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.
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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.
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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