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Delaunay Tessellations of Point Lattices fileDelaunay Tessellations of Point Lattices Theory,...
Transcript of Delaunay Tessellations of Point Lattices fileDelaunay Tessellations of Point Lattices Theory,...
Delaunay Tessellations of Point Lattices Theory, Algorithms and Applications
Achill Schürmann(University of Rostock)
( based on work with Mathieu Dutour Sikiric and Frank Vallentin )
Berlin, October 2013
ERC Workshop Delaunay Geometry
Polytopes, Triangulations and Spheres
Делоне
Boris N. Delone1890-1980
( Delone )French: Delaunay
Delone tessellations
Delaunay subdivisions of lattices
L = hexagonal lattice 2-periodic (m=2)
Delone star and DV-cell
( ) =�
∈ R : � � ≤ � − � ∈�
Up to translation, there is only on DV-cell in a lattice:
Delone star and DV-cell
( ) =�
∈ R : � � ≤ � − � ∈�
Up to translation, there is only on DV-cell in a lattice:
all Delone polyhedra incident to a given vertex
It is dual to a Delone star
How do I compute
the DV-cell(or Delone star)?
Computing DV-cells(first approach)
THM (Voronoi, 1908):
∈ ( )± �= +
Computing DV-cells(first approach)
THM (Voronoi, 1908):
⇒ ( ) ( − )
∈ ( )± �= +
Computing DV-cells(first approach)
THM (Voronoi, 1908):
⇒ ( ) ( − )
0
∈ ( )± �= +
Computing DV-cells(first approach)
THM (Voronoi, 1908):
⇒ ( ) ( − )
0
PLAN
• compute facets
• obtain vertices
∈ ( )± �= +
Computing DV-cells
STEP 1: Compute an initial vertex (an initial Delone Polyhedron)
(practical approach)
Computing DV-cells
STEP 1: Compute an initial vertex (an initial Delone Polyhedron)
(practical approach)
Computing DV-cells
STEP 1: Compute an initial vertex (an initial Delone Polyhedron)
(practical approach)
Computing DV-cells (continued...)
STEP x: Compute neighboring vertices (neighboring Delone Polyhedra)
• Compute facets of Delone polyhedron • For each facet find neighboring Delone polyhedron• Test if they are equivalent to one of the known ones
Computing DV-cells (continued...)
STEP x: Compute neighboring vertices (neighboring Delone Polyhedra)
• Compute facets of Delone polyhedron • For each facet find neighboring Delone polyhedron• Test if they are equivalent to one of the known ones
Computing DV-cells (continued...)
STEP x: Compute neighboring vertices (neighboring Delone Polyhedra)
• Compute facets of Delone polyhedron • For each facet find neighboring Delone polyhedron• Test if they are equivalent to one of the known ones
( up to translation, central inversion, ...)
Computing DV-cells (continued...)
STEP x: Compute neighboring vertices (neighboring Delone Polyhedra)
• Compute facets of Delone polyhedron • For each facet find neighboring Delone polyhedron• Test if they are equivalent to one of the known ones
( up to translation, central inversion, ...)
Well suited for exploiting symmetry!
Computational Results
obtained by Mathieu using polyhedral
Mathieu
Computational Results
obtained by Mathieu using polyhedral
Mathieu
IN: Complexity and algorithms for computing Voronoi cells of lattices, Math. Comp. (2009)
• computation of vertices for many different DV-cells of lattices (in particular for Coxeter-, Laminated and Cut-Lattices) • verified that Leech Lattice cell has 307 vertex orbits (Conway, Borcherds, et. al.)
Application I: Covering Constants
Application I: Covering Constants
µ( ) = sup∈ ( )
( )
Application II: Quantizer Constants
What happensif we vary the lattice?
Two views
Instead of varying the lattice we can equivalently vary the norm
Z � � =√
=
= Z � � =√
⇔
Quadratic Forms
Arithmetical Equivalence
Dictionary
Dictionary
Delone tessellations revisited
Delone tessellations revisited
Secondary cones
D R Z
∆(D) =�
∈ S> : ( ) = D�
Secondary cones
D R Z
∆(D) =�
∈ S> : ( ) = D�
Baranovskii Cones
Z
∆( ) =�
∈ S> : ∈ ( )�
DEF:
Baranovskii Cones
Z
∆( ) =�
∈ S> : ∈ ( )�
DEF:
THM:∆( )
Baranovskii Cones
Z
∆( ) =�
∈ S> : ∈ ( )�
DEF:
THM:∆( )
D ∆(D) =�
∈D∆(P)
Note:
Baranovskii Cones
Z
∆( ) =�
∈ S> : ∈ ( )�
DEF:
THM:∆( )
D ∆(D) =�
∈D∆(P)
Note:
⇒ ∆(D)
Application:Finding best lattice coverings
Application:Finding best lattice coverings
�∈ S> : ( ( )) ≤
�
THM (Barnes, Dickson; 1968): Among PQFs in the closure of a secondary cone
Θ
Application:Finding best lattice coverings
�∈ S> : ( ( )) ≤
�
THM (Barnes, Dickson; 1968): Among PQFs in the closure of a secondary cone
Θ
Note:
Voronoi’s second reduction
Voronoi’s second reduction
THM (Voronoi, 1908):
• there exist finitely many inequivalent secondary cones
• inclusion of faces corresponds to coarsening of subdivisions
• closures of secondary cones tesselate S>
Voronoi’s second reduction
THM (Voronoi, 1908):
• there exist finitely many inequivalent secondary cones
• inclusion of faces corresponds to coarsening of subdivisions
• closures of secondary cones tesselate S>
=> top-dimensional cones come from triangulations
Voronoi’s second reduction
THM (Voronoi, 1908):
• there exist finitely many inequivalent secondary cones
• inclusion of faces corresponds to coarsening of subdivisions
• closures of secondary cones tesselate S>
=> top-dimensional cones come from triangulations
Already known...
Already known...
IDEA: In higher dimensions, determine the best lattice coverings with a given group of symmetries!? (obtaining all Delone subdivisons with a given symmetry)
G-Theory?
G-Theory?
G-Theory?
IDEA: Intersect secondary cones with a linear subspace T
G-Theory?
IDEA: Intersect secondary cones with a linear subspace T
DEF: ∩∆(D)
T-secondary cones• T-secondary cones tesselate S> ∩
T-secondary cones• T-secondary cones tesselate S> ∩
�
:⇔ ∃ ∈ (Z) = � ⊆DEF:
T-secondary cones• T-secondary cones tesselate S> ∩
�
:⇔ ∃ ∈ (Z) = � ⊆DEF:
THM: ⊂ (Z)
T-secondary cones• T-secondary cones tesselate S> ∩
�
:⇔ ∃ ∈ (Z) = � ⊆DEF:
• Delone subdivision of a neighboring T-secondary cone can be obtained by a T-flip in repartitioning polytopes
THM: ⊂ (Z)
T-secondary cones• T-secondary cones tesselate S> ∩
�
:⇔ ∃ ∈ (Z) = � ⊆DEF:
• Delone subdivision of a neighboring T-secondary cone can be obtained by a T-flip in repartitioning polytopes
THM: ⊂ (Z)
Application to Lattice Coverings
What about the nice lattices?
What about the nice lattices?
Prominent Example:
(E )
Θ
What about the nice lattices?
Question: Are there local maxima??
Prominent Example:
(E )
Θ
What about the nice lattices?
Question: Are there local maxima??
YES! E Θ
Prominent Example:
(E )
Θ
What about the nice lattices?
Question: Are there local maxima??
YES! E Θ= , . . . ,
Prominent Example:
(E )
Θ
What about the nice lattices?
Question: Are there local maxima??
THM: A necessary condition for a local maximum is thatevery Delone polytope attaining the covering radius
is an extreme Delone polytopedim∆( ) =
YES! E Θ= , . . . ,
Prominent Example:
(E )
Θ
i-eutaxy and i-perfectness
DEF: Q is i-perfect if
DEF: Q is i-eutactic if
i-eutaxy and i-perfectness
DEF: Q is i-perfect if
DEF: Q is i-eutactic if
THM: Θ⇔
Bahavior of nice lattices
lattice covering densityZ global minimumA2 global minimumD4 almost local maximumE6 local maximumE7 local maximumE8 almost local maximumK12 almost local maximumBW16 local maximumΛ24 local minimum
Application: Minkowski Conjecture
Conjecture:
( ) = | · · · | ⊂ R det =
sup∈R
inf∈
( − ) ≤ −
= ( , . . . , )Z ( ) =
≤
Covering Conjecture
≤
Covering Conjecture
Local covering maxima among well rounded lattices are attained by T-extreme Delone Polyhedra and there are only
finitely many of them in every dimension.(with T = space of well rounded lattices)
THM:
≤
References
• Complexity and algorithms for computing Voronoi cells of lattices, Math. Comp. (2009)
• computation of covering radius and Delone subdivisions for many lattices• verified that Leech lattice cell has 307 vertex orbits (Conway, Borcherds, et. al.)
• A generalization of Voronoi’s reduction theory and its application, Duke Math. J. (2008)
• Inhomogeneous extreme forms, Annales de l'institut Fourier (2012)
• generalized Voronoi’s reduction for L-type domains to a G- and T-invariant setting• obtained new best known covering lattices and classified totally real thin number fields
• characterization of locally extreme forms for the sphere covering problem
http://www.geometrie.uni-rostock.de