Will it k-tile? Structural aspects of polytopes and lattices in - - PowerPoint PPT Presentation

Will it k-tile?

Structural aspects of polytopes and lattices in multiple tiling Alexandru Mihai, Melissa Sherman-Bennett, Dat Nguyen, Alexander Dunlap

Summer@ICERM

August 7, 2014

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1

Introduction to Multiple Tilings

2

k-tiling Polygons

3

A necessary condition for multiple tiling

4

A sufficient condition for multiple tiling with a lattice

5

A structure result for multiple tiling

6

Multiple tiling in three dimensions

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Section 1 Introduction to Multiple Tilings

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Starting Definitions

We say that a polytope P tiles Rd with a discrete set of translation vectors Λ if ÿ

λPΛ

1P`λ pvq “ ÿ

λPΛ

1P pλ ´ vq “ k @v R BP ` Λ

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Starting Definitions

We say that a polytope P tiles Rd with a discrete set of translation vectors Λ if ÿ

λPΛ

1P`λ pvq “ ÿ

λPΛ

1P pλ ´ vq “ k @v R BP ` Λ

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Starting Definitions

A polytope P is centrally symmetric about the origin if for all x P P, ´x P P. And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin.

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Starting Definitions

A polytope P is centrally symmetric about the origin if for all x P P, ´x P P. And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin.

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Starting Definitions

A polytope P is centrally symmetric about the origin if for all x P P, ´x P P. And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin.

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Starting Definitions

A polytope P is centrally symmetric about the origin if for all x P P, ´x P P. And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin.

Starting Definitions

A polytope P is centrally symmetric about the origin if for all x P P, ´x P P. And more generally a polytope is called centrally symmetric if there exist a translate of it which is centrally symmetric about the origin.

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Starting Definitions

A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric

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Starting Definitions

A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric P is the Minkowski sum of a finite number of line-segments

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Starting Definitions

A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric P is the Minkowski sum of a finite number of line-segments P is the affine image of some n-dimensional cube r0, 1sn The Minkowski sum is defined to be A ` B “ ta ` b | a P A and b P Bu

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Starting Definitions

A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric P is the Minkowski sum of a finite number of line-segments P is the affine image of some n-dimensional cube r0, 1sn The Minkowski sum is defined to be A ` B “ ta ` b | a P A and b P Bu

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Starting Definitions

A zonotope is a polytope P satisfying the following equivalent conditions: All of the faces of P are centrally symmetric P is the Minkowski sum of a finite number of line-segments P is the affine image of some n-dimensional cube r0, 1sn The Minkowski sum is defined to be A ` B “ ta ` b | a P A and b P Bu

3 2 1 1 2 3 3 2 1 1 2 3

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Polygons and the Fourier Transform

The Poisson Summation Formula tells us that given any ”nice” function f on Rd, we have ÿ

nPΛ

f pnq “ ÿ

ξPΛ˚

ˆ f pξq where by definition ˆ f pξq :“ ş

Rd f pxq e2πixx,ξydx, and where the dual

lattice is defined by Λ˚ :“ tx P Rd | xl, xy P Z, for all l P Λu.

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Polygons and the Fourier Transform

The Poisson Summation Formula tells us that given any ”nice” function f on Rd, we have ÿ

nPΛ

f pnq “ ÿ

ξPΛ˚

ˆ f pξq where by definition ˆ f pξq :“ ş

Rd f pxq e2πixx,ξydx, and where the dual

lattice is defined by Λ˚ :“ tx P Rd | xl, xy P Z, for all l P Λu. Thus, by the Poisson Summation Formula, we have k “ ÿ

λPΛ

1P pλ ´ vq “ 1 | det Λ| ÿ

mPΛ˚

ˆ 1P pmq e´2πixv,my Now we use the fact that Fourier series expansions are unique, so all the nonzero Fourier coefficients on the right must vanish, because k is

• constant. And this leads to the proof of a very interesting theorem.

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Theorem

A convex polytope P k-tiles Rd by translations with the lattice Λ if and

• nly if

ˆ 1P pmq “ 0 for all nonzero vectors m P Λ˚. Moreover, we have k “ VolpPq

| detpλq|.

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Theorem

A convex polytope P k-tiles Rd by translations with the lattice Λ if and

• nly if

ˆ 1P pmq “ 0 for all nonzero vectors m P Λ˚. Moreover, we have k “ VolpPq

| detpλq|.

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Observations in the Fourier Plane

Now what to do with these equations?

6 4 2 2 4 6 6 4 2 2 4 6 6 4 2 2 4 6 6 4 2 2 4 6

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Observations in the Fourier Plane

But how should we go about working with these curves? One of our initial conjectures was that the dual lattice Λ˚ would be found at the intersections seen in the Fourier transform.

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Observations in the Fourier Plane

But how should we go about working with these curves? One of our initial conjectures was that the dual lattice Λ˚ would be found at the intersections seen in the Fourier transform.

2.0 2.5 3.0 3.5 4.0 4.5 5.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0

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Observations in the Fourier Plane

Now plotting the gradient field of the Fourier Transform we confirm that intersections are really saddle points.

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Observations in the Fourier Plane

And we end up with a numerical approximation of the curves.

1 2 3 4 5 1 2 3 4 5 x y

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Section 2 k-tiling Polygons

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Motivation & Guiding Questions

Combinatorial characterization Ñ constructive understanding Irrational multi-tilers An algorithm which, given a polygon P, determines if there exists a lattice with which P multi-tiles

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Bolle Characterization

Some terminology: Given two parallel edges e and e1 “ e ` s, the vector s separating them is called the side-pairing of e (or e1). Can be thought of as the sum of the edges separating e and e1.

e e' s

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Bolle Characterization

Let Λ be a lattice and P a centrally symmetric convex polygon. Let e and e1 be a pair of parallel edges of P.

Definition

e is called Type 1 if e ` s “ e1 for some

• s P Λ

Type 2 if e “ ´ e1 P Λ and Aff(e)` s “Aff(e1) for some

• s P Λ and e is not Type 1.

e e' s

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Bolle Characterization

Let Λ be a lattice and P a centrally symmetric convex polygon. Let e and e1 be a pair of parallel edges of P.

Definition

e is called Type 1 if e ` s “ e1 for some

• s P Λ

Type 2 if e “ ´ e1 P Λ and Aff(e)` s “Aff(e1) for some

• s P Λ and e is not Type 1.

e e' s

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Bolle Characterization

Theorem (Bolle, 1991)

Given a lattice Λ, a centrally symmetric convex polygon P k-tiles with Λ if and only if every edge of P is Type 1 or Type 2.

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Bolle Characterization

Idea behind necessity: As one “leaves” one polygon translate by moving across an edge, one must enter another polygon translate.

e e' s e e' e e' s e' e

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Bolle Characterization

Idea behind necessity: As one “leaves” one polygon translate by moving across an edge, one must enter another polygon translate.

e e' s e e' e e' e' e

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Bolle Characterization

Easy facts: For a given lattice and a centrally symmetric convex polygon with 2n edges, if: All edges of P are lattice vectors Ñ all side-pairings are lattice vectors Ñ all edges are Type 1

§ Corollary: Can’t have all Type

2 edges

§ All lattice polygons have all

Type 1 edges

n ´ 1 pairs of edges are Type 2 Ñ nth pair of edges is Type 1

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Bolle Characterization

Easy facts: For a given lattice and a centrally symmetric convex polygon with 2n edges, if: All edges of P are lattice vectors Ñ all side-pairings are lattice vectors Ñ all edges are Type 1

§ Corollary: Can’t have all Type

2 edges

§ All lattice polygons have all

Type 1 edges

n ´ 1 pairs of edges are Type 2 Ñ nth pair of edges is Type 1

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Polygon Graphs

Let P be a centrally symmetric convex polygon that k-tiles with some lattice Λ.

Definition

The polygon graph associated with P is G “ pV , Eq where V is the set of vertices of P. For v, w P V put pv, wq P E if a) v and w are separated by a lattice vector side-pairing or b) v and w are the vertices of a Type 2 edge. Let E 1 be the set of edges satisfying a and E 2 be the set of edges satisfying b. E “ E 1 Y E 2.

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Polygon Graphs

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Characteristics of Polygon Graphs

2–regular Connected components are cycles, which are well-understood Combinatorial structure does not reflect full geometric structure

§ Affine span condition not

considered

§ 1 connected component Ñ

lattice polygon i i + 1 i + n 2

• 1

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Characteristics of Polygon Graphs

2–regular Connected components are cycles, which are well-understood Combinatorial structure does not reflect full geometric structure

§ Affine span condition not

considered

§ 1 connected component Ñ

lattice polygon j j - n 2 + 1 j + n 2

• 1

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Results

Lemma

A polygon graph G has one connected component if |E 1| ” 0 pmod 4q and two connected components if |E 1| ” 2 pmod 4q. In other words, a k-tiling polygon with Type 2 edges must have an odd number of pairs of Type 1 edges. In particular, for k-tiling n-gon, |E 2| ” 2 pmod 4q if n ” 0 pmod 4q and |E 2| ” 0 pmod 4q if n ” 2 pmod 4q.

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Results

Theorem

Let P be a centrally symmetric convex polygon that k-tiles with a lattice Λ. If P has more than one pair of Type 2 edges, all vertices of P are Λ-rational (i.e. under the transformation of Λ to Z2, they have rational coordinates). Remark: Follows from the affine span condition, which results in lines with rational slope intersecting.

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Results

Corollary

Let P be a centrally symmetric n-gon that k-tiles with a lattice Λ. If n ” 0 (mod 4), P can have Λ-irrational vertices only if it has 1 pair of Type 2 edges; if n ”2 (mod 4), P can have Λ-irrational vertices only if it has all Type 1 edges.

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k-tiling Algorithm

Theorem

Given an oracle to determine rationality, there is a polynomial-time decision algorithm to determine if there exists a lattice such that a given centrally symmetric polygon multi-tiles the plane.

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k-tiling Algorithm

Observation: There is an algorithm which, using our oracle, determines if a set of vectors generates a discrete subgroup. Step 1 Check if the set of side pairings generates a discrete subgroup. Step 2 Iterating through the edges: pick an edge e as a Type 2 candidate Check if e and the set of remaining side pairings generate a discrete subgroup. Check if e and e1 could satisfy the affine span condition

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Section 3 A necessary condition for multiple tiling

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Assumptions:

1 P is a zonotope in Rn:

Each cell Cd Ă Cd`1 has a unique opposite cell in Cd`1.

2 No assumption on the multi-set Λ of translation vectors. 3 P multi-tiles with Λ:

ÿ

λPΛ

1P`λpxq “ const

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Coloring cells: easy for codimension 1

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Coloring cells: Higher codimensions?

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Definition

A cell Cd Ă Cd`1 Ă ... Ă Cn´1 Ă P is called directly opposite to the cell C 1

d Ă C 1 d`1 Ă ... Ă Cn´1 Ă P if

1 For one k, C 1

k is the opposite cell of Ck in Ck`1; and

2 For every other k, C 1

k Ă C 1 k`1 “looks the same as” Ck Ă Ck`1.

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Coloring cells.

1 Start with a cell. Color it red. 2 Directly opposite cells have different colors. Summer@ICERM K-Tiling August 7, 2014

Caution: The data Cd Ă Cd`1 Ă ... Ă Cn´1 Ă P is very important. We are NOT coloring all parallel cells at the same time.

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A necessary condition for multiple tiling

Suppose P multi-tiles with a set of translations Λ. Then every cell has a Λ-translation that intersects a cell of different color.

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Example: two dimensions.

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Example: three dimensions. (Assuming 0 P Λ).

1 One among tv0, v1, v2, v1 ` v2 ` v3u is in Λ. 2 One among tv1 ` tv0, v2 ` uv0u is in Λ, for some t, u P R. 3 v2 ` av1 ` bv0 is in Λ for some a, b P R. Summer@ICERM K-Tiling August 7, 2014

Section 4 A sufficient condition for multiple tiling with a lattice

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The necessary condition above is VERY loose.

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Assuming:

1 P is a zonotope in Rn. 2 Λ is a lattice.

A sufficient condition for multiple tiling with a lattice

If every cell has a Λ-translation that intersects a cell directly opposite to it, then P multi-tiles with Λ.

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Comparing to the necessary condition.

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A characterization of multiple tiling in dimension 3

In dimension 3, the sufficient condition above is also necessary.

1 One among tv0, v1, v2u is in Λ. 2 One among tv1 ` tv0, v2 ` uv0u is in Λ, for some t, u P R. 3 v2 ` av1 ` bv0 is in Λ for some a, b P R. Summer@ICERM K-Tiling August 7, 2014

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Application: the belt condition.

Theorem (Minkowski, 1897; Venkov, 1954; McMullen, 1980)

A convex polytope P (single) tiles with a lattice if and only if

1 P is centrally symmetric. 2 All facets of P are centrally symmetric. 3 Every codimension-2 “belt” of P is of size 4 or 6. Summer@ICERM K-Tiling August 7, 2014

Conjecture

Suppose a convex polytope P in R3 multi-tiles with a lattice. Then the projection of P along any edge to a plane A is a polygon that multi-tiles A with some lattice.

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Conjecture

Suppose a convex polytope P in R3 multi-tiles with a lattice. Then the projection of P along any edge to a plane A is a polygon that multi-tiles A with some lattice. Answer:

1 If the edge has lattice direction: TRUE. 2 If not: FALSE. Summer@ICERM K-Tiling August 7, 2014

Section 5 A structure result for multiple tiling

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Why focus on lattices? Λ1 “ ˆ 1 π 2 ˙ Z2, Λ2 “ ˆ 1 ? 2 2 ˙ Z2 ` ˆ 0 1 ˙

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Λ1 “ ˆ 1

π 2

1 ˙ Z2 Ą Λ1 “ ˆ 1 π 2 ˙ Z2

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Assuming:

1 P is a zonotope in Rn. 2 Λ is the union of a finite collection of translated lattices

C “ tΛj ` θjuM

j“1.

Theorem

There exists a lattice Λ1 with which P multi-tiles. Moreover, every lattice in the collection C can be refined to be one such Λ1.

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Theorem (Gravin, Kolountzakis, Robins, Shiryaev, 2012)

If a three-dimensional convex polytope multi-tiles with a multi-set, then it multi-tiles with a finite union of translated lattices.

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Theorem (Gravin, Kolountzakis, Robins, Shiryaev, 2012)

If a three-dimensional convex polytope multi-tiles with a multi-set, then it multi-tiles with a finite union of translated lattices.

Theorem (My friends, yesterday)

If a three-dimensional convex polytope multi-tiles with a multi-set, then it multi-tiles with a lattice.

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Section 6 Multiple tiling in three dimensions

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Theorem (Bolle, 1991)

Let P be a convex, centrally-symmetric polygon. Then P multi-tiles by a lattice Λ iff, for each pair of opposite edges e, e1 of P, one of the following conditions holds:

1 e and e1 differ by a vector in Λ. 2

• e “

e1 is in Λ, and the affine spans of e and e1 differ by a vector in Λ.

e e′

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Proving Bolle’s characterization using Fourier analysis

(Kolountzakis, 2000) Assign opposite edges “valences,” which cancel iff P multi-tiles. µ0: arc-length measure on a segment µ “ µ` ` µ´. ÿ

λPΛ

µ ˚ δλ ” 0.

µ+ = +µ0 ∗ δ(···) µ− = −µ0 ∗ δ−(···)

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Proving Bolle’s characterization using Fourier analysis

Fact: if ÿ

λPΛ

µ ˚ δλ ” 0, then p µpτq ” 0 for all τ P Λ˚; that is, Λ˚ Ă pˆ µ “ 0q. Λ˚ “ tτ P R2 | xτ, λy P Z for all λ P Λu (dual lattice). Compare with Poisson summation formula.

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Proving Bolle’s characterization using Fourier analysis

Λ˚ Ă pˆ µ “ 0q If we choose nice coordinates, we can compute easily!

+µ0 ∗ δ(0,1/2) −µ0 ∗ δ(0,−1/2)

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Proving Bolle’s characterization using Fourier analysis

Λ˚ Ă pˆ µ “ 0q If we choose nice coordinates, we can compute easily! µ “ µ0 ˚ ` δp0,1{2q ´ δp0,´1{2q ˘

+µ0 ∗ δ(0,1/2) −µ0 ∗ δ(0,−1/2)

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Proving Bolle’s characterization using Fourier analysis

Λ˚ Ă pˆ µ “ 0q If we choose nice coordinates, we can compute easily! µ “ µ0 ˚ ` δp0,1{2q ´ δp0,´1{2q ˘ x µ0pτq “ C ¨ sinpπτ1q πτ1

+µ0 ∗ δ(0,1/2) −µ0 ∗ δ(0,−1/2)

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Proving Bolle’s characterization using Fourier analysis

Λ˚ Ă pˆ µ “ 0q If we choose nice coordinates, we can compute easily! µ “ µ0 ˚ ` δp0,1{2q ´ δp0,´1{2q ˘ x µ0pτq “ C ¨ sinpπτ1q πτ1 p µpτq “ C ¨ sinpπτ1q πτ1 ¨ sinpπτ2q

+µ0 ∗ δ(0,1/2) −µ0 ∗ δ(0,−1/2)

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Proving Bolle’s characterization using Fourier analysis

Λ˚ Ă pˆ µ “ 0q If we choose nice coordinates, we can compute easily! µ “ µ0 ˚ ` δp0,1{2q ´ δp0,´1{2q ˘ x µ0pτq “ C ¨ sinpπτ1q πτ1 p µpτq “ C ¨ sinpπτ1q πτ1 ¨ sinpπτ2q pˆ µpτq “ 0q “ rR ˆ Zs Y rpZzt0uq ˆ Rs

+µ0 ∗ δ(0,1/2) −µ0 ∗ δ(0,−1/2)

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Proving Bolle’s characterization using Fourier analysis

Λ˚ Ă pˆ µ “ 0q If we choose nice coordinates, we can compute easily! µ “ µ0 ˚ ` δp0,1{2q ´ δp0,´1{2q ˘ x µ0pτq “ C ¨ sinpπτ1q πτ1 p µpτq “ C ¨ sinpπτ1q πτ1 ¨ sinpπτ2q pˆ µpτq “ 0q “ rR ˆ Zs Y rpZzt0uq ˆ Rs

+µ0 ∗ δ(0,1/2) −µ0 ∗ δ(0,−1/2)

The point: Λ˚ Ă rR ˆ Zs Y rpZzt0uq ˆ Rs .

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Proving Bolle’s characterization using Fourier analysis

Λ˚ Ă rR ˆ Zs Y rpZzt0uq ˆ Rs Ă rR ˆ Zs Y rZ ˆ Rs So (group theory) either Λ˚ Ă R ˆ Z, so Λ Ą 0 ˆ Z; or Λ˚ Ă Z ˆ R, so Λ Ą Z ˆ 0.

still need affine span. . . e e′

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Theorem

P multi-tiles by a lattice Λ iff, for each four-legged frame of edges e, e1 in facet E, f , f 1 in facet F,

1 F and F 1 differ by Λ, or 2 AffpFq and AffpF 1q

differ by Λ, and

1

e and e1 differ by Λ,

• r

2

• e “

e1 “ f “ f 1 P Λ, and

1

Affpeq and Affpf q differ by Λ, or

2

Affpeq and Affpe1q differ by Λ.

Convex, centrally symmetric polytope P

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Generalizing Kolountzakis’s proof to three dimensions

ˆ µpτq “ C ¨ sinpπτ1q sinpπτ2qsinpπτ3q πτ3 pˆ µpτq “ 0q “ rZ ˆ R ˆ Rs Y rR ˆ Z ˆ Rs Y rR ˆ R ˆ pZzt0uqs

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Generalizing Kolountzakis’s proof to three dimensions

Λ˚ Ă pˆ µpτq “ 0q “ rZ ˆ R ˆ Rs Y rR ˆ Z ˆ Rs Y rR ˆ R ˆ pZzt0uqs

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Generalizing Kolountzakis’s proof to three dimensions

Λ˚ Ă pˆ µpτq “ 0q “ rZ ˆ R ˆ Rs Y rR ˆ Z ˆ Rs Y rR ˆ R ˆ pZzt0uqs

Lemma

Let G be a group. If G Ă rZ ˆ R ˆ Rs Y rR ˆ Z ˆ Rs Y rR ˆ R ˆ pZzt0uqs, then either G Ă Z ˆ R ˆ R or G Ă R ˆ Z ˆ R or G Ă R ˆ R ˆ Z.

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Generalizing Kolountzakis’s proof to three dimensions

Thus either Λ Ą Z ˆ 0 ˆ 0 or Λ Ą 0 ˆ Z ˆ 0 or Λ Ą 0 ˆ 0 ˆ Z. Again, more work is necessary for affine span conditions.

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Final thoughts

Even higher dimensions? Lattice structure takes center stage.

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Acknowledgments

We gratefully acknowledge Sinai Robins, Richard Schwartz, Michael Mossinghoff, Nhat Le, Tarik Aougab, Sanya Pushkar, and Emmanuel Tsukerman, as well as ICERM, Brown University, and the National Science Foundation.

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References

Ulrich Bolle. On multiple tiles in E 2. In Intuitive geometry (Szeged, 1991), volume 63 of Colloq. Math. Soc. J´ anos Bolyai, pages 39–43. North-Holland, Amsterdam, 1994. Nick Gravin, Mihail N. Kolountzakis, Sinai Robins, and Dmitry Shiryaev. Structure results for multiple tilings in 3d. Discrete and Computational Geometry, 50(4):1033–1050, 2013. Mihail N. Kolountzakis. On the structure of multiple translational tilings by polygonal regions. Discrete and Computational Geometry, 23(4):537–553, 2000.

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