the moment map and non–periodic tilings -...
TRANSCRIPT
![Page 1: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/1.jpg)
The moment map and non–periodic tilings
Elisa Prato
Universita degli Studi di Firenze
June 28th, 2012
joint work with Fiammetta Battaglia
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Delzant theorem
![Page 3: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/3.jpg)
Delzant theorem
Delzant theorem
![Page 4: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/4.jpg)
Delzant theorem
Delzant theorem
Delzant polytopes ⇐⇒ symplectic toric manifolds
![Page 5: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/5.jpg)
Delzant theorem
Delzant theorem
Delzant polytopes ⇐⇒ symplectic toric manifolds
Delzant polytope
![Page 6: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/6.jpg)
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
![Page 7: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/7.jpg)
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
![Page 8: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/8.jpg)
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
![Page 9: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/9.jpg)
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) = ∆
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Delzant construction
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Delzant construction
Delzant construction
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Delzant construction
Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
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Delzant construction
Delzant construction
∆ =⇒ M
explicit construction using symplectic reduction
idea
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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 ∆
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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
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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
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generalized Delzant construction
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generalized Delzant construction
natural question
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generalized Delzant construction
natural question
what if ∆ is any (not necessarily rational) simple convex polytopein (Rn)∗?
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generalized Delzant construction
natural question
what if ∆ is any (not necessarily rational) simple convex polytopein (Rn)∗?
generalized Delzant construction
![Page 21: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/21.jpg)
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
![Page 22: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/22.jpg)
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 ...
![Page 23: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/23.jpg)
generalized Delzant construction
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generalized Delzant construction
◮ the lattice L is replaced by a quasilattice Q
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generalized Delzant construction
◮ the lattice L is replaced by a quasilattice Q
◮ rationality is replaced by quasirationality
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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
![Page 27: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/27.jpg)
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
![Page 28: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/28.jpg)
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
![Page 29: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/29.jpg)
quasifold geometry
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quasifold geometry
quasilattice
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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
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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
![Page 33: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/33.jpg)
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
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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
![Page 35: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/35.jpg)
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
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rationality vs quasirationality
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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 }
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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
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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
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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
![Page 41: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/41.jpg)
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
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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
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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
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Penrose tilings
![Page 45: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/45.jpg)
Penrose tilings
2 examples non-periodic tilings of the plane
![Page 46: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/46.jpg)
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
![Page 47: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/47.jpg)
convex Penrose tiles
![Page 48: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/48.jpg)
convex Penrose tiles
◮ the thin rhombus
![Page 49: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/49.jpg)
convex Penrose tiles
◮ the thin rhombus
◮ the thick rhombus
![Page 50: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/50.jpg)
convex Penrose tiles
◮ the thin rhombus
◮ the thick rhombus
![Page 51: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/51.jpg)
convex Penrose tiles
◮ the thin rhombus
◮ the thick rhombus
◮ the kite
![Page 52: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/52.jpg)
convex Penrose tiles
◮ the thin rhombus
◮ the thick rhombus
◮ the kite
![Page 53: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/53.jpg)
geometric properties of the tiles
![Page 54: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/54.jpg)
geometric properties of the tiles
◮ the tiles are obtained from a regular pentagon with a verysimple geometric construction
![Page 55: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/55.jpg)
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
![Page 56: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/56.jpg)
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 :
![Page 57: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/57.jpg)
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
![Page 58: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/58.jpg)
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
![Page 59: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/59.jpg)
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
![Page 60: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/60.jpg)
rationality issues
![Page 61: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/61.jpg)
rationality issues
◮ the rhombuses of a given tiling are not simultaneously rationalwith respect to a same lattice
![Page 62: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/62.jpg)
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
![Page 63: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/63.jpg)
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
![Page 64: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/64.jpg)
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
![Page 65: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/65.jpg)
choice of quasilattice for the Penrose tilings
![Page 66: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/66.jpg)
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)
![Page 67: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/67.jpg)
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)
![Page 68: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/68.jpg)
choice of quasilattice for the Penrose tilings
![Page 69: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/69.jpg)
choice of quasilattice for the Penrose tilings
facts
![Page 70: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/70.jpg)
choice of quasilattice for the Penrose tilings
facts
◮ any rhombus, thick or thin, of a given rhombus tiling isquasirational with respect to Q
![Page 71: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/71.jpg)
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
![Page 72: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/72.jpg)
symplectic geometry of the rhombus tiling
![Page 73: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/73.jpg)
symplectic geometry of the rhombus tiling
we apply the generalized Delzant construction and we get
![Page 74: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/74.jpg)
symplectic geometry of the rhombus tiling
we apply the generalized Delzant construction and we get
=⇒ M = S2r ×S2
r
Γ
![Page 75: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/75.jpg)
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
![Page 76: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/76.jpg)
why? what are r , R and Γ?
![Page 77: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/77.jpg)
why? what are r , R and Γ?
thin rhombus
![Page 78: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/78.jpg)
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
![Page 79: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/79.jpg)
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
![Page 80: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/80.jpg)
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
![Page 81: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/81.jpg)
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
![Page 82: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/82.jpg)
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
![Page 83: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/83.jpg)
symplectic geometry of the kite and dart tiling
![Page 84: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/84.jpg)
symplectic geometry of the kite and dart tiling
=⇒ M =
![Page 85: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/85.jpg)
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 }
![Page 86: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/86.jpg)
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
![Page 87: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/87.jpg)
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
![Page 88: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/88.jpg)
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
}
![Page 89: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/89.jpg)
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
)
![Page 90: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/90.jpg)
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)]
![Page 91: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/91.jpg)
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
}
![Page 92: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/92.jpg)
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
![Page 93: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/93.jpg)
Ammann tilings
![Page 94: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/94.jpg)
Ammann tilings
Ammann tilings
![Page 95: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/95.jpg)
Ammann tilings
Ammann tilings
◮ they are three–dimensional generalization of Penrose rhombustilings
![Page 96: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/96.jpg)
Ammann tilings
Ammann tilings
◮ they are three–dimensional generalization of Penrose rhombustilings
◮ they provide a geometrical model for the physics of certainquasicrystals
![Page 97: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/97.jpg)
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
![Page 98: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/98.jpg)
Ammann tiles
![Page 99: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/99.jpg)
Ammann tiles
◮ the oblate rhombohedron
![Page 100: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/100.jpg)
Ammann tiles
◮ the oblate rhombohedron
◮ the prolate rhombohedron
![Page 101: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/101.jpg)
Ammann tiles
◮ the oblate rhombohedron
◮ the prolate rhombohedron
![Page 102: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/102.jpg)
Ammann tiles
the facets of these rhombohedra are so–called golden rhombuses:the ratio of their diagonals is equal to φ
![Page 103: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/103.jpg)
choice of quasilattice for the Ammann tilings
![Page 104: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/104.jpg)
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)
![Page 105: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/105.jpg)
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
![Page 106: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/106.jpg)
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
![Page 107: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/107.jpg)
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
![Page 108: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/108.jpg)
the face–centered lattice
![Page 109: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/109.jpg)
the face–centered lattice
◮ the vectors Ui have norm equal to√2
![Page 110: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/110.jpg)
the face–centered lattice
◮ the vectors Ui have norm equal to√2
◮ there are exactly 30 vectors in F having the same norm
![Page 111: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/111.jpg)
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
![Page 112: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/112.jpg)
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
![Page 113: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/113.jpg)
another important quasilattice in this setting
![Page 114: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/114.jpg)
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)
![Page 115: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/115.jpg)
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
![Page 116: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/116.jpg)
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
![Page 117: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/117.jpg)
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
![Page 118: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/118.jpg)
the simple icosahedral lattice
![Page 119: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/119.jpg)
the simple icosahedral lattice
◮ the vectors αi have norm equal to√
3−φ2
![Page 120: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/120.jpg)
the simple icosahedral lattice
◮ the vectors αi have norm equal to√
3−φ2
◮ there are exactly 12 vectors in P having the same norm
![Page 121: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/121.jpg)
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
![Page 122: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/122.jpg)
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
![Page 123: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/123.jpg)
symplectic geometry of Ammann tiles
![Page 124: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/124.jpg)
symplectic geometry of Ammann tiles
=⇒ M = S2r ×S2
r ×S2r
Γ
![Page 125: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/125.jpg)
symplectic geometry of Ammann tiles
=⇒ M = S2r ×S2
r ×S2r
Γ
=⇒ M =S2R×S2
R×S2
R
Γ
![Page 126: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/126.jpg)
why? what are r , R and Γ?
oblate rhombohedron
![Page 127: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/127.jpg)
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)
![Page 128: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/128.jpg)
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
![Page 129: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/129.jpg)
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
![Page 130: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/130.jpg)
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
![Page 131: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/131.jpg)
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
![Page 132: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/132.jpg)
visual aids
![Page 133: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/133.jpg)
visual aids
◮ all models are built using zometool R©
![Page 134: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/134.jpg)
visual aids
◮ all models are built using zometool R©
◮ all 3D pictures are drawn using zomecad
![Page 135: The moment map and non–periodic tilings - UniFIweb.math.unifi.it/users/eprato/exposeSouriau.pdf · The moment map and non–periodic tilings ... torus T = Rn/L. Delzant theorem](https://reader031.vdocuments.us/reader031/viewer/2022021622/5b9ef74409d3f2d0208c7851/html5/thumbnails/135.jpg)
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].