Crystal Topologies and Discrete MathematicsCrystal Topologies and Discrete Mathematics Workshop Real and Virtual Architectures of Molecules and Crystals Sep 30–Oct 1, 2004, MIS Leipzig

Crystal Topologies andDiscrete MathematicsWorkshop Real and Virtual Architectures of

Molecules and Crystals

Sep 30–Oct 1, 2004, MIS Leipzig

Olaf Delgado-Friedrichs

Wilhelm-Schickard-Institut fur Informatik, Eberhard Karls Universitat Tubingen

Department of Chemistry and Biochemistry, Arizona State University

Crystal Topologies and Discrete Mathematics – p.1/28

The role of topology

Materials of the samecomposition (e.g. pure carbon)can have different properties.Goal: Describe theirconformations qualitatively.

Potential applications:• taxonomy for crystals• recognition of structures• enumeration of possibilities• design of new materials

Crystal Topologies and Discrete Mathematics – p.2/28

Topology?

But what do we mean by a crystal topology?There are at least two possible versions:

intrinsic topology — the structure itself

ambient topology — its embedding into space

Any knot is intrinsically just acircle.

Crystal Topologies and Discrete Mathematics – p.3/28

Some recentenumerations

Numerical scan (O’KEEFFE et al., 1992).

Vector-labelled graphs (CHUNG et al., 1984).

Symmetry-labelled graphs (TREACY et al.,1997).

Tilings (DELGADO et al., 1999).

All these approaches produce many duplicates.

The last 3 are in some sense conceptuallycomplete.

Crystal Topologies and Discrete Mathematics – p.4/28

Crystal models

A hierarchy of models:

Atompositions inFaujasite.

Theatom-bondnetwork.

Networkdecomposedinto cages.

Crystal Topologies and Discrete Mathematics – p.5/28

Capturing all space

Here, the remainingspace is split up into“super cages” to forma tiling.

Tilings have beenproposed as modelsfor matter time andagain since antiquity.

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Platonic atomsPLATO thought that the elements fire, air, waterand earth were composed of regular, tetrahedra,octahedra, icosahedra and hexahedra (cubes),respectively.

ARISTOTLE later objected: most of these shapesdo not fill space without gaps.

Crystal Topologies and Discrete Mathematics – p.7/28

Snow balls

The diamond net as a spherepacking. KEPLER used these toexplain the structures of snowflakes.

Compressing evenlyyields what we now call aVoronoi tiling. Bothconcepts are still popular.

Crystal Topologies and Discrete Mathematics – p.8/28

Rubber tilesTwo tilings are of the same topological type, ifthey can be deformed into each other as if theywere painted on a rubber sheet.More formally: some homeomorphism betweenthe tiled spaces takes one into the other.

Crystal Topologies and Discrete Mathematics – p.9/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Crystal Topologies and Discrete Mathematics – p.10/28

Are these the same?A problem posed by LOTHAR COLLATZ (1910–1990).

Yes, they are!Crystal Topologies and Discrete Mathematics – p.10/28

Techniques

In order to represent tilings ina finite way, we start bydissecting tiles into trianglesas shown below.

A color-coding later helpswith the reassembly. Eachcorner receives the samecolor as the opposite side.

Crystal Topologies and Discrete Mathematics – p.11/28

Blueprints for tilings

A

A

A

A

AA

AA

AA

AA

A

A

A

A

AA

AA

AA

AA

C

C

C

C

CC

CC

CC

CC

C

C

C

C

C

C

C

C

CC

CC

CC

CC

C

C

C

C

B

B

B

B

BB

BB

BB

BB

B

B

B

B

B

B

B

B

BB

BB

BB

BB

B

B

B

B

AA

AA

A

A

A

A Symmetric pieces get acommon name,leading tocompactassemblyinstructions.

AC

B

Face and vertex degreesreplace particular shapes.The result is called aDelaney-Dress symbol.

C8/3

A

B

4/3

8/3

Crystal Topologies and Discrete Mathematics – p.12/28

Heaven & Hell tilings

Each edge separatesone black and onenon-black tile.

All black tiles arerelated by symmetry.

There are 23 types ofsuch tilings on theordinary plane.

(A.W.M. DRESS, D.H. HUSON. Revue Topologie Structurale, 1991)

Crystal Topologies and Discrete Mathematics – p.13/28

Heaven & Hell tilings

Each edge separatesone black and onenon-black tile.

All black tiles arerelated by symmetry.

There are 23 types ofsuch tilings on theordinary plane.

(A.W.M. DRESS, D.H. HUSON. Revue Topologie Structurale, 1991)

Crystal Topologies and Discrete Mathematics – p.13/28

All heaven and hell

Crystal Topologies and Discrete Mathematics – p.14/28

Simple tilings

A spatial tiling is simple ifit has four edges meetingat each vertex and oneface at each angle.

It is uninodal if all verticesare related by symmetry.

There are 9 types ofsimple, uninodal tilings in ordinary space.

(O. DELGADO FRIEDRICHS, D.H. HUSON. Discrete & Computational Geometry, 1999)

Crystal Topologies and Discrete Mathematics – p.15/28

Simple tilings

A spatial tiling is simple ifit has four edges meetingat each vertex and oneface at each angle.

It is uninodal if all verticesare related by symmetry.

There are 9 types ofsimple, uninodal tilings in ordinary space.

(O. DELGADO FRIEDRICHS, D.H. HUSON. Discrete & Computational Geometry, 1999)

Crystal Topologies and Discrete Mathematics – p.15/28

Simple tilings

A spatial tiling is simple ifit has four edges meetingat each vertex and oneface at each angle.

It is uninodal if all verticesare related by symmetry.

There are 9 types ofsimple, uninodal tilings in ordinary space.

(O. DELGADO FRIEDRICHS, D.H. HUSON. Discrete & Computational Geometry, 1999)

Crystal Topologies and Discrete Mathematics – p.15/28

Petroleum crackers

Of the 9 types ofsimple, uninodaltilings, 7 carryapproved zeoliteframeworks as of the"Atlas".

But how can weproduce all the otherframeworks?

SOD LTA

RWY RHO

FAU KFI CHA

Crystal Topologies and Discrete Mathematics – p.16/28

Petroleum crackers

Of the 9 types ofsimple, uninodaltilings, 7 carryapproved zeoliteframeworks as of the"Atlas".

But how can weproduce all the otherframeworks?

SOD LTA

RWY RHO

FAU KFI CHA

Crystal Topologies and Discrete Mathematics – p.16/28

Is diamond simple?

The diamond net has no simple tiling — butalmost. We just have to allow two faces insteadof one at each angle. The tile is a hexagonaltetrahedron, also known as an adamantane unit.

There are 1632 such quasi-simple tilings, whichcarry all 14 remaining uninodal zeolites.

Crystal Topologies and Discrete Mathematics – p.17/28

Is diamond simple?

The diamond net has no simple tiling — butalmost. We just have to allow two faces insteadof one at each angle. The tile is a hexagonaltetrahedron, also known as an adamantane unit.

There are 1632 such quasi-simple tilings, whichcarry all 14 remaining uninodal zeolites.

Crystal Topologies and Discrete Mathematics – p.17/28

Ambiguities

The tiling for an atom-bond graph is not unique.

We also need methods to analyze nets directly.

Crystal Topologies and Discrete Mathematics – p.18/28

Ambiguities

The tiling for an atom-bond graph is not unique.

We also need methods to analyze nets directly.

Crystal Topologies and Discrete Mathematics – p.18/28

Barycentric drawings

Place each vertex inthe center of gravityof its neighbors:

p(v) =1

d(v)

∑

vw∈E

p(w)

wherep = placement,d = degree.

Crystal Topologies and Discrete Mathematics – p.19/28

Tutte’s idea[TUTTE 1960/63]:

Pick and realize aconvex outer face.

Place restbarycentrically.

G planar, 3-connected⇒ convex

planar drawing.

Crystal Topologies and Discrete Mathematics – p.20/28

Tutte’s idea[TUTTE 1960/63]:

Pick and realize aconvex outer face.

Place restbarycentrically.

G planar, 3-connected⇒ convex

planar drawing.

Crystal Topologies and Discrete Mathematics – p.20/28

Periodic versionPlace one vertex, chooselinear map Z

d→ R

d.

Theorem:This defines a uniquebarycentric placement.

Corollary:All barycentricplacements of a net areaffinely equivalent.

Crystal Topologies and Discrete Mathematics – p.21/28

Periodic versionPlace one vertex, chooselinear map Z

d→ R

d.

Theorem:This defines a uniquebarycentric placement.

Corollary:All barycentricplacements of a net areaffinely equivalent.

Crystal Topologies and Discrete Mathematics – p.21/28

Periodic versionPlace one vertex, chooselinear map Z

d→ R

d.

Theorem:This defines a uniquebarycentric placement.

Corollary:All barycentricplacements of a net areaffinely equivalent.

Crystal Topologies and Discrete Mathematics – p.21/28

Stability

In a barycentric placement, vertices may collide:

If that does not happen, the net is called stable.

Crystal Topologies and Discrete Mathematics – p.22/28

Stability

In a barycentric placement, vertices may collide:

If that does not happen, the net is called stable.

Crystal Topologies and Discrete Mathematics – p.22/28

Ordered traversals

For a locally stable net:

Place start vertex,choose map Z

d→ R

d.

Do a breadth firstsearch.

Sort neighbors byposition.

⇒ unique vertex numbering⇒ polynomial time isomorphism test

Crystal Topologies and Discrete Mathematics – p.23/28

Ordered traversals

For a locally stable net:

Place start vertex,choose map Z

d→ R

d.

Do a breadth firstsearch.

Sort neighbors byposition.

(0,0)1

⇒ unique vertex numbering⇒ polynomial time isomorphism test

Crystal Topologies and Discrete Mathematics – p.23/28

Ordered traversals

For a locally stable net:

Place start vertex,choose map Z

d→ R

d.

Do a breadth firstsearch.

Sort neighbors byposition.

(0,0)1

⇒ unique vertex numbering⇒ polynomial time isomorphism test

Crystal Topologies and Discrete Mathematics – p.23/28

Ordered traversals

For a locally stable net:

Place start vertex,choose map Z

d→ R

d.

Do a breadth firstsearch.

Sort neighbors byposition.

(0,0)

(1,1)(−1,1)

(0,−2)

2 41

3

⇒ unique vertex numbering⇒ polynomial time isomorphism test

Crystal Topologies and Discrete Mathematics – p.23/28

Ordered traversals

For a locally stable net:

Place start vertex,choose map Z

d→ R

d.

Do a breadth firstsearch.

Sort neighbors byposition.

2 46

5

7 8

91

3

⇒ unique vertex numbering⇒ polynomial time isomorphism test

Crystal Topologies and Discrete Mathematics – p.23/28

Natural tilings(local version)

Definition:A tiling is called natural for the net it carries if:

1. It has the full symmetry of the net.

2. No tile has a unique largest facial ring.

3. No tile can be split further without violatingthese conditions or adding edges.

Note:

A natural tiling need not be unique for its net.

Crystal Topologies and Discrete Mathematics – p.24/28

Natural tilings(local version)

Definition:A tiling is called natural for the net it carries if:

1. It has the full symmetry of the net.

2. No tile has a unique largest facial ring.

3. No tile can be split further without violatingthese conditions or adding edges.

Note:

A natural tiling need not be unique for its net.

Crystal Topologies and Discrete Mathematics – p.24/28

Natural (quasi-)simple tilings

The 9 simple tilings areall natural.

Of the 1632 quasisimpletilings, 94 are natural.

Among these 103 tilings,no net appears twice.

All 21 uninodal zeolitesappear, except ATO.

ATO has a natural tilingwhich is not quasisimple.

AFI

ATO

Crystal Topologies and Discrete Mathematics – p.25/28

Some basic netsWhich are the spatial nets every school childshould know about? Here’s one suggestion:

The 5 regular nets and their tilings.(O. DELGADO FRIEDRICHS, M. O’KEEFFE, O.M. YAGHI. Acta Cryst A, 2002)

Crystal Topologies and Discrete Mathematics – p.26/28

Other scalesCellular structures occur in nature at all scales.How can we grasp their shapes and dynamics?

(Image: Doug Durian, UCLA Physics) (Image: Sloan Digital Sky Survey)

Crystal Topologies and Discrete Mathematics – p.27/28

Acknowledgements

Andreas Dress, Bielefeld/LeipzigGunnar Brinkmann, GentDaniel Huson, TübingenMichael O’Keeffe, TempeOmar Yaghi, Ann ArborAlan Mackay, London

Jacek Klinowski, CambridgeMartin Foster, Tempe

and many more...

Crystal Topologies and Discrete Mathematics – p.28/28