A new method to generate quasicrystalline structures : examples in 2D tilings

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A new method to generate quasicrystalline structures :

examples in 2D tilings

Jean-François Sadoc, R. Mosseri

To cite this version:

A

new

method

to

_generate

_{quasicrystalline}

structures :

examples

in

2D

_tilings

Jean-François

Sadoc

₍₁₎

and R. Mosseri

₍₂₎

(1)

Laboratoire de

_Physique

des _Solides, Université de Paris-Sud et CNRS, 91405

_Orsay,

France

(2)

Laboratoire de

_Physique

des Solides de

Bellevue-CNRS,

92195 Meudon Cedex, France

(Reçu

le 18

_juillet

1989, révisé et

_accepté

le 20 octobre

₁₉₈₉₎

Résumé. 2014

Nous

_présentons

un nouvel

_algorithme

_{pour la}

_génération

des structures

quasi-cristallines. Il est _{relié à la méthode de coupe} et

_projection,

mais il _permet une

_génération

directement dans

_l’espace

«

_physique

» E de la structure. La sélection des sites dans

_l’espace

orthogonal

est

_remplacée

_par un test directement dans une

_grille

de domaines

_{d’acceptance}

dans

l’espace

E. Cette méthode montre

_qu’il

y a une sorte de réseau cristallin

_sous-jacent

au

quasi-cristal. Nous illustrons la construction dans le cas 4D-2D avec les

_symétries

d’ordre _5,

_{8, 10}

et 12

qui

sont _{obtenues par}

_projection

de 4D à 2D. Par la même méthode d’autres _types de

quasi-cristaux avec une

_{symétrie plus}

_{basse, ayant} un _{réseau moyen,} sont construits. Nous

_présentons

un

_exemple

de

_symétrie

4. Les

points

de ce

_{quasi-cristal}

sont un sous-ensemble des

_points

du

quasi-cristal

ayant la

_{symétrie complète}

d’ordre 8. Abstract. 2014 We

present a new

_algorithm

for the

_generation

of

_{quasicrystalline}

structures. It is related to the cut and

_{projection method,}

but allows a direct

_generation

of the structure in the

«

_physical

» _{space E. The}

_orthogonal

_{space site selection is}

_replaced

_by

a direct check in a

_periodic

array of « _acceptance »

regions

in E. This method shows that there is a sort of

underlying

crystalline

lattice in

_{quasicrystals.}

We illustrate the construction in the 4D-2D cases with the 5-, 8-, 10- and 12-fold

_symmetries

which can be obtained

by

projection

from 4D to 2D.

Using

this

new method we also _generate

_{quasicrystals}

with a lower _symmetry which have

_simple

mean

lattices. We _present for instance a

_quasicrystal

with a 4-fold _symmetry. The

_points

of this

quasicrystal

are a subset of the

_quasicrystal

which has the whole 8-fold symmetry.

Classification

Physics

Abstracts 61.40

1. Introduction.

In this paper we

_present

a new

_algorithm

for the

_generation

of

quasicrystalline

structures,

closely

related to the cut and

_projection

_(C.P.)

method

_[1],

but which allows a direct

generation

of the structure in the «

physical

» _{space E. The}

orthogonal

_{space site selection is}

replaced

by

a direct check in a

_periodic

_array of «

_acceptance

»

_regions

in E as will be

_explain

below. This

_method,

which is valid for

_projection

_{from any}

dimension,

is

_presented

in the next

section.

We then illustrate the construction in the 4D-2D cases. There is a theorem

[2]

which allows one to determine the smallest dimension

_required

in order to obtain a 2D

_quasicrystal

with a

given

symmetry.

If the

_symmetry

is of order s the number of its

_{prime integers}

smaller than

s

_give

the dimension of the space. We are interested in

_{quasicrystals}

obtained

from

4D. Consider the different

_{possible symmetries}

which are not

_{crystalline :}

_{5, 7, 8, 9,}

_10,

_{11, 12,}

...

Other

_{symmetries give}

more than 4 numbers. So the

_5,

_{8, 10}

and 12

_symmetries

exhaust the

quasicrystalline symmetries

which can be obtained

_{by projection}

from 4D to 2D.

In the

_present

_paper we shall discuss

_mainly

the

_octagonal

and the

_dodecagonal

quasicrystals

with a few comments on the two other cases.

We also

present

in an

_appendix

a method which allows one to determine

_rapidly

_{the space}

on which it is

_interesting

to

_project

the 4D structure, and which

_explains

the determination of the

_projection

matrix. In the usual C.P. framework the determination of the

_physical

_{space is} done

_{by considering representations}

of the

_symmetry

_group

_operations

in the n-dimensional

space. Here we

_adopt

a more

geometrical approach by considering

the

_symmetry

of the Petrie

polygon

attached to the lattice.

2.

_Description

of the

_algorithm.

Let us first recall a few facts and definitions from the C.P. method. Let L be an n-dimensional lattice embedded in

_Rn.

The «

_physical

» _space

E,

in which the

quasiperiodic

tiling

is to be

generated,

is a

_{p-dimensional}

_{space. For}

_simplicity

we shall suppose that n =

2 p.

Let f2 be a bounded volume in

_Rn-

n is shifted

_along

_E,

_defining

a «

strip

». All vertices of L

which fall in the

_strip

are selected and

_{orthogonally mapped}

onto

_E,

_giving

the vertices of the

quasiperiodic

tiling.

For the « standard »

tilings,

5li is the L unit cell or the Voronoi

_region.

In the

_language

of the 2D case, such

_{quasiperiodic tilings}

are obtained

_{by mapping}

on E a

monolayer

surface selected in the

_strip

_(this

surface is tiled

_by

_{square faces when the lattice L}

is a cubic

_lattice).

The selection

_{algorithm usually proceeds}

as follows : the lattice vertices to

be selected are

_precisely

those which

_fall,

_{by orthogonal}

_projection

onto _{the space E’}

orthogonal

to

_E,

inside a « window » W which is the hull of the

_projection

onto E’ of

5li .,

_They

are

therefore

obtained

_by

a

_systematic

_inspection

of all the L vertices.

The modified version will

_proceed

_differently

for the site selection and in most cases will be more efficient.

_{Specifically,}

the standard C.P.

_{algorithm speed}

is

_proportional

to the n-th power of the size which is

inspected.

The modified version runs at

_only

the

_p-th

_{power of this} size.

The basic idea is to

_split

L into reticular

sub-lattices,

i.e. to _{express L} as the

_product

of two

complementary p-dimensional

lattices,

the base B and the fibre F :

B and F are

_respectively

_generated

_by

_{bl,

...,

b.p}

and

{f1,

...,

fp} .

The

simplest example

is,

if L is the

_hypercubic

lattice

Z",

to take as B and F the lattices

_respectively

generated by

(ei , ... ,

ep}

and

_{{ep +}

1, ...,

en}

where

{ei}

is the canonical basis in Rn. Now the site selection

(see Fig.1 ) .

Let us call

W {Q}

the intersection of

F { Q }

the

_strip

S

_{generated by translating}

llÎ

_along

E :

Fig.

1. - Scheme of the modified

projection

method. B is the base, F the fibre, E the

_physical

space and

A is the centre of an _acceptance domain.

’

The

_points

of

F {Q}

which are to be selected are

_precisely

those which are inside

W {Q} .

The entire

_procedure

can be done

_directly

on the

_physical

_{space E. If} II is the

orthogonal projection

onto

_E,

we define :

The Z {Q} form a

_periodic

_{array of «}

_acceptance

domains »

(A.D.),

which is

generally

not a

tiling.

Indeed

_neighbouring

A.D. could

_.overlap

in some cases. To each

_point

of U one

attaches a _{copy of the lattice}

V {Q}

and we select those

_points

of

V {Q}

which fall inside the

A.D.

Z {Q} .

The

_{quasiperiodic}

structure X reads

_{formally :}

The

_advantage

is that it is easy to calculate a

_priori

which

_point

of

V lo l

is closest to the

center of the A.D.

Z {O}.

It is then sufficient to test

_only

a small subset T of

V lo l surrounding

this

_point.

This set T is the maximal subset of

V lo l

which can fit into the A.D. All these

_steps

will be illustrated below in 2D

_examples.

3.

_{Lattices, honeycombs}

and

_polytopes

in R4.

It is

_{always possible}

to characterise the local order of a

_{simple crystalline}

structure

_by

one, or a

few,

polyhedron

(a

polytope

in

_{4D) [3] :}

this could be the coordination

shell,

the Voronoi cell

or interstices in the structure. If we are interested in 2D structures

_mapped

from

R",

such local

_polytopes

are

_mapped

in the

_plane

E inside

_{polygons using}

the cut and

polygonal

line in R4 selected

_by

the

_strip,

the

_plane

E must remain close to the

_polygonal

_line,

so this line

_gives

information on the

_position

of the

_{projection plane}

in R4.

There is a

_polygonal

line which

_strongly

characterizes a

_{polytope :}

the Petrie

_polygon

_[3].

The Petrie

_polygon

of a

_polyhedron

is a skew

_polygon

_{such that every} two consecutive

_sides,

but not

_three,

_belong

to a face of the

_polyhedron.

This definition could be extended to

polytopes

of

_higher

dimension : a Petrie

_polygon

of an n-dimensional

_polytope

is a skew

polygon

such that any

(n - 1 )

consecutive

sides,

but no n,

_belong

to a Petrie

_polygon

of a cell.

The Petrie

_polygon

of a

_{regular polytope}

has a

_symmetry

_{group which is the}

_largest

_sub-group

of the

_symmetry

_group of the

_polytope

_[3].

In all cases two consecutive

_edges

define

_entirely

a

unique

Petrie

_polygon.

3.1 SEARCH FOR THE 4D STRUCTURE LEADING TO THE OCTAGONAL QUASICRYSTAL. - We

are

_looking

for a

_regular

_octagon

in E. So in R4 we search for an

_octagonal

skew

_polygon.

The Petrie

_polygon

of the 4D cube is a

_good

candidate

(see

_Fig.

2).

Fig.

2. -

Schegel diagram

of the

_hypercube

with the

_octagonal

Petrie

_polygon

_(with

black

_edges).

The cubic 4D lattice

Z4,

or

_{honeycomb {4,}

_3,

_3,

_{4 }}

in Schlâfli

notation,

is a structure which is well characterized

_by

the

_{4,

3,

3 }

hypercube

which

is,

in this case, both the interstice

configuration

and also the Voronoi cell.

Consider a Petrie

_polygon

of a

_{given hypercube.}

It has 8

_edges

whose mid

_points

are in a

single plane.

It is clear that if we use this

_plane

as the E

_plane,

with a suitable

_acceptance

region

in E’ it will be

_possible

to select vertices of the Petrie

_polygon,

and to

_reject

other

vertices of the

_{hypercube. By projection}

on E and E’ the Petrie

_{polygon gives}

a

_regular

octagon.

This

_justifies

the choice of the 4D cubic lattice in order to derive the

_octagonal

quasicrystal.

There are three

_{possibilities}

for

_choosing

a local

_polytope

characteristic of the local order in

Z4 : the cubic

_cell,

the Voronoi cell of a vertex, or the coordination shell of the same vertex.

We choose the Petrie

_polygon

of the Voronoi cell because it defines a

_plane

E

_containing

the central vertex. The Voronoi cell is a

_hypercube

with

_edges

_parallel

to those of the cubic cell.

By

projection

on

E,

the Petrie

_{polygon gives}

an

_octagon.

With this choice there is a vertex at

the centre of the

_octagon.

’3.2 THE DODECAGONAL SYMMETRY. - We search for a

_polygonal

line which has twelve

vertices. The Petrie

_polygon

of the

_{3,

_4,

_{3 }}

_polytope

has this

_property.

It is

_represented

in

figure

3

_using

the

_Schlegel

_projection

of the

_polytope.

The

_{3,

_4,

_{3 }}

_polytope

is build from 24

Fig.

3. -

Schegel diagram

of the

_{{3, 4, 3 }}

polytope

with the

decagonal

Petrie

_polygon

_(with

black

edges).

polyhedron

(the

vertex

_figure

defined

_by

_Coxeter)

is

_{a {3, 4, 3 } : the {3, 3, 4, 3 }}

honeycomb.

It is a

_packing

of

_regular

cross

_polytopes

_{3,

_3,

_{4 } .}

This structure could also be described as a lattice : the Leech

_{A4 lattice}

[4].

It is obtained

_by

_begining

with a

_triangular

2D

lattice

_then,

in a third

dimension,

triangular

lattices are stacked up

_leading

to an f.c.c.

_lattice,

then in a fourth dimension f.c.c. lattice are stacked up. There are two _ways to _{stack up f.c.c.}

lattices : sites of the « _upper one » could be over tetrahedral

_interstices,

or over octahedral

interstices ;

it is the latter

_{configuration}

which leads to the

_{3,

_{3, 4,}

_{3 }}

_honeycomb.

This lattice could be defined

_using

a cubic

_{crystallographic}

cell,

but there are two

_{possibilities.}

The first one is built from the cubic cell of the f.c.c. 3D lattice

_{by adding}

a fourth dimension to

obtain a

_hypercubic

_{cell ;}

then the lattice is a

_{non-primitive}

2-face centered lattice

_(type

F in

crystallographic

notation).

The second

_possible

cubic cell could be derived from the first one.

Consider the four

_diagonals

of the

_{hypercube : they}

form an

_orthogonal

basis.

_Using

this basis

another cubic cell is

defined ;

then the lattice is a

_body

centered lattice

(type I).

If the

_{E space}

is chosen such that it contains the

_mid-points

of the

_edges

of the Petrie

polygon

of a

_{3,

_4,

_{3 }}

coordination

_polytope

of the

_{3,

_{3, 4,}

_{3 }}

_honeycomb,

a

_dodecagonal

quasicrystal

results from the cut and

_projection

method.

3.3 THE PENTAGONAL AND DECAGONAL SYMMETRIES. - There is

a

_{regular polytope}

whose Petrie

_polygon

has 5 vertices :

_{the {3,}

_3,

_{3 }}

_{in 4D space}

_{(Fig. 4),}

which is called the

_simplex.

Unfortunately

there is no

_{regular honeycomb}

whose cells or vertex

_figure

are

_only

{3,

3,

_{3 }}

polytopes.

Nevertheless,

if we consider a

_crystalline

4D structure in which

_regular

simplexes

occur

_{periodically,}

it will be a

_good

candidate in order to

_{give pentagonal}

quasicrystals.

Consider once

_again

the

stacking

of f.c.c. structures in a fourth dimension. But now we

stack up a new f.c.c. structure with its sites over half of the tetrahedral interstices of the first

f.c.c. structure, and so on. A

_honeycomb

with two

_types

of cells is obtained. Cells of the first

type

are

_simplices

and those of the second

_type

are

_{semi-regular polytopes}

whose vertices are on the mid

_edges

of a

_simplex.

For the same reason which allows in 3D the two dense

_h.c.p.

and the f.c.c. structures, in 4D there are several

_{possibilities}

of

_{stacking leading}

also to

Fig.

4. -

Schegel

diagram

of

_{the {3,}

3,

3 }

simplex

with the

_pentagonal

Petrie

_polygon

_(with

black

edges).

structures

_by

the C.P. method. This

_honeycomb

is also a lattice in a similar way to the 3D

example

where the f.c.c. lattice is a

_honeycomb

formed

_by

a

_packing

of tetrahedra and

octahedra. Notice that this lattice is a _{reticular 4D space in the cubic lattice}

Z5

which is used in the standard

_projection

method for

_obtaining

the Penrose

_tiling.

There is another

_simple

related lattice defined

_by

a rhombohedral unit cell. It is built

_by

a

basis of four vectors

_joining

the centre of a 4D

_simplex

to 4 of its 5 vertices.

_Adding

these 4

vectors

_gives

the

_opposite

of the fifth vector

_joining

the last vertex. So this lattice has the

{3,

3,

3 }

symmetry.

The two lattices are related : the first one consists of a selection of vertices of the second one. The second lattice leads to a

_decagonal

_symmetry

when the E

plane

is chosen as the

_plane

of mid

_points

of the

_edges

of the Petrie

_polygon

of the

_simplex

defined

_by

the five vectors

_(a

5 vector

_star).

This is a _{consequence of the central}

_symmetry

of

the lattice. The first lattice could lead to a

_pentagonal

_symmetry

if the E

_plane

is defined

_by

the Petrie

_polygon

of a

_simplicial

cell.

4. Génération of the

_{octagonal quasicrystal.}

The

_{octagonal tiling}

has

_already

been the

_subject

_{of many studies}

_(see

for

_example

Refs.

_{[5, 6]}

and has been invoked to describe the structure of CrNiSi

_alloys

_[7].

Nevertheless it is a _very

useful

_example

to

_present

this new method of construction.

4.1 THE PLANE OF PROJECTION E. - In the last section

we have concluded that

_octagonal

quasicrystal

could result from the

_projection

of a cubic

_{honeycomb {4,}

_3,

_3,

_{4 } .}

In order to

define the

_plane

on which we could

_project

the structure we have to consider the Petrie

polygon

of the coordination

_shell,

a

_{3,

_3,

_{4 }}

cross

_polytope,

whose vertices are first

neighbours

from the

_origin.

It has 8 vertices which are

_gathered

4

_by

4 in two

_planes

Pi

and

_P2.

These two

_planes

have

_only

one

_point

in common at the

_{origin ; they}

are

_completely

orthogonal. They

are described in the

_appendix,

in which we determine an « intermediate »

plane

E between the two

_{planes Pl}

and

_P2.

This

_plane

E has

_only

the

_origin

in common with

Pl

and

_P2.

We also show in the

_appendix

how to find a matrix M which

_changes

coordinates in the

This matrix could be written

with

Applying

this matrix to the 8 vertices of the Petrie

_polygon

and

_{keeping only}

the first two

coordinates we obtain a

_regular

_octagon.

The two sets of 4 vertices in the two

_planes

Pi

and

_{P2 give}

an

_octagon

drawn

_by

two _{squares rotated from}

_{’TT /4.}

It is this matrix which is used to

_project

the structure : it

_gives

new

_coordinates,

whose first two are coordinates in the

plane

E of the 2D structure.

4.2 BUILDING OF THE QUASIPERIODIC TILING.

4.2.1 Reticular

_planes

in the 4D

_crystal.

- The

{4,

3, 3,

4 }

honeycomb

is a cubic lattice in R4. So we can

_{apply simple crystallographic principles}

to it. The

_{plane Pl}

which contains

several vertices of the

lattice,

contains a whole 2D lattice which is a sublattice of the 4D

lattice. The 4 vertices of the Petrie

_polygon

in

_Pi

form a _{square, the}

_{origin being}

the centre of this square. So the whole 2D lattice is

simply

a _{square lattice. Now consider} a

_family

of reticular

_{planes parallel}

to the

_{plane Pi.}

There are translations which

_change

the

_plane

Pl

into another

_plane

of the

_family.

The

_{plane P2}

contains also a _square lattice which defines

these translations. It is

_possible

to associate a translation of the

_{plane P2}

with each

_plane

of the reticular

_family.

Now we can use the formalism

_presented

in section 2. The

_{plane P2}

becomes the base B and the reticular

_{plane Pl}

is a fibre F. Then the lattice Z4 is : Z4 = F

E9

B.

4.2.2

_Acceptance

domain. - Vertices of the 4D

lattice,

which are

_mapped

on the

quasiperiodic

tiling,

are selected

_by

a

_strip

S. The

_points

in a fibre which are to be selected are

those enclosed in an

_acceptance

domain defined

_by

the intersection of the

_strip

with this fibre.

The A.D. in F is the

_oblique

_projection

of the

_Voronoi

cell of the

_origin

onto the F

_plane.

The Voronoi cell is a

_hypercube

of unit side. The

_projection

on E’ is a

_regular

_octagon

obtained from the common

_part

of two _squares

_relatively

rotated from

_{ir/4. (The}

_square

_edge

_length

is . 1 +

B/2/2).

The

oblique

projection

on F is also a

regular

octagon

but

expended by

a factor

à ;

so the

_{edge length}

_{of the squares used} to build it are 1 +

à.

4.2.3

_Mapping

on the E

_plane.

-

Every lane

of the reticular

_family

_projected

on E

_gives

a

square lattice with an

_edge

_length

2/2.

We call

V {Q}

the lattice

_projected

from

F { Q } ,

but the

_origin

of these

_planes

are translated

_by

vectors _{of another square lattice of the}

same

_{edge length}

which is rotated

_by

_{’TT /4.}

This lattice is the

_projection

of the base B. As

explained

in section 2 the A.D.

W { Q }

in the fibre

F { Q }

is

_mapped

onto

Z {Q} lying

on E.

The set of

Z {Q}

forms a

periodic

_{array of}

_overlapping

_octagons

as discussed below.

Before

_projection

on

_E,

the selection of

_points

consists in

_taking

on each lattice

F {Q} only

vertices

_lying

within an A. D. W

{Q} .

The centre of this A. D. is the

_point

A which is common to the

_planes

E and

F {Q} .

A

_{generic point}

of the

_{plane F {Q}}

is

_{given by :}

a , {3, f

and m are

_integers

then ON is a

_point

_{of the lattice in R4. If}

_only

f

and m are

_integers

then N is the current

_point

into the

_{plane F { Q } .}

Consider in E the two vectors _{qI and q2 with coordinate in R4 :}

which are used to characterize the matrix M

_{(see appendix).}

_Any

_point

C of E is

_{given by :}

with coordinates in the basis

_(fl,

_{f2, bl,}

_{b2 ) :}

Now we write the condition for a

_point

A to be

_{simultaneously}

in E and in

F {Q} :

which

_{gives x and y}

when f

and m are

_given

_(f

and m determine which

_plane

F {Q}

is

_{taken) :}

We want to _{compare the} vector OA with the translation OI of the

_origin

of the lattice

y0}

_(Flol

_mapped

on

_E).

The translation in B is

_expressed

in the new basis

(ql,

q2, q3,

q4 )

(see appendix) :

Then in E with the basis

_{(ql, q2 )}

it remains

which is

_just

the half of

OA,

then OA = 2 01.

4.2.4 The

_{algorithm for selecting}

vertices. - We consider in E

a _square lattice

_U,

which is the

projection

of the lattice in

B,

whose

_parameter

is a =

B/2/2.

points,

centres of the

_acceptance

domains. With each translation OP of U we associate a

translation 2 OA of r.

_Surrounding

all r vertices we

_reproduce

the A.D. and so we have the

periodic

array of A.D.

Z {Q} .

Its in-circle has a diameter

₍₁

+

..Ji/2)

or

(Nf2

+

_{1 )}

a. As this value in

_greater

than 2 _a, A.D.

_overlap

when

_they

are

_reproduced

at each r vertices. The

lattice U and the

_periodic

_array

Z {Q}

are shown in

figure

5. Then consider a _square lattice

V {Q}

with a

_{parameter a,}

but rotated

_by

_{ir /4}

with

_respect

to U. We

put

the

_origin

of this

lattice on a vertex P of U and select all its vertices

_falling

within the A.D. associated with

OA = 2 OP in r. These selected

points

are

_points

of the

_{quasiperiodic}

_tiling.

The whole

quasiperiodic tiling

is obtained

_by

_repeating

this

_operation

for all vertices in U.

_Figure

6 shows how some

_points

of this

_tiling

_{appear in the A.D. for} an

_octagonal

_{quasicrystal.}

Fig. 5.

Fig. 6.

Fig.

5. -

Periodic array of

octagonal

acceptance domains on the

_physical

space E. The U lattice is also drawn.

Fig.

6. -

Some

points

of the

octagonal tiling

in their acceptance domain and the

octagonal tiling.

In

_practice

it is not _necessary to test all the

_positions

of the

V lol

vertices with

_respect

to

the

_acceptance

domain

Z {Q} .

Indeed one can determine

_explicitly

which

_point

in

V lol

is closest to the A.D. centre

_{(see chapter 7)}

and then test

_only

a limited

_part

of

V {Q }

centered on that

_point.

In the

_present

case this set contains 9

_{points only.}

This

_greatly

reduces the number of

_{computer steps}

involved in the

_{tiling generation.}

More

_precisely

this number of

_steps

_grows as the second power of the

tiling

linear size

_compared

to the fourth power with the usual cut and

_projection

_algorithm.

5. Génération of the

_{dodecagonal quasicrystal..}

5.1 PROJECTION PLANE. - We have

already justified

the choice of the

_{3,

_{3, 4,}

_{3 }}

4D

possible

to have a cubic cell with several

_points

in each cell. We use the cell derived from an

f.c.c. cell and add one dimension.

The four basis vectors of the unit cell

_expressed

in the cubic basis are :

The coordination

_polytope

is a

_{3,

_4,

_3}

with 24 vertices. This

_polytope

has a Petrie

polygon

with 12 vertices. These 12 vertices can be

_gathered

6

_by

6 in two

_planes

F and B. These two

_planes

have one common

_point

at the

_origin

_(centre

of the coordination

_shell),

and the 6 vertices form a

_{regular hexagon.}

We consider an « intermediate »

_plane

E between the two

_planes

and then the Petrie

polygon

is

_projected

onto it. A

_{regular dodecagon}

is obtained.

In the

_appendix

we show

_briefly

how to determine the E

_plane,

and how to obtain the

matrix which

_changes

the coordinates in the cubic basis into coordinates in a new

_orthogonal

basis

_(qi,

_{q2, q3,}

_{q4 ).}

The first two vectors of this basis characterize the

_plane

_E,

the other two

define an

_{orthogonal plane}

E’.

This matrix can be written :

Applying

this matrix to the 12 vertices of the Petrie

_polygon

and

_{keeping only}

the first two

coordinates we obtain a

_dodecagon.

The two sets of 6 vertices in the two

_planes

F and B

_give

a

dodecagon

drawn

_by

two

_hexagons

rotated

_by

_{’TT /6.}

It is this matrix which is used to

_project

the structure : it

_gives

new

_coordinates,

and the first two are coordinates in the

_plane

E of the

2D structure.

5.2 BUILDING OF THE QUASIPERIODIC TILING.

5.2.1 Reticular

_planes

in the 4D

_crystal.

- The

{3,

3, 4,

3}

honeycomb

is a lattice in R4. The

plane

F,

which contains several vertices of the

_lattice,

contains all a 2D lattice which is a

sublattice of thé 4D lattice. The 6 vertices of the Petrie

_polygon

in F form a

_hexagon

the

_origin

being

the centre of this

_hexagon.

So the whole 2D lattice is

_simply

a

_hexagonal

lattice.

Consider now a

_family

of reticular

_{planes parallel}

to the

_plane

F. There are translations which

change

the

_plane

F into other

_planes

of the

_{family. They}

are in the

_plane

B,

and

_they

also form

a

_hexagonal

lattice.

All

_points

of the

_{quasiperiodic}

_tiling

are

_points

of the lattice F

_mapped

on

_E,

or of other

planes

of the reticular

_family

F

_{lo l ,}

but there are

_only

a small number of

_points

in each

_plane

F {Q}

which contribute to the

_{quasiperiodic}

_tiling.

5.2.2

_Mapping

on the E

plane.

-

Every

plane

of the reticular

_family

_projected

on E

_gives

a

hexagonal

lattice with an

_{edge length}

a = {(3 + J3)/6} 1/2.

_They

are lattices

V {Q}

projected

from

_{F {Q} ,}

but the

_origin

of these

_planes

are translated

_by

vectors of another

hexagonal

lattice of the same

_{edge length}

which is rotated

_by

’TT /6.

This lattice is the

On each lattice

F lol

_only

vertices inside an A.D. are

_kept.

The centre of this A.D. is the

point

A which is common to the

_planes

E and

F lol

_{(Fig. 7).}

Fig.

7. -

Periodic array of

dodecagonal

acceptance domains.

With the same

_procedure

used in the

_octagonal

case it is

_possible

to show that 01 = 2 OA.

The Voronoi cells of the

_{3,3,4,3}

_honeycomb

are

_{3,4,3}.

It results that the

acceptance

domain in F and then in E are limited

_by

a

_regular

_dodecagon.

This

_dodecagon

is

on a circle of radius d =

(1

₊

J3)

a/2

where a is the

_parameter

of the

mapped

lattice

V{Q}.

5.2.3 The

_{algorithm for selecting}

vertices. - We consider in E

hexagonal

lattice

_U,

with

parameter

a. We then consider another

_{hexagonal lattice}

(r)

with

_parameter

2 a and the

Fig.

8. - A

piece

of the

_{dodecagonal tiling.}

same orientation. This lattice is the lattice of all A

points,

centres of the

_acceptance

domains.

Surrounding

all r vertices we

_reproduce

the

_A.D.,

and construct the

Z lol

_{array. A.D.}

overlap

when

_they

are

_reproduced

at each r vertices

_{(Fig. 7).}

Then consider a

_hexagonal

lattice

V { Q }

with a

_parameter

a, but rotated

by

’TT’ /6

with

respect

to U. We

_put

the

_origin

of this lattice on a vertex P of U and select all its vertices

_falling

within the A.D. associated with OA = 2 OP in

Z {Q} .

These selected

points

are

_points

of the

_{quasiperiodic tiling.}

The whole

quasiperiodic

tiling

is obtained

_{by repeating}

this

_operation

for all vertices in U.

Figure

8 shows this

_tiling

with a choice of

_edges.

In the

appendix

we describe the different

types

of tiles encountered in this

_tiling.

Looking

at a

_tiling

obtained from the

_projection

of a cubic structure in 4 or

_5D,

it is usual to

see cubes in

_{perspective ;}

notice that in this case it is

possible

to see

_part

of the octahedra in

perspective,

with

_triangular

faces.

_They

are

_projected

from the surface selected in R4

by

the

strip.

In this surface there are also

_parts

of tetrahedra : in fact all four vertices of

3D-tetrahedra,

but the surface is formed

_by

two faces of each tetrahedron.

_Consequently

there is

an

_ambiguity

on this

_choice,

because it is also

_possible

to consider on the surface the other two

triangular

faces,

and then to have another choice for

_{drawing edges}

in the

_{quasiperiodic}

_tiling.

6. Génération of the

decagonal

quasicrystal.

We use the rhombohedral lattice in R4 defined

_by

the

simplicial

star. This lattice can be considered as the

_product

of two

2D-lattices,

the first one is defined

_by

two of its basis vectors,

the second

_by

the other two. These four vectors

_expressed

in an

_orthogonal

basis are

[5] :

The fifth vector in the

_simplicial

star is :

It is

_easily

shown that a

_pentagon

is obtained

_by

_projection

on the

_plane

defined

_by

the first

two coordinates

_(which

is the E

_plane)

of these five vectors

_(vl,

_{V2, V3, V4,}

_{v5 ).}

The

method to obtain a

_quasicrystal

is then the same as that

_presented

for the

_octagonal

and

the

_dodecagonal

cases. The two 2D-lattices F and B are

_mapped

on E where

_they

_give

two

different rhombic lattices.

The unit cell of the lattice V obtained

_{by projection}

of the lattice F

_(basis

_Vl,

_V2)

is a

rhombus with an

_angle

2

_{ir 15}

at the

_origin

of the basis. The unit cell of the lattice U obtained

by

projection

of the lattice B

_(basis

_v3

_v4)

is also a rhombus but with an

_angle

’TT /5

at the

_ongin.

The lattice of

_acceptance

domains is similar to the lattice V but

_expended

by

a factor

5.

The

_acceptance

domains is a

_non-regular

deca on

(Fig. 9)

which could be obtained from a

_{regular decagon}

inscribed in a circle of

_{radius 2 ( r}

+

_{2 )/5}

_by

an

_affinity

of a

Fig.

9.

_Fig.

10.

Fig.

9. -

Periodic array of

decagonal

acceptance domains and the first

_points

selected

_by

these A.D. This shows how the local 10-fold _{symmetry appears.}

Fig.

10. - A

_piece

of the

_decagonal

_{quasilattice.}

7.

Quasicrystais

derived from

periodic

lattices.

In section 2 we showed that the

_algorithm

_consists,

in a first

_step,

in the determination of

which

_point

in V { Q } is closest to the centre of the

_acceptance

domain. If we

_stop

at this

_step

and consider the

_{configuration}

of

_points

which are then

generated,

we find a _very

_intersecting

new

_{tiling, closely}

related to the final one. It is also

_{quasiperiodic}

but it is now _{very easy} to

calculate a closed formula for the vertex coordinates. Furthermore it has a _{very natural}

underlying

average

periodic

lattice

(the

centres of the

_A.D.).

In the

_following

we

_present

the case related to the

_octogonal

_{quasicrystal. Among}

the vertices of _{the square lattice}

V {Q} falling

into an

_acceptance

_domain,

we select

_only

one

_{point :}

the closest to the centre of the domain. It is clear that the lattice of

_acceptance

domain is an _average structure _{which may}

be called the

_labyrinth

or the

_octagonal

_pivot.

In

_figure

11 the

_labyrinth

is shown : there are 3 kinds of tiles : _{square, kite} and

_trapezoid.

This

_{quasiperiodic}

structure is a subset of the

_{octagonal quasicrystal,}

but the 8-fold

symmetry

has been broken into a 4-fold

_symmetry.

It _{has very}

_{interesting geometrical}

properties

which are

_presented

elsewhere

_[8].

We calculate

_explicit

coordinates for all its vertices.

Consider a

_point

1 of the lattice U

(the

_projection

of the

_base)

and the

_{corresponding point}

A of the lattice of

_acceptance

domain.

_They

have coordinates

_(L,

_{m )}

and

_{(2 l,}

2

_{m ).}

Let T be the rotation matrix

_by

_{?r /4,}

which transforms the lattice U into the lattice V

_(the

projection

of a

fibre)

The

_point

1

_expressed

in V is

T-1.

OI

_(we

write as if the vector were a column

matrix).

In

Fig.

11. - The

_{octagonal pivot}

with three _types of tiles : a _square, a kite and a

_trapezoid.

translation of V. The

_point

of the shifted V lattice closest to A is

_{Int { T-1.}

_{(OA -}

_s)}

where Int

_{(x )}

is the closest

_integer

to x. In the V basis the

_point

is :

This can be written

This allows a _very efficient

_generation

of the

_octagonal

_pivot

_quasicrystal

with a

_computer,

but

also

_greatly

increases the

_efficiency

of the

_algorithm

used to obtain the

_octagonal

_{quasicrystal.}

From the above

_expression

one

_easily

derives the average square lattice

_{by replacing}

the

« int »

_operator

by

the

_identity

one.

We have

_presented

the

_example

of the

_{octagonal pivot}

4-fold

_{quasicrystal,}

but the derivation of new

_{quasicrystals}

_(with

_average

_lattices)

derived from the

_dodecagonal

and

decagonal quasicrystals proceeds

in a _{similar way.}

8. Conclusion.

We

have,

by

this

_method,

an

_algorithm

which

_generates

_{quasicrystals directly}

in the

_physical

space. This

algorithm

is

_clearly

related to

_crystalline

methods

_using

lattices to describe

structures. The existence of an

_underlying

lattice

_(the

_{array of}

_acceptance

domains)

indicates

that one can consider

_{quasicrystals}

as

_crystals

with an

_evolving

motive in each unit cell. This motive has not a constant number of

_points

in the

_generic

_case, and so does not

_correspond

to

what is

_usually

called an _{average lattice. If} we restrict to one

_point

in each motive we

_get

new

quasiperiodic

tilings

of lower

_symmetry.

This

_algorithm

also

_presents

some

_advantages

for the

_computer

_generation

of a

_{quasicrystal.}

algorithms already

well known

_{using grid}

methods which are efficient. For instance the Amman

_{procedure using non-periodic grid,}

or

_procedures

_{using periodic}

_N-grid.

The

comparison

of the

_efficiency

between these different

_procedures

is still an _open

_question.

Note that the different methods are not

_{fully equivalent,}

even in their

_generalized

versions. Indeed the

_grid

method

_{only provides tilings}

with rhombuses

_{(or rhombohedra)}

while the

C.P. method leads to

_point

_{configurations}

_{which may} or _may not

_{provide simple}

_tilings.

There are several extensions or

_applications

of this new

_description.

Phason fields which are

related to a shift of the

_strip,

in the cut and

_projection

method,

are for instance in the

homogeneous

case related to a

_{simple displacement}

of the

_origin

of A.D.

_periodic

_array.

More

_generally

it

_might

be fruitful to discuss the relative role of

_phasons

and

_phonons

with

respect

to the

_splitting

of the lattice as base and fibre. The diffraction

_patterns

of these

tilings

are also obtained

_using

a similar

_{technique applied}

to the

_reciprocal

_{space. The main} difference is that in this case

_acceptance

domains have to be

_{replaced by}

their Fourier transform : in

_large,

small

_acceptance

domains defined in the

_reciprocal

_{space will select}

Bragg

reflection with

_high

intensities. This method of

_description

_{may also prove} to be

interesting

for the

_analysis

of excitation

_spectra

in

_{quasicrystals.}

Appendix.

A

_plane

in R4

_{passing through}

the

_origin

can be characterized

_by

its intersection with a

_sphere

centered at the

_origin.

This

_sphere

S3 is cut

_by

a

_{plane along}

a

_great

circle. The

_position

of this circle defines

entirely

the

_plane.

Then we

_{systematically}

use the fibration

_properties

of

_S3

_by

_great

circles

[9].

A _space can be considered as a fibre bundle if there is a

sub-space

(the fibre)

which can be

reproduced by

a

_displacement

so _{that any}

_point

_{of the space} is on a fibre and

_only

one. For

example

the Euclidean space

R3

can be considered as a fibre bundle of

_straight

lines,

all

perpendicular

to the same

_plane.

If fibres are 1-D

lines,

in a 3-D space, it is

_possible

to determine a

_point

on a fibre

_by

one

parameter,

then there remain two

_parameters

to characterise the fibre itself. So there is a 2-D space in which a

_point

characterizes a fibre. This space is called the base.

Successive toric

_layers

_appear

_naturally

if

_S3

is described

_{by using}

toroidal coordinates :

a torus is defined

_by

a

_{constant 4>}

_parameter.

Each of these tori could be considered as a _{2-D space}

_equivalent

to a

_rectangle

with

opposite

sides identified two

_by

two

_(or

a _square in the

_particular

case of the

_spherical

_torus).

All

_diagonals

of these

_rectangles

have the same

_length

and

_give

a

_great

circle of

83

after the identification of the sides which close this line. So it is

_possible

to draw on a torus a

whole

_family

of

_great

_circles,

_{which appear} as

_{parallel lines}

on the

_{equivalent rectangle.}

All these

_great

circles drawn on a whole

_family

of toric

_layers

defined

_by

their two common axis

form a fibre bundle. This is the

_Hopf

fibration of

_S3.

The base is a

_2-sphere,

but this

_sphere

is not embedded in

_S3.

If it were, it would have two

common

_points

with a

_fibre,

as a circle cuts a

_sphere

in two

_points.

This is not

_possible

since

only

one

_point

on the base characterises a fibre.

If a

_point

on the base is defined

_{by spherical}

coordinates

_{e o}

_{and 03A6o,}

then the toric coordinates of

_points

on the

_{corresponding}

fibre are : 0 = w ₊

₈₀

_{and çb,0/2.}

_So a circle

on the

base,

defined

_by

a

_{constant 00,}

_represents

a torus in

_S3.

vertices into several sets of similar fibres. The 16 vertices of the

_{4,

_3,

_{3 }}

cube are

_gathered

into 4 fibres

_{(fc; )}

_containing

4 vertices. All these fibres are on the same

_spherical

torus so

_they

could be drawn

_{schematically}

on a _{square surface whose sides have} to be identified. A fibre

_(f)

on the

_spherical

torus which is at

_equal

distance of two fibres

_{(fci fc2) containing}

cube vertices defines a

_plane

which is the

_plane

E that we consider for

_{projection :}

the 4 vertices on the two

close fibres are

_mapped

on this

_plane

as two _squares rotated

_by

_Tr/4.

It is

_possible

to find two vectors from the

_origin

to two

_points

on f which are

_orthogonal,

and then two other vectors

_{completely orthogonal}

to this

_plane.

This defines 4 vectors

ql, q2, q3 > q4°

For instance

leading

to the matrix M used for the

_octagonal

_{quasicrystal.}

The

_{3,

_4,

_{3 }}

_polytope.

In this case there are 4 fibres

_containing

6 vertices. The 4 fibres are

defined

_by

a

_regular

tetrahedron on the base. The

_plane

of

_projection

is defined

_by

a fibre

which is at

_equal

distance of two such

_hexagonal

fibres.

This leads to the matrix M used to build the

_dodecagonal

_{quasicrystal.}

Projection

of

cube.

_Figure

12 shows the

_projection

of a

_{4,

_3,

_{3 } hypercube}

of the Z4

lattice in

_R4

on the

_plane

defined above. This

hypercube

has a vertex at the

_origin.

After

mapping

the faces are of two

_{types :}

_square or rhombus which are the tiles of the

quasiperiodic

structure.

Fig.

12. -

Projection

of

the {3,

3,

4 }

polytope. - Figure

13 shows the

_projection

of a

_{3,

_3,

4 }

_polytope

which is the cell of the

_{{3, 3, 4, 3 }}

_honeycomb

on the above defined

_plane.

It has a vertex at

the

_origin.

After

_mapping,

faces of

_{the {3, 3, 4 }}

_polytope

are of three

types :

equilateral

triangles,

isocel

_triangles

with a small

base,

and isocel

_triangles

with a

_large

base. These are

the tiles of the

_dodecagonal

_non-periodic

structure

_presented

here.

Fig.

13. -

Projection

of

_{a {3, 3,4}}

_polytope

on a

_plane.

For

_clarity

of the

_figure

some bonds from the

external square vertices to other vertices are omitted.

References

[1]

DUNEAU M., KATZ A.,

Phys.

Rev. Lett. 54

₍₁₉₈₅₎

2688 ;

ELSER V., Acta

_{Cryst. A}

42

₍₁₉₈₅₎

36 ;

KALUGIN P. A., KITAEV A. Y. and LEVITOV L. _C., J.

_Phys.

_Lett.

46

₍₁₉₈₅₎

L601.

[2]

See in Du cristal à

_l’amorphe,

Ed. C. Godrèche

_(Editions

Françaises

de

_Physique)

1988,

Introduction à la

_{Quasi-Cristallographie by}

D. Gratias.

[3]

COXETER H. S. _M.,

_{Regular Polytopes}

Dover.

[4]

CONWAY J. H. and SLOANE N. J. A.,

_{Sphere packings,}

lattices and groups

(N.

Y.

_Springer)

1988.

[5]

BENKER F. P. M.,

Report

S2-WSK-04 _sept. 1982

Dept.

of Math.

_University

of

_Technology,

Eindoven.

[6]

SOCOLAR J.,

Phys.

Rev. B 39

₍₁₉₈₉₎

10519.

[7]

WANG N., CHEN F. H. and Kuo K. Y.,

Phys.

Rev. Lett. 59

₍₁₉₈₇₎

1010.

[8]

SIRE

C.,

MOSSERI R. and SADOC J. F., J.

_Phys.

France

_{(déc. 1989).}