The minimal nonfullerene Voronoi p olyhedra

Bor Plestenjak

Department of Theoretical Computer Science

IMFM University of Ljubljana Ljubljana Slovenia

email borplestenjakmatuniljsi

Tomaz Pisanski

Department of Theoretical Computer Science

IMFM University of Ljubljana Ljubljana Slovenia

email tomazpisanskimatuniljsi

Ante Graovac

The Rug jer Boskovic Institute

HR Zagreb POB Croatia

email graovacolimpirbhr

Abstract

When placing n p oints uniformly on a sphere their asso ciated

Voronoi regions dene a p olyhedron that is for most n a fullerene

In this pap er the exceptional minimal nonfullerene p olyhedra are dis

cussed

In this pap er we study uniform distributions of p oints on a unit sphere

To each distribution of n p oints with unit vectors R fr r r g on a

1 2 n

unit sphere we may asso ciate an energy function

X

E R

2

d

ij

ij

where d jr r j is the usual Euclidean distance in D space Distri

ij i j

butions with minimal energy are sought Since nding a global minimum

is dicult we used an approximate numerical metho d It is based on the

standard gradient metho d combined with the random walk metho d

in order to escap e from a lo cal minimum Thus we use a sort of simulated

annealing metho d

To each p oint distribution we may asso ciate a Voronoi diagram in the

following way The sphere is cut into n regions each region f b eing asso ci

i

ated with a p oint r The region f contains those p oints on the sphere that

i i

are closer to i than to any other p oint of the distribution R The collection

of all regions obtained in this way denes the Voronoi diagram V R on the

sphere Until now Voronoi diagrams have b een usually studied in the plane

or in the space The reader is referred to for further reading

The Voronoi diagram V R is therefore a graph emb edded in the sphere

The b oundary b etween two regions f anf f is a segment of the great circle

i j

of a unit sphere p erp endicular to and bisecting the one connecting the p oints

r and r Let e denote the numb er of such segments These segments are the

i j

edges of the Voronoi diagram There is only a nite numb er v of p oints in

the sphere where more than two regions meet These p oints are the vertices

of the Voronoi diagram The Voronoi diagram has thus v vertices e edges

and n regions faces By the Euler p olyhedral formula we have

v e n

Note that Voronoi diagrams can b e considered as topological polyhedra with

curved faces

As in the case of the Voronoi diagrams in the plane the most common

situation arises when all of its vertices are trivalent although cases with higher

valencies could o ccur In the latter case a p erturbation of a single p oint in

R may remove some higher valencies and thus results in a dierent Voronoi

diagram This means that the higher valencies are top ologically unstable

In the trivalent Voronoi diagram the Euler formula combined with the hand

shaking lemma e v gives the following identities

v n

e n

The problem of a uniform placing of n p oints on a unit sphere can now b e

rephrased in terms of p olyhedra The minimal energy p oint distribution R

denes the minimal Voronoi diagram and thus the minimal nface polyhedron

In the further text V denotes a Voronoi p olyhedron with n faces having the

n

minimal energy

If for a given n a single minimum exists and if the corresp onding p oly

hedron is trivalent we may exp ect that the numerical calculation will yield a

stable solution that will correctly describ e the top ological typ e of the p olyhe

dron This reasoning can b e partially reversed If numerical exp eriments give

always the Voronoi diagrams of the same top ological typ e we may conclude

that the likeliho o d of this solution of b eing the correct minimum is high

Our preliminary exp eriments indicate an interesting fact that minimal

Voronoi p olyhedra are indeed trivalent with a single exception found Even

more almost all of them are top ologically equivallent to fullerenes ie triva

lent p olyhedra whose faces consist of hexagons and exactly p entagons

We observed only a small numb er of exceptions Since no fullerenes with

v vertices exist it is clear that the minimal

Voronoi p olyhedra will b e nonfullerene for n

Our exp eriments indicate that even in the cases of higher n where fullerenes

exist we found two cases v and v for which the minimal Voronoi

p olyhedra are nonfullerene

Although the complete results on fullerenes obtained with our exp eri

ments will app ear elsewhere for the sake of completeness we identify

the remaining minimal Voronoi p olyhedra up to n as the fullerenes

V V V V V V

12 14 15 16 17 19

V and V where notation is the same as in Atlas

20 22

Tables A A of reference

Conjecture Only a nite number of the minimal nonful lerene Voronoi

polyhedra exists

Here we present the minimal energies and drawings of the ab ove excep

tions together with their structure of faces The results are collected in Table

Beside the minimal energies we present there also their normalized values

E is dened as The average p er pair of particles energy

E

E

n

Table The list of the minimal nonful lerene Voronoi polyhedra V with n

n

faces and v vertices The numbers n n n n and n denote the numbers of

2 3 4 5 6

and gons respectively In the next two columns the corresponding

minimal E and average E energies on the unit sphere are given

n v n n n n n Energy E

2 3 4 5 6

We can observe that the values of the average energy E follow approx

p

n rule Indeed the least square tting has given the approxi imately the

mation

05116

E n

52

Therefore the maximal energy E b ehaves approximately as n what

can b e estimated from simple physical arguments as well

The drawings of the minimal nonfullerene Voronoi p olyhedra together

with their Schlegel diagrams are given in Figure

Figure The drawings of the minimal nonful lerene Voronoi polyhedra to

gether with their Schlegel diagrams

V faces vertices

3

V faces vertices

4

V faces vertices

5

V faces vertices

6

V faces vertices

7

V faces vertices

8

V faces vertices

9

V faces vertices

10

V faces vertices

11

V faces vertices

13

V faces vertices

18

V faces vertices

21

From the ab ove drawings the following prop erties of the ab ove minimal

p olyhedra are deduced

V is the wellknown graph It has two vertices on the p oles and

3

three parallel edges as meridians

V is the complete graph K on the sphere as the regular tetrahedron

4 4

V is the trigonal prism Its graph is the Cartesian pro duct K C

5 2 3

V is the cub e K K K

6 2 2 2

V is the p entagonal prism K C

7 2 5

The next graph V is the only nontrivalent graph among al l minimal

8

p olyhedra It can b e obtained from a bichromatic cycle C by adding

8

two vertices b and w and edges in such a way that white vertices of

C are joined to w and all black vertices of C are joined to b

8 8

V has a pronounced pair of antip o dal vertices surrounded by three

9

p entagons dening a fold axis of rotation

V is a generalized Petersen graph GP It can b e also obtained

10

from V by truncating b oth of its fourvalent vertices

8

V is showing a lower symmetry than the previous p olyhedra

11

V is the largest in the series of nonfullerenes by mathematical ar

13

gument Namely for n the fullerenes are ruled out since

the smallest fullerene is the do decahedron which happ ens to b e V

12

Since fullerenes on v vertices exist for v k k k see for

instance a fullerene on vertices do es not exist hence V cannot

13

b e a fullerene

V has two opp osite twisted quadrilaterals whose centers dene a

18

fold alternating axis of improp er rotation This prop erty is also valid

for V Note that V is of lower energy as dened in than all

10 18

nonisomorphic fullerenes on vertices

There are nonisomorphic fullerenes on vertices It turns out

that a nonfullerene V is obtained when minimizing the function

21

This and the previous p olyhedron are the only known p olyhedra where

nonfullerenes b eat fullerenes of the same size V has a distinguished

21

edge whose midp oint is xed by each of the symmetries of the p olyhe

dron The distinguished edge lies on the b order of two p entagons and

each of its endp oints is surounded by three p entagons This prop erty

can also b e found in other two cases V and V with the same p oint

11 13

group C Note that the face opp osite to the distinguished edge is a

2v

quadrilateral in V and V but is a hexagon in V

18 21 11

The ab ove p olyhedra could b e characterized by their p oint groups and

related graphs by their automorphisms groups The connection b etween ge

ometrical symmetry and graph automorphisms is an involved problem but

the rst results app eared recently In the Table the p oint groups

and the orders jAutV j of the automorphisms groups of the minimal non

n

fullerene Voronoi p olyhedra V are given Further the numb ers of vertex

n

edge and face orbits of the automorphisms groups AutV are given Let us

n

recall that a vertex orbit of AutG of a graph G is the subset of vertices of

G which is invariant under the action of AutG In an analogous way the

edge and face orbits are dened

Table The orders jAutV j of the automorphism groups numbers of ver

n

tex edge and face orbits and point groups of the minimal nonful lerene Voronoi

polyhedra V with n faces and v vertices

n

n v jAutV j vertex orbits edge orbits face orbits Point group

n

D

3d

T

d

D

3d

O

h

D

5d

S

4

D

3h

S

4

C

2v

C

2v

S

4

C

2v

Although the problem of nding the minimal conguration of p oints on

a unit sphere is as old as the Thompson mo del of the atom and since then

has b een studied in such diverse elds as stereo chemistry b otany virology

information theory oce assignment problem etc resulting

in large numb er of pap ers here we fo cused for the rst time attention to a

situation in which the p oints are asso ciated with the faces of the p olyhedra

As our computer exp eriments indicate that fullerenes minimize the energy

function in almost all cases we restricted our attention here to the

known exceptions to this nding The smallest exceptions arise from

the mathematical arguments of nonexistence of corresp onding fullerenes

The last two exceptions although not b eing prohibited by mathematical

arguments to b e fullerenes turned out to b e nonfullerenes

All V except V are trivalent p olyhedra It is easy to see that trivalent

n 8

p olyhedra are maintaining their top ological prop erties even when the min

imal conguration of p oints are slightly distorted due to numerical errors

No V contains a face of size larger than For n only quadrilaterals

n

p entagons and hexagons are candidates for the faces

It would b e interesting to know top ological prop erties of the minimal

p olyhedra in advance without actually computing them at least for some

cases It would b e also interesting to know which top ological prop erties

remain invariant under numerical errors typ e of surface on which the p oints

are placed etc Esp ecially it would b e interesting to study the inuence of

2

the choice of the energy function For instance instead of the r p otential

a more usual Coulomb r p otential or even more general function r can

b e considered The work on these questions is in the progress

Finally the most imp ortant question that we address in a followup pa

p er is the systematic study of larger minimal p olyhedra V their relation

n

to fullerenes and esp ecially to those exp erimentally detected up to now in

chemical lab oratories

Acknowledgements

This work was supp orted by the Ministry of Science and Technology

of Slovenia Grant No P and the Ministry of Science and

Technology of Croatia Grant No The authors would like to thank

Professor Patrick W Fowler for his remarks ab out the manuscript One of

the authors BP would like to thank Professor Jorg M Wills for fruitful

discussion and for p ointing out imp ortant references Another author AG

also expresses thanks to Dr Marko Razinger and coworkers of the National

Institute of Chemistry of Slovenia on hospitality and helpfull discussions

during the preparation of this work

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