delaunay tessellations of point lattices - freie universität · pdf filedelaunay...

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