Topology of the phase in aperiodic crystals

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Topology of the phase in aperiodic crystals

Maurice Kléman

To cite this version:

Topology

of

the

_phase

in

_{aperiodic crystals}

Maurice Kléman

Laboratoire de

Physique

des Solides

(*),

Bât. 510, Université de Paris-Sud, 91405

_Orsay

Cedex,

France

(Received

2 March 1990,

accepted

in

_{final form}

1 S June

₁₉₉₀₎

Résumé. 2014 Les

_degrés

de liberté relatifs à la

phase

des cristaux

_{apériodiques}

sont loin d’être

compris.

Dans cet _article, nous étudions en

_grand

détail les

_propriétés

_{géométriques}

et

topologiques

de

_{l’espace représentatif}

de la

_phase

pour des

symétries quasicristallines

usuelles

(icosaédrique

et

_pentagonale)

et _pour les cas d = 3,

d~ =

1 et d = 4,

d~ =

2. On montre _que

l’espace

de la

_phase

a pour revêtement universel un cristal

_d’espace

courbe de courbure

gaussienne négative ;

le groupe fondamental de

l’espace

de la

phase

est un sous-groupe du groupe

d’automorphismes

de ce cristal courbe. Nous le calculons dans

chaque

cas

_envisagé.

Les résultats

ne

_dépendent

en aucune manière du choix et de la « surface

_atomique

»

_{(le motif) qui}

décore la cellule de base du cristal

hypercubique

d’où le

_quasicristal

est

_engendré.

En

_{conséquence,}

ils ne

dépendent

pas non

_plus

du fait que les

phasons

sont continus ou discrets. Ces recherches constituent un

_premier

pas nécessaire dans l’étude de la nature

_{géométrique}

et

_topologique

des déformations du _type «

_phason

». On discute en outre la nature de

_{l’homomorphisme}

entre

l’espace

de la

_phase

et le groupe

qui

classe les dislocations. Finalement nous

_indiquons

sans entrer dans les détails que le groupe fondamental de

l’espace

de la

_phase

classe les défauts

_topologiques

du type

phason.

Abstract. 2014 The

_{phase degrees}

of freedom of

_{aperiodic crystals (quasicrystals)}

are far from

_being

understood. In this paper we

_study

in _great detail the

geometrical

and

topological properties

of the

_{« phase}

space » of

quasicrystals,

for usual

quasicrystalline symmetries (icosahedral

and

pentagonal cases)

and for the cases d = 3,

d~ = 1

and d = 4,

d~ =

2. It is shown that the universal

covering

of the

phase

space is a curved

crystal

of

negative

Gaussian curvature, whose group of

automorphisms

contains therefore the fundamental group of the

phase

space as a

_subgroup.

This

fundamental group is calculated in each case. The results do not

_depend

in any way on the choice

of the « atomic surface »

(the motif)

which decorates the

_hypercubic

cell of the

high-dimensional

crystal

from which the

_quasicrystal

is

_generated.

As a _{consequence, it does} not

_depend

on the fact

whether the

_phasons

are continuous or not. These

_{investigations}

constitute a

_prerequisite

for any

further

_study

of the

_topological

and

geometrical

nature of

_phason

strains. Moreover, the

homomorphism

between the fundamental group of the

phase

space and the group which classifies the dislocations is discussed. It is indicated, as a

_preliminary

result of a _{research in progress, that}

the fundamental group of the

phase

space classifies the

topological

phasons.

Classification

Physics

Abstracts 61.42 - 61.70

1. Introduction.

The

_{crystallographic description}

of an

_{aperiodic crystal (quasicrystal)}

makes use of a

d-dimensional Euclidean

_crystal

Zd

c

Ed (Ed :

d-dimensional Euclidean

space),

in which a

dl -dimensional

planar

cut

_PII

is

_performed.

This cut has some irrational orientation with

respect

to the lattice. A restricted set of vertices of

Zd

is

_projected

on

_{Pli ; they}

constitute the

quasicrystal

in

_{question. Pli}

is indeed

_thought

of as the

_physical

space,

with dll

= 3 or

dl

= 2

according

to the _case. The

hyperlattice

is a

_hypercubic

_lattice,

with d =

6,

dll

= 3

(resp. d

=

5,

dll

=

2)

for the icosahedral

crystal (resp.

the

pentagonal

crystal).

Pjj

is a

subspace

of

Ed which

is

globally

invariant under the icosahedral group, which is a

subgroup

of the

_{hyperoctahedral}

group in d = 6

(resp.

the

pentagonal

_{group, which is} a

subgroup

of the

_{hyperoctahedral}

group in d =

5).

For _more

detail,

_see references

[1-3].

The

_quasicrystal

which is obtained

_by

the above process

depends

of course on the restricted

set of vertices of

Zd

which is selected. A most usual choice is to select the vertices which

belong

to a d-dimensional

_{strip parallel}

to

_Pli

_{and whose breadth spans} a unit cell of the

_high

dimensional

_crystal

as in

_figure

1. A more

_general

construction

_employs

the device of the

so-called « atomic surface » S. This is a

_{dl = d - d}

dimensional manifold made of

_{equal pieces}

(motifs)

attached to all the

_equivalent

vertices of the

_{hyperlattice [4].}

S is therefore invariant

by

the translations of the

_hyperlattice

and has a

_{global icosahedral}

_symmetry

_(resp.

pentagonal symmetry).

The

_quasicrystal

is the set of the zero dimensional intersections of the

motifs and of

_Pli.

Note that the two methods are

_equivalent

if the motif is a _copy of

Pi,

the

_{orthogonal complement}

of

_Pli

in

Ed

_(Ed

=

pl,

_x

P J.. ).

Fig.

1. - The case d = 2,

d.L

= 1 which illustrates the construction of the

strip

of selected vertices.

Whatever the method

_used,

the structure of the

_{quasicrystal depends globally}

on the exact

position

of

_P)j,

which can be moved

parallel

to itself in

Ed.

The related

change

of structure will be referred to as a

_{global phase shift (GPS),}

and is

_{easily analyzable}

as a sum of

_independent

localized

_phase

shifts

(LPS),

also called

_phasons,

when the

_displacement

of

_Pli

is small.

The

_physical

nature

_{of phasons (an example}

is

_given

in

_Fig.

2 for

_{the dp}

= 2

case)

is far from

being

understood,

but it is

_by

all means clear that

_{they play}

an

_important

role in the structural

Fig.

2. -

Localized

phase

shifts.

They

are

_certainly

akin to the discommensurations of incommensurate structures and to

stacking

faults. It is also of fundamental interest to know whether the

_phasons

are continuous or discontinuous. All models constructed up to now for

perfect quasicrystals figure

out the

phasons

as discontinuous

_(as

in

Fig. 2),

which means that the atoms of the

_quasicrystal

may suffer some

_jump

when

_Pp

is

_continuously

moved in

Ed.

But Levitov has

_recently

shown

[9]

that,

with certain choices of

S,

the atoms move

_continuously

when

_Pp

is moved.

GPSs have

_recently

received some consideration in a seminal paper of Frenkel et al.

_[10],

in

which these authors consider the effect of a motion of

_{Pi along}

a circuit in

_P..L

which

_brings

it

back to the same

_{position ; they}

conclude that there is some

_{indeterminacy}

on the

_position

of

the final vertices. This

_{indeterminacy}

is,

according

to

_them,

related to the

_multiple,

topologically non-equivalent,

ways of

performing

such

circuits,

and takes its

origin

in the

irrationality

of the cut, i.e. in the

_{non-crystallographic}

_symmetry

of the

_{aperiodic crystal.}

The

_topological

nature of the

_phase

can indeed be understood

through

the

_study

of

_loops

and

_paths

in

_P..L’

This is

_certainly

a first

_approach

to the

_physical

nature of

_phasons.

The intersection of

_Pp

and

_P..L

is

_generically

a

_point

which we note

_A..L’

_{Reciprocally,}

a

_point

A..L

in

_P..L

defines a

_physical

_space

_Pp

and the

_quasicrystal

it

_carries,

whatever the method of

construction which is used. It is therefore of the utmost interest to consider the

_topology

of the

_subspace

U of

_P..L

which

contains all the

_inequivalent

_A..L

s. This

subspace

is

clearly

the

perpendicular

projection

on

_P..L

of the unit cell of the

_hyperlattice

in

Ed ;

it is a

quasicrystals

which have the same

_phase

but differ

_by

some

_global

translation. We shall

consider,

in a first

_step,

that two _{q.-c.s which differ}

_by

a

_global

translation are different. All the

_{A 1}

s which lead to _q.-c.s

_{differing only by}

a

_global

translation form a dense but discrete

subset of measure zero in TR or RI.

As a

_preliminary

_step

to the

_{understanding}

of

_phasons,

we

investigate

in this _paper the

topological

nature of the closed circuits in

_P.i.

We are also able to

_classify

them

_(as

elements of a certain _group of

_homotopy)

and conclude as Frenkel et al.

_[10]

to their multivaluedness. However we do not think that this multivaluedness

_implies

a non-deterministic nature of the

quasicrystalline

structure.

_Finally

we stress the fact that the

topology

of the

_loops

in

_question

does not

_depend

on the

_precise

choice of

S,

i.e. on the

_properties

of

_continuity

of the

_phasons.

In a

_forthcoming

_paper, we shall use the

_present

results to

_investigate

the nature of defects in

quasicrystals,

especially phasons.

Our final discussion deals in an intuitive manner with this

topic.

The

present

paper is in a sense more

_especially

devoted to the

_{geometrical properties}

of

P.i

« tiled » with TRs or RIs. A

_preliminary

version of some of our results

_[11]

was

_published

in 1988.

2.

_Topology

of

_phase

_{shifts ;}

nature of the

_problem.

The

geometry

of the

_{orthogonal projection}

U of the unit cell C of the

_{hypercrystal (living}

in

Ed)

onto

_P.i

has been described several times

_[10].

U

_(which

is a TR or a RI in the

_physical

examples

we shall

consider;

we discuss also some other

examples)

is.made of the

«

_{silhouetting »}

faces of the unit cell in

Ed,

i.e. those which are

_tangent

to

_{a dll}

-dimensional

« ray »

parallel

to the

_general

direction

_Pli.

Some of the vertices of C fall therefore inside U. Consider now a

_particular

_C,

the

_{corresponding}

_U,

and the set of

_projections

of all the other Cs onto

_P.i.

The vertices V of the

_{hypercrystal project along}

a dense

(non-compact)

set of

points V.i

of

_{P1. Any}

vertex V is

_always

in

_{silhouetting position}

for some of the unit cells

which are incident to it. Therefore the set of all Us is also dense in

_P.i.

.

Let us

_hang

in each cell

_Ci

of the

_hypercubic

_lattice,

after the manner of Frenkel et al.

[10],

a

_d.i

dimensional manifold _o-i, which

_projects

one-to-one on

_{Ui ;}

the set of all these

_equal

« motifs » builds a sort of

_special

atomic surface

_1:a,

_(since

the intersections of

_1:a

with

Pj

are

_{points Ai)}

which can be made

_locally

continuous over all the cells

_(for

an

illustration,

see Ref.

[10]). Since Ui

is

_{homotopically}

equivalent

to

_{Di, 1:a}

can be considered as an

unfolding

of

_P.i

tiled with Us.

Any

closed

_{path (any loop)}

in

_P.i

is

_homotopic

to the null

_path

in

_P.i’

but its

_{images (its}

lifts)

in

_1:a,

which are _many, have a richer

_topology.

When

_traversing

the set of

_points

A.i

of a

_loop

in

_P.i’

it is

_enough

to know to which

_Ui

_A.i

_belongs

to

_get

an

_unambiguous

image

of the

_loop

in

_1:a.

_Furthermore,

because of the

_{irrationality,}

_any two

_points

Ai,

Ay

belonging

to _any two different cells

_Ci,

_Cj

_project

on

_points

_Aii,

_Ajj

in

P.i

which are carried

_{unambiguously by}

two different

_Ui,

_Uil

even if

_{A.i i =}

_{A .ij.}

1:a

is therefore some

_covering

of the

_Ui

s

_(in

the mathematical sense of this term ; see

_Massey

[12])

in which

_loops

traced in

_P.i

are covered either

_{by loops,}

or even

_by

_open

_{paths (j oining}

an intersection of a

_Pli

with

_1:a

to another

_{intersection).}

There is a

_{particular covering}

of the

Ui

s, the so-called universal

covering,

in which all the

loops

in

1:a

are even _open

_paths.

This

paper will reach the

topology

of

loops

and

paths

on 1:a

through

the

study

of the fundamental

geometrical

and

_{topological properties}

of these

_coverings.

It is remarkable that this

_topology

does not

_depend

on the

_{precise shape}

of the motif from which

_1:a

is

built,

but

_only

on the Us

and on their relations of

_incidence,

which are the

_projections

in

_Pi

of the relations of

3. Primitive icosahedral

_{crystal ;}

d =

6, dh _ 3.

3.1 THE TRIACONTAHEDRON TR AND ITS LAWS OF INCIDENCE IN

_P.L.

The unit cell of a

primitive hypercubic

lattice

_projects

on

_P.L

_{(d.L = 3)}

_along

a triacontahedron

_TR,

i.e. a

polyhedron

with 32

_vertices,

30

_equal

rhombic

faces,

and 60

_{edges (Fig. 3a).}

The faces are the

projections

of the «

_{silhouetting »}

2-faces of the

_{hypercube [10].}

Two

_opposite

and

_parallel

faces are related

_by

a

_{vector yi}

_(i

=

1, 2,

...,

15 )

in the

hypercubic

lattice ;

yi is also the vector

which

_joins

the centers of two

_{adjacent hypercubes.}

_{The yl}

s

_project

on

_{P.L along}

vectors

’Y.L s which

_join

2

_opposite

and

_parallel

rhombuses on a TR

_{(Fig. 3b),}

as well as the centers of 2

_adjacent

TRs. The TR is invariant under the icosahedral group

Y,

and in

_particular

has a

center of

_symmetry.

Let us

study

in some more details the

geometry

of a TR. A TR is divided into 6

_equivalent

zones, made of

_equal

rhombuses

_(with

an

_angle

Cf) =

63.4349, tg

_cp =

2)

with one

_edge

direction in common. There are 10 rhombuses in each zone. One

_recognizes

that in

_figure

3b.

5 of these 6 zones are

_generated

_by

the 5 rhombuses

_{which join}

at the central

_point

_{I ;}

the 6th

one is the waist made of the rhombuses which are second

neighbors

to I.

Vertices are of two kinds :

_V3

vertices have a 3-fold

_symmetry

axis

_joining

them to the

center of the TR

_(there

are 20

_{V 3}

s

_meeting

at rhombus

_angles

_{ç5 =} 7r -

rp) ; V 5

vertices have

a 5-fold

_symmetry

axis

_(there

are 12

_{V 5}

s, with rhombus

_{angles Cf)).}

Each

_edge

carries a

V3

at one

_extremity,

and a

_{V 5}

at the other.

Let us consider the 6 directions in

_P.L

which are the

_projections

of the 6 cubic

_edges

in

E6 ;

their

_{(unnormalized)}

_components,

in a suitable coordinate frame

_simply

related to the

icosahedron,

are

_{(Fig. 3c)}

Fig.

3.

_{- a)}

A bird’s eye view of a triacontahedron TR ;

b)

its

Schlegel diagram

with faces identified

by

the _symmetry translations of the parent

hypercube ; c)

definition of the vectors a, b, c, d, e and f.

In the

_{Schlegel diagram}

of

_figure

3b,

the direction f

_points

_upwards,

and is normal to the

plane

of the

_drawing

at I. It is also the common

_edge

direction of the 6th zone alluded to

above.

Since there are 6 directions at each vertex

_A.l.

_projected

from a vertex A in

Ed,

each vertex is also Z = 12 coordinated in

P.l.’

. It is then easy to show that each

edge

in

P.l.

carries 10 faces

_belonging

to 10 different TRs. For

_example

the

_{edge along}

the f direction in

I,

which is a

_particular

_{A -L’}

carries the faces

(a, f ), ..., (e, f ) (which

are not

_represented

in

(a, f)

is cp ; therefore 1 is a

_V3

for the TRs to which these 5 faces

_belong.

It is easy to see that

there are 5 such

_TRs,

each of them

carrying

the 1 vertex and 3 faces which are

(a,

f), (c, f)

and

(a, c )

for one of

them,

the other

being

obtained

by permutation.

Note that

these 5 TRs

_overlap

around the

_edge

If,

and that

_they

close after a 4 7r _{rotation about this} axis. Therefore the dihedral

_angle

of a TR

_along

an

_edge

is 4

_{7r /5}

in

_P1..

The

angle

in 1 of the 5 rhombuses of

_type

_(-

a,

f )

is ’P ; 1 is therefore a

_VS

for the TRs to

which these faces

_belong.

There are 5 such

_TRs,

and each of them shares a face with the TR which is defined

_by

the set of

_{edges t}

=

(-

_{a, -}

b, -

_{c, -}

d, -

e )

in I. For

example

the TR

which has with t the common face

_(-

a, -

b )

is defined

by

the sequence of

edges

11

=

(-

_{a, -}

b, f,

-

e,

_c)

marging

in I.

Similarly

t2 =

(-

b, -

_c,

f,

- _a, +

d),

t3

=

(-

c, - d,

f,

-

b, e),

etc... The 5 TRs _{tl, t2,}

_t3,

_t4,

_t5

have common faces. For

_example

t, and

_t3

share the

face

_{(- b, + f ).}

The natural order of TRs about f is _{tl, t3,}

ts, t2, t4.

These 5 TRs close after a 4 TT rotation about

f,

since the dihedral

angle

of each of

them is 4

_{7r /5.}

It is a remarkable result that the N = 32 TRs

(see

[10])

which meet at a vertex

Ai

divide

_naturally

into two sets, those of the

_VS

_type,

which are in contact 2

_by

2 and close a

4 7r solid

_angle

around

_Al.’

and those of the

_V3

_type,

which

_obey

the same

_property.

In both cases we have 5 TRs about an

edge.

It is

possible by

combinatorial

analysis

to show that there

are in fact 20

_V3

s and 12

_VS

s

_meeting

at each vertex. To

_summarize,

the set of all TRs

projected

from the Cs divide into two

_independent

_equal

sets, each of them such

that

a)

there

are 20 TRs at each

_V3

and 12 TRs at each

_{V5 ; b)}

any vertex is either a

_V3

or a

V5 ; c)

there are 5 TRs about each

_{edge. Finally,}

the two

_independent

sets are so related that

each

_V3

vertex of one set is a

_VS

of the other set, and vice versa.

3.2

_{H6 ,}

THE UNIVERSAL COVERING OF TR. - Let us now unfold the TRs of one of the two

equal

sets discussed above in such a _way that the

overlaps disappear,

without anyway

creating

any

void,

and

letting

subsist the relations of incidence between them. In so

_doing,

_any

closet

path

in

_Pl

is covered

by

an _open

_path,

so that we obtain the universal

_covering

of TR. It is a 3-dimensional

_crystal

_H6

whose unit cell is a deformed TR

(but

still invariant under the same

group

Y).

This

« crystal » (it

is a

_crystal

in the sense that the deformed TR tiles

Hl

without

_overlappings

or

_vacancies)

lives in a space of

negative

Gaussian curvature, as we

shall see.

The identification of the 2-faces of the TRs 2

_by

2

_yield

_{the group}

_H6

of translations of

H6 ,

, which has 15

generators

A ij

(i, j

=

1, 2,..., 6 ;

see

Fig. 3b),

where i

and j

number the

two zones to-which the face

_{(ij ) belongs.}

By definition,

_Aij

sends the face

_{(ij )}

on the

_opposite

face

_{U i),}

in the same _way as the

_{corresponding}

translation

_operator

_{’Y 1. k does in}

Pi,

or _{yk in}

Ed.

We shall therefore also use the notation

_{’Yk = Aij.}

_By

definition

Aji

=

Ai} 1.

The

A ij’s

obey

12 relations

_{rk (A jj)}

=

1,

k =

l, 2, ...,12 ;

each of them is obtained

by rotating

a TR around some of its

_{edges by}

an

_angle

of 4 7r which scans the 5

possible

positions

of the TR about this

_edge.

We

repeat

this

_operation

for each of the 12 different and

Note that in these formulas the

_operators

act to the

_{right :}

in rl, for

example,

A 12

acts

_first,

then

_{A 13,}

A _16,1 etc... The same convention will be used below all

throughout.

The _{group of translations}

_Hb

is

_finally

defined

_by

its

_generators

and relations

and is an infinite _{group, since the number of}

_relations,

which is

_12,

is smaller than the number

of generators,

which is 15. The

_crystal

_{H 6}

is therefore

_infinite,

but it is not

euclidean,

since the translations are not commutative. It is

_clearly

of

_negative

curvature, which can also be seen in the fact that the number of TR cells about each

_edge,

which is

_5,

is

_larger

than what could be accommodated in euclidean space, where 2.5 cells fill an

_angle

of 2 Ir.

However

_{H 6}

is not a

manifold ;

the

neighborhood

of the

_{V 5}

vertices is not a

3-ball,

but a

torus _{of genus 4. The} vertex

_figure

of such vertices in

_H6

is indeed made of F = 12

_pentagons,

which are the

_planar

sections of the 12 TRs at

_{V 5 ;}

the number of

_edges

is E = 12 _x

5

x 1 =

_30,

and the number of vertices is

V = 12 x 5 x 12,

since each vertex of the

2 5

vertex

_{figure belongs}

to 5 TRs. Therefore the Euler characteristic is

and

not y =

2

_{(characteristic}

of a

_sphere),

as it should be for a Riemannian manifold. The same

_type

of calculation

_{yields X (V3)}

= 2.

H 6 -L

is a

_crystal

in a _{space of}

negative

_curvature,

with

_{singular points}

at each other vertex. This

_{intriguing pathology}

does not

_{play probably}

_any

important

role,

but it was of interest to mention it.

Note that

_{H6 is}

_by

_{definition the fundamental group of the} triacontahedron with faces identified

_by

_{the "Y J.. k} s

Therefore this group is related to the _group

_{of loops}

on

_1:a.

It is in fact a

_lârger

_{group ;} some of

its elements

_represent

indeed

_paths

in

_1:a

which

_join

a

_point

M on a

_{quasi-crystal}

_Q

to another

point

M’ on

another,

_{equivalent, quasi-crystal}

_Q’,

obtained from the first

_by

a

global

hyperlattice

translation which

_brings

Q

on

Q’

and M on M’. The

corresponding path

in

1:a

projects

on

_{P 1- along}

a

_path

which

_{joins M 1-}

to

_{M i .}

3.3 CLASSES OF LOOPS ON

_1:a.

In order to select the elements of

_H6

which

represent

the classes of

_loops

on

1:a,

we

employ a

method

inspired by

the

theory

of

_coverings

of

_graphs

and

2-complexes,

which we

_developed

in a recent

_{publication [14]. Although}

it is not the

_only

way

to reach the results we are

going

to

_present,

it has the

_advantage

to

_give

a

_geometrical

meaning

to the

_projection

of the

_hyperlattice

in

_{P 1- .}

Instead of

_focusing

our attention on

Ht ,

, which is a

crystal,

let us consider its order

parameter

space, i.e. the space

B6

whose first

_homotopy

group is

precisely

H6.

As indicated

above,

TR with faces identified could be used as such a

_B6,

but there is a much

_simpler

choice.

Let us consider a

_bouquet

of 15 oriented circles

_Aij’

all attached to a common

_point

P,

with 12 disks attached to them such

that, along

the

_perimeter

of any of these

disks,

one traverses

the circles

_Aiy

in the order

_given

_by

one of the

_{relations ri}

= 0

(Eq. (2)).

It is clear that _any

loop

on such a

_B6

is an element of the group

H6,

_{and that the group}

H6

is

_entirely

and

non-trivially

represented

by

the

loops

on

_B6.

_B6

is called a

_2-complex

and is a 2-dimensional

group,

by

definition,

is

trivial ;

furthermore

B6

is such that the

vicinity

of any of its vertices is

homeomorphic

to the

_vicinity

of

P,

i.e.

_showing

_up 30 oriented

_edges

_Aij

_traversing

_P,

with disks attached in the same way. All the

loops

r of

B6

become open

paths

f

in

B6.

The terminal

_{points Pi}

and

_Pj

of such

_{paths join equivalent points}

on

B6

and the

ts

are classified

_by

the elements of

H6.

Therefore

B6

shows up the same

_symmetries

as

H6,

and can in fact be

easily

embedded in

it ;

it is constructed of the

edges

of the

_reciprocal

lattice

_(with

nodes for

_example

at the centers of the

TRs,

edges joining

centers of

_adjacent

TRs)

and disks

_suitably

attached. In that sense, there is no difference between

_B6

and

H 6 -L ;

B6

is in

fact,

so to

_speak,

the

_{Cayley 2-complex}

of _{the group of translations}

H6 of H 6 -L

.

The

_{topological properties}

of

_loops

_{on 1:a}

are more

easily

discussed when

_considering

2-complexes

(B6,

B6)

rather than

_{H 6 1}

itself and a TR with faces identified. In

_fact,

the set of

edges y1

s, which can be used as

_elementary

_segments

_along

which the

_loops

in

1:a

can be

_traced,

constitute a

_covering

_graph

of the

_graph

of

_edges

in

_B6,

and it suffices to

attach suitable disks to

_{the ’Yi}

s to

_get

a

_{covering 2-complex}

of

_B6,

which we note hereafter

K6. Similarly,

È6

is also the universal

_covering

of

_K6,

which indeed

_stays

_midway

between

B6

and

_{B6. The -yi}

s are commutative

_operators,

and

_K’6

is an abelian cover of

B6.

According

to

_{covering theory [12],}

_{the fundamental group}

irl(iff)

of _any cover

6

of a manifold

_{(graph, 2-complex, etc...) 6}

is a

_subgroup

of

_7r,(19).

If, furthermore,

lrl(ià)

is an invariant

_subgroup

of

_7Ti(S),

then the

_quotient

_{group H} =

7TI(g)/1TI(g)

is a

group of

automorphisms

of

6.

This

property

is

_clearly

fulfilled in the

_example

above,

with 19 =

B6,

S

=

B6 ;

_we have indeed

11’1 (B6)

=

1,

and

Hf =

_7T

1 (B6),

the group of translations of

B6 (easily

embedded in

_{H 6}

Consider now the commutator

_{subgroup K15}

of

_lrl(1%6);

it is

_{generated by}

the elements

Cij

= _yi

yy

yl

1 y i

and the related

2-complex

has the group of

symmetry

H =

H6/Kls,

whose

_presentation

is

This group is

isomorphic

to the free abelian group with 10

generators

Zlo,

but is not the _group of

_{automorphisms}

of

_Ia,

_(which

is

_undoubtedly

_Z6).

It is indeed

_possible

to show that there

are some

_{supplementary}

relations between

_{the yl}

s ;

they

can be deduced from the

explicit

expressions

of

_{the yl}

s which are

_given

in Frenkel et al.

_[10].

_Using

their

_notations,

we find 4

independent supplementary

relations

₍₁₎

from which we obtain

_by

_permutation

24 relations between

_{the y}

_{i s, of the form :}

but we need

_only

4 of

them,

sl, s2, S3, s4, say, to construct _{the groups}

_Ha

and

ir 1 (£,,)

which follow

_(this

is evident in the case _{of the abelian group}

_{Ha ;}

the normal

_subgroup

of

_{equation (8)}

contains all the 24 relations

_{by conjugation}

and

_{multiplication}

of si

(i

=

1, 2, 3,

4 ) by

suitable commutations

cij).

The abelian group

is therefore a _{group of}

_{automorphisms}

of

_.!a,

and the

_{corresponding 2-complex}

is made of the

graph

of

_{the yl}

s with suitable

_rj-disks

_attached,

or of the

_graph

of the ’Y 1- i S in

Pi ,

with the same

_r,-disks

attached. The fundamental group

(the

group of the

loops)

of

.!a

is therefore the invariant

_subgroup

of

_{Hf =}

_{’TT 1 (B6) generated by}

the commutators

Ckt and the elements Sm :

4.

_Pentagonal

_symmetry.

The penrose

tiling.

4.1 THE RHOMBIC ICOSAHEDRON RI AND ITS LAWS OF INCIDENCE IN

_{P ..L .}

The

_silhouetting

projection

RI

_{(Fig. 4a)}

of the 5-cube onto

_{P..L’ along}

the

_Pli

directions,

has 40

_edges

of

_equal

length,

20 rhombic faces which fall into two

_types

RI

_{(tg cp}

=

T - 2,

_’P =

57.36)

and R2

(tg f/J =

r2 1 tb =

78.12)

and 22 vertices which fall into three

_types,

_{V5, 14}

and

_13.

There are :

a)

2 vertices

_V5,

with five-fold

_symmetry,

_{diametrically opposed}

on

_{RI ;}

the 5

_merging

rhombuses are of the RI

_{type ;}

b)

10 vertices

_14,

with 4

angles ’P, f/J,

7r -

_{f/J, f/J}

meeting

in this

order at each vertex, and

_c)

10 vertices

_I3,

with 3

_angles

’TT’ - ’P, ir -

_f/J,

’TT’ - - ’P

meeting

in this

order at each vertex. _{The group of}

_symmetry

of RI is

_D5

h.

It is known from Frenkel et al.

[10]

that 22 RIs meet at _any vertex

_{A 1}

in

_P..L.

All the vertices in

_Pi

are

_equivalent

from the

point

of view of the set of

_merging

RIs. These RIs meet face

_by

face but

_overlap

in

_P..L.

Each face can be constructed on 2 of the directions

_(out

of

₁₀₎

±

_{a j}

_{(j =}

0, 1,

...,

4 )

which are the

projections

of the base vectors of the

hypercube

onto

P ..L.

The 3

_components

_{of the ai} s _are, in some coordinates frame in

_P..L

and

_{project along}

the 5 directions of a

_regular

_pentagon

on any

plane perpendicular

to the direction

_{0,

_0,

_{1 },}

which is the 5-fold direction of the RI.

We notice that a .. a . ₁ -

T - 1

- cos

(

r -

),

a

.. a . _{+ 2} =

T

cos _{p .}

We notice that

_{8j . ay}

₊ 1

_{= =}

cos

(7T - t/J ),

_{8 j . 8 j +}

_{2 = -3 =}

cos cp. °

The

_counting

of the faces is then a

_question

of combinatorics. There are 40 faces :

2013 10 RIs meet in

A 1

with

_angles

cp; these are the faces

_{(ay, 8j+2)}

and

_{(- 8j, -}

_8j+2).

Together they

define 2 RIs

_meeting

at a

_{V 5’}

with no face in common.

2013 10 R2s meet in

with

angles

with

_angles

in

_{A 1}

.

Each vertex

_{A 1}

is

_clearly

a

_{V 5}

vertex for only two

_Rls,

which are built from the first set of Rls.

Fig.

4. -

The rhombic icosahedron RI.

_a)

a bird’s eye view ;

b)

its

Schlegel diagram.

The Rls of the second set can be

_coupled

two

_by

two in

_adjacent

rhombuses,

say

{ ( - a 1, a 3 ), (a3, - ao)}

and form 10 I3s with a R2

_belonging

to the first set of

R2s,

say

(-

ah -

a3).

There are therefore 10 I3s.

Finally, combining

2 R2s of the second set of R2s with 1 R2 of the first set of R2s and 1 RI of the first set of

R 1 s,

one obtain 10 I4s.

There are

_therefore,

as

_expected,

22 RIs _per vertex, each vertex is 10 times a

_14,

10 times a

13,

and twice a

_{V 5.}

Finally,

there are 8 faces

_{meeting along}

each

_edge,

since each

_aj

builds a face with _any of the

other ±

_{ay s.}

4.2 THE UNIVERSAL COVER

H 5

OF A RI. - We

_reproduce

in

_figure

_4b,

the

_{Schlegel diagram}

of the RI with faces identified

_by

the same translations which

_bring

a 2-face of the

_hypercube

on an

_opposite

2-face. Each face

_belongs

to two zones i

_{and j (1 -- i, j -- 5 ),}

and is labelled

Bii* Bij

is also the

operator

which

_brings

the face

_{(i,j )}

onto the

_opposite

face

(j, i ).

Then

_Bij

1 =

_Bji.

We affect a

symbol

_Bij

or

_Bji

to a face

by considering

the direction of

the vector

_product

i

_{1B j (i j)}

of the unit vectors carried

_by

the arrows which orient the

_edges

where each relation realizes a rotation about an

_{edge j =}

1, 2, 3, 4, 5 ;

8

_operators,

i.e. 8 Rls are met

_during

this

rotation ;

all the

_edges

of a

_{zone j}

are made

_equivalent

in the process-

(2).

The group of translations

_Ht

of the

_crystal

_Hi

which realizes the universal cover of RI has the

_following

_presentation

Again,

since the number of relations

_(which

is

₅₎

is smaller than the number of generators

(which

is

_10),

_{the group is infinite. However the} 3-dimensional _{space which carries}

Ht

is not an

_hyperbolic

_space, but a

_variety

with

_{singular points,}

since it can be

_proved,

in the same _way as above for

_{H6 ,}

that the vertex

_figure

of a vertex

_(all

the vertices are

_equivalent)

is

not a

_2-sphere,

but a torus with 5 holes. Its Euler characteristic is

indeed X = -

8,

since we

have

4.3 SPECIALIZING TO THE CLASS OF ISOMORPHISM y = 0. In the de

Bruijn’s

construction

of a Penrose

_{tiling [ 15],}

the relevant domain in

_{P-L of inequivalent points}

_{A -L}

is not the full

_RI,

but

_only

a set of 4

_{planar pentagonal}

intersections

_Pi,

_{P2, P3,}

_P4

of RI with

_2-planes

perpendicular

to the 5-fold axis. These

_planes

are indicated in

_figure

_5,

_by

their intersections

with the RI. The

_{corresponding tilings}

are said to

_belong

to the « class of

_isomorphism

y = 0». The

pentagons

belonging

to two

_adjacent

RIs are themselves in contact

_along

these intersections. Note that two

_edges

like

_af3

and

_{a’ /3’}

in

_figure

_5,

which

_belong

to two

different

_pentagons

of the

RI,

are

_equivalent

under the translation

_B12.

_{B12 brings}

in fact one

P3 belonging

to the RI under consideration into an

_adjacent

_{RI ;}

we note

_bl2

the restriction of the

_operation

_BI2

to a translation

_acting

on a

_{pentagon ;}

_{repeating systematically operations}

of that

_type

with all the

_bij

s, it is

possible

to construct the covers of the 4

_pentagons,

which are

undoubtedly

4

_hyperbolic

_{crystals {5,10}}

_sitting

in 4

_{hyperbolic planes}

₍₃₎

_(there

are

Fig.

5. - Domain of

inequivalent

A 1.

s for the class of

_isomorphism

y = 0 ;

Schlegel diagram.

The

edges

of the 2 _pentagons

_Pl

and

_{P4 (resp. P2}

and

_P3)

are

_equivalent

under the elements of the group of

hyperbolic

translations

_(b13’

b14, b24, b2s, b3s ; u

=

1)

(resp.

(bl2,

b23, b34, b4s, bsl ; v

=

1)).

(2)

The identifications between our

_Bij

s _{and the ’Yt} s of Frenkel et al.

_[10]

are as follows :

10

_pentagons

at each

_vertex).

Each cell

_{of H 5}

is foliated

with

_{4 {5, 10}}

s, and the set of all its cells is foliated with an infinite ensemble of

_{similar (5, 10}}

s, which we note

_Hi

_(y

=

0).

Consider the

_{4 (5, 10}}

s

_belonging

to a

_given

RI.

_They

divide into two

_{types ;}

some

translations

_bij

_bring

_Pi

_along

a

_pentagon

of the

_{hyperbolic plane}

which

_prolongs

P4,

while some others

_{bring P2 along}

the

_{hyperbolic plane prolonging P3,}

but nô

_opération

brings Pl

or

_P4,

_along

the

hyperbolic plane prolonging P3

or

_P2.

A translation which

_brings

Pl

into a

_pentagon

_adjacent

to

_{P4 brings P2}

and

_P3

into

_pentagons

_belonging

to other

hyperbolic planes.

It is easy to

_figure

out

_them,

how the

_bij

s

divide,

and to calculate the

relationships

which link

_by

a natural extension of Maskit’s

_procedure

to the

_complex

«

_{polygon »}

made of the 4 faces

_{Pi, P2, P3, P4,}

with

_edges

and vertices

_suitably

identified. One finds in fact two relations between the

_operators

_blJ,

_specifically

The

meaning

of u - 1 is

that,

by applying b 35

to

_Pl,

_say, then

_{b - 13}

to P4 plus

the

_image

b35(PI)

to which it is now

_glued,

then

b 141,

etc..., one finds the 10

_pentagons

which close space

about the vertex from which we have started the

_{procedure (here a ).}

Note that the unit cell

(the

fundamental

_domain)

_{of a {5, 10}}

is made of two

_pentagons.

It is easy to

_verify

that the Euler characteristic of the fundamental domain

is X - -

2,

as it should be for a

{5, 10 },

since V = 1

(all

the vertices of the two

pentagons

Pl

and

P4 identify

in the

process),

E = 5

(the

10

edges identify by pairs)

and F = 2

(two

pentagons).

The relation v - 1 refers to

P2

and

_P3.

We can also

_interpret

the

_operators

u and v as the

_generators

of the invariant

subgroup

of H 5

which leave the

_hyperbolic

2-dimensional

_crystals

invariant under the 10-fold

rotations about the

_edges

of the Rls.

_Finally,

_{the group of}

_symmetry

of the

_hyperbolic

foliation

_{H 2 -L}

_{(y =0)}

is the

_quotient

_{group of}

_Ht

where the

_bij

s in

_{equation (12)}

have been

_{replaced by}

_Biy

s.

4.4 SPECIALIZING TO THE CLASSES OF ISOMORPHISM _{’Y =1=} 0. - Pavlovitch and Kléman

[17]

have shown that for the

_generalized

Penrose

_tiling

the domain

_{of unequivalent A 1.}

s is made

of 5

_planar

sections of the

_RI,

2

_pentagons

and 3

_{decagons (Fig. 6).}

The

_edges

of these

polygons identify

under the translations

_bij

s. As

_above,

we can build the universal cover of

this

_domain,

_using

Maskit’s

_procedure,

and calculate its group of translations. We find the

same relations than for

H -L 5

_{(Eq. (10)),}

and

consequently

the same _{group of} translations :

In a sense, this is not a

_surprise,

because the foliation

Hi

_{(’Y #= 0)}

is embedded

_generically

in

_{H5 .}

This was not so for

_{H 2}

(y =

0).

Each foil of

_{H 2}

_{(’Y #= 0 )}

is a

_tiling

of

_{decagons (6}

at each vertex, as it can be

_shown)

and

pentagons

(2

at each

_vertex),

the sum of

_{polygons being}

8

_(there

are 8 RIs

meeting along

an

edge).

The Euler characteristic of the fundamental domain

(which

is made of 2

_pentagons

and 3

_{decagons) is y = -}

10,

since the set of the 5

_polygonal

sections of the

RI,

with

_edges

and vertices

_suitably

identified,

contains = 5

independent

vertices, F

= 5

faces,

and E = 20

independent edges.

Its genus

is g

=

6,

i.e. it is also a torus with 6 holes.

4.5 TOPOLOGY OF THE PHASE.

_{- ,¡’a}

is a 3-dimensional surface. If

_Pjj

is moved at

’Y =

constant, i.e.

conserving

the class of

isomorphism,

then any vertex A

_belonging

to

Pil

moves in a 2-dimensional cut of

_$a,

_belonging

to a foliation which we note

,’a ( ’Y).

The fundamental _group

_{of any}

foil in

_{£a ( Y )}

is related to _{the fundamental group}

_{of any}

foil in

_{Hi (’Y).}

We have discussed elsewhere

[18]

in some details the

_symmetries

and fundamental groups of 2-dimensional

_{hyperbolic crystals}

_{« with manifold » : the group of}

translations

_{depends only}

on _{the genus g of the}

_crystal,

and has the

_following

standard

representation

where the

_Ai

s form a set of

_{2 g geherators.}

In this

_{representation,}

the unit cell

(the

fundamental

_domain)

is a

_regular

4

_g-polygon,

_{and any}

_generator

_{Ai brings}

an

_edge

of this

unit cell to the

_{opposite edge.}

But

_{equation (15)}

is

_only

one

_{possible representation}

of the

fixed

_point

free group, which acts on the fundamental domain of the

_{hyperbolic tiling of genus}

g ; we have seen above that the natural fundamental domains which are of interest in our case are not

_{regular polygons.}

It is therefore useful to look for

_{representations}

whose

_generators

are more

_directly

related to the natural fundamental domains.

We consider

_only

the case y =

0,

for

simplicity.

The two foils of the atomic

surface.

1:a(Y

= 0 ;

Pi, P4)

and

_{X ,(Y = 0;}

_{P2, P3)}

are

_différent,

and

_they

are the

_images

in

ES

of

_hyperbolic

surfaces of different curvatures

_{(related by}

inflation

_symmetry),

but tiled the

same way,

along

a {5, 10},

the domain of

_acceptance

in each foil

_containing

_{2 pentagons}

(which

build a fundamental domain of the

_tiling).

These two

_pentagons

can be

_{replaced by}

the

fundamental domain of the dual

_{tiling ( 10, 5},}

a

_decagon

which can be shown to have the

same area than the

_{2 pentagons.}

Therefore this

_decagon

is an

_equally

valid

_acceptance

domain. The

_edges

of the

pentagons

are the translations which

_generate

_{thé {10, 5}}

and it is of interest to use them as new

_generators

_(rather

than the

_bij

s)

for the

_hyperbolic

group. In

fact,

we define the

_following

_generators

_(for

the case

_{P2, P4)}

The

equation

u = 1

(Eq. (12))

reads now :

must be found. We obtain it

_{by introducing}

first another set of

_(pentagonal)

_generators

which

_obey

the

_{(clearly evident)}

relation :

Express

now the ai s in function of

_{the ai}

s. We have

_by

définition

Fig.

7.

_{- {10,5}}

and related generators

(the

denomination of the generators is at variance with the text, but the reader will

easily

make the relevant

identifications).

According

to

_{equation (19),}

we have therefore

Since,

for

_example,

_{a 5} =

a3 a5

according

to

(20),

we

have,

after

(21) :

Hence the second relation _{between the ai s}

It is easy to convince _{oneself that the ai s} act in the

_{hyperbolic plane,}

and are

_along

5

pentagonal

directions. For

_example

_a

_projects

in

_{P 1-}

to _{Y2 - Y9} =

(1,

0, 0, 0,

1

),

which is a

pentagon

over an

_edge

of the same

_pentagon.

Note further

than,

by eliminating

a 5, say, between

equation

(17)

and

equation

(22),

one recovers a relation of the

_type

displayed

in the group

representation

of

_{equation (15),}

with some

_change

in the

_séquence

of

edges, only.

Therefore the two relations u =- 1

_{(Eq. (17))}

and u’ == 1

_{(Eq. (22))}

are the

relations of the _group of translations of the

_{hyperbolic crystal}

_{{5, 10}}

where the ai s are linear combination of _Yi s.

According

to the

_reasoning

done for the icosahedral case, the fundamental group of

$a(Y =

0 ;

P1,

P4)

is the commutator

_subgroup

of

_{H t 2(y}

=

0).

There are no

_{supplementary}

relations between the yi s _{involved in the ai s}

_(Y2,

_{y3, ’Y6, ’Y7,}

_y9).

Similar

results,

mutatis

mutandis,

hold for

_{Ht(y -}

_{0 ;}

_{P2, P3),}

which is a _group

_isomorphic

to

_{Ht (P1,}

_{P 4) ;}

here too the

’Y; s

_(Yi,

_{’Y 4’ ’Y 4, y8,}

’Y10) .

form a set of

_independent

vectors.

If the

_{representative point}

_A-L

of

_Pp

is allowed to move

_through

all the 3-dimensional

domain

RI,

the fundamental _group of the

_loop

in

_$a

is now the commutator

_subgroup

of

Ht,

with some

_{supplementary}

relations between the

_Bij

s due to the

_{supplementary}

relations which exist between the

_10,yi

s. There are 5 such relations which we do not write.

5. Some

_{simple pedagogical examples.}

We show here how the above

_concepts

_apply

_{in 2 very}

_simple

cases which have also been

considered

_by

Frenkel et al.

_[10].

5.1

_{H5 ,}

THE CRYSTAL FOR THE CASE d =

3,

d 2. -

In this

very

pedagogical

case, the

1-d «

_{quasi-crystal »}

is

_periodic.

There are

_{3 y}

_{i s, which} can be taken as _{y i -}

_{1, -}

_{1, 0 ;}

y2 = -

1, 0, 1 ;

y3 =

0,

1, -

1. Hence

YI + Y2 + Y3 = 0. The

projection

of the unit cube in

P..L

is an

_{hexagon (6 )}

which tiles the

_plane.

Therefore

_{H3 -L}

_{{6, 3},}

which is the universal

cover

_{of {6} ;}

the related

_2-complex

je, 13

is at the same time the universal cover and the

abelian universal cover

iT’ = je,3

of the

_{bouquet B3,}

made of 3 circles with 2 disks attached. We have indeed

The fundamental _group of

_1:a,

_{which is also the fundamental group of}

_{(T ’}

is therefore trivial. This is in

_conformity

with the fact that this «

_{quasi-crystal »}

is in fact a true

_{periodic crystal.}

5.2

_{H 4 ,}

THE CURVED CRYSTAL FOR THE CASE d =

4,

d.L =

2. - This

pedagogical example

has also been discussed

_by

Frenkel et al. The

_projection

of a 4-cube in

_P.L

is a

_regular

_octagon.

Hence

_{H 4 8,}

_8},

the

_{hyperbolic crystal}

_{of genus g} =

2,

is the universal _cover of the

projection

of the 4-cube in

_{P .L.}

The related

_2-complex

_H4

is the universal cover of a

bouquet

B4

whith 4 circles and one disk attached which realizes the

_unique

relation

The abelian cover

_T’1

is universal because there are no

_{supplementary}

relations between the

independent

vectors _{y i s. Therefore the}

_symmetry

_group

_{of J" is ’Y i ; r}

=

1, kij = 0 >

and its fundamental _group

_{7ri((T )}

is the commutator

_subgroup

_{of 7T 1 ( $4) = ’Y i ; r = 1 ).}

Note that in the 4 --> 2 case

Hi

has

only

one sheet : any 4-cell of the

_hypercubic

lattice can be reached

6. Discussion.

As we shall see in some detail a

_forthcoming

_{paper, the} fundamental _group

irl(£.)

of

1:a,

the

_special

atomic surface which has been introduced

_by

Frenkel et al.

_[10],

classifies the

phason-type topological

defects. In a Penrose

_tiling,

a mismatch is such a

_topological

defect

(Fig. 2) ;

in the

_{d = 3, d.L = 2}

case studied in section

5.1,

the fundamental group

’TT’

(1:a)

is

trivial,

which is in accordance with the fact that the 1-dimensional

quasi-crystal

is

periodic.

In this last _case, the fundamental group

Irl(U) _ Z3

is

isomorphic

to _{the group of}

topological

dislocations of the 3-dimensional cubic lattice in

Ed.

This is not either the effect of

a

_coincidence,

but the illustration of a more

_general

_{property ;}

we summarize these results as

follows.

We have in this paper introduced two order

_parameter

spaces. The first one,

U,

is the

projection

in the

_{perpendicular}

_space

_P.L

of the unit cell of the

_{hyperlattice.}

Each of its

_points

A.L

represents

a

_{quasi-crystal}

which is made of the set of atoms at the intersection of

Pli

and of the set of atomic surfaces in the sense of Levitov

_[9].

_Pli

intersects U in

Ai-

All the

_points

of U lift to different

_{Pl s,}

which differ one from the other

by

a

phase

shift

and a

_{displacement ;}

the

_Pp

s which differ one from the other

_by

a trivial

_phase

shift form dense

sets in U. The fundamental group of U

(whose

universal cover is

_{H d )@}

’TT’ ₁

(U),

classifies therefore

_phonon

_{topological (dislocations}

of

_Burgers’

vectors b

_equal

to a

_symmetry

translation in the d-dimensional

_lattice)

defects and

phason

topological

defects,

according

to

the methods of the

_{topological theory}

of defects

_[19].

But

_{phason topological}

defects are

specifically

defined

_by

the

_loops

in

_1:a,

which are classified

_by

_7rl(£a),

an invariant

commutator

_subgroup

of

_{ir 1 (U).}

The

_quotient

_{ir 1 (U) / Ir 1 (-Va)}

is therefore an abelian group,

which here is

_Zd,

_{i.e. the group of dislocations in}

Ed.

The

_{phason topological}

defects are

represented by

loops

in

_1:a,

and the dislocations b

_by

open

paths,

which

_project

in

P .L along

open

paths

which

_bring

a

_point

_A.L

i in

Ui

on an

equivalent point

Ajj

in

Uj.

These open

paths

in

_1:a

have therefore a clear

_meaning,

since

_they

can be classified in U as

closed

_{loops, by folding}

of the

_path

_A.Li

_{i Ai j.}

But _{other open}

_paths

in

_1:a,

those which for

example bring

a

point

Ai

to a

_point

_Aj

_belonging

to

the

same

_Pli

do

_{certainly project along}

a

loop

in

_P.L’

but the cover of such a

_path

in

_{H d}

does not

_{join equivalent}

_points

in

Vi

and

_Uj.

Therefore

_they

are not

_represented

in

_irl(U).

These

_paths,

which

_classify

the

_{physical possible Burgers’}

vectors

_{bll ,}

i.e. the

_projections

of the d-dimensional

_Burgers’

vectors

_b,

are therefore not included in the

_present

discussion of the

_{topological properties}

of defects in

_{quasi-crystals. Similarly}

the

_{perpendicular}

_Burgers’

vectors

_{b.L =}

b - b _il are not included either. Therefore the

_phasons

which we have here in

mind are indeed the LPSs we mentioned in the first section.

To end this

discussion,

let us

_emphasize

one

_again

the fact that the

_present

_topology

of the

phasons

does not

_depend

at all on the

_specific

atomic surface which is

_chosen,

and in

particular

on the continuous or discontinuous

physical properties

of the

_phasons.

Acknowledgments

We are

_particularly

indebted to Dr. Carlo

_Ripamonti

for numerous discussions and

encouragement

in the course of this work and to Prof. Louis Michel for discussions and a