MIMS Technical Report No.00036 (201110031)

LECTURE ON TOPOLOGICAL CRYSTALLOGRAPHYTOSHIKAZU SUNADA

Commemorating the fourth centennial anniversary of the publication of
Kepler’s pamphlet“New-Year’s gift concerning six-cornered snow” (1611)
Abstract. This is an expository article on modern crystallography
based on discrete geometric analysis, a hybrid field of several traditional
disciplines: graph theory, geometry, theory of discrete groups, and probability, which has been developed in the last decade. The mathematicalpart relying on algebraic topology is fairly elementary, but may be still
worthwhile for crystallographers who want to learn how mathematics iseffectively used in the practical science. A brief history of crystallography is also explained.

1. Introduction
The main purpose of this expository article is twofold; first, to tell a brief
history of crystallography, a practical science originated in the classificationof the observed shapes of crystals, and second, to provide the reader with amathematical insight into modern crystallography developed in the last two
decades in order to bridge the gap of knowledge between mathematicians
and crystallographers.
The mathematical tools employed here are adopted from elementary algebraic topology. More specifically, the routine theory of covering spaces andhomology groups is effectively used in the study of the 3D networks associated with crystals. Actually many notions and tools invented by crystallographers may be interpreted in these algebraic-topological terms establishedby the first half of the 20th centuty (see Appendix). Moreover the enumeration of crystal structures, an area of considerable scientific interest for many
years, is a simple byproduct of the classical theory of covering spaces as far
as their topological structures are concerned (see Section 9 and 10 ).
In the latter part of the lecture, I shall explain a minimum principle
for crystals in the framework of discrete geometric analysis ([39]), whichcharacterizes canonical placements1 of crystal structures, a way to place2000 Mathematics Subject Classification. Primary 05-01; Secondary 05C63 31C20,
39A12, 74E15.
Key words and phrases. topological crystal, canonical placement, discrete geometricanalysis, discrete Abel-Jacobi map.
1
This notion is the same as standard realizations introduced in [27], and coincides withthe archetypical representations (Eon [23]); see Remark in Section 12).
1

2Topological Crystallographythem in space so as to have the most symmetric microscopic shapes. In
its concrete construction, the well-known idea of orthogonal projections inharmonic integrals is employed. In spite of its pure-mathematical nature,
the notion of canonical placements turns out to fit with a systematic designof crystal structures.
Canonical placements turn up in asymptotic behaviors of random walks
on topological crystals, the abstraction of crystal structures, and are closely
related to a discrete analogue of Abel-Jacobi maps in algebraic geometry
as explained in Section 15. As a matter of fact, my initial motivation behind topological crystallography have arisen from these seemingly irrelevant
subjects (see [28] and [30]).
This article is based on my lecture for the Summer Challenge project at
High Energy Accelerator Research Organization (KEK) in Tsukuba. A fullaccount of topological crystallography will be given in the forthcoming book[40]. See Eon [23] for an overview from crystallographer’s standpoint.
Acknowledgement I am grateful to D. M. Proserpio and J.-G. Eon forfruitful discussions who also provided me with relevant references in chemical crystallography. I also thank Hisashi Naito and my daughter Kayo for
producing the beautiful CG images of several hypothetical crystals.

2. Legacy of Ancient Greece in crystallography
Crystals, the word derived from the Greek meaning “ice”, are the moststable form for all solids. They are found throughout the natural world, andhave been always recognized as being distinct from other forms of matter.
The links between morphology of crystals and mathematics are ancient.
It is said that geometry in ancient Greece started from the curiosity about
the shapes of crystals2. Actually legend has it that Pythagoras (about 569
BC-about 475 BC) derived the notion of regular polyhedra from the shapeof a crystal. This legend is not entirely baseless because the southern part in
Italy where Pythagoras dwelled and established his school3 produces Pyrite
crystal (an iron sulfide with the formula FeS2 ) whose shape is roughly cubic,
octahedral, or dodecahedral.
Euclid’s Element, written around 300 BC and consisting of thirteen volumes, ends up with the classification of regular convex polyhedra; that is, it
is shown in the final volume that there are exactly five regular convex polyhedra; say, tetrahedron, cube, octahedron, dodecahedron, and icosahedron(Figure 1). According to Plato’s dialogue Timaeus written around 360 B.C.,
the first mathematician who established the classification is Theaetetus, a2

Poincaré says “If there were no solid bodies in nature, there would be no geometry”.
Pythagorean school in Crotone, a Greek colonial town on the south-eastern coast of
Italy, was a religious order in which Pythagoras was a sort of cult figure.
3

Toshikazu Sunada

3

contemporary of Plato4. In today’s viewpoint, the classification of regularpolyhedra is closely related to that of finite subgroups of the orthogonalgroup O(3), point groups in crystallography ([33]).

Figure 1. Regular convex polyhedra
Pappus (290-350) of Alexandria testifies in his Collection that Archimedes(287 BC-212 BC) proposed, in a now-lost work, the notion of semi-regularpolyhedron as a generalization of regular polyhedron, and found 13 such
solids5. By this reason, semi-regular polyhedra are sometimes referred to as
Archimedean solids.

Figure 2. Semi-regular convex polyhedra
The discovery of these solids by Archimedes seems to have been madewith purely mathematical motivation. After more than 2000 years, however,some of semi-regular polyhedra are going to show up in material science asgraphic representations of certain chemical compounds6.
4A regular convex polyhedra, called also Platonic solids, are characterized by the property that its faces are congruent regular polygons, with the same number of faces meetingat each vertex.
5A polyhedron is said to be semi-regular if it has regular faces and a symmetry groupwhich is transitive on its vertices. Precisely speaking there are two infinite series of convex
prisms and convex antiprisms satisfying this condition.
6The network model of Fullerene C , a compound of carbon atoms whose existence60
was confirmed in 1990, is the 1-skeleton of the truncated icosahedron, one of semi-regularpolyhedra.

Topological Crystallography

4

3. Kepler’s contribution to crystallography
Since the period of Hellenism, geometry had taken its own path, andthe study of crystals had not been the central theme in mathematics. An
exception is the work of Johannes Kepler (1571–1630) on snowflake. Kepler’s short pamphlet entitled New-Year’s gift concerning six-cornered snow(“Strena Seu de Nive Sexangula” in Latin) is considered the first work on
the problem of crystal structures7 though he did not refer to the atomistic
viewpoint at all that dates back to ancient Greece8. Kepler’ motivation
came from the question about why snow crystals always exhibit a six-foldsymmetry.

Figure 3. Johannes Kepler
Among other things, he pondered two questions in this pamphlet, which
are eventually to break new ground in very diverse fields of science, especiallyin mathematics and crystallography.
(1) He speculated that the densest packing of equally sized spheres isattained by the hexagonal arrangement, which later became known as the
Kepler conjecture9.

Figure 4. Hexagonal sphere packing
7This pamphlet was dedicated on the occasion of the new year of 1611 to his friend
and patron, the scholar and imperial privy councilor Johannes Matthäus Wackher von
Wackenfels.
8
The atomism was advocated by Leucippus and Democritus in the 5th century BC.
9
In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture.

Toshikazu Sunada5

The hexagonal arrangement is closely related to what crystallographerscall the face-centered cubic lattice (fcu10) illustrated in Figure 5, where thenodes (vertices) are located at centers of equal spheres, and the line segments(edges) indicate that two spheres located at their end points touch eachother.

Figure 5. The face-centered cubic lattice(2) In connection with the hexagonal arrangement, Kepler noticed thatthe honeycomb structure maximizes the number of wax walls each bee shareswith his neighbor, thereby allowing bees to collaborate in constructing theshared walls of a cell. The hexagon also turns out to be most efficient interms of exploiting the maximum of space using the minimal amount of
wax11.
Later we will see that a different kind of minimum principle characterizes
the honeycomb structure.
Kepler’s observation includes indirect pointers to the law of constant angles for a six-sided snow crystal; thus he is regarded as a forerunner of
N. Steno (1669), M. W. Lomonosov (1749) and Romé de l’Isle (1783), thediscoverers of the law.
Needless to say, Kepler is renowned as an astronomer who discovered
the three laws of planetary motion. His essay demonstrates that Kepler
had been thinking of not only the law of the vast universe but also thesmallest aspects of nature. It should be emphasized, however, that he wasan astrologer and theologian as well, and always sought an interpretation of
the harmony of forms as God’s choice ([35]). Namely he asks the snowflakes
the same question as he did the planets; which form follows God’s order.
It is worthwhile to mention that, in Harmonice Mundi (1619) which includes his discovery of the third law of planetary motion, Kepler accomplished a complete classification of semi-regular polyhedra ([11]). This
achievement tells, though his study on geometric figures can not be separated
from his theological dogmatism12, that he was one of the most outstanding
mathematicians of his day.
10

We use the three-letter names for crystal ctructures propsed by [34].
Pappus had already noticed that bees have the foresight at their disposal allowing
them to understand that a hexagonal partition is more effective than a triangular or asquare partition.
12
It is said that Kepler is the last astronomer of the Renaissance, and not the first of
the new age.
11

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Topological Crystallography

4. Crystallographic groups
It is only in the 19th century, more than 2,000 years later since the timeof ancient Greece, that mathematics began to play a serious role in crystallography; that is, group theory became matured enough to be applied tothe morphology of crystals13.
As a prehistory of the morphology, there have been observations by people
mentioned above about the angles between the faces of a crystal. That is,
the faces could be of different sizes, depending upon the conditions underwhich the crystal have grown, but the relations them remain fixed.
As a reasoning of this fact, Huygens thought of crystals, say the mineral
calcite, as being built from small ellipsoidal units (1690). Using this picture,
he explained that the resulting faces always preserve the same relationships.
The science of crystallography started when Haiiy published his Essaid’une Théorie sur la Structure des Cristaux (1784) in which he developed
Huygen’s concept to explain the law of angles between faces. Since then,scientists had studied many physical properties (for instance, optical activity
discovered by Aragon in 1811) in connection with the morphology of crystals.
The study of morphology led the early crystallographers to a simple classification of all crystals in terms of symmetry. It is German crystallographer
J. F. C. Hessel who, for the first time, investigated the possible types ofsymmetry for a crystal (1830). He found that there are 32 types of groupsof symmetry. His book Krystallonomie und Krystallographie stating thisconclusion was published in Leipzig in 1831, and has not been read seriously enough by other scientists at that time. A. Bravais in 1849 and A.
Gadolin in 1867 rediscovered the same 32 symmetry groups by repeating thederivation. Their work was well-known before L. Sohncke, a crystal physicist, found Hessel’s earlier book in 1891. It should be pointed out that thenotion of groups has been seldom used till 1860’. C. Jordan is the first who
recognized that the classification of crystal symmetry is described in termsof subgroups of O(3) (1869). As for 3-dimensional crystallographic groups14(space groups), after fundamental works by C. Jordan and L. Sohncke, all
230 isomorphism classes have been determined in 1891 by E. S. Fedorov andA. Schoenflies independently.
From a mathematical viewpoint, it is natural to study crystallographic
groups in general dimension. In 1900, D. Hilbert posed a question on finiteness of isomorphism classes of crystallographic groups in his famous address
at the second International Mathematical Congress held at Paris. L. Bieberbach (1910-12) and G. Frobenius (1911) solved affirmatively the problem.
13

The notion of “group” originated in the work of Evariste Galois (1811-1832) on theinability to generalize the quadratic equation to algebraic equations of degree greater thanor equal five. Nowadays the idea of group is one of the great unifying ideas of mathematics.
14A subgroup G of the congruence transformation group of Rd is said to be a d-dimensional crystallographic group if it is discrete and has a compact quotient G\Rd .

Toshikazu Sunada

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5. Discovery of crystal structures
Crystallographic groups introduced to describe macroscopic symmetry of
crystals have been a basic tool in classical crystallography even after Max
von Laue’s discovery of crystal structures by the diffraction of X-rays (1912).
The X-ray technique is based on observing the scattered intensity of an
X-ray beam hitting a sample. What was found by Laue is that crystals are
solids composed of atoms arranged in an orderly repetitive array, and hence
confirmed the anticipation that classical crystallographers had conceived
long before. Just after Laue’s discovery, W. L. Bragg and W. H. Bragg
greatly simplified von Laue’s description of X-ray interference, and solved
the structures of diamond, sodium chloride (NaCl) and Zinc sulfide (ZnS)
in 1913. Today, crystallographers can produce a 3D picture of the densityof electrons within the crystal from which the mean positions of the atoms
in the crystal can be determined, as well as their mutual interactions.

Figure 6. Covalent bonding

It should be worthwhile to point out that there are several ways of interactions. A typical one is covalent bonding, which is a form of chemicalbonding that is characterized by the sharing of pairs of electrons between
atoms (the term “covalent bond” dates from 1939). When we draw a pictureof a crystal structure (and any molecule), we join two atoms by a line if theyinteract each other. This line is called a bond.
Let us exhibit a few examples of crystal structures. Figure 7 illustrates
the structures of Diamond and Lonsdaleite.

Figure 7. Diamond and Lonsdaleite8Topological Crystallography

Diamond is an allotrope of carbon which is formed and synthesized at
high-pressure and high-temperature conditions, and is known to be less stable than graphite15 though the conversion rate from diamond to graphite is
negligible at ambient conditions. Silicon and germanium adopt similar typesof crystal structure. On the other hand, Lonsdaleite (called also hexagonal
diamond) is a carbon allotrope formed when meteorites containing graphite
strike the Earth, which was first identified in 1967 from the Canyon Diablo
meteorite.
The copper crystal is a typical one having the fcu structure. Silver andgold also have the fcu structure. The honeycomb lattice, a 2D crystal
structure, appears as the crystal structure of Graphene16, an allotrope ofcarbon.
Figure 8 is what crystallographers call the body-centered cubic lattice(bcu). It is known that the caesium chloride (CsCl) forms a crystal whose
structure is bcu.

Figure 8. The body-centered cubic lattice
We will observe later that the crystal structures mentioned here, say Diamond, Lonsdaleite, fcu, bcu and honeycomb lattice, share a special featurein view of geometry.
In the 20th century, crystallographer’s interest shifted naturally from themorphology of crystals to the study of microscopic crystal structures. As
the learned reader may easily conceive, the tool available for this purpose
is graph theory, a field studying mathematical structures used to model
pairwise relations between objects from a certain collection. Actually oneof central themes in modern crystallography is to enumerate (hypothetical)
crystals by means of graph theoretic descriptions of crystal structures. Once
we find a hypothetical crystal, a systematic prediction of its physical properties for appropriate atoms can be carried out by first principles calculations.
The prediction by the computer power encourage (or discourage) materialscientists to synthesize the hypothetical crystals.
15
Graphite as a mineral is one of the allotropes of carbon, and the most stable form
under standard conditions. It has a layered, planar structure. In each layer, the carbon
atoms are arranged in a honeycomb lattice. Weak van der Waals forces hold the layers
together.
16
The term graphene was coined by Hanns-Peter Boehm (1962). The Nobel Prize inPhysics for 2010 was awarded to Andre Geim and Konstantin Novoselov “for groundbreaking experiments regarding the two-dimensional material graphene”.

Toshikazu Sunada

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Before proceeding to graph theoretic treatment of crystals, let us describea brief history of the interchange of ideas between chemistry and graphtheory.
6. Graphs associated with chemical compounds
A primitive idea of “bond” has been already conceived from as early as the
12th century. It supposes that certain types of chemical species are joinedby a type of chemical affinity. By the mid of 19th century, it was becoming
clear that chemical compounds are formed from constituent elements, andthe idea of bond had been developed as the theory of valency based on
the theory of radicals. Especially the theory of chemical structure by theGerman chemist August Kekulé, in which he took into account the specificcombining power (or valences) of specific atoms, provided dramatic new
clarity of understanding on chemical compounds (1858).
Here came a fusion of mathematical and chemical ideas. That is, graphtheory17 and chemistry started to interact each other.
Mathematically, a graph is represented by an ordered pair X = (V, E) of
the set of vertices V and the set of all directed edges E. Note that each edge
has just two directions, which are to be expressed by arrows.
For an directed edge e, we denote by o(e) the origin, and by t(e) the
terminus. The inversed edge of e is denoted by e. With these notations, wehave o(e) = t(e), t(e) = o(e).

Figure 9. Directed edge
The set of directed edges with origin x ∈ V is denoted by Ex ;Ex = {e ∈ E| o(e) = x}.
The number of elements in Ex is said to be the degree of x, and is denotedby deg x.
The usage of graph theory in chemistry, which is natural in view of theidea of valency, is traced back to 1864 when the Edinburgh chemist Crum
Brown proposed to represent chemical compounds by graphs where each
atom is represented by a vertex of the graph, and each edge of the graphrepresents a bond. In 1874, Carl Schorlmmer linked chemistry with trees.
Meanwhile French chemist L. Pasteur observed that the relative spatial arrangement of atoms within molecules is vital in understanding of their chemical properties (1861); thus he may be described as the first stereochemist.
In stereochemistry, molecules are represented as graphs realized in space.
17

The notion of graphs was implicitly used by Leonhard Euler (1707-1783) when hesolved the problem of “Königsberg seven bridges” (1736).

10Topological Crystallography

It is interesting to point out that the term “graph” was used for the firsttime by Sylvester in a note appeared in the scientific journal Nature (1878)
where a relationship between chemistry and algebra was suggested.
To avoid a confusion, we shall use the terminology “network” (or simply“net”) from now on when we are handling a graph placed in space (plane).
7. Fundamental finite graphs and building blocks
After confirmed that atoms and molecules have physical existence, network representations of molecules also broke away from fictitious entity. It
is thus a matter of course for crystallographers to employ graph theory forthe study of crystal structures. Strangely enough, however, the systematic
approach by means of graph theory has not been done until 1950’s though,by this time, plenty of crystal structures have been understood and represented by networks. The reason may be due to the self-explanatory fact thatthe ideal model of a crystal is an infinite graph, which is not an object thatthe ordinary graph theory handles.
It is A. F. Wells who initiated the intensive research by treating crystals
as periodic 3D nets in space ([41], [42]). In his influential book [42] (1977),
Wells treated uniform nets, and tried to investigate them systematically.
Here “uniform” means that the net is a regular graph, and that the shortest
circuit containing any two edges through any vertex shall have the same
number of edges.
The net associated with a crystal is not just an infinite graph realized inspace (or plane), but, as observed by crystallographers ([8]), a graph witha translational action which becomes a finite graph when factored out. We
shall take up the term “a Bravais lattice” to express a group of translations
leaving the crystal invariant. The finite graph obtained by factoring out iscalled the fundamental finite graph (or quotient graph). This observation
has been used in a systematic enumeration of crystal structures (see [24] for
instance).
To illustrate the concrete way to get the fundamental finite graph X0 =(V0 , E0 ) from a net X = (V, E) of a crystal, we employ Figure 10. Here as a
Bravais lattice, we take the lattice group generated by two vectors depictedby arrows. We first label vertices and edges of X; say, A, B, C, D for vertices
and a, b, c, d, e, f for edges in the honeycomb lattice, in such a way that thesame label is assigned to two edges (resp. two vertices) if one is obtained
from another by a translation belonging to the Bravais lattice. Then wegather the labels of vertices and join them by labeled edges, keeping theadjacency relation. Thus one gets the fundamental finite graph18.
This construction allows us to assign a vector v(e) to each directed edge
e in X0 as follows. Choose a direct edge e0 in X which corresponds to e.
18What we explained here is a special case of the construction of quotient sets associatedwith equivalence relations.

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Figure 10. How to get the fundamental finite graph
Since e0 is a directed line segment in space (or plane), it represents a vectorv(e) which does not depend upon the choice of e0 . Obviously v(e) = −v(e).
The system of vectors[{v(e)}e∈E0 ={v(e)| e ∈ E0x }x∈V0completely determines the net X. In this sense, {v(e)}e∈E0 deserves to becalled the building block19 of X.

Figure 11. A building block
Figure 11 tells how to get the crystal from the building block. For example, travel on the graph X0 along AbBdCeDf C, and sum up the vectors
corresponding to directed edges in the trail20−−→ −−−→ −−→ −−−→
AbB + BdC + CeD + Df C.
Drawing such vector sums in plane for all trails starting from A, we obtain
the honeycomb X.
Again from the construction of the fundamental finite graph, we obtaina map from X onto X0 in a natural manner, which we call the canonicalmap (precisely speaking, the canonical map is a morphism of graphs which
19Crystallographers use the term “a labelled quotient graph” for X with a building0
block. Mathematically a building block is nothing but a 1-cochain on X0 with values in
Rd .
20In Section 8, “trails” will be called “paths”.

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Topological Crystallography

brings vertices to vertices and edges to edges in such a way that the adjacency
relation between vertices and edges is preserved.
One should remark that the choice of Bravais lattice is not unique. Two
Bravais lattices (represented by arrows) are shown in the Figure 12. The
corresponding fundamental finite graphs are depicted on the left and right.
We usually take the maximal one as a Bravais lattice.

Figure 12. Two Bravais lattice and their fundamental finite graphs
We now take a look at Diamond and Lonsdaleite. For Diamond, if wetake the maximal Bravais lattice, the fundamental finite graph is the graph(A) in Figure 13 consisting of two vertices joined by four parallel edges21.
For Lonsdaleite, its fundamental finite graph is the graph (B). In both cases,
the building blocks are related to the regular tetrahedron.

Figure 13. Fundamental finite graphs for Diamond and Lonsdaleite
Figure 14 is the crystal structure of the 3D kagome lattice22, which iscomprised of corner-sharing tetrahedra. The fundamental graph for the
maximal Bravais lattice is given on the right. The 3D kagome lattice turns
out to have a special feature as seen in Section 13.
It is interesting to point out that 1-skeletons of many semi-regular polyhedra are hidden in crystals. For instance, one can find the cubooctahedron in
the face-centered cubic lattice, and truncated tetrahedron in the 3D kagome
lattice.
So far we have traced the path that crystallographers have taken. Now let
us ask a mathematician “What’s the mathematical nature of crystal structures ?”. His immediate answer would be “Topologically they are infinitefold abelian covering graphs over finite graphs. Crystals are their periodic
21The honeycomb lattice is considered the 2-dimensional analogue of the diamond
crystal22This is a 3D version of the ordinaly kagome lattice; see Section 13.

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Figure 14. 3D kagome lattice and its fundamental finite graphrealizations in space”. This simple answer is a sort of folklore in the community of mathematicians, and hence it is not attributed to anybody. It is
no wonder, however, that Henri Poincaré (1854-1912), founder of algebraic
topology, could have easily conceived this answer if he would have witnessed
Laue’s discovery, which was coincidentally accomplished in the last year of
Poincaré’s life.
The main aim from now on is to explain the nature of this answer byadopting routine tools in algebraic topology.
8. Homology
The homology group (of a cell complex in general) is an algebraic systemconstructed from figures (cells)23. We are not going to be involved in its
general theory, but shall confine ourselves to the case of graphs. The reader
who is familiar with algebraic topology may skip the definitions of homology
groups and covering graphs.
Given a finite graph X0 = (V0 , E0 ), we consider a sum over vertices of thefollowing formXax x(ax ∈ Z).
x∈V0

We call such a sum a 0-chain (with coefficients in Z). This sum and the
product ax x (integer × vertex) are formal so that one should not worry
about their meaning. The set of 0-chains is denoted by C0 (X, Z) with which
we equip the structure of a free abelian group in the following way.
X(1)
ax x = 0 if and only if ax = 0 for every x ∈ V0 ,
x∈V0

(2)

X

x∈V0

ax x ±Xx∈V0bx x =

X(ax ± bx )x.

x∈V0

Thus C0 (X, Z) is a free abelian group with the Z-basis V .
Likewise we define the abelian group C1 (X0 , Z) consisting of finite formalsums over directed edgesXae e(ae ∈ Z),e∈E0
23

Historically the germination of homology theory is seen in the work by Riemann andPoincaré on algebraic functions.

14

Topological Crystallography

which we call a 1-chain (with coefficients in Z). In this turn, however, we
impose the relation e = −e so that a way to express a 1-chain as above is
not unique24. If we fix an orientation E0o of X; that is, a subset E0o ⊂ E0
E0 , then E0o is a Z-basis of C1 (X0 , Z), so
satisfying E0o ∩ E0o = ∅, E0o ∪ E0o =X
that any 1-chain is expressed asae e in a unique way.
e∈E0o

Roughly speaking, the homology group of a graph is introduced to describe the adjacency relation between vertices and edges in an algebraic
way. To be exact, we define the boundary operator (homomorphism) ∂ :C1 (X0 , Z) −→ C0 (X0 , Z) by putting ∂(e) = t(e) − o(e) and extending it toa homomorphism, where we should note that ∂ e = ∂(−e), and hence thedefinition of ∂ is compatible with the relation e = −e. The 1-homology
group H1 (X0 , Z) is defined to be the kernel of ∂:H1 (X0 , Z) = Ker ∂ (= {α ∈ C1 (X0 , Z)| ∂α = 0}).
An element of H1 (X0 , Z) is said to be a homology class or a 1-cycle. H1 (X0 , Z)
is a free abelian group of finite rank. The rank of H1 (X0 , Z), symbolicallyb1 (X0 ), is called the (first) Betti number. The Betti number is an indicatorof the “complexity” of finite graphs.
Homology classes are represented by closed paths. Here a path in X0 is asequence of edges c = (e1 , . . . , en ) with t(ei ) = o(ti+1 ) (i = 1, . . . , n − 1). Ift(en ) = o(e1 ), then c is said to be closed.

Figure 15. A path
For a path c = (e1 , . . . , en ) in X0 , the symbol hci expresses the 1-chain
e1 + · · · + en . If c is closed, then ∂(e1 + · · · + en ) = 0, and hence hci is ahomology class. An important fact we frequently use is that every homology
class is represented by a closed path.
We shall enumerate several facts of the homology group.
Fact 1 There exists a Z-basis of H1 (X0 , Z) consisting of homology classes
represented by closed paths.
Fact 2 Let T ba a spanning tree25 in X0 . Then the number of undirectededges in X0 not contained in T is equal to the first Betti number b1 (X0 ).
24We are considering the factor group of the free abelian group with the Z-basis E
0modulo the subgroup generated by {e + e| e ∈ E0 }.
25Among all subtrees in a graph X , we can find a maximal one with respect to inclu0
sion, which we call a spanning tree of X0 .

Toshikazu Sunada

15

Fact 3 When X0 is a plane graph26 without vertices of degree one, X0
divides the plane into a finitely many bounded connected small regions; say,
D1 , . . . , Dk . One can prove k = b1 (X0 ).
Fact 4 (The Euler relation) |V0 | − |E0 |/2 = 1 − b1 (X0 ). This is equivalent
to Euler’s polyhedral theorem27.
Given a system of vectors {v(e)}e∈E0 , we consider the homomorphismb : H1 (X0 , Z) −→ Rdv
defined bybv(d = 2, 3)³X´
Xae e =
ae v(e).

e∈E0

e∈E0

b is well-defined, and that, if {v(e)}e∈E0 is a
One can easily check that vb (H1 (X0 , Z)) coincides with thebuilding block of a crystal, then the image v
Bravais lattice. The converse holds:
Theorem 8.1. A system of vectors {v(e)}e∈E0 in Rd is a building block of
b is a lattice group in Rd
a crystal if and only if the image of v
The proof is given in the next section.
So far we have handled the homology group with coefficients in Z. Ina similar fashion, H1 (X0 , R), the 1-homology group with coefficients in R,
is defined by replacing Z by R in the definition. The group H1 (X0 , R) is
actually a vector space over R, and H1 (X0 , Z) is a lattice group in H1 (X, R).
9. Topological crystals
Recall that, given the net X associated with a crystal and a Bravais
lattice, we have the canonical map ω : X −→ X0 . A special feature of ω isthat it is a regular covering map28 whose covering transformation group isisomorphic to Zd (d = 2, 3).
For the convenience of the reader, let us give a graph-theoretical description of maps between graphs, and also covering maps. Let X = (V, E),
X0 = (V0 , E0 ) be connected graphs. A morphism ω : X −→ X0 is a correspondence of vertices and edges preserving the adjacency relation among
them. More precisely, it¡ is a pair¢ ω =¡(ωV ,¢ωE¡) of maps
¢ ωV :¡V −→¢ V0 , ωE :E −→ E0 satisfying o ωE (e) = ωV o(e) , t ωE (e) = ωV t(e) for every
26A plane graph is a graph drawn on the plane in such a way that its edges intersect

only at their endpoints, while a planar graph is a graph that can be embedded in the plane.
27This theorem due to Euler (1750) asserts v − e + f = 2 where v, e, f are the numberof vertices, edges and faces of a convex polyhedron. One can make use of this formula
to prove that there are just 5 regular convex polyhedra. Historically Euler’s theorem isconsidered to be the starting point of “topology”.
28A primitive idea of covering maps for general spaces is seen in Riemann’s study ofalgebraic functions and their integrals (1851). The book Lehrbuch der Topologie by H.
Seifert and W. Threlfall published in 1934 gave a rigorous set-up and popularized thenotion of covering maps.

16

Topological Crystallographye ∈ E, and ωE (e) = ωE (e). By abuse of notations, we write ω(x) for ωV (x)and ω(e) for ωE (e) for simplicity.
If both ωV and ωE are bijective (one-to-one and onto), the morphism ω =(ωV , ωE ) is said to be an isomorphism. An isomorphism ω of X onto itself iscalled an automorphism. The totality of automorphisms of X, symbolically
Aut(X), is a group in a natural manner.
A morphism ω : X −→ X0 is said to be a covering map if(i) ω : V −→ V0 is surjective,(ii) for every x ∈ V , the restriction ω|Ex : Ex −→ E0,ω(x) is a bijection.
The group G(ω) = {σ ∈ Aut(X)| ω ◦ σ = ω} is called the coveringtransformation group of the covering map ω. A covering map ω is said tobe regular if, for any x, y ∈ V with ω(x) = ω(y), there exists an element
σ ∈ G(ω) such that y = σx.
Let d be any positive integer. A d-dimensional topological crystal29 X isa regular covering graph over a finite graph, say X0 , whose covering transformation group L is a free abelian group of rank d. After the case of the
net associated with a crystal, we call X0 the fundamental finite graph. X(or ω : X −→ X0 ) is said to be a topological crystal over X0 . We also call
L an abstract Barvais lattice.
In a word, a topological crystal is an abstraction of a real crystal, the notion obtained by forgetting how the net are placed in space, which, however,still retains all the information on the connectivity of atoms in the crystal.
The following is well-known in the general theory of covering maps:
Theorem 9.1. (1) There exists a (unique) topological crystal ω ab : X0ab −→X0 such that the dimension of X0ab is b1 (X0 ). Its abstract Bravais lattice is
H1 (X0 , Z).
(2) For any topological crystal ω : X −→ X0 , there exists a regular bcovering map ω1 : X0ab −→ X such that ω ◦ ω1 = ω ab (ω1 is called a subcoveringmap). The dimension of X is less than or equal to b1 (X0 ). In this sense,
X0ab is the maximal one30 among all topological crystals over X0 . We shallcalle X0ab the maximal topological crystal over X0 .
(3) The covering transformation group H of ω1 : X0ab −→ X is a subgroupof H1 (X0 , Z) such that the factor group H1 (X0 , Z)/H is isomorphic to theabstract Bravais lattice of topological crystal ω : X −→ X0 and hence freeabelian. Conversely, for a subgroup H of H1 (X0 , Z) such that H1 (X0 , Z)/H
is free abelian, there exists a topological crystal X whose abstract Bravais
lattice is H1 (X0 , Z)/H.
29The reason to adopt the term “topological crystal” is to emphasize its abstract na-

ture and at the same time to keep the word “crystal” in order to make it clear that weare addressing the problem of crystals, not the problem of general graphs. Chemical crystallographers adopt the term periodic graphs for the underlying topology of crystal
structures.
30In [4], the term “minimal net” is used for the maximal abelian covering over a finite
graph.

Toshikazu Sunada17

A subgroup H of H1 (X0 , Z) such that the factor group H1 (X0 , Z)/H isfree abelian will be called a vanishing subgroup.
Roughly speaking, the theorem above tells that there is a one-to-one correspondence between the family of topological crystals X over X0 and thefamily of vanishing subgroups H of H1 (X0 , Z).
As an example of X0 , let us consider the 3-bouquet graph B3 (Figure16)31. Let E0o = {e1 , e2 , e3 } be an orientation of B3 . Then, e1 , e2 , e3constitute a Z-basis of H1 (B3 , Z). The maximal topological crystal overB3 is the (topological) cubic lattice. On the other hand, the triangularlattice is a topological crystal corresponding to the vanishing subgroup H =Z(e1 + e2 + e3 ).

Figure 16. The cubic lattice and the triangular latticeb (α) = 0},
We now give a proof of Theorem 8.1. Let H = {α ∈ H1 (X0 , Z)| v
b , and ω : X −→ X0 be the topological crystal corresponding
the kernel of v
to H. Define the map Φ : V −→ Rd by setting
Φ(x) = v(ω(e1 )) + · · · + v(ω(e)),
where c = (e1 , . . . , en ) is a path in X joining a reference point x0 and x (onecan prove that Φ(x) does not depend upon the choice of c). Then we extend
Φ to a piecewise linear map from X into Rd . The reader may notice thatthe definition of Φ is a generalization of what we have illustrated in Section
7 for the honeycomb lattice. The crystal which we wanted is defined as theimage of Φ.
One can checkv(ω(e)) = Φ(t(e)) − Φ(o(e))and(1)

Φ(σx) = Φ(x) + ρ(σ),

where σ is an arbitrary element of the abstract Bravais lattice L = H1 (X0 , Z)/H,b (α) =and ρ : L −→ Rd is the injective homomorphism characterized by v
31The n-bouquet graph is the graph with a single vertex with n loop edges.

18

Topological Crystallography

ρ(µ(α)). Here µ : H1 (X0 , Z) −→ L = H1 (X0 , Z)/H is the canonical homomorphism.
10. Enumeration of topological crystals
If we forget how crystals are placed in space, then the enumeration of crystals reduces to that of topological crystals, or equivalently the enumerationof finite graphs X0 and vanishing subgroups H of H1 (X0 , Z).
If we restrict ourselves to the class of finite graphs X0 such that deg x ≥ 3
for every x ∈ V0 , then there are only finitely many X0 with a fixed Betti
number. Thus one can enumerate X0 in principle at least.

Figure 17. Several graphs with b1 = 3

Figure 18. Several graphs with b1 = 4
It should be pointed out that, given an integer d with 1 ≤ d < b1 (X0 ),there are infinitely many vanishing groups H with rank H1 (X0 , Z)/H = d,from which it follows that there are infinitely many d-dimensional topological crystals over X0 . Hence to enumerate topological crystals, we need to
introduce a sort of “magnitude” of vanishing subgroups
H. For this sake,X
we first define the norm kαk1 of a 1-chain α =ae e by setting
kαk1 =

Xe∈E0o|ae |,e∈E0owhere it should be noted that kαk1 does not depend upon the choice ofan orientation E0o . Then for a Z-basis S = {αd+1 , . . . , αb } of a vanishingsubgroup H, we puth(S) = max(kαd+1 k1 , . . . , kαb k1 ),Toshikazu Sunada

19

and define the height of H byh(H) = min h(S).
S

Consider two sets
R1 = {S| h(S) ≤ h},
R2 = {H| h(H) ≤ h}.
Clearly R1 is a finite set. The correspondence
S 7→ H generated by S
gives a surjective map of R1 onto R2 . Thus we get the following:
Theorem 10.1. For any positive number h and positive integer d with1 ≤ d < b1 (X0 ), there are only finite number of vanishing subgroups H of
H1 (X0 , Z) satisfying(1) rank H1 (X0 , Z)/H = d,(2) h(H) ≤ h.
11. Canonical placements
Topological crystals are purely mathematical objects “living in the logicalworld, but not in real space” in the sense that they are constructed on thebasis of pure reflection32, thereby being not visible even for the 3D case if
we would leave it intact.
The issue of topological crystals becomes more interesting if we seek themost natural way to place them in space. For instance, the diamond crystal
is one of the placements in space of the topological diamond (the maximal
topological crystal over the graph (A) in Figure 13). Figure 19 illustrates
another placement of the topological diamond (“graphite-like” placement).

Figure 19. The graphite-like placement of the diamond crystal
Actually there are infinitely many ways to realize a given topologicalcrystal in space. Then a natural question is raised: “what is the mostcanonical placement, especially what are the characteristics possessed by
32This is the so-called Platonic view; that is, we mathematicians insist that mathematical entities are abstract in not being spatiotemporally located, and hence lie outside areal world.

20

Topological Crystallography

the diamond crystal that differ from all other placements of the topologicaldiamond ?”The diamond crystal is more symmetric than its graphite-like placement.
This observation motivates us to seek the most symmetric placements ingeneral. As a candidate of such placements, we shall introduce the notionof canonical placements (see Remark below).
We first define the notion of general placements in view of (1). Let X =(V, E) be a d-dimensional topological crystal with an abstract Bravais latticeL, and let X0 = (V0 , E0 ) be its fundamental finite graph. A piecewise linear
map Φ : X −→ Rd is said to be a placement if there exists an injective
homomorphism ρ : L −→ Rd satisfying(1) Φ(σx) = Φ(x) + ρ(σ)(x ∈ V, σ ∈ L),(2) ρ(L) is a lattice subgroup of Rd .
The image Φ(X) is the cystal33 associated with the topological crystal X.
Its Bravais lattice is ρ(L). The building block {v(e)}e∈E0 of Φ(X) is givenby v(ω(e)) = Φ(t(e)) − Φ(o(e)). As in the case of nets, the building block
determines Φ uniquely (up to translations).
To define the canonical placement of X, we shall resort to a certain minimal principle, having in mind Euler’s remark “Since the fabric of the Universe is most perfect and the work of a most wise creator, nothing at all
takes place in the Universe in which some rule of maximum or minimum
does not appear”.
In order to formulate the minimal principle, let us think of a crystal as asystem of harmonic oscillators; that is, each edge is supposed to represent
a harmonic oscillator whose energy is the square of its length (Figure 20).
This simple model of a crystal leads us to the notion of energy functional
defined as34¡¢−2/d X
(2)
E(Φ) = vol Dρ(L)kv(e)k2 ,e∈E0which is regarded as thetotal potential energy of the system¡ “normalized”¢
per unit cell. Here vol Dρ(L) stands for the volume of the unit cell Dρ(L)associated with the Bravais lattice ρ(L).
¡¢−2/d
The presence of the term vol Dρ(L)in (2) makes E(Φ) a size-freequantity.
The canonical placement is defined to be a placement Φ minimizing E(Φ).
33Strictly speaking, Φ(X) could be pathological in the sense that different vertices may

be mapped by Φ to the same point, the image of an edge may collapse, or the image of
two edges may overlap.
34A real crystal (crystalline solid) is also physically regarded as a system of harmonic
oscillators under an appropriate approximation of the equation of motion, but the shapeof energy is much more complicated.

Toshikazu Sunada

21

Figure 20. A system of harmonic oscillators
Theorem 11.1. ([30],[27]) (1) There exists a unique canonical placement
Φ of E up to homothetic transformations35 of Rd .
(2) A placement Φ is canonical if and only ifX(3)v(e) = 0(x ∈ V0 ),
e∈E0x

(4)

X

hv(e), ni2 = c (constant)e∈E0

for any unit vector n ∈ Rd ,One can see that (5) is equivalent to there being a positive constant csuch that, for any x ∈ Rd ,X­®
(5)x, v(e) v(e) = cx.
e∈E0Namely (5) is equivalent to saying that {v(e)}e∈E0 is a tight frame, a notion
introduced in wavelet analysis.
The condition (3) in the theorem tells that the crystal as a system of
harmonic oscillators is in equilibrium in the sense that the total force acting
on any “atom” from its nearest neighbors vanishes. A placement satisfying(3) is what we called a harmonic realization in [27] because (3) is equivalent
to saying that Φ is “harmonic” with respect to the discrete version of the
Laplacian ([39], [28], [30]); see Remark (2) in the next section.
The proof for the characterization (3), (4) in Theorem 11.1 relies on thefollowing fact:
For a symmetric matrix S of size d with positive eigenvalues,(6)

tr S ≥ d(det S)1/d ,where the equality holds if and only if S is a scalar matrix, i.e. S = λId
with λ > 0. The inequality (6) reduces to the inequality of arithmetic and
35A homothetic transformation of Rd is an affine transformation of the form

M (x) = cU x + a(c > 0, a ∈ Rd ),where U is an orthogonal matrix, i.e. U ∈ O(d). A congruent transformation is the casethat c = 1.

22Topological Crystallographygeometric means:1(a1 + · · · + ad ) ≥ (a1 · · · ad )1/d .
d
The existence of canonical placements will be shown in the next section.
In the case of maximal topological crystals, we have the following explicit
inequality
E(Φ) ≥ 2b1 (X0 )κ(X0 )−1/b1 (X0 ) ,where κ(X0 ) is the number of spanning trees in X0 . The equality holds if
and only if Φ = Φab (up to homothety).
The following theorem tells that the canonical placement has the property“extrinsic symmetry =intrinsic symmetry”, and hence has maximal symmetry among all placement36.
Theorem 11.2. ([30]) Let Φ : X −→ Rd be the canonical placement. Then
there exists a homomorphism¡ κ : Aut(X)that¢ −→ M (d) suchd(1) when we write κ(g) = A(g), b(g) ∈ O(d) × R , we haveΦ(gx) = A(g)Φ(x) + b(g)(x ∈ V ),¡¢(2) the image κ Aut(X) is a crystallographic group,
12. Construction of canonical placements
The canonical placement of the maximal topological crystal X0ab is constructed in the following way. First we provide H1 (X0 , R), the 1-homology
group with real coefficients, with a natural inner product (which allows us
to identify H1 (X0 , R) with the Euclidean space Rb , b = b1 (X0 )). For thissake, we start with an inner product on C1 (X0 , R), the group of 1-chainswith real coefficients.
For e, e0 ∈ E0 , we set(e0 = e)1he, e0 i = −1 (e0 = e),0(otherwise)
which extends to an inner product on C1 (X0 , R) in a natural manner; say,
DX
E
X
X
ae e,
be0 e0 =ae be0 he, e0 i.
e∈E0

e∈E0

e,e0 ∈E0

Restricting this inner product to the subspace H1 (X0 , R) (= Ker ∂ ⊂ C1 (X0 ,R)), we get an Euclidean structure on H1 (X0 , R), which will be the space
where the maximal topological crystal X0ab is placed.
36Roughly speaking, maximal symmetry means that no structural deformation of its
periodic arrangement of atoms in a crystal will make the structure more symmetrical thanit is

Toshikazu Sunada

23

Let Pab : C1 (X0 , R) −→ H1 (X0 , R) be the orthogonal projection, and put
b ab : H1 (X0 , Z) −→vab (e) = Pab (e), regarding each edge as a 1-chain. Since v
H1 (X0 , R) coincides with the injection, it follows from Theorem 8.1 thatvab (e)}e∈E0 is a building block of a placement Φab of the maximal topologicalcrystal X0ab . One can prove that Φab is canonical.
We now go over to the general case. Let X be a topological crystal overX0 corresponding to a vanishing subgroup H of H1 (X0 , Z). Let HR be thesubspace of H1 (X0 , R) spanned by H, and HR⊥ the orthogonal complementof HR in H1 (X0 , R):
H1 (X0 , R) = HR ⊕ HR⊥ .
Then dim HR⊥ = rank L = d. By choosing an orthonormal basis of HR⊥ , weidentify HR⊥ with the Euclidean space Rd .
Let P :¡ H1 (X0¢, R) −→ HR⊥ = Rd be the orthogonal projection. If we putv(e) = P vab (e) , then we find that {v(e)}e∈E0 gives the building block of
the canonical placement of X.
To sum up, we have the following commutative diagram for the canonicalplacements.
Φab

X0ab −−−−→ Rb




ω1 y
yP
X

−−−−→ Rd
Φi.e., P ◦ Φab = Φ ◦ ω1 , where ω1 : X0ab −→ X is the subcovering mapof ω ab : X0ab −→ X0 . To coin a phrase, the canonical placement of a
topological crystal is obtained by projecting down the canonical placementof the maximal topological crystal onto a suitable hyperplane.
Finally we shall give a practical procedure by means of matrix computations to materialize the logically constructed object, thereby giving a powerful approach to the systematic design of (hypothetical) crystal structures.
(1) Give a finite graph X0 , and a vanishing subgroup H of H1 (X0 , Z).
Take a Z-basis α1 , . . . , αb (b = b1 (X0 )) of H1 (X0 , Z) such that αd+1 , . . . , αb(d ≤ b) comprise a Z-basis of H. We then have the d-dimensional topologicalcrystal X corresponding to H.
(2) Compute the square matrix of size b¶µ¡¢
A11 A12A = hαi , αj i =A21 A22
where A11 (resp. A22 ) is a square matrix of size d (resp. b − d).
Put Γ = A11 − A12 A−1
22 A21 , which turns out to be a positive¡ definite¢dsymmetric matrix. Take vectors a1 , . . . , ad ∈ R such that Γ = hai , aj i ,which is to be a Z-basis of the Bravais lattice of the crystal Φ(X).

24

Topological Crystallography¡¢(3) Compute b(e) = t he, α1 i, . . . , he, αb i (e ∈ E0 ), and a(e) = A−1 b(e) =dX¡¢t a (e), . . . , a (e) . Then putting v(e) =ai (e)ai , we obtain a building1bi=1

block {v(e)}e∈E0 of the canonical placement Φ : X −→ Rd .
Thus the most important part in the procedure is to take a Z-basis of
H1 (X0 , Z) (preferably a Z-basis represented by closed paths) with which
everything reduces computation of matrices.
Remark 1 The canonical placement explained here is what we called thestandard realization in [27] (2000), which derives from the asymptotic behaviors of random walks on topological crystals. Originally the standard
realizations were introduced as an analogue of the Abel-Jacobi maps in algebraic geometry.
The reader may find a quite a bit analogy with the classical theory of
harmonic integrals in the above discussion. Actually everything we explained
here is a discrete version of the method of orthogonal projections developedby Weyl, Hodge, and Kodaira. A protoidea in the graph setting is alreadyseen in Weyl’s work in 1920s applied to the problem of electric circuits (see[39]).
Remark 2 Crystallographers also sought canonical ways to place periodic
graphs in space, and proposed the notions of archetype embeddings (Eon [22],1999), archetypical representations (Eon [23], 2011), equilibrium placements
or barycentric placements (Delgado-Friedrichs and O’Keeffe [12], 2003), andbarycentric drawings (Delgado-Friedrichs [14], 2004)37.
The equilibrium placement coincides with the harmonic realization introduced in [27] (2000) as a special case of harmonic maps ([20], [31]), andhence is characterized as a minimizer of the energy functional with a fixedhomomorphism ρ. Delgado-Friedrichs and O’Keeffe [12] rediscovered thisfact. The first person in crystallography who seems to have mentioned theidea of equilibrium placements is H-J. Klein [26] (1996). The archetypeembeddings introduced by Eon is a special equilibrium placement of the
maximal topological crystal (minimal net) which is constructed in the same
way as Φab . Actually, reading off the terminology in [23] (being not an easytask), one may find that the archetypical representations are identical to thecanonical placements.
13. Examples
We are now ready to produce examples of canonical placements basedon the general recipe given in the previous section. An interesting aspectis that “beautiful” shapes of various crystal structures are characterized by
37Delgado-Friedrichs constructed a poweful algorithm (called SYSTRE) for thebarycentric drawing. The program is available at http://www.gavrog.org/. An input
data for SYSTRE is a finite graph together with an initial building block (with values in
Zd ), whereas the input data of our algorithm is a finite graph and a vanishing group.

Toshikazu Sunada

25

the minimal principle. This reminds us of the isoperimetric inequality whichcharacterizes the round circle, the most symmetric closed curve.
(1) Figure 21 illustrates the canonical placements of the quadrangle lattice,
hexagonal lattice, triangular lattice, and kagome lattice38. They are called
the square lattice (sq1), honeycomb lattice (hcb) (or regular hexagonal lattice), regular triangular lattice (hx1), and regular kagome lattice (kgm),
respectively. Taking a look at their building blocks, we can easily check thatthese placements are actually canonical.

Figure 21. Classical 2-dimensional lattices
Here the regular kagome lattice is the canonical placement of the topological crystal over the graph depicted in Figure 22 with the vanishing group
H = Z(e1 + e2 + e3 ) + Z(f1 + f2 + f3 ).
It is known that some minerals, for instance jarosites (KFe3+ 3 (OH)6 (SO4 )2 )and herbertsmithite (ZnCu3 (OH)6 Cl2 ), contain layers with kagome latticearrangement of atoms in their crystal structure.

Figure 22. A fundamental finite graph of the kagome lattice
Figure 23 is a tiling of pentagons with picturesque properties that has
become known as the Cairo pentagon39. Its 1-skeleton is the canonical placement of a topological crystal over the finite graph depicted on the right.
Figure 24 is the canonical placement associated with the rhombille tiling(kdg) which has an action by translations of the lattice group generated by
two vectors (represented by arrows). The quotient graph is the one on theright.
38The term “kagome” derives from two separate Japanese words, meaning the patternof holes (“me”, literally “eyes”) in a basket (“kago”).
39This is also called Macmahon’s net or mcm.

26

Topological Crystallography

Figure 23. Cairo Pentagon

Figure 24. Dice pattern lattice(2) Some of real crystals are canonically placed. Here are six examples:(i) Diamond (dia)(ii) Lonsdaleite (lon)(iii) Cubic lattice (pcu)(vi) Face-centered cubic lattice (fcu)(v) Body-centered cubic lattice (bcu)(vi) 3D kagome lattice (crs)

Figure 25. ThSi2 structure
ThSi2 structure. Figure 25 is the canonical placement of what crystallographers call the ThSi2 structure, which is realized in a compound ofThorium and Silicide, and found in a number of other materials. As an
abstract graph, ThSi2 structure is the maximal topological crystal over thefinite graph depicted on the right.
(3) We shall exhibit several hypothetical crystals.
(i) The K4 crystal is the canonical placement of the maximal topologicalcrystal over the complete graph K4 ([38]).
The K4 crystal is a purely mathematical object at present, not having
been known so far to exist in nature as a pure crystal like diamond, but

Toshikazu Sunada

27

Figure 26. CG image of the K4 crystal
possessing remarkable mathematical properties by which the K4 crystal deserves to be called the diamond twin (see Section 14 and [38]).
The K4 crystal, which I rediscovered by chance in the midst of my study ofrandom walks, is called (10, 3)-a ([42]) or srs net ([13]) by crystallographers(for more on the history of srs see [25]). The name srs comes from the factthat it occurs in the compound SrSi2 as the Si substructure. H. S. M.
Coxeter [10] called it “Laves’ graph of girth ten”. It is believed that the
crystallographer who discovered for the first time this crystal structure asa hypothetical crystal is Fritz H. Laves (1933). The K4 crystal has a close
relationship with the gyroid, an infinitely connected triply periodic minimal
surface discovered by Alan Schoen in 1970. In [9], the K4 crystal is called
the triamond net.
(ii) (tfa) Figure 27 is the canonical placement of the maximal topologicalcrystal over the graph depicted on the right.

Figure 27. 3D example(iii) Figure 28 is the canonical placement of the topological crystal overthe graph depicted on the right (called K33 ) corresponding to the vanishingsubgroup H = Z(e1 + e2 + e3 + e4 + e5 + e6 ).
(iv) 3D kagome lattice depicted in Figure 14 is usually regarded as a
3D analogue of the kagome lattice in crystallography since the 2D kagome
lattice is a network of corner-sharing triangles while 3D kagome lattice iscomprised of corner-sharing tetrahedra.
There is another 3D analogue of the kagome lattice which is defined to be
the canonical placement of the topological crystal over the graph depicted28

Topological Crystallography

Figure 28. 3D example

Figure 29. Another 3D kagome latticeon the right in Figure 29 with the vanishing subgroup
H = Z(e1 + e2 + e3 + e4 ) + Z(f1 + f2 + f3 + f4 )(compare with the description of the kagome lattice given in (1)).
(v) The case that X0 is the graph depicted on the right in Figure 30 withthe vanishing subgroup
H = Z(e1 + e2 + e3 + e4 ) + Z(f1 + f2 + f3 + f4 ).

Figure 30. 3D example
14. Open problems
As mentioned in Section 2, the curiosity of Greek mathematicians about
the beautiful shapes led to the classification of regular convex polyhedra.

Toshikazu Sunada

29This is also the case of Kepler who established the complete classification ofsemi-regular polyhedra. Namely an eager desire to explore beautiful forms
has been the pulling force in geometry.
The open problems proposed here are concerned with the classificationof crystal structures with big symmetry. We are thus following the long
tradition of geometry40.
An isotropic41 crystal is a crystal with the property that there is no distinction in any direction and position in the sense that there is a congruent transformation preserving the crystal and exchanging any two directededges. An isotropic crystal as an abstract graph has constant degree.
In the 2-dimensional case, such crystals are the classical lattices mentioned
at the beginning of the previous section; that is,(i) Square lattice,(ii) Honeycomb lattice,(iii) Regular triangular lattice,(iv) Regular kagome lattice,
together with three series of pathological crystals (see Figure 31 for an example of isotropic pathological 2D crystal42).

Figure 31. A pathological isotropic crystal
Here are (non-pathological) 3D examples known heretofore:(i) K4 crystal,(ii) Diamond,(iii) 3D kagome lattice (in our sense),(iv) Cubic lattice,degree 3
degree 4
degree 4
degree 640Here we take for granted the belief that beauty is bound up with symmetry. However
human’s aesthetic sense is not so simple to take the belief above as a complete agreement.
Someone says cynically that “symmetry is death” since, if something has symmetry, it isstatic, unchanging, frozen as in death. Needless to say, aesthetic sense depends heavily
upon our cultural background. For instance, compare the asymmetric feature of Japanese architecture with the so-called Greco-Roman tradition that strictly obeys the rule ofsymmetry.
41The term “isotropic” is used in a different context in crystallography. That is, anisotropic crystal is a crystal which has the same optical properties in all directions.
42This figure is placed in my book [37] published in 2006, and is called a Pythahorian
lattice because it is related to rational solutions of the equation x2 + y 2 = 1. See [19](2009) (and [17], [18]) for a related question in which a weaker version (edge-transitivity)of isotropy is discussed.

30

Topological Crystallography(v) 3D kagome lattice,(vi) Body-centered cubic lattice,(vii) Face-centered cubic lattice,degree 6
degree 8
degree 12

Problem 1 Classify all 3D isotropic crystals including pathological ones43.
One can consider a stronger version of isotropy. An isotropic crystal issaid to be strongly isotropic if, for any two edges with the same end point,
there is a congruent transformation preserving the crystal and exchanging
these two edges while other edges (with the same end point) are fixed.
The K4 and diamond crystals are the only two 3D crystals having thestrong isotropic property (this is the reason why we call the K4 crystal the
diamond twin). In two dimension, the honeycomb lattice is the only example. A typical example in Rd is the d-dimensional diamond, the canonicalplacement of the maximal topological crystal over the graph consisting of2 vertices joined by d + 1 parallel edges. The maximal topological crystal
over the complete graph Kn also yields an example, whose dimension is(n − 1)(n − 2)/2. Besides the 4-dimensional diamond, the maximal topological crystal over the graph in Figure 28 gives a 4D example.
Problem 2 Classify all strongly isotropic crystals of general dimension.
15. Discrete Abel-Jacobi maps
The canonical placement Φab of the maximal topological crystal X0ab constructed in Section 12 has a special feature, which is properly interpretedby analogy with classical algebraic geometry. The reader may usefully consult the reference [2], [3], [6], [29], [39] for more about “discrete algebraic
geometry”, an active field still in a state of flux.
For the convenience of the reader, we shall start with a brief review of a
relevant part in classical algebraic geometry.
Given an algebraic curve44 S with a reference point p0 , we denote by Ω1 (S)the space of holomorphic 1-forms on S, andthe first homology group¡ think¢of∗1
H1 (S, Z) as a subgroup of the dual space Ω (S) by using the pairing map
Z([α], ω) =ω,αwhere [α] stands¡ for the¢ homology class of a 1-cycle α. Since H1 (S, Z) is a∗lattice group of Ω1 (S) , the factor group¡¢∗
J(S) = Ω1 (S) /H1 (S, Z).
43This problem is interesting in view of architectural design.
44Precisely speaking, we are considering a non-singular complex projective algebraic

curve, or equivalently a closed Riemann surfarce.

Toshikazu Sunada31

is a complex torus, which we call Jacobian (torus). The Albanese map Φ :S −→ J(S) is the holomorphic map defined by the pairing
Z p(7)(Φ(p), ω) ≡ω (mod H1 (S, Z)).
p0

On the other hand, the Picard group is defined as the factor group
Pic(S) = Div0 (S)/Prin(S),where Div0 (S) is the group of divisors with degree 0, namelynXoX
Div0 (S) =ap p ∈ C0 (S, Z)|
ap = 0 ,p∈Sp∈Sand Prin(S) is the group of principal divisors, which is defined to be thesubgroup of Div0 (S) consisting of divisorsX(8)
ordp (f )p.
p∈S

associated with meromorphic functions f on S. Here ordp (f )Xis the order of apole or zero of f . A classical result in complex analysis tells
ordp (f ) = 0,p∈S0and hence the divisor (8) belongs to Div (S).
The Abel-Jacobi map Ψ : S −→ Pic(S) is a holomorphic map defined byΨ(p) ≡ p − p0

(mod Prin(S)).

Abel’s theorem, one of the culmination of classical algebraic geometry,asserts that the correspondence p − p0 7→ Φ(p) yields an isomorphism of
Pic(S) onto J(S). Thus under the identification between Pic(S) and J(S),the Abel-Jacobi map coincides with the Albanese map.
Having the above review in mind, we are now going to deal with ananalogue of Jacobian torus in discrete set-up. Let X0 = (V0 , E0 ) be a finite
graph with a reference point x0 . Define the flat torus J(X0 ) by setting
J(X0 ) = H1 (X0 , R)/H1 (X0 , Z),
where the flat metric is the one induced from the inner product introducedin Section 12. We call J(X0 ) the Jacobian torus45, which has the structureof abelian group at the same time.
Now let Φab : X0ab = (V ab , E ab ) −→ H1 (X0 , R) be the canonical placement of X0ab with Φab (x0 ) = 0. SinceΦab (αx) = Φab (x) + α(α ∈ H1 (X0 , Z)),

we obtain a piecewise linear map Φab
0 : X0 −→ J(X0 ). This is what weregard as an analogue of Albanese map.
45This is also called the Albanese torus.

32

Topological Crystallography

To see a resemblance between a holomorphic Albanese map and its graphtheoretic analogue, we introduce the notion of harmonic 1-form46 as ananalogue of holomorphic 1-form. A harmonic 1-form on X0 is defined to bea function ω : E0 −→ R satisfying
(9)
(10)

ω(e) = −ω(e)X
ω(e) = 0(e ∈ E0 ),(x ∈ V0 ).

e∈E0x

Then
(11)

(Φab
0 (x), ω) ≡ ω(e1 ) + · · · + ω(en )¡¢mod H1 (X0 , Z)

for every harmonic 1-form ω and x ∈ V0 , where (e1 , . . . , en ) is a path in X0
joining x0 and x. The pairing on the left hand side is defined by³X´
Xae e, ω =ae ω(e).
e∈E0

e∈E0

The right hand side of (11) is regarded as an analogue of line integral along
a curve.
To go further, we introduce the notion of discrete Jacobian, a direct discrete analogue of Jacobian in algebraic geometry.
The homology group H1 (X0 , Z) with integral coefficients is an integral
lattice in H1 (X0 , R) in the sense that hα, βi ∈ Z for every α, β ∈ H1 (X0 , Z).
Denote by H1 (X0 , Z)# the dual lattice of H1 (X0 , Z) in H1 (X0 , R); that is,
H1 (X0 , Z)# = {α ∈ H1 (X0 , R)| hα, βi ∈ Z for every β ∈ H1 (X0 , Z)}.
Since the lattice H1 (X0 , Z) is integral, we have H1 (X0 , Z) ⊂ H1 (X0 , Z)# .
Then the discrete Jacobian J(X0 ) is defined to be the factor group
H1 (X0 , Z)# /H1 (X0 , Z),
which is identified with a finite subgroup of the Jacobian torus J(X0 ).
Now we shall observe Φab
0 (V0 ) ⊂ J(X0 ). For any e ∈ E0 and α ∈H1 (X0 , Z), we find hPab (e), αi = he, Pab (α)i = he, αi ∈ Z, and vab (e) =Pab (e) ∈ H1 (X0 , Z)# . Therefore Φab (x) ∈ H1 (X0 , Z)# for every x ∈ V0ab ,which immediately leads to Φab
0 (V0 ) ⊂ J(X0 ).
We shall call the restriction Φab
0 |V0 : V0 −→ J(X0 ) the discrete Albanesemap.
Theorem 15.1. (1) Φab0 (V0 ) generates J(X0 ).
(2) The order |J(X0 )| coincides with the number of spanning trees κ(X0 )in X0 .
The proof of (1) is carried out as follows. Take a spanning tree T of
X0 , and let e1 , . . . , eb , e1 , . . . , eb (b = b1 (X0 )) be all edges not in T . Thenvab (e1 ), . . . , vab (eb ) constitute a Z-basis of H1 (X0 , Z)# . This is because
46The notion of harmonic forms is originally introduced in Riemannian geometry. It isrelated to cohomology theory of manifolds.

Toshikazu Sunada

33

we may take a Z-basis of H1 (X0 , Z) consisting of circuits c1 , . . . , cb in X0such that ci contains©ei , and hci , vab (ej )iª= hci , Pab (ej )i = hPab (ci ), ej i =hci , ej i = δij (namely vab (e1 ), . . . , vab (eb ) is the dual basis of {c1 , . . . , cb }}).
From this fact, our assertion immediately follows.
See [29] for the proof of (2).
The discrete Jacobian J(X0 ) has a natural graph structure defined asfollows. Let E0o be an orientation of X0 . The proof of Theorem 15.1 tellsthat {vab (e)}e∈E0o generates J(X0 ). Thus the map i : E0o −→ J(X0 ) definedbyi(e) = vab (e) mod H1 (X0 , Z),¡¢yields a connected graph X J(X0 ), E0o ; that is, the Cayley-Serre graphassociated with the map i (see the remark below), which we denote byJ(X0 ) by abuse of language. It is easy to see that the graph structure
does
upon the choice of an orientation. If we assign the edge
¡ ab ¡not depend¢ ¢Φ0 o(e) , e of J(X0 ) to each e ∈ E0o , then, forgetting the orientation, we
get a morphism Φab
0 : X0 −→ J(X0 ).
Remark Let G be a group, and i : A −→ G be a map of a finite set A into
G such that i(A) generates G. We define the Cayley-Serre graph X(G, A) in
the following way (cf. [36]). The set V of vertices is just G. Directed edges
are the pairs (g, a), g ∈ G, a ∈ A. The origin and terminus of the edge (g, a)
are defined to be g and gi(a), respectively. Forgetting orientation, we get aconnected regular graph X(G, A) of degree 2|A|. The graph X(G, A) has anatural free G-action given by g(g 0 , a) = (gg 0 , a). Thus given a subgroup H
of G, one can consider the quotient graph X(G, A)/H, called the Schreiergraph, whose vertices are right cosets of H. In particular, X(G, A)/G is the|A|-bouquet graph. In other words, the Cayley-Serre graph X(G, A) is aregular covering graph over a bouquet graph.
Conversely, a regular covering graph over a bouquet graph is identified
with a Cayley-Serre graph.
We define the group of divisors of degree zero in the discrete category by
settingnXoX
Div0 (X0 ) =ax x ∈ C0 (X0 , Z)|
ax = 0 ,
x∈V0

x∈V0

and the group of principal divisors by¡¢Prin(X0 ) = (∂ ◦ ∂ ∗ ) C0 (X0 , Z) ,where ∂ ∗ is the ajoint of ∂. Obviously Prin(X0 ) is a subgroup of Div0 (X0 ).
Define the discrete Picard group by
Pic(X0 ) = Div0 (X0 )/Prin(X0 ).
The order |Pic(X0 )| turns out to be equal to κ(X0 ).
Now we are ready to introduce a discrete version of Abel-Jacobi map.
Imitating the case of algebraic curves, we define the discrete Abel-Jacobi

34

Topological Crystallographymap Φaj
0 : V0 −→ Pic(X0 ) byΦaj
0 (x) ≡ x − x0(mod Prin(X0 )).

An important fact is that the discrete Abel-Jacobi map has the following“universal” property (cf. [3]).
Theorem 15.2. Let A be an additive group. Given a harmonic function
f : V0 −→ A with f (x0 ) = 0, there exists a unique homomorphism ψ :Pic(X0 ) −→ A such that ψ ◦ Φaj
0 = f.
In the above, a function f : V0 −→ A with values in A is said to beharmonic if
X £ ¡
¢
¡
¢¤
f t(e) − f o(e) = 0.
e∈E0x

The discrete Albanese map is a harmonic function on V0 with values in
J(X0 ).
The following theorem, a discrete analogue of Abel’s theorem, tells thatthe discrete Abel-Jacobi map is essentially the same as the discrete Albanesemap.
Theorem 15.3. (A discrete version of Abel’s theorem) There existsaba unique isomorphism ϕ of Pic(X0 ) onto J(X0 ) such that ϕ ◦ Φaj
0 = Φ0 .
The proof goes as follows. Use the universal property of discrete AbelJacobi maps to find a unique homomorphism ϕ : Pic(X0 ) −→ J(X0 ) satababisfying ϕ ◦ Φaj
0 = Φ0 . Since Φ0 (V0 ) generates J(X0 ), we conclude thatϕ is surjective. That ϕ is an isomorphism is a consequence of the fact
|J(X0 )| = κ(X0 ) = |Pic(X0 )|.
16. Appendix
For the convenience of the reader, we shall give a “dictionary” of terminology used in crystallography ([22]) and mathematics.
(i) Cycle space = homology group H1 (X0 , R)(ii) Cocycle space = the orthogonal complement of H1 (X0 , R) in the 1chain group C1 (X0 , R) with respect to the canonical inner product(iii) Labelled (quotient) graph= a finite graph with a building block(iv) Cyclomatic number= the first Betti number(v) Minimal net = the maximal abelian covering graph(vi) Edge space= 1-chain group C1 (X0 , Z)(vii) Equilibrium placement = harmonic realization(viii) Archetypical representation = standard realization (canonical placement)Toshikazu Sunada

35

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Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University