→ U
[(z1, z2)] 7−→ [τ(z1, z2)]
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
}
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
)
it induces the homeomorphism
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
}
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
}
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
)
it induces the homeomorphism
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 are three–dimensional generalization of Penrose rhombustilings
◮ they provide a geometrical model for the physics of certainquasicrystals
Ammann tilings
Ammann tilings
◮ they are three–dimensional generalization of Penrose rhombustilings
◮ 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 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
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)
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)
the quasilattice F is known in the physics of quasicrystals as theface–centered lattice
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)
the quasilattice F is known in the physics of quasicrystals as theface–centered lattice
fact
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)
the quasilattice F is known in the physics of quasicrystals as theface–centered lattice
fact
◮ any rhombohedron, oblate or prolate, of a given Ammanntiling is quasirational with respect to F
the face–centered lattice
the face–centered lattice
◮ the vectors Ui have norm equal to√2
the face–centered lattice
◮ the vectors Ui have norm equal to√2
◮ there are exactly 30 vectors in F having the same norm
the face–centered lattice
◮ the vectors Ui have norm equal to√2
◮ there are exactly 30 vectors in F having the same norm
◮ these 30 vectors point to the vertices of an icosidodecahedron
the face–centered lattice
◮ the vectors Ui have norm equal to√2
◮ 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
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)
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)
the quasilattice P is known in the physics of quasicrystals as thesimple icosahedral lattice
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)
the quasilattice P is known in the physics of quasicrystals as thesimple icosahedral lattice
fact
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)
the quasilattice P is known in the physics of quasicrystals as thesimple icosahedral lattice
fact
◮ up to a suitable rescaling, P has the property of containing allof the vertices of the Ammann tiling
the simple icosahedral lattice
the simple icosahedral lattice
◮ the vectors αi have norm equal to√
3−φ2
the simple icosahedral lattice
◮ the vectors αi have norm equal to√
3−φ2
◮ there are exactly 12 vectors in P having the same norm
the simple icosahedral lattice
◮ the vectors αi have norm equal to√
3−φ2
◮ there are exactly 12 vectors in P having the same norm
◮ these 12 vectors point to the vertices of an icosahedron
the simple icosahedral lattice
◮ the vectors αi have norm equal to√
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
symplectic geometry of Ammann tiles
=⇒ M = S2r ×S2
r ×S2r
Γ
symplectic geometry of Ammann tiles
=⇒ M = S2r ×S2
r ×S2r
Γ
=⇒ M =S2R×S2
R×S2
R
Γ
why? what are r , R and Γ?
oblate rhombohedron
why? what are r , R and Γ?
oblate rhombohedron
◮ symplectic reduction yields M = S3r ×S3
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)
why? what are r , R and Γ?
oblate rhombohedron
◮ symplectic reduction yields M = S3r ×S3
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)
◮ considerS1 × S1 × S1 = { exp (p, p, s, s, t, t) ∈ T 6 | p, s, t ∈ R } ⊂ N
why? what are r , R and Γ?
oblate rhombohedron
◮ symplectic reduction yields M = S3r ×S3
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)
◮ considerS1 × S1 × S1 = { exp (p, p, s, s, t, t) ∈ T 6 | p, s, t ∈ R } ⊂ N
◮ then M = S2r ×S2
r ×S2r
Γ , with Γ = NS1×S1×S1
why? what are r , R and Γ?
oblate rhombohedron
◮ symplectic reduction yields M = S3r ×S3
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)
◮ considerS1 × S1 × S1 = { exp (p, p, s, s, t, t) ∈ T 6 | p, s, t ∈ R } ⊂ N
◮ then M = S2r ×S2
r ×S2r
Γ , with Γ = NS1×S1×S1
prolate rhombohedron
why? what are r , R and Γ?
oblate rhombohedron
◮ symplectic reduction yields M = S3r ×S3
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)
◮ considerS1 × S1 × S1 = { exp (p, p, s, s, t, t) ∈ T 6 | p, s, t ∈ R } ⊂ N
◮ then M = S2r ×S2
r ×S2r
Γ , with Γ = NS1×S1×S1
prolate rhombohedron
◮ same, with R = 14√
2(3−φ)instead of r
visual aids
visual aids
◮ all models are built using zometool R©
visual aids
◮ all models are built using zometool R©
◮ all 3D pictures are drawn using zomecad
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].