Algebraic theory of non-periodic tilings of the plane by two simple building blocks : a square and a rhombus

simple building blocks : a square and a rhombus

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Beenker, F. P. M. (1982). Algebraic theory of non-periodic tilings of the plane by two simple building blocks : a square and a rhombus. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 82-WSK-04). Eindhoven University of Technology.

Document status and date: Published: 01/01/1982

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ONDERAFDELING DER WISKUNDE DEPARTMENT OF MATHEMATICS

EN INFORMATICA AND COMPUTING SCIENCE

Algebraic theory of non-periodic

tilings of the plane by two simple

building blocks: a square and a

rhombus by F.P.M. Beenker AMS Subjectclassification 05B45 T.H. - Report 82-WSK-04 September 1982

Contents

Chapter 1. Introduction and notation

Chapter 2. Tetragrids and GR-patterns

2.1. Skeletons of parallelogram tilings: a heuristic preparation

2.2. Definition of a tetragrid

2.3. Rhombus patterns associated with regular tetragrids

2.4. Types of vertices in a GR-pattern

2.5. A geometrical interpretation of GR-patterns on the basis of a cubic lattice

Chapter 3. The set V

3.1. The location of the vertices of a GR-pattern in the set V

3.2. The probability distribution of vertex-types in a GR-pattern

Chapter 4. A new parameter for the tetragrid 4.1. Shift-equivalence

4.2. Some transformations of the parameters and their effect on the tetragrids and the GR-patterns

4.3. Non-periodicity of GR-patterns

Chapter 5. On singularity and symmetry 5.1. Singular tetragrids

5.2. GR-patterns associated with singular tetra-grids

5.3. Symmetries of tetragrids

Chapter 6. Deflation and inflation 6.1. Introduction Page 1 7 7 8 10 13 15 17 17 21 25 25 27 31 32 32 33 38 41 41

6.2. The similitude ratio p 41

6.3. Algebraic definition of deflation and inflation 42 6.4. Geometrical description of deflation and

6.5. A part of the forcings of the vertices of

a GR-pattern 50

6.6. Relation of GR-patterns to sequences of zeros

and ones generated by special rewriting rules 53

Chapter 7. The question of the existence of local joining conditions

Chapter 8. PR-, GR- and AR-patterns

References

..

55

60

Chapter 1. Introduction and notation

Some years ago R. Penrose found a pair of plane shapes, called "kites" and "darts", which, when matched according to certain simple rules, could tile the entire plane, but only in a non-periodic way. The precise shapes are illustrated in figure 1.1.

Fig. 1.1. The kite and the dart.

The condition for joining the pieces together is simply that arrows have to match; adjacent pieces must have the same arrow in the same direction on the common edge.

B~ dissecting kites and darts into smaller .pieces and putting them together in other ways Penrose found an other pair of tiles with properties similar to those of kites and darts. This is the pair of rhombuses shown in figure 1.2, (cf. Penrose[ 7]). All edges have length equal to 1. N.G. de Bruijn called them the thick and the thin rhombus.

Fig. 1.2. The thick and the thin rhombus.

Again the joining condition is that arrows have to match.

One could just start with some kites and darts or thick and thin rhombuses around one vertex and then expand radially. Each time one adds a piece to an edge one has to select one of the two shapes. Some-times the choice is forced, someSome-times it is not. Later one might get in a position where no piece can be legally added, and be forced to go back and make the other choice somewhere. If the whole plane is tiled we call it an AR-pattern (AR stands for arrowed rhombus). A piece of an AR-pattern with thick and thin rhombuses is given in figure 1.3.

Fig. 1.3. A piece of an AR-pattern.

The above backtracking procedure is not easily seen to lead to a tiling of the whole plane. Penrose, however, found a systematic way to obtain such a tiling. He found two remarkable operations,

called "inflation" and "deflation". By an ingenious subdivision rule for the separate kite and dart or thick and thin rhombus a tiling is turned into a new one. By deflation the tiles have a smaller side-length, (

~

+

~

)-1 golden ratio) times the older one. Infla-tion is the inverse process.

Starting from a single piece, we can repeat deflation indefini-tely, and cover arbitrary large portions of the plane. By a standard process of selecting partwise constant subsequences and by

diagona-lization one can obtain coverings of the whole plane.

Penrose also considered a simpler construction that produces some (but not all) of these coverings of the plane. He started with a finite tiling with the property that its deflation, followed by a blowing up with a factor ~ +

TIs,

leads to a pattern that contains

a translational copy of the original one. So there is an operation, composed of deflation, blow up and shift, that actually extends the original tiling. Repeating this operation, we get further extensions and the sequence of repetitions leads to a covering of the whole plane. Let us call such a tiling a PR-pattern. CPR stands for Penrose rhombus). It is obvious from the construction that at most countably many

patterns are constructed this way. It is clear that a PR~pattern is also an AR-pattern. The converse need not to be true.

All the work of Penrose is purely geometrical. N.G. de Bruijn presented an algebraic theory of Penrose's non-periodic tilings of the plane (see de Bruij~ 2] ). As building blocks he used the thick and thin rhombuses. In his algebraic description he introduced rhom-bus patterns produced by so-called "pentagrids". These patterns are built up from thick and thin rhombuses. Let us call them GR-patterns

(GR stands for grid rhombus). It can be proved that every GR-pattern is an AR-pattern. De Bruijn proved that every AR-pattern is a GR-pattern (cf. de Bruijm 2, section 15]). In the same way one can prove that every PR-pattern is a GR-pattern. Hence, in short notation, de Bruijn found

(1. 1) PR C GR :: AR,

where the first inclusion is a strict inclusion.

One of the most remarkable things to be noticed in the descrip-tion of de Bruijn is the equality in (1.1). AR-patterns, defined by the simple arrow-conditions, (see figure 1.2), are produced by penta-grids. Another remarkable fact of the kite- an? dart patterns and thick and thin rhombus patterns is the golden ratio. Again and again the golden ratio appears. The proportion of the thick and thin rhombuses in a PR-pattern equals the golden ratiO, the "inflation-value" equals the golden ratio', the proportion of the areas of the thick and thin rhombus equals the golden ratio, etc.

In this report we present an algebraic theory of a tiling of the plane by the two rhombuses given in figure 1.4. All sides have length equal to 1. One rhombus is a square, henceforth called "the square". The other rhombus has angles 450 and 1350 • This rhombus is

Fig. 1.4. The square and the rhombus.

The ratio 1 +

V2

plays the same role as the golden ratio in the thick-and thin-rhombus patterns.

As the basis of our algebraic description we shall take the tetragrids, (this in accordance with the idea of the algebraic des-cription of the tilings with the thick and thin rhombuses gtven by de Bruijn). A tetragrid is a figure in the plane, obtained by super-position of 4 ordinary grids, obtained from each other by rotation over angles of multiples of 1T/4 (combined with certain shifts). Here

~

we used the term "ordinary grid" for the set of points whose distance

•

to a fixed line is an integral multiple of a fixed positive number.

This report is organized as follows.

We start with the tetragrids. A tetragrid is described by four reals Yo' 1 1, Y2 ' 13 (representing the shifts in four directions). A tetra-grid is called singular if there is a point in the plane where three or more grid-lines intersect, otherwise regular. A regular tetragrid determines a GR-pattern, which is a special kind of tiling with squares and rhombuses. Singular tetragrids can be obtained as limits of regular tetragrids, but depending on the way we approach the limit we get

different GR-patterns (sometimes 2, sometimes 8 different patterns). Nex4 the set V is introduced. For its definition we refer to chapter 3. The set V has the feature that the type of a vertex in a GR-pattern is made visible by means of a corresponding point in the

set V.

The four real parameters Yo' ..• , Y3 define a single complex parameter $. If two regular tetragids have the same parameter $ then the corresponding GR-patterns are obtained from each other by a shift. Even if we have two tetragrids with parameters $1 f iJJ

such that

W

l - ~2

=

o

(mod Z[

n]),

where

n

=

exp(~i/4) I then the

GR-patterns corresponding to

W

1 and ~2 are obtained from each other by shifts. Accordingly, shift-equivalence of regular GR-patterns can be described in terms of the parameter ~. Likewise, symmetries of GR-patterns can be described in terms of the parameter ~. In chapter 5 we give a complete survey of all GR-patterns with symmetry.

Furthermore, by observing the effect of some transformations of the parameter ~ on the tetragrids and the corresponding GR-patterns, we are able to prove that every GR-pattern is non-periodic.

In chapter 6 we determine operations called inflation and de-flation for the tetragrids and the corresponding GR-patterns. There are various ways to define inflation and deflation for the tetragrids. We have chosen for the one with the simplest geometrical effect on the GR-patterns. By using the set V we obtain an inflation and defla-tion which has a unique geometrical interpretadefla-tion for the separate square and rhombUS. For details we refer to chapter 6. This unique geometrical interpretation of inflation and deflation gives us the opportunity to define AR-patterns and PR-patterns b~ilt up by the square and the rhombus. We find the following inclusion

(1. 2) PR C GR c AR ,

where all the inclusions are strict. Unfortunately, it is impossible to give joining-conditions for the square and rhombus which would necessarily enforce non-periodic tilings of the plane. We refer to chapter 7.

As an illustation a piece of a GR-pattern is given in figure 1.5.

Notation. The letters C, R, Z have the usual meaning of complex plane, real line, set of integers, respectively.

The letter j always represen~ an element of the set {O,l,2,3}.

3 "For all J' n will mean II for 0 _{, ••• ,} 3" _i E stands for E _{j j=O'} We always put

( 1.3)

n

=

exp(~i/4),

p = 1 +

n

+

n

7

=

1 +

12.

Il" n

Chapter 2. Tetra~rids and GR-patterns

2.1. Skeletons of parallelogram tilings; a heuristic preparation

To understand the idea of tetragrids we repeat section 3 of de Bruijn[2]. This section has mainly the purpose of a heuristic preparation for the next section.

"If we have somehow tiled the plane by meanS of parallelograms such that every two adjacent parallelograms have a full edge in common, we can characterize that tiling completely by what we shall call a skeleton.

Consider an edge of any parallelogram in the tiling. Then the tiling contains a strip (infinite in both directions) of pairwise adjacent parallelograms each one of them having two edges equal and parallel to the edge we started from. Orienting that edge arbitrarily, we get a vector that plays the same role for all parallelograms of the strip. We connect the midpoints of the parallel edges, and thus we get a curve that stays inside the strip. We can do this for every edge in the pattern. The edge determines a strip, and to the strip we attach a curve and a vector.

Next we erase all parallelograms, just keeping th~ curves plus the vectors that belong to-them. Now we distort the plane with the curves topologically, without distorting the vectors. Let us call the resulting structures plus vectors a skeleton. The fun is that on the basis of the skeleton we can still build up the original parallelogram pattern (apart from a shift). Corresponding to the intersection of any two curves we draw (in a new plane) a parallelogram defined by the vectors belonging

to these curves. Having don~ this for all intersection points we note that the parallelograms nicely fit together, and form the original pattern" .

We note the duality between the skeleton and the parallelogram

pattern. An intersection pOint in the skeleton corresponds to a parallelo-gram, and a mesh in the skeleton plane corresponds to a vertex of a

parallelogram (we use the term mesh for the connected components of what is left when we remove the skeleton from the plane). The skeleton and the parallelogram pattern are each other~s topologically dual, (cf. figure 2.1).

Figure 2.1. A of a parallelogram pattern and its skeleton.

2.2. Definition of a tetragrid

Let~I"'1 Y3 be real numbers. In the complex plane we consider four grids. For j = 0,1,2,3 the j-th grid is the set

(2. 1) or, elaborated, (2.2) {Z E e l Re(zn-j) + y. E Z} , J Q-th grid: {z E C Re(z) + y 0 l-st grid: {z E C ~h

(

Re (z) E Z} f + Im(z» 2nd grid: {z E C Im(z) _{+ Y} 2 E Z} I 3rd grid: {z E C ~h(-Re(z) + Im(z»

+ Y₁ E Z}

,

+ Y

3 E Z}

.

The tetragrid determined by Yo' Y1f Y2' Y

3 is the union of (2.1) for

Fig. 2.2. A piece of a tetragrid.

Atetragrid is called regular if no point of C belongs to more than two of the four grids, otherwise it is called singular.

Given Yo"'" Y3' we associate with every point z E C four integers Ko(z) , .... , K

3(Z) where

(2.3)

(for notation see chapter 1).

Notice that the K. (z) are constant in each mesh! Hence, to each mesh

J

we can associate a vector (k

o' k1, k2, k3), i.e. the common value of (Ko(Z) " " 1 K

3(Z» for all z in the mesh.

Let rand s be integers with 0 ~ r

<

s ~ 3 and let k and k be integers. Then the point z determined by the equations

o (2.4) Re(zn -r ) + Y

=

k r r -s Re(zn ) + y

=

k s s r s

is the point of intersection.. of a line of the r-th grid and a line of the s-th grid. In a small neighborhood of Zo the vector (Ko(Z) , ... / K

3(Z» takes four different values, the four vectors we get from the formula

(2.5) _{(K (z ) , ... ,K}

3(Z » + E1(8 , ... ,83 ) + E2(8 / .• ,,83 ),

by taking (e

1, £2)

=

(0,0), (0,1) f (1,0), (1,1), respectively. Here 0ij is Kronecker's symbol: 1 if i

=

j, 0 if i ~ j.

2.3. Rhombus patterns associated with regulartetragrids

The skeleton of a parallelogram tiling built up from ~quares and rhombuses contains four different vectors. The orientation of these

vec-a 1 2

tors may be given by the argument of the complex numbers

n ,

n , nand 3

n • Hence a vertex of a rhombus pattern built up from squares and rhom-buses is described by

(2.6)

4

where (k_o"'" k3) e Z . Which (k

o"'" k3) do we have to take for a rhombus pattern which is associated with a regular tetragrid? In the previous section we have seen that every intersection point of two grid-lines corresponds with the four vectors given by (2.5). By using (2.6) we assign to these vectors four points in the complex plane. Note that these four pOints (with £1 and £2 taken from the set

{O,l})

form the vertices of a square or a rhombus.

Assuming the tetragrid (given by Yo"'" Y3) to be regular, we can attach a square or a rhombus to every intersection point of the tetragrid. We will show in a moment that they form a tiling of the plane by squares and rhombuses. First we show what the correspondence between an inter-section point of two grid-lines and the square and the rhombus actually is. By using (2.5) and (2.6) we find the following correspondences given in figure 2.3.

I

T

D

Fig. 2.3. From an intersection point of two grid-lines to a square or a rhombus.

Now it is easily seen what vectors we have to attach to the four grids (2.1). This is shown in figure 2.4. There the vectors of the tetragrid are given by arrows. The intersection pOints of the tetragrid are numbered. The corresponding square or rhombus is assigned with the same number.

6

1

•

Fig. 2.4. A piece of a tetragrid and the corresponding piece of the pattern.

In order to prove that the squares and rhombuses constructed from the intersection points of a regular tetragrid (given by Yo"'" Y3) form a tiling of the plane, we first examine the function f(z) given by

(2.7) _{z e C.}

Since the K.'s are constant over each mesh, this fez) is constant over

J

each mesh. If 1-1 is a mesh, we denote "par abus de langage", the constant value of f over 1-1 by f(lJ). Note that the set of all points fez) is the set of the vertices of the squares and rhombuses.

Some properties of the function fare:

1) f(z) is constant in every mesh of the tetragrid. 2) Let1J1 and 1-12 be meshes of the tetragrid, then we have

(2.8) _{f("l) ,..}

₌

_{f("2) I-' -} 11 1-'1 - "'2' - 11

Since K

J. (zl) E Z and K. (Z2) _{2 3 ]} E Z for all j I from the Q-l.inear

indepen-dence of 1"

n, n

I

n

we conclude that

This implies that]Jl

=

]J2'

3) fez) - 2z is bounded.

Proof. for all j we define

(2.9) A. (z)

=

K. (z) - Re(zn-j ) - y] ..

J J

Then we have 0 ~ Aj(Z)

<

1 and

fez) :::

r:.

Re(zn-j)nj ₊ Lj(A j + yj)nj

=

] 2z L. (A. j

=

+ + yj)n • J J

o

From this we deduce that fez) - 2z is bounded.

o

Theorem 2.1.The squares and rhombuses constructed from the intersection pOints of a regular tetragrid (given by YO"'" Y3) by means of (2.5) form a tiling of the plane.

Proof. We orient the boundary of the square and rhombus in the usual counterclockwise fashion, (cf. figure 2.5).

L7

Fig. 2.5. The orientation of the boundary of the square and rhombus.

Definition 2.1. Let K be an oriented 'closed curve in the complex plane and let P be a pOint inside K. The winding-number of K around P is the number of times that the closed path K winds around P.

Let P be a point in the rhombus plane which does not lie on a side. From definition 2.1 and the positive orientation of the boundary of the square and rhombus it follows that each boundary of a square or rhombus has winding-number 0 or 1 around P. We hav~ to prove that there is

exactly one square or rhombus whose boundary has winding-number 1 around P. Let M satisfy

V

z€C

I

f (z) - 2z

I <

M ] t

(cf. property 3 of the function f).

We consider a large square S in the tetragrid plane such that the image of the boundary of S under the mapping g(z)

=

2z is a closed curve C around P and such that the distance of P to C is at least 2M. It is easily seen that C has winding number 1 around P.

Let E be the set of intersection points of the tetragrid inside S. By a small variation of the boundary of S we find a closed curve in the rhombus plane which consists of sides of squares and rhombuses correspon-ding to the points of E. This curve is positively oriented and because

I

f(z) - 2z

I

<

M we may conclude that it winds exactly once around P. To each point of E corresponds a positively oriented square or rhombus which winds exactly once around P or not al all. Together they wind exactly once around P. Hence, there is exactly one element s € E such that P lies in

the interior of the square or rhombus corresponding to s. This completes

the proof; the complex plane is exactly covered once.

o

Notation. A rhombus pattern associated with a tetragrid will be called a GR-pattern, (GR stands for grid rhombus) .

2.4. Types of vertices in a GR-pattern.

In this section we give all the possible types of vertices which occur in a pattern. It is easy to check that the vertices in a GR-pattern can be of 6 different types (apart from rotations) according to

the 6 different types of meshes in a tetragrid. These six different types of meshes and corresponding vertices are given in table 2.1. At the left side the meshes, at the right side the corresponding vertices. The 6 types of vertices are called vertex of type 1, vertex of type 2,

• • • f vertex of type 6.

Table 2.1. The 6 different types of vertices.

, vertex of type 1

, vertex of type 2

I vertex of type 3

, vertex of type 5

, vertex of type 6

It may be observed that each one of these types has an inflectional symmetry. It is easy to understand that the shape of the tetragrid forces a configuration of squares and rhombuses around a given vertex. These forcings are given in chapter 6.

2.5. A geometrical inter~retation of GR-patterns on the basis of a cubic lattice

There is an other way to look at the GR-patterns. We intersect the regular four-dimensional cubic lattice by certain two-dimensional planes and we look at the cubes which have points in common with the plane. Projecting the centres of those cubes onto that plane we get the vertices of the GR-pattern.

Let Yo"'" Y3 be reals. We assume that the tetragrid defined by these yls is regular. We consider the four-dimensional cubic lattice. Each cube can be indexed by four integers k_o"'" k

3, such that the interior

of the cube is the set of all points (x , ••. , x

3) with k - 1

<

x

<

k , .•• ,

o 0 0 0

k₃- 1

<

x3

<

k

3· Let us call that interior "the open unit cube of the vector k". Now consider the two dimensional plane given by the equations

(X j - y.) (-1) j Re (n j )

₌

0, J (2.10) E. (x. - y.) <-l)jrm(nj ) == O. ] ] ]

Theorem 2.2. The vertices of a GR-pattern produced by a regular tetra-grid (with parameters Yo"'" Y3) are the points

4 where (k

o"'" k3) runs through those elements of Z whose open unit cube has a non-empty intersection with the plane given by (~.10).

Proof. From (2.10) we have that the vector (x

o - Yo"'" x3 - Y3) is perpendicular to the vectors (1, -~/2, 0, ~/2) and (0, -~/21 1, -~/2). Consequently (x_o - Yo"'" x3 - Y3) is a linear combination of the vectors (/2, 1, 0, -1) and (0, 1,

12,

1), i.e.

for certain reals a,

a.

If we define the complex number z by

then we find for all j

If (xo "'" x3) lies in the cube of ko "'" k3 we obtain that

- r

- j

1

k. - Re(zn ) + y. • J J 2 3 According to section 2.3, k o+ kln + k2n + k3 n is a vertex of the GR-pattern produced by the regular tetragrid with parameters Yo"'" Y3'

The same argument works the other way around. Note that regularity of the tetragrid guarantees that if k.

=

rRe(zn-j ) +

y.l

I then we have

J J

k.

=

Re(zn ) + y. for at most two j '5, so we can manage to vary z a

] J

Chapter 3. The set V

3.1. The location of the vertices of a GR-pattern in the set V

In this. section we introduce the set V. This set has the feature that the type of a vertex in a GR-pattern is made visible by means of a corresponding point in the set V.

Let Yo"'" Y3 be reals. We assume that the tetragrid defined by these y's is regular. We have seen that every vertex of the GR-pattern corresponding with this tetragrid can be written as k + ••• + k3 n • 3 Now

o

we start the other way round. Let kO/'" k3 be integers. We ask ourselves the question whether there is a mesh in the tetragrid where KoCZ)

=

k~,

••. , K₃Cz)

=

k3 (see (2.3». In other words, we wish to know when the follOWing assertion is true:

(3.1)

To answer this question we define the set V. The set V is a subset of C given by

(3.2)

Theorem 3.1. Condition (3.1) for k O / ' " k3 is equivalent with

(3.3) L

J. (-1)j(k. - Y,)l1

j

€ V.

J J

Proof. Assume that condition (3.1) is satisfied. By setting

A.

=

k. - Re(zn-j ) - Y, in (3.1) we find

J J J

(i) 0

<

Ao

<

1, ••• , 0

<

A3

<

1 and

(ii) E, (-1) j (k

-

_{y.) n}j = E, (-1) (A.-j . Re(zl1-j »nj

J j J J J

=

E. (-1)j

A.n

j . J J = Hence E J. (-l)j(kj - y.)n j €

v.

J , .

Conversely, if E. (_1)J (k. - y.)nJ c V then there are reals y ,.,.,

J J J 0

By arguments similar to those used in the proof of theorem 2.2 we deduce

We can draw the set V in the complex plane. V is the i~terior of the octagon with vertices

n

2, 1 +

n

2, 1 +

n

2 -

n

3, 1 -

n

3, 1 -

n - n

3,

3 2

-n - n , -n

and

-n

+ n. In figure 3.1 we have depicted V.

:2

-n

+

n

-n

3

-n - n

Fig. 3.1. The set V.

•

Using theorem 3.1 we easily see whether a point k o + ••• + k 3

n

3 . ~s a vertex of a GR-pattern or not. However, we can deduce more. By using this set V, we can also study the following question:

o

"If ko + ••• + k3n3 is a vertex of a GR-pattern, which of the neigh-bors of this point still satisfy (3.3)1"

The term neighbor is used for the points we get by addition of ~l, +n,

2 3

+n I ~n , or, what is the same thing I increasing or decreasing one

of the k. by 1. In this manner we can find and depict the

J (in the

sense of section 2.4) of the vertex in the set V. The result, as given in figure 3.2, is as follows:

If E_j (-l)j(k

j yj)n

j

lies in one of the regions marked by a 3

number j in figure 3.2, then the point k

o+"'+ k3n of the GR-pattern is a vertex of type j. For every vertex (except for a vertex of type 1) there are eight different orientations. Hence, for every type of vertex there are eight corresponding regions in V, indexed by a, •.. ,h. In table 3.1 a specification of the vertex corresponding to a number in figure 3.2 is given. 4h ₄b sg 6g ₆c 4f ₄d Sc 6f _{Se Sd} 6e

Fig. 3.2. The set V divided with octagonal symmetry into 41 subsets.

Table 3:' L A specification of the vertices in a GR-pattern corresponding to the numbers in figure 3.2.

*2

a

-*2

b

-¥2

0

*-

2d

*'

2

e .

~f

*2

g

¥

2h

~/

* 3

b

~

3°

.

~

3

d

*3

e _

* - 3

f

~3g

t-

3

h

-¥-

4 a

~

4 b

-k

4° - ) ( 4 d

-*

4 e

~4f

.

~4g.

>f-

4h .

-y

Sa

---r(

Sb

-l(

5°

~

Sd

~

Se

)l-:f

:r

Sg

r

Sh

y

6

a

-{6

b

.'.

-<

60

---\ 6

d

A

6

e .

}-6

f

>-6

g ·

j-

6

h

3.2. The probability distribution of the vertex types in a GR-pattern

By observing the set V one gets the strong suspicion that the proba-bility of a vertex type in a GR-pattern (i.e. the fraction of all vertices which belongs to this type) is directly proportional to the area of its corresponding region in the set V. This turns out to be the case. We will prove this on the basis of uniform distribution of sequenc~s. For a

detailed study on uniform distribution of sequences we refer to Kuipers and Niederrei tert 5] •

First we give some definitions.

Let a

=

(a

1, a2) and ~

=

(b1, b2) be two vectors with real components; that is, let a, b € R2. We say that a < b (a

~

b) if a.< b. (a.

~

b.)

- - - - J J J J

for j

=

1,2. The set of points x € R2 such that a

~

x < b will be denoted

by [a,£). The 2-dimensional unit cube 12 is the interval

[~,1),

where

a

=

(0, 0) and 1

=

(1, 1). The fractional part of ~ is {~}

=

({Xl}' {x₂}).

Let (x )1 n

=

1,2, ..• , be a sequence of vectors in R2 For a subset

-n

.E of I2, let A(EiN) denote the number of points {x }, 1

~

n

~

N, that lie

-n

in E.

Definition 3.1. The sequence (x ), n = 1,2, ..• , is said to be uniformly -n distributed mod 1 in R2 if A([~, b) iN) (3« 4) lim = (b 1- al ) (b2- a2) N-l><» N

for all intervals [ ~, ~)

_-

c r2 with a ~ b. Definition 3.2. Let (z ), n

=

1,2, ... , be

n .

Let Re.(.z.}. ... lcg,nd 1m (z )

=

y . Then the

n n n n

a sequence of complex numbers. sequence (z ) is said to be

n

uniformly distributed mod 1 in C if the sequence «x ,

y» ,

n

=

1,2, ..• ,

n n

is uniformly distributed mod 1 in

~2.

Theorem 3.2. The sequence given by

(3.5)

Proof. According to Kuipers and Nlederrei ter[ 5, pg. 18, 48J the

sequence (3.5) is uniformly distributed mod 1 in C if and only if for

2

every lattice point (n

1, n2) 1E:.z , (n1, n2)

=I

(0, 0) (3.6) where (3. 7) I k =-K 1 1

U' his criterion is called the Weyl-cri terion) •

Rewriting the general term in the series (3.6) we find

(3.8)

=

0,

From this we conclude that the seriein (3.6) is a product of two

geometrical series with common ratio unequal to 1 but with modulus equal

...

to 1. Hence, all the partial sums are uniformly bounded. Thus, we have

•

proved that (3.6) holds.

o

Remark. Similar to the proof of theorem 3.2 one can prove that ~[n] with the usual enumeration is uniformly distributed mod 1 in C. A related state-ment is that Z[ n] is dense in C.

Let S be a circle with radius R

>

0 and with the origin as its centre. Let V. be one of the 41 subregions of the set V. We search for the number

J

of (k

o"'" k3) IE:

~4

which satisfy condition

~(R),

given by

2 3 i i k _{+ k2n} - k n - k n IE: W. ( =V. +E.(-I)y.n), 0 I 3 _{J J} ~ ~ (3.9) tf? (R) : Ik 2 3

'"

+ k2n + kIn + k3n

I

2R. 0

We compare this amount with the number of (k

l , k3)E ~2 which satisfy condition ~(R) as given by

(3.10) If' (R)

Because of theorem 3.2. it is clearly seen that the number of

(k1f k3) € Z2 which satisfy If'(R) is (asymptotically) directly

proportio-nal to the area of V. as R + 00. We will show that this number is

asympto-J

tically equivalent to the number of (k

o"'" k3) which satisfy ~(R) as

R + 00; the number we are searching for.

Theorem 3.3. Let Yo"'" Y

3 be reals and let _{i i} V. _J be one of the 41 subregions ₄

of the set V. Let W. == V. + 4. (-1) Y.n. The number of (k , ••• , k3) € Z

J J ~ ~ a

which satisfy ~(R) is asymptotically equivalent to the number of (k

1, k3)

€

~2

which satisfy If'(R), as R + 00. In short notation

(3.11) _{.:ff(k k' )<P(R) '" #(k k )'¥(R).}

0 " ' " 3 l' 3

Proof. I f _{(k 1f} k3) satisfies If'(R) then, because the diameter of Wj is less then 1, there are uniquely determined k

o' k2 € ~ such that (3.12) From (3.12) we deduce W . j (3.13)

I

_{k a + k2 n} 2 _{+ k 1 n + k 3 n} 3

I

_<"

_{2R + m,} where (3.14)

m

:== max

Ixl.

X€W j

Hence (kat ••• , k3) satisfies ~(R + ~m). Conversely, if (k

o"'" k3) satisfies ~CR

-

~m) then it is easily seen that (k

1, k3) satisfies qr(R).

Since m is a fixed positive number, the number of vertices of type j between the ci~les with radius R + ~m, R - ~m, respectively, is

OCR} ,

(R + 00). This is small in comparison with the total number of vertices of type j, which is 0(R2), (R + 00). In other words, in short notation, we

~(R - ~m) c ~(R) c ~(R + ~m), and (3.15)

From (3.15) we deduce (3.11), which completes the proof.

From theorem 3.3 we may conclude that the probability of a vertex type in a GR-pattern equals the area of its corresponding region in V divided by the total area of V. An easy calculation yields that the area

o

of V equals 2 + 2/2 = 2p, where p

=

1 + 12. In table 3.2 the probability of each type of vertex in a GR-pattern is given. In this table no distinc-tion is made between a vertex type and its seven rotadistinc-tions. Note that

the probability for type 2a is one eight of the probability for type 2, etc.

Table 3.2. The probability of the vertex types in a GR-pattern.

vertex of type corresponding area probabili ty

1 2p -3 = -14 + 1012 _P -4

₌

17

-

1212 Z 0.029 2 2p -4 = 34 - 2412 p -5 =-41 + 2912 ~ 0.012 3 4p -3 = -28 + 2012 2p -4

=

34 - 2412 z 0.059 4 4p -2 = 12

-

812 2p -3 =-14 + 1012 ..., 0.142 5 4p -1 = -4 + 412 2p -2 = 6 - 412

-

~ 0.343 6 2 _P -1 = -1 + 12 ~ 0.414

From table 3.2 we easily derive the fraction of squares and rhombuses in a GR-pattern. We know the probability of each vertex type and we know the number of squares and rhombuses belonging to each vertex type. From

-1

this we easily derive that the fraction of squares equals p and the fraction of rhombuses equals 1 - P -1

If the area of the square equals 1 then the area of the rhombus equals

~/2. From the equality

(3.16) p-1 = (/2 - 1) -1

=

(/2 - 1)· (area square)

=

(1 - (/2 - 1)1~/2 = (1 - P -1 ). (area rhombus)

we deduce that if we spoot with an arrow at a GR-pattern then hitting a square has the same probability as hitting a rhombus.

Chapter 4. A new parameter for the tetragrid

From the four real independent variables Yo"'" Y3 we pass to a single complex variabLe given by

(4.1)

By rewriting E. (-l)jk.nj

(2.10) in the form E. (-l)jx.nj

=

W

or (3.3) in the form

J J

J J

- W

E V, we see that the GR-pattern associated with y , •.• , o

Y3 in the regular case depends on W only. As we will see, some properties like shift-equivalence, symmetry and singularity of tetragrids and cor-responding GR-patterns can be expressed in terms of the complex parameter

4.1. Shift-equivalence

In this section we examine shift-equivalence of tetragrids and their corresponding GR-patterns.

Definition 4.1. Two tetragrids are said to be shift-equivalent if they can be obtained from each other by a parallel shift.

From this definition we infer that the tetragrids determined by

*

*

Yo'·'" Y3 and Yo"'" Y3' respectively, are shift-equivalent if and only if there exists Z E C with

o

(4.2)

•

f ,1.*

*

*

We orm W from Yo"'" Y3 by (4.1) and similarly 0/ by Yo"'" Y3' Now

shift-equivalence can be seen to depend on wand w* only,

Theorem 4,1. The two tetragrids are shift-equivalent if and only if

*

\jJ-WEr:[n].

Proof. Assume the two tetragrids to be shift-equivalent. Then, according to (4.2) there is _{• 0} Z E C with

Re(z n

If we put' m,

=

Re(z

~-j)

+ y, -

y~

J 0 J J then

*

l/J - l/J m, E: 2'.:. J Hence

*

Conversely, if l/J - l/J E ~[n] then we may write

l/J - l/J

From (4.1) we infer

*

m ,E 2'.:,

J j =: 0, ... ,3.

By arguments similar to those used in the proof of theorem 2.2 we deduce

which leads to (4.2).

- m,

J

o

In this theorem~we,have obtained a'result concerning the tetragrids

*

in the case ~ - l/J € 2'.:[

nJ.

Next we are going to investigate the

corres-ponding GR-patterns. Here we use a result obtained in the previous chapter:

Z[ n] is dense in Ci see section 3.2.

*

Theorem 4.2. We consider two regular tetragrids determined by l/J and l/J.

Now,

*

(i) the two tetragrids produce the same GR-pattern if and only if l/J=l/J I

*

(ii) their are shift-equivalent if and only if l/J-l/J ~ ~l n] •

Proof.

*

(ii)

tetragrids produce the same GR-pattern.

*

Conversely, if ~ and ~ produce the same GR-pattern, we have

*

*

~

=

~

.

For i f ~ ~ ~ we would get a contradiction by theorem 3.1, taking k. € Z such that

J

This is possible since z[ n] is dense in ~.

*

If ~ - ~ E z[n] then we have by the second part of the proof of theorem 4.1

=

m. + Re(z

n-

j)

J 0

for some m. € Z, Z €~. In the next section we prove that a shift

J 0

by z in the z-plane has no influence on the GR-pattern (see (Tl)). o

Furthermore, we prove that the m.'s shift the GR-pattern by an

ele-J

*

ment of Z( n] , {see (T2». We conclude that if ~ - ~ Ii: z( n] then the GR-patterns are shift-equivalent.

Finally we assume the GR-patterns to be shift-equivalent. It

*

remains to prove that ~ - ~ € ~n] . According to (2.6) every vertex

of a GR-pattern is an element of ~[n]. Hence, the shift-vector (i.e. the vector that has to be added to the points of the ~ - pattern in

*

j

order to get the ~ - pattern) has necessarily the form

E.n.n ,

**

*

**

J J

nj E Z. If we take Y

j

=

Yj - nj then the ~ - pattern coincides with

the ~- pattern ( a proof of this statement is given in the next section).

**

**

*

By (i) we have ~

=

~ ,Since y. = y - n we conclude that

J j j

*

~

-

~ € z[ n] •

4.2. Some transformations of the parameters and their effect on the tetragrids and the GR-patterns

o

In this section we examine the effect of some transformations of the parameter vector (y " .• , _o Y3)on~ and on the point sets G and R. Here G stands _. for the tetragrid, considered as a point set in the complex plane, and

where K. (z) is given by (2.3). In the case that the tetragrid is regular,

J

R is the set of rhombus vertices of the corresponding GR-pattern.

In the following we use some obvious notations for transformed sets in the complex plane: G - z stands for {z - z

I

z € G} ,

o 0

G

= {;

I

Z E G}, -G

=

{-z

I

Z E G} 1 etc.

We start with the parameter vector (Yo"'" Y3)' From this vector we

*

pass to a vector Y in several different ways:

*

(Tl ) Let z € C. We define Y by 0

*

+ Re(z n -j) Y_j

=

Y_j j

=

0, ... ,3. 0

*

*

*

Now _1/1

₌

_1/11 G = G - z 0 ' R

=

R. Proof.

*

(-l)\~nj

(-l}jy.nj (-l)jRe(z n )nj:::: (i) 1jJ :: k. _{== k. + k.} J _{J J J J} 0 == k. (-l)jY.nj _"" 1jJ. J _J If z

*

*

+ Re(zn-j ) {O, •.• ,3}

(ii) E G then y. E t; _{for some} j E

*

. J z ) n -j)

have Y_j + Re(zn-J ) E!;**y. ₊ Re «z + E Z .. Z +

J 0

*

We conclude that G :::: _{G - z}

.

0

*

*

*

* .

(iii}If u E R then u = E j Kj(Z)n

J _{for some z E ~G. But}

*

.

E. K. (z) nJ = ) )

*

hence R

=

R.

*

(T2) Let no' n_l, n₂, n3 be integers. If we define y by

*

Y_j = Y_j + n. j :::: _{0, •.• ,3,} )

*

(-l)j n. nj

*

*

then 1jJ = _1/1 +

Z.

G :::: G, R :::: R + ) _J

.

We Z E G. 0

o

Proof. The first two assertions are trivial.

*

*

*

* '

If u E: R then u :=

r:

K. (z)n J for some z E

C-

G, but j J

*

j We conclude that R

=

R + E n.n .

*

(T3) If we pass from y to y

*

y =-y j 4-j j J by

*

* -

*

2 3 then ~

=

~, G

=

G, R

=

R + (n + n + n ). Proof. (i)

o

(ii) If z € G* then

y~

+ Re(zn-j) E Z for some j E {0, •.. ,3} . But an

easy calculationJYieldS

y~

+ Re(zn-j) E Z

~

Y4 . + Re(i n-(4-j» E Z.

*

_

J -J

We conclude that G

=

G.

(iii) If u*€ R* then u*

=

E.

K~(z)nj

for some Z E c\G. Since·z E c\G

. J J

we have Re{zn-J ) + y. ~ Z for all j. Using the equality

J

r-al == -ral + 1 if a ~ Z we find

*

j

r

-j

*1

j

r

~ 3

r

-

- j ~

E. K.(z)n

=

E. Re(zn ) + y. n == y + Re(z~ - E. 1 -Y.-Re(zn ~

J J J J 0 J= J

=

r. K. (i)

il

j + (n + n 2 + n 3) • J J

*

2 3

From this we infer that R == R + (n + n + n ).

*

(T4) If we define the vector y by

*

0, .•. ,3, Y_j == -y. j ₌₌ J * *

_*

then _~

₌

-Ij) G == -G, R = -R + ( 1 + n +

o

2 3

n

+

n ).

Proof. The first two assertions are trivial.

*

*

*

*

j

If u E R then for some Z E ~G we have u

=

E_j Kj(Z)~ . By

arguments similar to those used in (T3) we find

*

.

_!jrRe(z~

-j)

- y.ln

j E .

{-f

Re (-Z11 -j) +

y.1

+ 1}nj

E

_j K. (z)nJ ::::

=

=

J J J J -E. K.(-z)nj (1 2 3

=

+ + n + n + n ). J J

*

2 3 0 From this we conclude that R :::: _{-R + (1} +

n

+ ₁₁ + n ).

(T5) If we take

*

*

*

*

Yo :::: Y1' Y1 :::: Y₂, Y2 :::: Y3, Y3

=

-y 0

*

-1

*

-1

*

-1 3

then 1jJ :::: -n 1jJ, G ::::

n

G, R

₌

n R + n

.

(hence, this cyclic transform is connected with rotation) .

Proof.

(i) -11 -1 1jJ.

(ii) If z E G

*

then-yo

*

+ Re(zn ) E ~ for some j E

{Of .•. ,3}.

An

J

easy calculation yields

*

( . +1)

Y_j + Re (zn ) E _{Z ~ y. 1} + Re«zn)n- J ) E Z,

J+

where _{Y :::: -Yo'} From this we infer that G

₌

11 -1 G. 4

*

*

~G we have

(iii) If u E R then for some Z E

*

*

.

= E.

r

Re (zn

y~lnj

u

=

E.

K.(z)nJ ) ₊ : : J J J J E. rRe( (zn)

n-

j ) ₊

y.1T'l

-1 ~j -1

=

- n

J J

*

-1 3

4.3. Non-periodicity of GR-patterns

As we have mentioned before in chapter 1 and suggested in the title of this report, none of the GR-patterns are periodic. By non-periodicity of a GR-pattern we mean that there is no shift which leaves the pattern invariant. (This is a "stronger" definition of non-periodicity than the one used by Penrose and Gardner. By non-periodicity they mean that the pattern does not possess a period-parallelogram).

Theorem 4.3. A GR-pattern is non-periodic

Proof. We restrict ourselves to the regular case; the proof for the singular case is similar.

Assume a GR-pattern to be periodic. According to (2.6) every vertex

j

of this pattern has necessarily the form~. n.n with n.

J. ) . J

mation (T2) this means that we have ~ (-l)Jn.nJ

=

0 and

2j 3 J

the Q-linear independence of 1, n, n and n we conclude

E Z. By trans for-j ~. n.n

1

O. From J J that this is impossible.

o

Chapter 5. On singularity and symmetry

5.1. Singular tetra2rids

In section 2.2 we have given the following definition of a singular tetragrid:

(5.1)

A tetragrid is called sin2ular if there is a point z

e

C which belongs to three or four grids. In the latter case it is also

said to be exceptionally singular.

The question whether a tetragrid, defined by reals Yo"'" Y3 is

singular, can be answered by means of the complex parameter ljJ, (see (4.1».

Theorem 5.1. A tetragrid, defined by reals Yo"'" Y3' is singular if and only if its parameter ljJ has one of the forms

(5.2) .r:J. + p, r:J.n + p, r:J.t) 2 + p, r:J.n 3 + p,

with r:J. € R, P e Z[n].

Proof. Assume the tetragrid to be singular. According to (5.1) there are three grid lines passing through one single point. In case there are four lines passing through this point we select three of them. It is easily seen that one of these three lines is the bisector of the other two. By means of a shift and/or a rotation we can manage that the intersection point becomes 0 and the axis of symmetry the imaginary axis. This means that after these transformations we have

Yo' Yl and Y₃ are integers.

Applying transformation (T2) of section 4.2 we get to Yo

=

_Y1

=

_Y3

=

O.

Hence, according to (4.1), after this last transformation we have

thus, the parameter ljJ is purely imaginary.

If we denote the parameter in the original state by ljJ, after the

**

transformation by ~ we find by using transformations (Tl), (T2) and (T3) of section 4.2

for some j E {O, ... ,3} ,

We have seen that ~** is purely imaginary, ~**

=

ai, a E Rt say, hence

From this we conclude that

n-j~

is congruent mod

Z[n]

to a purely imaginary number.

The same arguments work the other way around.

o

In the proof of the previous theorem we have seen that if in a tetra-grid three lines pass through one single point then one of the lines is the bisector of the other two. Because of this symmetry this line contains infinitely many points of threefold intersection, and infinitely many points of twofold intersection. Between two points of threefold inter-section there is at least one and there are at most two points of twofold intersection, (cf. figure S.l.a). Let us call this line a singular line of the grid. The singular line is a line of symmetry for the whole tetra-grid. In an extreme case there are two singular lines in the tetratetra-grid.

,

This case is depicted in figure S.l.h. In the exceptionally singular case there are four singular lines passing through o~e single point, (there is at most one point of fourfold intersection in a tetragrid). In this case we have an eightfold symmetry, (cf. figure 5.1.c). Apart from shifts there is only one such tetragrid, namely the one given by ~ = O.

5.2. GR-patterns associated with Singular tetragrids

Thus far we have considered GR-patterns associated with regular tetra-grids only. Next we are going to investigate whether it makes sense to ascribe a GR-pattern to a singular tetragrid. Therefore, we consider a singular tetragrid given by the parameters

Y~O)

, ... ,

Y~O).

We perturbe

(a) _{(b) ( c)}

Figure 5.1. A part of a tetragrid with, (a) one singular line, (b) two singular lines, (c) four singular lines.

this tetragrid by varying the parameters a'little. In this manner we get a tetragrid with parameters Yo"'" Y3'

Let us use the term j-line for all lines of the form Re (Zl1-j)

=

constant. Let us assume that the imaginary axis is a O-line of the unperturbed

grid and that some l-line and some 3-line of that grid intersect on this O-line. It follows that this O-line is an axis of symmetry and that the l-lines and 3-1ines are arranged in pairs which intersect each other on that axis.

(0) (0) (0)

Without loss of generality we may assume y := y

=

y = O. For

0 1 3

the time being we shall consider tetragrids which are singular but not

(0)

exceptionally singular, hence y 2 'E: z •

We now consider a pair of a i-line and a 3-1ine intersecting on the singular line. T he following theorem holds:

Theorem 5.2. In the perturbed grid, the intersection point lies on the left of the perturbed O-line if

(5.3) Re(1j!) = Yo + l:i( y 3 - Y 1) (n + n -1 )

is negative, and on the right expression (5.3) is positive.

a-line Re (z) + y = a, -1 0 1-line Re(zn ) _{+ Y1}

₌

0, -3 3-line Re(zn ) ₊ Y

o.

3

The intersection paint s of the perturbed i-line and 3-line lies on the left of the perturbed a-line if and only if

Re(s)

<

-y •

o

An easy calculation yields

which completes the proof.

o

We conclude that if the perturbation moves ~ to the left then the intersection paints of i-lines and 3-1ines, which were lying on the unper-turbed a-line, all appear on the left of the perunper-turbed O-line. We have depicted this situation in figure S.2.a. Similarly we get the situation shown in figure S.2.b if ~ moves to the right. Hence, we can consider the singular tetragrid as the limit of a sequence of regular tetragrids in two different ways. ' The two GR-patterns corresponding with these two limits are different.

(a) (b)

Figure 5.2. a. ~ approaching from the left,

In section 2.3 we have associated with every regular tetragrid a GR-pattern by giving a one-to-one correspondence between the meshes of the tetragrid and the vertices of the pattern. But in a singular tetra-grid we can still associate a pOint

L.

K. (z)nj to each mesh, and we can

. J J

connect these pOints L. K.(z)nJ in a way corresponding to the edges of

J J

the meshes (cf. figure 2.4). However, we do not get just squares and rhombuses, but also hexagons (corresponding to threefold int?rsections), and possibly a regular octagon (corresponding to the fourfold inter-section) .

The figures formed by a small variation of the three lines of a threefold intersection are of two different types. If we perturbe a

singular tetragrid by moving ~ to the left (cf. figure S.2.a) the hexagon is filled as shown in figure S.3.a. By moving ~ to the right (cf. figure 5.2.b) we get the situation shown in figure S.3.b.

(a) (b)

Fig. 5.3. The hexagons corresponding with figure S.2.a, 5.2.q respectively_

In this way we see that a pattern corresponding to a singular tetra-grid, built up from squares, rhombuses and hexagons, can be filled in two ways to form a pattern built up from squares and rhombuses. Such a

is also called a GR-pattern. One of them is obtained by taking ~~e limit of the GR-pattern of the perturbed tetragrid with ~tending to its limit from the left. The other one is obtained if ~ approaches from the right. These two patterns are mirror twins. In the middle they have an infinite chain of hexagons (either all as in figure 5.3.a, or all as in figure 5.3.b) alternated with one or two squares. See figure 5.4.

Apart from this chain, the GR-patterns are symmetric with respect to this chain.

Fig. 5.4. A singular line, the perturbed singular line and its corresponding chain of hexagons and squares.

We now consider the exceptionally singular tetragrid (~

=

0, say). We have seen that in this case we get a regular octagon. Now the question arises in what way this octagon is filled by squares and rhombuses when the four lines through the fourfold intersection point are varied a little. The answe~ to this question depends on which one of the 8 angles formed by the lines Re(zn

=

0 contains ~. This means that there are 8 different ways to approach ~

=

0, and these 8 are obtained from each other by rotation. Hence, to the exceptionally Singular tetragrid there

•

correspond 8 different GR-patterns. All these are congruent, (cf. figure 5.5). In figure 5.5 an example of four lines almost passing through one point and its corresponding octagon filling is given.

Fig. 5.5. Four lines almost passing through one point and the octagon filling corresponding to it .

On each side of the octagon of figure 5.5 there grows an infinite chain of hexagons and squares as indicated in figure 5.4.

5.3. Symmetries of tetragrids

In this section we investigate symmetries of tetragrids, whether they are singular or not. Symmetries of regular tetragrids carryover at once to the corresponding GR-patterns. For singular tetragrids the con-struction of section 5.2 may distort the symmetry.

The symmetries we consider (with the notation of section 4.2) are of the kind where some rotation turns G into something that is either shift-equivalent to G or G. That means either (cf. theorem 4.1 and trans-formations (T3) and (T5) of section 4.2)

(5.4)

(5.S)

with j

=

0,1,2,3.

(It is easily shown that the cases with j= 5,6,7, j

=

4,5,6,7, respecti-vely, are analogues of the cases mentioned in (5.4), (5.5), respectively. In other words, "G and something" of the second paragraph of this section are interchanged).

According to theorem 4.1 it will suffice to indicate just one element from every class of mutually congruent tetragrids. This means that we may reduce $ modulo Z[ 1"1] •

We first consider (5.4). 2$ E Z[ 1"1], gives $

=

~E.

n.nj

J J

tetragrid is congruent to the

The case with j

=

4, i.e. the relation with n. E Z. If all n. are even then the

J J

one with $

=

O. If three of the n. are even,

J

nl""~ n

3, say, we get $

=

~. By rotation we get the other four cases $

=

~nJ. If two of the n. are even we get two essential different cases:

J

if n

2 and n3 are even then $ get 1jJ

=

~(1

+ 1"12). The other

=

~(1 ~ 1"1) I if n

1 and n3 are even then we cases are reduced to these by rotation. If one of the n. is even then we find $ =

~(1

+ n + 1"12); by rotation we get

J

the other four cases. Finally, if all the n. are odd then the tetragrid

J₂ 3 is congruent to the one with $

=

~(1 + 1"1 + 1"1 + 1"1 l.

If j 1= 4 then we get the relation (1 - nj 1lJ! E z[ n]. In z[ n] the

. 2'

factor (1 .. nJ ) divides 1 .. n J and therefore 2. So (5.4} implies 2$ E Z[ n] and this case is investigated above.

We now turn our attention to (5.5). In case j

=

0 we get the relation

lJ! .. !/J E Z[ n] I hence $ .. 1jJ = n

1 = n3, whence ImlJ! = ~n1/2

lJ! (in case j = 0)

(5.6 )

n.nj with n, E Z. We easily derive n

=

0,

J J 0

+ ~n2' From.this we find as general form for

In case j

=

2 we substitute in (5.5) 1J!

=

n~. For ~ the following relation holds

~

..

fEn

-1Z[ 1"1] . Noticing that 1"1 -1Z[ 1"1]

=

Z(

n]

we conclude that the case j

=

2 is reduced to the case j

=

0 by a rotation.

In case j

=

1 we substitute in (5.5) 1jJ

=

(1 +1"1 )~. For ~ we find the relation

~

-

~

E (1 + n)-lZ [nJ

=

~(1

- tl + 1"12 + n3)Z( 1"1]' Thus, we

may write ~ - ~

=

~(1

- n + n2 + n3)L. n.nj with n. € Z for all j. In

] ] ]

the same way as we did for j

=

0 we now derive n3

=

n

2 and no

=

-nl, whence Im(~)

=

~ n1/2 + ~(n2 - n

l). From this we find that the general form of ~ is also given by (5.6). As general form for ~, in case j

=

1, we find

(5.7)

where a ~ R, nl' n

2 ~ Z.

In the same way as we treated the transformations of section (4.2) one can easily find that the case with j

=

1 is essentially different from the cases j

=

0,2. (The multiplication ~

=

(1

+n)'

implies a

*

transformation of the parameter vector y, given by y~ = y. - y. l'

J ]

J-where Y-₁ = -Y₃).

Similarly to the case j

=

2, it fo~lows that the case j

=

3 is reduced to the case j = I by a rotation.

(5.8)

Summarizing we have the following cases of symmetry

~ :: 0, ~ =: ~, ~ = ~ (1 + 11),

l/!

=~(l+n)f 2 2 2 3 ~

=

~(1 + 11 + n ), ~

=

~(I + 11 + 11 + 11 ),

2 3

l/!

e (1 + n)R, ~ e (1 + n)R + ~(n + 11 ).

These cases are essentially different, apart from the fact that the latter six can be equivalent to one of the others in exceptional cases.

In (5.8) the cases with

~

= 0,

~

=

~f ~

=

~(1

+ n2) and

~

=

~(1

+ 11 +

n

2) are all singular (cf. theorem 5.1). The other values of ~ correspond with regular tetragrids.

Chapter 6. Deflation and inflation

6.1. Introduction

One of the most beautiful properties of the kite- and dart-patterns of R. Penrose and the corresponding thick- and thin-rhombus patterns is the existence of a geometrically simple subdivision operatio? called deflation, and its inverse, inflation.

By deflation, by an ingenious subdivision rule for the separate kite and dart or thick and thin rhombus a tiling is turned into a new one, where the pieces have smaller side-lengths, -~ + ~/5 times the original ones. This construction can already be applied to a finite set of kites and darts or thick and thin rhombuses that covers just a part of the plane. Inflation is the same process carried the other way.

In this chapter we investigate whether it is possible to define a deflation and inflation with similar properties for a GR-patterri. This turns out to be possible.

6.2. The similitude ratio p

~f there exists something like an inflation then this operation turns a given GR-pattern into a new one, where the pieces are similar to the original ones, but with greater side-lengths. Let us denote the similitude ratio by p. As we have seen in section 2.3, every vertex of a GR-pattern is an element of Z[ n] • We demand that the inflated- and deflated pattern have the same property.

Consinering an inflated and a deflated pattern as a GR-pattern

multi--1

plied by p or p ,respectively, (cf. de Bruijn[2t section 14]), we find that p should satisfy the following three conditions:

(6.1) p

>

1 t P E: z[ n] p -1 E: z[ n] •

-1

From (6.1) we deduce that p and p must be of the form