michel deza and mathieu dutour sikiric- fullerenes: applications and generalizations

8/3/2019 Michel Deza and Mathieu Dutour Sikiric- Fullerenes: applications and generalizations

1/94

Fullerenes: applications andgeneralizations

Michel Deza

ENS, Paris, and ISM, Tokyo

Mathieu Dutour Sikiric

Rudjer Boskovic Institute, Zagreb, and ISM, Tokyo

p.1/9


2/94

I. General

setting

p.2/9


3/94

Denition

A fullerene

is a simple polyhedron (putative carbonmolecule) whose vertices (carbon atoms) are arranged in

pentagons and

hexagons .The

edges correspond to carbon-carbon bonds.

exist for all even

except

.

isomers

for

,

,

, . . . ,

.preferable fullerenes, , satisfy isolated pentagon rule.

"

! #

$

% ,"

& #

$

% are only icosahedral (i.e., with

symmetry $

%

or $

) fullerenes with

'

vertices

p.3/9


4/94

buckminsterfullerene("

) 0

1

2

3

4

truncated icosahedron,soccer ball

5 7

6 )

1

8

)

3

4

elongated hexagonal barrel 5 7

9@

1

8

)

A

4

p.4/9


5/94

23

45

34

3489

4590

23781267

1560

12

15

1560

15

2378

3489

45

34

4590

1267

1223

Dodecahedron

#

$

%

B

C

D

C #

the smallest fullereneBonjour

Graphite lattice

E

B

F

the largest (innite)fullerene

p.5/9


6/94

Small fullerenes

24,G

!

H

26,G

%

28,G

28,I

H

30,G Q

P

% 30, S R 30,G

S

R

p.6/9


7/94

A T U V

p.7/9


8/94

What nature wants?

Fullerenes or their dualsW

appear in architecture andnanoworld:

Biology: virus capsids and clathrine coated vesicles

Organic (i.e., carbon) Chemistry

also: (energy) minimizers in Thomson problem (for

unit charged particles on sphere) and Skyrme problem(for given baryonic number of nucleons); maximizers, inTammes problem, of minimum distance between

points on sphere

Simple polyhedra with given number of faces, which are thebest approximation of sphere?

Conjecture: FULLERENES

p.8/9


9/94

Gravers superfullerenes

Almost all optimizers for Thomson and Tammesproblems, in the range

X

'

'

X

are fullerenes.

For Y X

, appear`

-gonal faces; almost always for

Y

.

However, J.Graver (2005): in all large optimizers the X

-and

`

-gonal faces occurs in 12 distinct clusters,corresponding to a unique underlying fullerene.

p.9/9


10/94

Isoperimetric problem for polyhedra

Lhuilier 1782, Steiner 1842, Lindelf 1869, Steinitz 1927,Goldberg 1933, Fejes Tth 1948, Plya 1954

Polyhedron of greatest volumea

with a given numberof faces and a given surfaceb

?

Polyhedron of least volume with a given number of

faces circumscribed around a sphere?Maximize Isoperimetric Quotient for solids

$

c

e

d

f

g

h

i

'

(with equality only for sphere)

p.10/9


11/94

Isoperimetric problem for polyhedra

polyhedron $

c

p

upper boundTetrahedron

q

!

r

s

q

!

r

Cubeq

!

s

X

t

q

!

Octahedronq

r

s

X

s

`

Dodecahedronq u

v

w

g

y

x

P

w

s

`

X X

q u

v

w

g

y

x

P

w

Icosahedronq u

C P

r

s

s

X

IQ of Platonic solids(

C

r

P

: golden mean )Conjecture (Steiner 1842):

Each of the X

Platonic solids is the best of all isomorphic

polyhedra (still open for the Icosahedron) p.10/9


12/94

Five Platonic solids

p.11/9


13/94

Goldberg Conjecture

$

c

$

'

$

c

!

s

t

Conjecture (Goldberg 1933):

The polyhedron with

facets with greatest

$

c

is afullerene (called medial polyhedron by Goldberg)

polyhedron $

c

p

upper bound

Dodecahedron

#

$

%

q u

v

w

g

y

x

P

w

s

`

X X

q u

v

w

g

y

x

P

w

Truncated icosahedron ! #

$

%

s

X

s

X

Chamfered dodecahed. & #

$

%

s

s

Sphere

p.12/9


14/94

II. Icosahedral

fullerenes

p.13/9


15/94

Icosahedral fullerenesCall icosahedral any fullerene with symmetry

$

% or $

All icosahedral fullerenes are preferable, except

#

$

%

I

, whereI

(triangulation number )

with

' '

. $% for

or

(extended icosahedral group); $

for

(proper icosahedral group)

! #

$

%

=

-dodecahedrontruncated icosahedron

"

& #

$

%

=

-dodecahedronchamfered dodecahedron p.14/9


16/94

Icosadeltahedra

Call icosadeltahedron the dual of an icosahedral fullereneW

#

$

% orW

#

$

Geodesic domes: B.Fuller

Capsids of viruses: Caspar and Klug, Nobel prize 1962

3489

45

4590

34

1237

15

1560

O

3459 1450

O

1267

2348

12

2378

23

1450

45

34

3489

2348

1267

1237

1256

2378

23121256

1560

15

3459

4590

DualW

! #

$

% ,

pentakis-dodecahedronGRAVIATION (Esher 1952)omnicapped dodecahedron

p.15/9


17/94

Icosadeltahedra in Architecture

j

k

Fullerene Geodesic dome l

m

k

n

o

g

k

One of Salvador Dali houses l

l

k

o

k

Artic Institute, Bafn Island

m

k

o

z

{

k

Bachelor ofcers quarters, US Air Force, Korea |

|

k

o

g

k

U.S.S. Leyte }

m

k

o

i g

k

Geodesic Sphere, Mt Washington, New Hampshire ~

m

k

o

{

k

US pavilion, Kabul Afghanistan

m

k

o

v g

k

Radome, Artic dEW

k

o

i z

k

Lawrence, Long Island l

m

k

o

g

{

k

US pavilion, Expo 67, Montreal l

m

k

o

z

k

Gode du Muse des Sciences, La Villete, Paris l

m

k

o

z

k

Union Tank Car, Baton Rouge, Louisiana

Alternate , Triacon and Frequency (distanceof two

X

-valent neighbors) are Buckminster Fullerss terms

p.16/9


18/94

W

& #

$

% ,

=

(

@ 0

1

2

4

,1

4

=1

4

p.17/9


19/94

W

C & #

$

% ,

Bonjour

W

C & #

$

% as omnicappedbuckminsterfullerene ! #

p.18/9

T i l i h i l l


20/94

Triangulations, spherical wavelets

Dualt

-chamfered cube

, %Dual

t

-cham. dodecahedronW

P C # ,

, $

%

Used in Computer Graphics and Topography of Earth

p.19/9


21/94

III. Fullerenes in

Chemistry and Biology

p.20/9

F ll i Ch i t


22/94

Fullerenes in Chemistry

Carbon and, possibly, siliciumb

are onlyt

-valentelements producing homoatomic long stable chains or nets

Graphite sheet : bi-lattice

-Voronoi( ), innitefullereneDiamond packing : bi-lattice

G

-complex, -centering ofthe lattice f.c.c.=

Fullerenes : 1985 (Kroto, Curl, Smalley): ! #

$

%

trunc. icosahedron, soccerball, Nobel prize 1996.But Ozawa 1984. Cheap ! # : 1990.1991 (Iijima): nanotubes.Full. synthesized by now: ! # , # , ! , & , & , & .Fullerene alloys, stereo organic chemistry

Carbon: semi-metal p.21/9

All t f b


23/94

Allotropes of carbon

Diamond : cryst.tetrahedral, electro-isolating, hard,transparent. Rarely Y

X

carats, unique Y

:Cullinan

. M.Kuchner: diamond planets?

Graphite : cryst.hexagonal, soft, opaque, el. conductingFullerenes : 1985, spherical

Nanotubes : 1991, cylindrical

Carbon nanofoam : 1997, clusters of aboutt

atomslinked in graphite-like sheets with some

`

-gons(negatively curved), ferromagnetic

Amorphous carbon (no long-range pattern): synthetic;coal and soot are almost such

White graphite (chaoite): cryst.hexagonal; 1968, inshock-fused graphite from Ries crater, Bavaria p.22/9



24/94


Carbon(VI) : cr.hex.??; 1972, obtained with chaoiteSupersized carbon : 2005, 5-6 nm supermolecules(benzene rings "atoms", carbon chains "bonds")

Hexagonal diamond (lonsdaleite): cryst.hex., very rare;1967, in shock-fused graphite from several meteorites

ANDR (aggregated diamond nanorods): 2005,Bayreuth University; hardest known substance

graphite: diamond: p.22/9


25/94

La"

&

rst Endohedral Fullerenecompound

(

0

Exohedral Fullerenecompound (rst with a single

hydroxy group attached)

p.23/9

A quasicrystalline cluster (H Terrones)


26/94

A quasicrystalline cluster (H. Terrones)

In silico: from ! #

and

#

I

H

(dark); cf. 2 atoms inquasicrystals p.24/9

Onion like metallic clusters


27/94

Onion-like metallic clusters

Palladium icosahedral -cluster

Outer shell Total # of atoms # Metallic cluster

-

-

-

-

-

-

Icosahedral and cuboctahedral metallic clusters p.25/9

Nanotubes and Nanotechnology


28/94

Nanotubes and Nanotechnology

Helical graphite Deformed graphite tube

Nested tubes (concentric cylinders) of rolled graphite;use(?): for composites and nanowires p.26/9

Other possible applications


29/94

Other possible applications

Superconductors: alcali-doped fullerene compounds

"

! # at

, . . . ,

"

! # at

but still too low transitionI

HIV-1 : Protease Inhibitor since derivatives of ! #

arehighly hydrophobic and have large size and stability;2003: drug design based on antioxydant property of

fullerenes (they soak cell-damaging free radicals)Carbon nanotubes

? superstrong materials

? nanowires! already soon: sharper scanning microscope

But nanotubes are too expensive at present p.27/9

Chemical context


30/94

Chemical context

Crystals : from basic units by symm. operations, incl.translations , excl. order

X

rotations (cryst. restriction).Units: from few (inorganic crystals) to thousands

(proteins) atomsOther very symmetric mineral structures: quasicrystals ,fullerenes and like, icosahedral packings (no

translations but rotations of order X

)Fullerene-type polyhedral structures (polyhedra,nanotubes, cones, saddles, . . . ) were rst observed

with carbon. But also inorganic ones: boron nitrides,tungsten, disulphide, allumosilicates and, possibly,uorides and chlorides.

p.28/9

Stability


31/94

Stability

Minimal total energy: $

-energy and

the strain in the -system.

Hckel theory of $

-electronic structure: every eigenvalue

of the adjacency matrix of the graph corresponds to anorbital of energy

. : Coulomb parameter (same for all sites)

: resonance parameter (same for all bonds)The best

$

-structure: same # of positive and negative

eigenvalues

p.29/9

Skyrmions and fullerenes


32/94

Skyrmions and fullerenes

Conjecture (Sutcliffe et al.):any minimal energy Skyrmion (baryonic density isosurfacefor single soliton solution) with baryonic number (the

number of nucleons)

`

is a fullerene

&

.Conjecture :there exist icosahedral minimal energy Skyrmion for any

X

with integers

'

,

(not any icosahedral Skyrmion has minimal energy).

Skyrme model (1962) is a Lagrangian approximatingc

G

(a gauge theory based on

b

group). Skyrmions arespecial topological solitons used to model baryons.

p.30/9

Life fractions


33/94

Life fractions

life: DNA and RNA (cells)

-life: DNA or RNA (cell parasites: viruses)

naked RNA, no protein (satellite viruses, viroids)DNA, no protein (plasmids, nanotech, junk DNA, ...)

no life: no DNA, nor RNA (only proteins, incl. prions)

Atom DNA Cryo-EM Prion Viruses

size 0.2-0.3 s

s

X

X

t

nm B-19, HIV, MimiVirion: protein capsid (or env.spikes) icosadeltahedron

W

#

,I

(triangulation number)

p.31/9

Digression on viruses


34/94

Digression on viruses

life

-life . . . viroids . . . non-lifeDNA and RNA DNA or RNA neither DNA, nor RNA

Cells Viruses Proteins, incl. prionsSeen in 1930 (electronic microscope): tobacco mosaic.

of seawater has s

million viruses; all seagoingviruses s

`

million tons (more 20 x weight of all whales).Origin : ancestors or vestiges of cells, or gene mutation?Or, evolved in parallel with cellular forms fromself-replicating molecules in prebiotic RNA world"

Virus: virion, then (rarely) cell parasiteVirion: capsid (protein coat), capsomers structureNumber of protein subunits is

I

, but EM resolves onlyclusters-capsomers (

I

vertices ofW

#

), including

pentamers ( X

-valent vertices) at minimal distance

p.32/9


35/94

1954, Watson and Crick conjectured: symmetry cylindricaland icosahedral: dual

$

, $

% fullerenes

AIDS: icosahedral, but

? Plant viruses? Chirality? nm:

typical molecule;

Parvovirus

-

,t

Mimivirus;150 minimal cell (bacterium Micoplasma genitalium);90 smallest feature of computer chip (= diam. HIV-1).

Main defense of multi-cellular organism, sexualreproduction, is not effective (in cost, risk, speed) butarising mutations give some chances against viruses.

p.33/9

Capsids of viruses


36/94

Capsids of viruses

j

k

Fullerene Virus capsid (protein coat) l

m

k

n

o

g

k

Gemini virus l

l

k

o

k

turnip yellow mosaic virus |

m

k

o

z

k

hepatitis B, Bacteriophage

|

l

k

o

k

HK97, rabbit papilloma virus l

|

k

o

k

human wart virus

l

k

o

g

k

rotavirus }

m

k

o

i g

k

herpes virus, varicella ~

m

k

o

k

adenovirus

m

k

o

vg

k

infectious canine hepatitis virus, HTLV-1

m

k

o

g

k

Tipula virus

k

?

o

g

k

HIV-1

k

?

o

g

k

iridovirus

p.34/9

Some viruses


37/94

Some viruses

IcosadeltahedronW

#

$

% ,the icosahedral structure of

the HTLV-1

Simulated adenovirusW

P # #

$

% with its spikes

X

-dodecahedron P # #

$

%

p.35/9


38/94

IV. Some

fullerene-like3-valent maps

p.36/9

Mathematical chemistry


39/94

y

use following fullerene-like -valent maps:Polyhedra

P

!

for t

,`

or

( S

t

,

,

t

)Aulonia hexagona (E. Haeckel 1887): plankton skeleton

Azulenoids

P

on torus

; so, P

azulen is an isomer

of naftalen

,

,

p.37/9


40/94

' -surfaces


41/94

(

)1

0

)

2

) 3 4

: putativecarbon, 1992,

(Vanderbilt-Tersoff)

(

51

0

2

Bonjour

(

)1

0

2

Bonjour(

0

$

2

, $

6

& 3

$

& 3

(

7

6

8

6

8

7

2

Unit cell of(

)

0

5

2

:

& 3

- Klein regular map(

9

@

2

p.39/9

' -surfaces


42/94

(

)1

0

)

2

Bonjour

(

51

0

2

%

) A

, $ 4 5

(

)1

0

2

Bonjour(

0

$

)

2

, $

6

#

$

4

(

7

6

8

6

8

7

2

(

)1

0

5

2

: $

Unit cell of $ 5

: $ #

- Dyck regular map(

4

@

2

p.40/9

More ' -surfaces


43/94

B

C1

D

E

F

G

H

I E C

, P Q H R R

B

I

D

E

F

BonjourB

P

Q

D

P

S

H

I E

F

, G H E

P

Q

T

U E

H

U C

B

P

V

T

P W

T

W

V

F

Unit cell ofP

Q

H

C

:G

H

U E

- Dyck regular map

B

X

Y

F

p.41/9

Polycycles (with Dutour and Shtogrin)


44/94

A nite(

0

8

2

-polycycle is a plane

-connected nite graph,such that :

all interior faces are (combinatorial) -gons,

all interior vertices are of degree 8 ,

all boundary vertices are of degree in`

0

8

a

.

a(

&

0

#

2

-polycycle

p.42/9

Examples of ' -polycycles


45/94

$

#c

b

#

@

0

#

@e

d

0

#

@

d

f

;

$

b

@

0

@e

d

0

@

d

f

0

%

7

g

h

(h

$

%p

i

q

,%s

r ,%s

t )

Continuum for any

"

&

.But# A

proper(

&

0

#

2

-polycycles, i.e., partial subgraphs ofDodecahedron

$

3

: polyhexes=benzenoids

Theorem(i) Planar graphs admit at most one realization as

(

0

#

2

-polycycle(ii) any unproper(

0

#

2

-polycycle is a(

0

#

2

-helicene

(homomorphism into the plane tilingu

@w

v

by regular -gons)

p.43/9

Icosahedral fulleroids (with Delgado)


46/94

#

-valent polyhedra with

$

(

0

i

x

2

and symmetryy

ory

orbit size 3 5 # 5 5 )

# of orbits any ) ) )

-gonal face any#

&

h

i

b

(

0

i

2

$

(

)

6

3 5

0

i

2

with"

)

,

3

i

b

(

0

i

2

$

(

3 5

0

)

i

2

with"

)

, $ &

&

p.44/9

-fulleroids


47/94

of Sym j

k

l

m

lo

n

m m

k

l

m

n

m

z

{

k

l

m |

n

m

z

}

k ~

k

n

m

z

k k

m

n

l

k

m m

n

m

k

| m m

n

z

k

m

n

|

k

m m

n

z

}

k

m

n

m m

p.45/9

First ' -sphere icosahedral


48/94

(

y

2

$

%

(

(

y

2 2

; $ 3 5

p.46/9

Second ' -sphere icosahedral


49/94

(

y

2

$

(

(

y

2 2

; $ 3 5

p.47/9

' -sphere icosahedral


50/94

(

y

2

$

%

(

(

y

2 2

; $ 5 5

p.48/9



51/94

(

y

2

$

%

(

(

y

2 2

; $ ) 4 5

p.49/9



52/94

(

y

2

$

(

(

y

2 2

; $ ) 5

p.50/9



53/94

7

(

y

2

$

@

(

(

y

2 2

; $ 5

p.51/9



54/94

(

y

2

$

%

(

7

(

y

2 2

; $ & 3 5

p.52/9



55/94

(

y

2

$

7

(

(

y

2 2

; $ 3 5

p.53/9

All seven -isohedral ' -planes


56/94

A(

&

0

2

-plane is a#

-valentplane tiling by

&

- and -

gons.A plane tiling is 2-homohedral if its facesform 2 orbits under groupof combinatorial automor-phisms

h

.It is 2-isohedral if, more-

over, its symmetry group isisomorphic toh

.

p.54/9


57/94

V. -dimensional

fullerenes (with Shtogrin)

p.55/9

-fullerenes


58/94

(

d

)

2

-dim. simple (

-valent) manifold (loc. homeomorphicto

) compact connected, any

-face is&

- or 3

-gon.So, any -face,

#

, is an polytopal -fullerene .

So,

$

0

#

0

or&

only since (Kalai, 1990) any&

-polytopehas a#

- or -gonal

-face.

All nite#

-fullerenes : plane

#

- and space -fullerenes

Finite -fullerenes; constructions:h

(tubes of ) 5

-cells) and (coronas)Ination-decoration method (construction

,

)

Quotient fullerenes; polyhexes&

-fullerenes from& # # #

p.56/9

All nite -fullerenes


59/94

Euler formula$

d

f

6

$

7

"

5

.

But $

(

)

d

!

2

if oriented

d

! if not

Any

-manifold is homeomorphic to7

with ! (genus)handles (cyl.) if oriented or cross-caps (Mbius) if not.

g 5 )

(

2

(

2

)

(

2

surface7 7

7

%

7

) 5 5 3

"

51

0

$

)

"

9

"

A

"

5

0

$

)

0

#

-fullerene usual sph. polyhex polyhex elliptic

p.57/9

Smallest non-spherical nite -fullerenes


60/94

Toric fullerene

Klein bottle

fullerene projective fullerene p.58/9

Non-spherical nite -fullerenes


61/94

Elliptic fullerenes are antipodal quotients of centrallysymmetric spherical fullerenes, i.e. with symmetry

,7

,

7

,

,

@ ,

, ,y

. So, 5

(

2

.

Smallest CS fullerenes

7

(

ys

2

,

@ 7

(

@

2

,

@

(

2

Toroidal fullerenes have $ 5

. They are described byS.Negami in terms of

#

parameters.

Klein bottle fullerenes have

$

5

. They are obtainedby quotient of toroidal ones by a xed-point freeinvolution reversing the orientation.

p.59/9

Plane fullerenes (innite -fullerenes)


62/94

Plane fullerene : a#

-valent tiling of

7

by (combinatorial)&

- and 3

-gons.

If $ 5

, then it is the graphiteu

3

@

v

$

p

$

3 #

.

Theorem: plane fullerenes have 3

and $ .

A.D. Alexandrov (1958): any metric on

7

ofnon-negative curvature can be realized as a metric ofconvex surface on

@

.Consider plane metric such that all faces becameregular in it. Its curvature is

5

on all interior points

(faces, edges) and"

5

on vertices. A convex surface isat most half

7

.

p.60/9

Space -fullerenes (innite -fullerene)


63/94

Frank-Kasper polyhedra (isolated-hexagonfullerenes):

7

(

y

2

,

7

(

2

,

7

(

@

2

,

7

( 2

Space fullerene : a -valent tiling of

@

by them

Space 4-fullerene : a -valent tiling of

@

by anyfullerenes

They occur in:

ordered tetrahedrally closed-packed phases ofmetallic alloys with cells being atoms. There are

5

t.c.p. alloys (in addition to all quasicrystals)

soap froths (foams, liquid crystals)hypothetical silicate (or zeolite) if vertices aretetrahedra (or

h

) and cells

7

better solution to the Kelvin problem p.61/9

Main examples of space fullerenes


64/94

Also in clathrate ice-like hydrates: vertices are

7

,hydrogen bonds, cells are sites of solutes (

, , . . . ).

t.c.p. alloys exp. clathrate #

m

#

m

#

m

#

m

k

-

I:

l

1 3 0 0k

II:

l

2 0 0 1

III:}

3 2 2 0

5 8 2 0

7 2 2 2

-

6 5 2 1

n

-

15 2 2 6

n

{

(Bergman) 49 6 6 20

(Sadoc-Mossieri) 49 9 0 26 p.62/9

Frank-Kasper polyhedra and


65/94

Mean face-size of all known space fullerenes is in` & 6

(

2

0

&

6

(

h

2 a

. Closer to impossible&

( ) 5

-cell on#

-sphere) means energetically competitive with diamond.

p.63/9

New space -fullerene (with Shtogrin)Th l k hi h i b

d

( 2


66/94

The only known which is not by 7 , 7 , 7 and 7 .By

7 ,

7 and its elongation

@

(

2

in ratio 9

b

b

)

;so, smallest known mean face-size

& 5A )

& )

(

2

.

All space -fullerenes with at most 9

kinds of vertices:h

,

,

,

and this one (Delgado, OKeeffe; 3,3,5,7,7). p.64/9

Kelvin problem


67/94

Partition

@

into cells of equal volume and minimal surface.

Kelvins partition Weaire, Phelans partitionWeaire-Phelan partition (A15) is 0.3% better thanKelvins, best is unknown

In dimension

, best is honeycomb (Ferguson, Hales) p.65/9

Projection of 120-cell in 3-space (G.Hart)


68/94

(

& # #

2

:

3 5 5

vertices,

) 5

dodecahedral facets,

h

$

) 5 5

p.66/9

Regular (convex) polytopesA regular polytope is a polytope whose symmetry group


69/94

A regular polytope is a polytope, whose symmetry groupacts transitively on its set of ags.The list consists of:

regular polytope groupregular polygon

%

i

y

7

(

2

Icosahedron and Dodecahedron

@

) 5

-cell and 3 5 5

-cell

-cell

i (hypercube) and

i (cross-polytope) i

i (simplex)h

i =

(

6

)

2

There are#

regular tilings of Euclidean plane: $

7 ,# 3

and 3 #

, and an innity of regular tilings 8 of hyperbolic plane.

Here 8

is shortened notation for(

2

. p.67/9


70/94

-dim. regular tilings and honeycombs


71/94

Do Ico

63 36

600- 336

24- 344

435* 436*

Ico 353

Do 120- 534 535 536

443* 444*

36 363

63 633* 634* 635* 636*

p.69/9

-dim. regular tilings and honeycombs


72/94

24- 120- 600-

24-

600-

120- 5333 5334 5335

p.70/9

Finite -fullerenes


73/94

$

d

6

7

d

@

$

5

for any nite closed#

-manifold,no useful equivalent of Euler formula.

Prominent -fullerene: ) 5

-cell.

Conjecture : it is unique equifacetted -fullerene(

$

7 )

A. Pasini: there is no -fullerene facetted with (

y

2

( -football)Few types of putative facets:

7 ,

7 (hexagonalbarrel),

7 ,

7

( 2

,

@

(

2

(elongated

Dodecahedron),

@ 7

(

@

2

,

@

(

2

(elongated

7

)

p.71/9

constructions of nite 4-fullerenes


74/94

3-faces are

to ) 5

-cell

3 5 5

7

$

"

)

h

& 3 5

6

5

7 ,

@

(

2

#

d

(

2

# 5

(

2

7 ,

7 ,

(two)decoration C(

) 5

-cell) 5 3 5 5

7 ,

7 ,

7

( 2

decoration D( ) 5

-cell) 3 ) 3 5 5

7 ,

7 ,

@ 7

(

@

2

indicates that the construction creates a polytope;otherwise, the obtained fullerene is a

#

-sphere.h

: tube of ) 5

-cells

: coronas of any simple tiling of7

or

7

,

: any -fullerene decorations

p.72/9

Construction of polytopal -fullerene


75/94

Similarly, tubes of ) 5

-cells are obtained in

p.73/9

Ination method


76/94

Roughly: nd out in simplicial

-polytope (a dual -fullerene

) a suitable large(

d

)

2

-simplex,containing an integer number

of small (fundamental)simplices.

Constructions ,

:

= 3 5 5

-cell;

$

5

, 3 5

, respectively.

The decoration of

comes by barycentric homothety

(suitable projection of the large simplex on the newsmall one) as the orbit of new points under thesymmetry group

p.74/9

All known -fullerenesExp

)

:& # # #

(regular tiling of

by ) 5

-cell)


77/94

Exp : (regular tiling of by cell)

Exp

: (with 3

-gons also): glue two& # # #

s on some ) 5

-cells and delete their interiors. If it is done on onlyone

) 5

-cell, it is g@

(so, simply-connected)

Exp#

: (nite&

-fullerene): quotient of& # # #

by itssymmetry group; it is a compact -manifold partitionedinto a nite number of

) 5

-cells

Exp#

: glue above

Pasini: no polytopal&

-fullerene exist.

All known

-fullerenes come from usual spheric fullerenes orfrom the regular

-fullerenes :&

,& #

=Dodecahedron,& # #

= ) 5

-

cell,& # # #

, or 3

, 3 #

=graphite lattice, 3 # #

p.75/9

Quotient -fullerenes


78/94

A. Selberg (1960), A. Borel (1963): if a discrete group ofmotions of a symmetric space has a compact fund. domain,then it has a torsion-free normal subgroup of nite index.So, quotient of a

-fullerene by such symmetry group is anite

-fullerene.Exp 1: Poincar dodecahedral space

quotient of ) 5

-cell (on@

) by the binary icosahedralgroup

y

of order ) 5

; so,

-vector(

&

0

) 51

0

3

0

)

2

$

7

(

) 5

d

2

It comes also from

7

$

by gluing of its oppositefaces with

right-handed rotation

Quot. of

@

tiling: by

7 :(

)

0

3

0

3

0

0

)

2

Seifert-Weber space

and by

7

:

(

0

9

0

4

6

4

$

6

0

4

2

Lbell space p.76/9

Polyhexes7

7


79/94

Polyhexes on , cylinder, its twist (Mbius surface) andare quotients of graphite

3 #

by discontinuous andxed-point free group of isometries, generated by resp.:

translations,a translation, a glide reection

a translation and a glide reection.

The smallest polyhex has $ )

: on7

.The greatest polyhex is

3 # #

(the convex hull of vertices of 3 #

, realized on a horosphere); it is not compact (i.e. with notcompact fundamental domain), but conite (i.e., of nitevolume) innite -fullerene.

p.77/9


80/94

VI. Some special

fullerenes (with Grishukhin)

p.78/9

All fullerenes with hexagons in ring


81/94

; 30

V

; 32

Y

; 32

V

; 36

; 40 p.79/9

All fullerenes with pentagons in ring


82/94

7 ; 36 Y ; 44

Q

; 48

V ; 44

p.80/9

All fullerenes with hexagons in ring


83/94

@

; 32 @ ; 38

; 40

p.81/9

All fullerenes with pentagons in ring


84/94

@

; 38

innite family:triples in

,

"

) 5

, from

collapsed#

innite family:

V

V

B

Q

F

,

I

,

Q

if

odd

elongations ofhexagonal barrel

p.82/9

Face-regular fullerenes&

& &


85/94

A fullerene called if every -gon has exactly -gonalneighbors; it is called 3

if every 3

-gon has exactly 3

-gonal neigbors.

i 0 1 2 3 4 5# of

&

2 1 1# of

3

4 2 8 5 7 1

28,

7 32,

@

All fullerenes, which are 3

p.83/9

All fullerenes, which are


86/94

36,

7

44,

(also

) 48,

52,

(also

) 60,

(also

) p.84/9

All fullerenes, which are


87/94

40,

Bonjour56,

(also

)68,

Bonjour68,

(also

)

72,

80,

(also

) 80, (also

)

p.85/9

ullerenes as isom. subgraphs of half-cub


88/94

All isometric embeddings (of skeletons), for - or-fullerenes or their duals, are:

!

"

#

$

#

%

!

"

#

$

&

'

%

&

!

"

#

$

#

')

(

!

"

#

$

Conjecture (checked for 0 1

2 3

): all such embeddings,for fullerenes with other symmetry, are:

&

4

5

!

"

#

$

#

%

(

68

7

!

"

#

$ @

9

%

5 &

4

&

!

"

#

$

(

@

A

6@

7

!

"

#

$

# B

@

A A

6

!

"

#

$

# &

Also, for graphite lattice (innite fullerene):2

5

!

=

D

C

"

E

5

and

F

&

!

=

%

C

"

#

E

5

. p.86/9


89/94

VII. Knots and zigzags

in fullerenes(with Dutour and Fowler)

p.87/9

Triply intersecting railroad in G H I PR Q


90/94

p.88/9

Tight S with only simple zigzags

T U

t bit l th i t t


91/94

group -vector orbit lengths int. vectorV W

X

Y

aW

b

6V

c

V d

eg

f

a V

h

3,4V

b

i

d

p r

q

a s

t

3,3,3V

u

s W

Xv

Y

ad

w x

10V

t

s W

p y

q

ad

w x

1,3,6V

t

s

p

f

V V

VW

h

1,2,4,4i

V

t

andV

w x

d d

e

V V

w

12V

w w

V

e

Y

V V

b

V

i

b

6,6V

w w

andV

wx

i

a

i

W

X

V d

w c

15V

w

Conjecture: this list is complete (checked for 0 1

3 3

).

It gives Grnbaum arrangements of plane curves. p.89/9

First IPR fullerene with self-int. railroad


92/94

&

4

&

7

!

; realizes projection of Conway knot

2

!

%

p.90/9

Parametrizing fullerenes S

Id th h g f t it f t gi


93/94

Idea: the hexagons are of zero curvature, it sufces to giverelative positions of faces of non-zero curvature.

Goldberg (1937) All D

of symmetry ( , ) are given by

Goldberg-Coxeter construction

'

.Fowler and al. (1988) All

D

of symmetry 4

B , 4

& or 6

are described in terms of

parameters.

Graver (1999) All

can be encoded by3

integerparameters.

Thurston (1998) All D

are parametrized by3

complex

parameters.Sah (1994) Thurstons result implies that the number offullerenes

is 0

.

p.91/9

Intersection of zigzags


94/94

For any 0 , there is a fullerene with two zigzags havingintersection 0

p.92/9

michel deza and mathieu dutour sikiric- fullerenes: applications and generalizations

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