Mathieu Dutour and Michel Deza- Goldberg-Coxeter construction for - or -valent plane graphs

8/3/2019 Mathieu Dutour and Michel Deza- Goldberg-Coxeter construction for - or -valent plane graphs

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Goldberg-Coxeter

construction

for - or -valent plane graphsMathieu Dutour

ENS/CNRS, Paris and Hebrew University, Jerusalem

Michel Deza

ENS/CNRS, Paris and ISM, Tokyo

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Goldberg-Coxeter for the Cube

1,0

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1,1

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2,0

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2,1

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History

Mathematics: construction of planar graphs

M. Goldberg, A class of multisymmetric polyhedra,Tohoku Math. Journal, 43 (1937) 104108.

Objective was to maximize the interior volume of thepolytope, i.e. to find

-dimensional analogs of regularpolygons.

search of equidistributed systems of points on the spherefor application to Numerical Analysis.

p.3/4


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History

Biology: explanation of structure of icosahedral viruses

D.Caspar and A.Klug, Physical Principles in the Con-struction of Regular Viruses, Cold Spring HarborSymp. Quant. Biol., 27 (1962) 1-24.

symmetry capsid of virion

gemini virus

hepathite B

, laevo HK97, rabbit papilloma virus

, laevo rotavirus

herpes virus, varicella

adenovirus

?

, laevo HIV-1

p.3/4


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History

Architecture: construction of geodesic domesPatent by Buckminster Fuller

EPCOT in Disneyland.

p.3/4


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History

Mathematics:

H.S.M. Coxeter, Virus macromolecules andgeodesic domes, in A spectrum of mathematics; ed.

by J.C.Butcher, Oxford University Press/AucklandUniversity Press: Oxford, U.K./Auckland New-Zealand, (1971) 98107.

p.3/4


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History

Chemistry: Buckminsterfullerene

(football, Truncated Icosahedron)

Kroto, Kurl, Smalley (Nobel prize 1996) syn-

thetized in 1985 a new molecule, whose graph is

.

Osawa constructed theoretically

in 1984.

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I. ZigZags

and

central circuits

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Central circuits

A 4valent plane graph G

p.5/4


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Central circuits

Take an edge of G

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Central circuits

Continue it straight ahead ...

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Central circuits

... until the end

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Central circuits

A selfintersecting central circuit

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Central circuits

A partition of edges of G

CC=4 , 6, 82 p.5/4

Zi Z


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Zig Zags

A plane graph G

p.6/4

Zi Z


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Zig Zags

Take two edges

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Zi Z


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Zig Zags

Continue it leftright alternatively ....

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Zig Zags


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Zig Zags

... until we come back

p.6/4

Zig Zags


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Zig Zags

A selfintersecting zigzag

p.6/4

Zig Zags


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Zig Zags

A double covering of 18 edges: 10+10+16

z=10 , 162zvector p.6/4

Notations


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Notations

ZC-circuit stands for zigzag or central circuit in

- or

-valent plane graphs.

The length of a ZC-circuit is the number of its edges.

The ZC-vector of a

- or

-valent plane graph

is thevector

where is the number of

ZC-circuits of length

.

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II. Goldberg-Coxeter

construction

p.8/4

The construction


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The construction

Take a

- or

-valent plane graph

. The graph

isformed of triangles or squares.

Break the triangles or squares into pieces:

3valent case

k=5

l=2

l=2

k=5

4valent case p.9/4

Gluing the pieces


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Gluing the pieces

Glue the pieces together in a coherent way.

We obtain another triangulation or quadrangulation ofthe plane.

Case

-valent Case

-valent p.10/4

Gluing the pieces


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Gluing the pieces



:

-valent

:

-valent p.10/4

Gluing the pieces


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Gluing the pieces



p.10/4

Gluing the pieces


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Gluing the pieces



p.10/4

Final steps


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Final steps

Go to the dual and obtain a

- or

-valent plane graph,which is denoted

and called

Goldberg-Coxeter construction.

The construction works for any

- or

-valent map onoriented surface.

Operation

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Example of


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Example of

p.12/4

Example of


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Example of

p.12/4

Example of


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Example of

p.12/4

Example of


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p

p.12/4

Example of


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p

p.12/4

Example of


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p

p.12/4

Example of


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p

p.12/4

Properties


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p

One associates

(Eisenstein integer) or

(Gaussian integer) to the pair

in

- or

-valent case.

If one writes

instead of

, then onehas:

If

has

vertices, then

has

vertices if

is

-valent,

vertices if

is

-valent.

If

has a plane of symmetry, we reduce to

.

has all rotational symmetries of

and all

symmetries if

or

. p.13/4

The case


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Case

-valent Case

-valent

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The case


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Case

-valent Case

-valent

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The case


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Case

-valent

is called leapfrog

(=Truncation of the dual)

Case

-valent

is called medialBonjour

p.14/4

Goldberg Theorem


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is the class of

-valent plane graphs having only

-and

-gonal faces.

The class of

-valent plane graphs having only

- and

-gonal faces is called Octahedrites.

Class Groups Construction

,

Tetrahedron

,

Cube

,

Prism

!

,

Dodecahedron

Octahedrites "

,

Octahedron

p.15/4

The special case


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Any ZC-circuit of

corresponds to

ZC-circuits of

with length multiplied by

.

If the ZC-vector of

is

, then the ZC-vector

of

is

.

p.16/4

The special case


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Any ZC-circuit of

corresponds to

ZC-circuits of


.

If the ZC-vector of

is


of

is

.

p.16/4

The special case


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Any ZC-circuit of

corresponds to

ZC-circuits of


.

If the ZC-vector of

is


of

is

.

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III. The

-product

p.17/4

The mapping


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We always assume

if

if

is bijective and periodic with period

.Example: Case

,

:

operations:

p.18/4

The

-product


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Definition 0 (The

-product)If

and

are two elements of a group,

and

; we define

by

and

.Set

if

and

if

;then set

By convention, set

and

.For

,

, one gets the expression

A similar notion is introduced by Norton (1987) inGeneralized Moonshine for the Monster group.

p.19/4

Properties


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If

and

commute,

Euclidean algorithm formula

if

if

If

and

do not commute, then

.

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IV. ZC-circuits

in

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Position mapping, -valent case


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A zigzag in

0GC (G )5,2

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Put an

on it

Orientation

p.22/4



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e

1 2 3 4 5

6

7

A directed

edge e

A position p=3

Encoding:

a pair ( e, 3)

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1 2 3 4 5

6

7

k=5

l=2

e

Go to the

Dual !

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l=2

k=5

1

6

7

5432

3 4 521

is

1f

Image of ( e, 3)

( f , 5)

1 e

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l=2

k=5

1

6

7e

5432

is

2 3 4 5

6

7

f2

Image of ( e, 6)

( f , 1)

2

p.22/4

Iteration


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Position mapping is denoted

=

or

=

. So, one defines Iterated position

mapping as

.

is the set of directed edges of

.

is a

permutation of

.

For every ZC-circuit with pair

denote

the smallest

, such that

.

For any ZC-circuit of

one has:

=

-valent case

=

-valent case

The [ZC]-vector of

is the vector

where is the number of ZC-circuits with order . p.23/4

The mappings and


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and

are the following permutation of

with

and

being the first and second choice.Example of Cube

Dual Cube

p.24/4

The mappings and


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24 Directed Edges

p.24/4

The mappings and


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L(e)

L(L(e))

e

Successive images of e by L

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The mappings and


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R(e)

R(R(e))

e

Successive images of e by R

p.24/4

Moving group and Key Theorem


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is the moving groupIn Cube: a subgroup of

.

For

, denote

the vector

with multiplicities

being the half of the number ofcycles of length in the permutation acting on the set

.In Cube:

Key Theorem One has for all

- or

-valent plane graphs

and all

of

p.25/4

Solution of the Cube case


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and

do not commute

.

=

normal subgroup of index

of

.

is of order

.

divisible by

Elements of

have order

. Elements of

have order

.

Cube

has [ZC]=

if

and [ZC]=

,

otherwise

p.26/4

Possible [ZC]-vectors


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Denote

the set of all pairs

with

.

Denote

the smallest subset of

, which

contains the pair

and is stable by the twooperations

and

Theorem: The set of possible [ZC]-vectors of

is equal to the set of all vectors

,

with

.Computable in finite time for a given

.

p.27/4

Examples


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Dodecahedron

of order

. Order ofelements different from

are

,

or

.Possible [ZC] are

or

or

.

Klein Map

=

of order

"

. Order ofelements different from

are

,

,

or

.Possible [ZC] are

or

.

Truncated Icosidodecahedron

has size "

p.28/4


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V. action

p.29/4

action?


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is the set of pairs

. One has

and

The matrices

and

generate

.

We want to define

, such that

(i)

is a group action of

on

(ii) If

, then the mapping

satisfies

This is in fact not possible!

p.30/4

action


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is generated by matrices

and

all relations between

and

are generated by therelations

and

We write

p.31/4

action (continued)


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By computation

Group action of

on

.

If preserve the element

in

,then for all pairs

:

and

have the same [ZC]-vector. This define a

finite index subgroup of

p.32/4

Conjectured generators


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Graph

Generators of

Dodecahedron

,

,

Cube

,

Octahedron

,

,

p.33/4


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VI. Remarks

p.34/4

transitive


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is the set of all directed edges of

.

: all rotations in automorphism group

.

its action on

is free.

action of

and

on

commute.

If

is transitive on

, then its action onZC-circuit is transitive too and

defined by

, is an injective groupmorphism.

is normal in

.

p.35/4

Extremal cases


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non-trivial

restrictions on

.

transitive on

=

(

-valentcase) or

(

-valent case).

is formed of even permutation on

or

directed edges.

In some cases

.

We have no example of

-valent plane graph

with

. p.36/4

commutative


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commutative

is either a graph

, agraph

or a

-hedrite.

Class

(Grunbaum-Zaks): Goldberg-Coxeter of the

Bundle

Class

(Grunbaum-Motzkin)

Class

-hedrites

(Deza-Shtogrin)

No other classes of graphs or

-hedrites is known toadmit such simple descriptions.

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VII. Parametrizing

graphs

p.38/4

Parametrizing graphs


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

Goldberg (1937): All

,

or

of symmetry (

,

),

(

,

) or (

,

) are given by Goldberg-Coxeterconstruction

.

Fowler and al. (1988) All

of symmetry

,

or

are described in terms of

parameters.Graver (1999) All

can be encoded by

integerparameters.

Thurston (1998) The

are parametrized by

complexparameters.

Sah (1994) Thurstons result implies that the Nrs of

,

,

,

,

. p.39/4

The structure of graphs


78/83

triangles in

The corresponding trian-gulation

The graph

p.40/4

Tightness

A railroad in a

-valent plane graph

is a circuit of


79/83

p g p

hexagons with any two of them adjacent on oppositeedges.

They are bounded by two zigzags.A graph is called tight if and only if it has no railroads.

If a

- (or

-)valent plane graph

has no -gonal faces

with

=

(or

) and

then

is tight. p.41/4

- and railroad-structure of graphs


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All zigzags are simple.The -vector is of the form

with

the number of railroads is

.

has

zigzags with equality if and only if it is tight.If

is tight, then

(so, each zigzag is aHamiltonian circuit).

All

are tight if and only if

is prime.

There exists a tight

if and only if

is odd.

p.42/4

Conjecture on

are described by two


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complex parameters. They exists if and only if

and

"

.

with one zigzag The defining triangles

!

#

$

exists if and only if

%

&

'

!

(

)1

2

3 4

$

,% 5 (.

If % increases, then part of

$

amongst

!

!

$

goes to 100%

p.43/4

More conjectures

All

with only simple zigzags are:


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,

and

the family of

with parameters

and

with

and

"

They have symmetry

or

or

Any

with one zigzag is a

.

For tight graphs

the -vector is of the form

with

or

with

Tight

exist if and only if

, they arez-transitive with

iff

and, otherwise,

iff

p.44/4


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The End

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Documents

Mathieu Dutour and Michel Deza- Goldberg-Coxeter construction for - or -valent plane graphs