TECHNISCHE HOGESCHOOL EINDHOVEN EINDHOVEN UNIVERSITY OF TECHNOLOGY

NEDERLAND THE NETHERLANDS

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

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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

inflation 47

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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

64

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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. Now the question arises how to obtain a tiling of the whole plane.

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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, sometimes 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

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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 45 0 and 1350 This rhombus is henceforth called "the rhombus".

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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 2 , res"ectively,

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such that Wl - ~2 = o (mod Z[ n]), where n = exp(~i/4) I then the GR-patterns corresponding to W1 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 interpretation 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" E stands for E

, , i j j=O' We always put

( 1.3) n = exp(~i/4), p = 1 + n + n7 = 1 + 12. Il" n

If x E R, then fxl (the roof of x) is the least n E ~ with n ~ x.

z[ n] denotes the ring of all E. n.n j with n , ... ,n3 E Z. J J 0

p.g. 1.5. A piece of a GR-pattern.

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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).

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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 Eel 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) + Y2 E Z} I 3rd grid: {z E C ~h(-Re(z) + Im(z

+ Y1 E Z} ,

+ Y3 E Z} .

The tetragrid determined by Yo' Y1f Y2' Y3 is the union of (2.1) for j = 0,1,2,3, (cf. figure 2.2).

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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) , .... , K3 (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 K3 (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) -r Re(zn ) + 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) , ... / K3 (Z takes four different values, the four vectors we get from the formula

(2.5) (K (z ) , ... ,K3 (Z + E 1 (8 , ... ,8 3 ) + E2 (8 / . ,,83 ), o a a or r as s

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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 (ko "'" k3) e Z .

Which (ko"'" 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.

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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

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(2.8) f("l) = f("2) - 11 - 11 ,.. I-' 1-'1 - "'2'

Since KJ. (zl) E Z and K. (Z2) E Z for all j I from the Q-l.inear indepen-

23] 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)n j + 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.

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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 zC 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 GR-pattern. It is easy to check that the vertices in a GR-pattern can be of 6 different types (apart from rotations) according to

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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

I vertex of type 4

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, 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 k 3- 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 (nj ) = 0, J

(2.10) E. (x. - y.)

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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 (xo

- Yo"'" x3 - Y3) is perpendicular to the vectors (1, -~/2, 0, ~/2) and (0, -~/21 1, -~/2). Consequently (xo - 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, ko

+ 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 little in. order to get a point in the interior of the cube. 0

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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 3 + + k3 n Now can be written as k

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~, . , K3 Cz) = 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 kO/'" k3 is equivalent with

(3.3) LJ. (-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, j . Re(zl1- j n j - = (-1) (A.-J j J J J

= E. (-1)j A.n j .

J J

=

Hence EJ. (-l)j(k j - y.)n

j v.

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

J J J 0 Y3 with 0 < Yo < 1, .. , 0 < Y3 < 1 such that

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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 n2 , 1 + n2 , 1 + n2 - n3 , 1 - n3 , 1 - n - n3 ,

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 k 3 . o + + 3n ~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:

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If Ej (-l)j(k j yj)nj 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.

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*2a -*2b -20 *- 2d *' 2e . ~f *2g 2h

~/ *3b ~ 3 . ~ 3d

*3e _ *-3f ~3g t- 3h -- 4a ~ 4b -k 4 -)( 4d -* 4e ~4f . ~4g. >f- 4h . -y Sa ---r( Sb -l( 5 ~ Sd

~ Se )l-:f :rSg r Sh y 6a -{6b .'. -< 60 ---\ 6d

A 6e . }-6f >-6g j-6h

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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 = (a1 , a 2 ) 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 = (b1- a l ) (b2- a 2) N-l>

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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 (n1, 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 (kl

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

-23-

(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"'" Y3 be reals and let V. be one of the 41 subregions i i J 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 (k1 , k3) ~2 which satisfy If'(R), as R + 00. In short notation

(3.11) .:ff(k k' )

-24-

~(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 rotations. 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 -4 17 1212 Z 0.029 = P = -2 2p -4 34 - 2412 -5 =-41 + 2912 0.012 = p ~

3 -3 -28 + 2012 2p -4 = 34 - 2412 0.059 4p = z

4 -2 12 812 2p -3 =-14 + 1012 0.142 4p = -

.....,

5 -1 412 2p -2 6 - 412 0.343 4p = -4 + = -~

6 2 -1 -1 + 12 0.414 P = ~

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 = (1 - P ). (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.

-25-

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.n j

(2.10) in the form E. (-l)jx.n j = 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 Z E C with

0

Re(z n o

* } + y, - y E ~, J j j = 0, ... ,3.

-26-

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.

* (1) If l/J = l/J then it follows immediately from tneorem 3.1. that the two

(ii)

-27.,..

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

* * ~ = ~ . For if ~ ~ ~ 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 Yj = 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 " . , Y3)on~ and on the point sets G and R. Here G stands o . for the tetragrid, considered as a point set in the complex plane, and

(4.3)

-28-

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) Yj = Yj j = 0, ... ,3. 0

* * * Now 1/1 = 1/11 G = G - z 0' R = R.

Proof. * (-l)\~nj (-l}jy.n j (-l)jRe(z n )n j :::: (i) 1jJ :: k. == k. + k. J 0 J J J J

== k. (-l)jY.n j ""

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 Yj + 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 = Ej Kj(Z)nJ 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

* Yj = Yj + n. j :::: 0, . ,3, )

* (-l)j n. nj * * then 1jJ = 1/1 + Z. G :::: G, R :::: R + ) J

. We Z E G.

0

o

-29-

Proof. The first two assertions are trivial. * * * *' If u E: R then u := r: K. (z)nJ 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. Sincez 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) ilj + (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, Yj == -y. j == J

* * * then ~ = -Ij) G == -G, R = -R + ( 1 + n +

o

2 3 n + n ).

-30-

Proof. The first two assertions are trivial. * * * * j If u E R then for some Z E ~G we have u = Ej 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}n j Ej K. (z)nJ :::: = = J J J J -E. K.(-z)n j (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) -1

-11 1jJ.

(ii) * * If z E G then-yo + Re(zn ) E ~ for some j E {Of .. ,3}. An J

easy calculation yields

* ( . +1)

Yj + Re (zn ) E Z ~ y. 1 + Rezn)n- J ) E Z, J+

where From this we infer that G -1 Y :::: -Yo' = 11 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)n J ) + :: J J J J E. rRe( (zn) n- j ) + y.1T'l -1 ~j -1 = - n J J

* -1 3 From this we conclude that R = 11 R + 11 o

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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

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

first transformation (i.e. the shift and rotation) by ljJ* and after the last

-33-

** 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 R t 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 tetragrid.

,

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)

-34-

(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 013 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!) -1 = Yo + l:i( y 3 - Y 1) (n + n )

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

Proof. The perturbed O-line, 1-line and 3-1ine are given by

-35-

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 perturbed 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, b. ~ approaching from the right.

-36-

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)n j 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.

-37-

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

-38-

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)

with j = 1,2,3,4 or

-39-

(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""~ n3 , 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 n2 and n3 are even then $ get 1jJ = ~(1 + 1"1 2). The other

= ~(1 ~ 1"1) I if n1 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 J 2 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

-40-

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 = n2 and no = -n l , 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/! 2 =~(l+n)f

223 ~ = ~(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 + n2 ) are all singular (cf. theorem 5.1). The other values of ~ correspond with regular tetragrids.

-41-

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

(6.2)

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From (6.2) we deduce that p is a unit in Z[/2] . According to Hardy and Wrightl4, theorem 243] the only units of ~[/2] are given by

(6.3)

From (6.1) and (6.3) we infer that

(6.4)

for some n tN. It is reasonable to expect an inflation with ratio of similitude

1 + 12, which is the minimal possibility. After all, n~fold application of this minimal inflation gives an inflation with ratio of similitude (1 + 12)n. We note that

1 12 1 3 P = + :::: + n - n , (6.5)

-1 -1 + 12 -1 + 3 p = = n - n .

6.3. Algebraic definition of deflation and inflation

By trying to define a deflation and inflation similar to the method -1

used by de Bruij~ 2, section 14] , we were led to consider the set -p V. -1 It is easily seen that the set V contains -p V but the latter set does

not have the same center of symmetry as V. From this it immediately follo-wed that the deflation defined in this algebraic way would not be very elegant (in a geometrical sensel. For instance, because of the asymmetric

-1 position of -p V in V the orientation of a square or rhombus began to play a role. Moreover this asymmetric situation led to 16 different kinds of subdivisions of squares, and 19 different kinds of subdivisions of rhombuses into smaller squares and rhombuses, depending on the orientation. So, we were far from an elegant geometrical interpretation of inflation and deflation. Therefore it seemed worth-while to get symmetry first.

An easy calculation yields that the centre of V is given by :2 3 ~(1 - n + n n l. We define the set V by

{6.6) V = V - ~(1 - n + n2 3 n ) I

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then the origin is the centre of V. "'" The parameter $ is defined in the same way:

(6.7)

If we denote the parameter $ given by the four reals Yo"'" Y3 by $y then from (6.7) it follows that

(6.8) E. (-1)j(! 3 ~)nj p$y = + n - n ) (y , = J J

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is singular if and only if its parameter Wo has the form

(6.11)

for some 1 EO {O I' I 3}, i3 IR and mj Z fer all j. From (6.8) and (6.11) we find

(6.12)

Hence,

(6.13) '" 2 3 !fiy = Wy + ~(1 - n + n - n ) =

= s*nl + (m

- m + m3) + (-m + m - m2 + l)n + 0 1 0 1

(-m 1)n2 + (m m3)n 3

+ + m - m - - m + I 1 2 3 0 2

* -1 where e = -Sp IR. From (6.13) and theorem 5.1 we conclude that the tetragrid with

parameters Yo"'" Y3 is singular. The same argumentswork the other way round, which completes the proof. 0

From this theorem it follows that regularity and singularity are invariant properties under the deflation- and inflation operation.

In some special cases, the $6 pattern is shift-equivalent with the ~y- pattern:

Theorem 6.2. Let k Z and let Yo"'" Y3 be reals. We consider the tetragrid with parameters o~k) f"" o;k) which are obtained from Yo"'" Y3 by a k-fold application of transformation (6.9). The$dk) -pattern is shift-equivalent with the $ -pattern if and only if y

(6.14)

Proof. Follows immediately from relation (6.8) and theorem 4.2. 0

Next we will show that a vertex of the inflation pattern (i.e. the

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p$o - pattern) is also a vertex of the original pattern (the $y - pattern)

Theorem 6.3. Ej k j nj is a vertex offue p~o - pattern if and only if

(6.15)

. -1 . Proof. E. k.n J is a vertex of the p$~ - pattern if and only if E. k. P nJ

J J \J J J is a vertex of the $

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As one can see, figure 6.1 is rotation-symmetric. From this we conclude that the inflation and deflation defined by (6.12) do not depend on the orientation of the squares and rhombuses. Furthermore, from figure 6.1 and theorem 6.3 we infer that the type of vertex (in the sense of section 2.4) after deflation in almost all the cases is specified by the original type of vertex. A survey of these transition-relations is given in table 6.1.

Table 6.1. Transition-relations between vertices after deflation and inflation.

deflc tion defl. tion defla tion

In.fli: t:lon J.n:flc t:lon J.nfla tion

,vertex of vertex of disappeared vertex of type 1,2,3 type 1 type 5 or 6

~ertex of vertex of type vertex of vertex of vertex of vertex of ~ype 4: 1 or 2: type 5: type 3: type 6: itype 4: 1*

-- *or*- -l * Y ~ *or~ -r( ;;k -( ~

~ I -K *or : ~ -< 't--)<

*or* ~ ~ ~ +- *or A:- T I A --

~ *or-* * ,}- ~

I >r *or~ :r ~ >- -k )( *or~ ~ JL -k /"-

;

J

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I knowing these transition-relations and the properties of the we manage to give a geometrical description of deflation and

ation which has a very elegant form.

, ~-4. Geometrical description of deflation and inflation

By using table 6.1 one,easily finds that there are only six candidates for the effect of a deflation on a square and only one candidate for the effect of deflation on a rhombus. We have to investigate whether these deflations actually occur in a deflation pattern of a GR-pattern. It will turn out that of the six candidates for the square, five drop out, so we are left with just one.

First, in figure 6.2 we give the six candidates for deflation of the square, and the deflation of the rhombus.

1)

4)

2) 3)

5) 6)

Fig. 6.2. The six candidates for deflation of the square, and the deflation of the rhombus.

In chapter 3 we have seen that by means of the set V (which is divided in 41 subsets) we can answer the question what the neighbors are of a vertex of a GR-pattern. We apply this criterion to the candidates for deflation

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of the square and rhombus of figure 6.2. For instance, in candidate 6) of figure 6.2 we have the following configuration of vertices.

The corresponding regions for the vertices a) and b) in the set V are Sh and sb, respectively. From figure 3.2 we infer that it is not possible to pass from region Sh to region Sb by means of a step of length 1 in ~ nO-direction. Hence, in a GR-pattern vertex b) cannot be a neighbor of vertex a). From this we conclude that.andidate 6) of a square, as given in figure 6.2, will no~ occur in a GR-pattern.

In the same way we obtain that candidates 2) , ... , S) of a square will not occur in a GR-pattern. Remains one candidate for deflation of a square, viz. candidate 1), and one for a rhombus. We know that there exists a deflation for a square and rhombus, so the one deflation which is left occurs indeed.

Summarizing we conclude that we have found a geometrical interpretation of the deflation of the square and rhombus as depicted in figure 6.3.

Fig. 6.3. The deflation of the square and rhombus.

The orientation of the deflation of the square is uniquely determined by the position of corner (A). By observing the deflation it follows that

~very square in a GR-pattern has exactly one vertex of type 6; that is corner (A).

There is an easy way to indicate corner (A). As is easily seen from

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figure 6.3, the sides of a square and rhombus in a GR-pattern can be provided with arrows as depicted in figure 6.4. These arrows uniquely determine corner (A).

'---

-50-

in the original pattern. By observing the original pattern we can orient the arrows.

In this manner we have found a unique geometrical description of the deflation and inflation of a GR-pattern. In comparison with the description of de Bruij~ 2, section 14] our treatment of this subject is just the reverse. In figure 6.6 an example of a deflation-inflation GR-pattern is given.

6.5. A 2art of the forcings of the vertices of a GR-2attern

In the previous section we have seen that the sides of the squares and rhombuses of a GR-pattern can be provided with arrows such that every two a~acent building-blocks have the same arrow in the same direction on the common edge. One can imagine that this arrow condition, together with the condition that only the six types of vertices of section 2.4 may be used, forces a configuration of squares and rhombuses around the several types of vertices. Such a configuration is called a forcing of a vertex. Some of these forcings are quite extensive as one can see in the figures given below.

We do not claim these forcings to be ~aximal, and it does not seem to be easy to find the maximal forcings in all cases. After all the matter of forcings is much harder than in the Penrose case (kites and darts or thick and thin rhombuses). In our case the areas in which a part of the tiling is forced are not always simply connected. Some of our pictures show holes in the form of hexagons; these hexagons can be tiled in two different ways {cf. figure 5.3), aad therefore do not belong to the forced area.

Forcing of a vertex of type 1.

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Fig. 6.6. An inflation-deflation pattern.

Forcing of a vertex of type 2.

Forcing of a vertex of type

-52-

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Forcing of a

vertex of type 5

Forcing of a vertex of type 4

Forcing of a

vertex of type 6.

6.6. Relation of GR-patterns to sequences of zeros and ones generated by special rewriting rules

In [ 1 ] , de Bruijn dealt with doubly infinite sequences of zeros and ones. Then "deflation" is obtained by replacing each 0 by 10 and each 1 by 100. Not every sequence is the deflation of another one, but there exist sequences s which have inflations of all orders, by which we mean the follo-

. Th . th d fl t' f ( 1 ) th' (1 ) . the Wing. e sequence s is e e a ion 0 a sequence s, is S is (2) deflation of s 1 etc. The sequences with inflation of all order with

respect to rule 0 ~ 10, 1 ~ 100 occur in our singular GR-patterns! Consider the singular tetragrid with an infinite chain of squares

and filled hexagons as in section 5.2. For convenience we write the squares

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and hexagons from left to right instead of from bottom to top. We cut the hexagons into a left part HL and a right part Hp' We denote the square by

h

S and the hexagon by H. Now we study deflation. Deflation of H gives HRSSEL I and deflation of a S gives HRSHL A doubly infinite sequence of H's and SiS deflates into a doubly infinite sequence of H SSH 's and R L HRSHL'S. In this sequence each HRSSHL and each HaSHL is preceded by an HL. So if for each HRSSEL and each HRSHL we take away the HL on 'the right and add it on the left we still have the same infinite sequence. So instead of HRSSHL we have HSS and instead of HRSHL we have HS. Hence, the deflation of the GR-pattern causes in the central chain of H's and SIS just the same thing as the rewriting rule H ~ HSS, S ~ HS (this corresponds to H = 1, S = 0).

The H-S-sequence occuring in singular tetragrids have inflations of all order, and indeed, every a-S-sequence with this property occurs in some singular GR-pattern. This is easily verified by selecting the tetra-grid parameters so as to check with the algebraic formula of [1, section 6 and 7] for those sequences. (Notice that, without loss of generality, the singular line is the real axis, hence Y2 = O. A zero in the 0-1-se-sequence corresponds with an intersection point of t;e ost_ and 2nd grid. A one in the 0-1-sequence corresponds with an intersection point of the lst_ and 3rd_ grid on the real line).

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

Let us consider a tiling of any convex part of the plane by squares and rhombuses. The edges have not been oriented in any way, and no condi-tions on the situation around a point (like we have in GR-patterns) have to be satisfied.

The question arises whether this tiling can be a GR-pa~tern or a part of a GR-pattern.

In the case of the thick and thin rhombuses such questions can be answered (see de Bruij~ 2]) by local inspections. The edges of the pattern have to be provided with red and green arrows in such a way that all thick rhombuses show one and the same picture (red arrows leaving from an angle of 720 and green arrows pointing towards the opposite 720 angle) and all thin rhombuses show one and the same picture (red arrows pointing

o towar~one of ~~e 144 angles and green arrows pointing towards the opposite 1440 angle). Whe~~er a given arrow-less pattern of thick and thin rhombuses can be arrowed according to these specifications can be checked by local inspections (de Bruijn, oral communication). There is a criterion for a rhombus to be "happy", and whether a rhombus in the pattern is happy or not can be determined by investigating all pieces with

distan~e ~3 to the given rhombus. (The distance between two pieces in a tiling is~ of course, the minimal number of edges tilat have to be crossed when drawing a curve, avoiding the vertices, from the interior of the one piece to the interior of the other one). The final result is that a thick-and-thin-rhombus pattern in which all the pieces are happy is a pattern that can be obtained from a grid.

It seems that for our present case of the squares and rhombuses it is impossible to find such local conditions by means of which we can check that a given tiling is a GR-pattern.

In order to express what we are able to show, we define, for a given tiling T, for any piece x of T, and for any positive integer k, the tiling SeT, x, k). It is a sub-tiling of T, consisting of all pieces of T which have (in T) a distance to x which is at most k. The tiling T is assumed to be a tiling of the plane or a convex part of the plane by means of (arrow less squares and rhombuses.

This S(T, x, k) is a subtiling of T. We are interested in the question whether T can be embedded in a GR-pattern, and we would like to answer this

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question by investigating whether the smaller patterns SeT, x, k) can all be embedded. It will be the main result of this chapter that this cannot be done if we limit k to a finite interval.

In order to give a precise formulation, we define the predicate EGR(T) for tilings T: EGR(T) is true if and only if T can be embedded into some GR-pattern.

Our result can now be stated as follows. We are able to construct, for each positive integer k, a tiling T such that for all pieces x of T we have EGR(S(T, x, k I but nevertheless EGR(T) fails. This is a counter-example to the conjecture "if EGR(S(T, x, k for all x, then EGR(T)".

The T's we construct are tilings of bounded convex parts of the plane. We strongly conjecture that our technique can also be used to construct, for each k, a counterexample T which tiles the whole plane, but we do not claim to have proved this part.

In connection with the above-mentioned counterexample we first dis-cuss the vertex condition and the arrow condition. The vertex condition for a tiling T (not necessarily a tiling of the whole plane) means that every vertex (apart from those on the boundary of T) is one of the types 1, . , 6. The arrow condition says that it is possible to arrow the edges of the pieces (in the w~y indicated in the previous chapter) such that adjacent pieces have the same orientation on the common edge.

Notice that if a tiling satisfies the arrow condition then it need not to be true that the tiling satisfies the vertex condition, and conversely; cf. figure 7.1.

f /

Ii' I'

/

/' !' "-/

(I) (II)

Fig. 7.1. Tiling I satisfies the arrow condition but not the vertex condition. Tiling II satisfies the vertex condition but not the arrow condition.

It is easily seen that the arrow condition is not a local condition

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(in the sense that there is a number k such that it suffices to investi-gate SeT, x, k) for all pieces x of T). By placing an arbitrarily number of squares in the middle of figure 7.2 we manage that SeT, x, k) can be arrowed for all x, but T cannot be arrowed.

7.2. A set T which cannot be arrowed.

The vertex condition is trivially local in this sense. It is not hard to see that the conjunction of the two conditions is local, (actually k = 3 is sufficient). Therefore on behalf of our main counterexample, the property of T to satisfy both the vertex condition'and the arrow con-dition cannot be sufficient for EGR(T) .

Construction of the counterexample e

,In the previous chapter we have seen that a GR-pattern may be consi-dered as being built up by arrowed squares and rhombuses. These'arrowed squares and rhombuses will be used in the construction of the counter-example.

In section 2.5 we have found the following forcing of a vertex of type 1, as given in figure 7.3. We denote this forcing by ~ .

o

(1)

(7) (3)

Fig. 7.3. Forcing of a vertex of type 1.

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From table 6.1 we infer that if we deflate a GR-pattern the only vertices which turn into a vertex of type 1 are vertices of type 1,2,3 and some of the vertices of type 4. By using the set V we easily find that, in a GR-pattern, there are exactly four possibilities for filling the places (1) , .. , (8) in figure 7.3, (apart from rotation) I and these four possi-bilities indeed occur in every GR-pattern. These possibilities are shown in figure 7.4.

Fig. 7.4. The four possible configurations around ~ I o

(apart from rotation).

From figure 7.4 we conclude everything we need: By cutting and affixing in the configurations of figure 7.4 we can construct a finite set T of

-59-

squares and rhombuses that just covers a part of the plane and for which T is not a part of a GR-pattern. We have depicted the set T in figure 7.5.

Fig. 7.5. The set T.

From this we conclude that EGReT) fails. However, for every piece x of T it is easily seen by inspection that EGR(S(T, x, 3 holds. Notice that if we have a small part of a GR-pattern and we deflate this part then, after blowing up the deflated pattern, we get a greater part of a GR-pattern. Hence, if we deflate T (which is possible since T is built up in accordance with the arrow condition) and we blow up the deflated pattern with a factor p, then we get a new pattern, denoted by T(l), for which holds that EGR(T(! fails, whereas EGR{S{T(l), x, k 1 holds for some 1 3.

By repeating this process n for a fixed positive k ,

n holds. We get the counterexample arrows.

, T(n) h th t~mes we construct a set suc at, (n) , (n) EGR(T ) fa~ls whereas EGR(S(T I x, k

n from these sets T(n) by omitting all

From this counterexample we conclude that, if we limit k to a finite interval, it is not possible to answer the question whether a set T can be embedded in a GR-pattern by investigating whether the smaller patterns SeT, x, k) can all be embedded. Furthermore, from the construction of the counterexample we conclude that the property of a set T to satisfy both the vertex condition and the arrow condition cannot be sufficient for EGR(T)

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

Thus far we only have considered the GR-patterns. For these patterns we have given a geometrical definition of deflation and inflation (see section 6.4). For doing this, we needed the concept of the arrowed square and rhombus of figure 6.4. We have seen that every GR-pattern may be considered as been built up by arrowed squares and rhombuses. However, we can construct a larger family of patterns than the family of GR-patterns, called the AR-patterns (AR stands for arrowed rhombus). These patterns are built up by the arrowed squares and rhombuses in accordance with the arrow condition of the previous chapter. A piece of an AR-pattern is given in figure 8.1.

Fig. 8.1. A piece of an AR-pattern.

The geometrical definition of deflation and inflation of a GR-pattern is completely determined by the arrows. Hence these definitions give the deflation and inflation of an AR-pattern as well. By means of verification it is easily seen that a pattern, obtained by blowing up a deflation of an AR-pattern with a factor p, is also an AR-pattern. If we inflate an AR-pattern then in the inflation pattern we can orient the arrows by looking at the original pattern. (That is in fact the way we have defined the arrows in the previous chapter). However, the inflation of an AR-pattern need not to be an AR-pattern. For instance, the inflation of the pattern given in figure 8.1 is the empty set. It will appear that the property that the inflations of an AR-pattern, blown up by a factor p-l, are again AR-patterns gives a characterization of the GR-patterns among the AR-patterns. To speak in terms of de Bruijn{ 1] : An AR-pattern which has infinitely many

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predecessors is a GR-pattern. This result is formulated in the next theorem.

Theorem 8.1. An AR-pattern for which holds that every k-th inflation pattern, k blown up with a factor p , is again an AR-pattern, for every kEN, is a

GR-pattern.

Proof. Let

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We can find a regdar tetragrid which generates a GR-pattern X that coincides with ~(-n) in z and the neighbors of z . (This can

o 0 be established by taking an arbitrary regular tetragrid, and veri-fying that all types of vertices occur at least once in its GR-pattern). Therefore, the n-th deflation of $(-n) and the n-th deflation of X coincide inside a circle with centre 0 and radius ~tan(~/8). The

(-n) -n n-th deflation of is P and the n - th deflatio:r: of X has, according to section 6.3, the form p-n~, where ~ is again a GR-pattern. We find that ~ and ~ coincide inside a circle with centre 0 and radius

~pntan(~/8) and therefore in Izl < R. 0 Let be an arbitrary AR-pattern which has the stated property. Let ~1'

~2/" be GR-patterns generated by regular tetragrids such that ;n and coincide inside the circle with centre 0 and radius n. Let y , , Y3 on n be the parameters of the tetragrid which produces ~ . Take a fixed vertex

n ~. k.n j of . For n suffidently large it is a vertex J ] of ~ . This implies n

that there exists z C such that rRe(z n ) + YJ.

n1 = k .. According to

n. n J section 4.2, transformation (T1) f ~ stays invariant if we

n by Re (z n -j) + y. . So we may assume that I Y. I ~ I k. I + 1

n In In J

replace y. In from the start.

Hence, the sequence (Y.) ~1 is bounded. It follows that there is a sub-In n ... sequence (Yo1"'" y 31 ), (Y02 '"'' y 32 ),, converging to some (Yo'"'' Y3) If this Y produces a regular GR-pattern then it is easy to check that it coincides with $. If it produces a singular GR-pattern then its corres-ponding tetragrid is the limit of a sequence of regular tetragrids, and we get one of the singular patterns corresponding to the singular tetra-grid, (cf. section 5.2).

With this theorem we have characterized the GR-patterns in the set of AR-patternsbut we have not yet found an easy way to construct a GR-pattern without using the . Therefore we observe the AR-patterns in the way Penrose did. By means of deflating and blowing up the deflation pattern with a factor PI we construct a sequence of finite configurations of squares and rhombuses. We do this in such a way that the process con-verges to a square - rhombus which covers the whole plane. There-upon we shall prove that this limit-pattern is a GR-pattern.

o

We consider a union U(o) of squares and rhombuses which satisfies the arrow condi~ion and which contains the origin O. We deflate 0(0) and we blow-up the deflation with a factor Pi we denote this blown-up pattern

-63-

(1) (1). Co) . (1) by U . We demand that U conta~ns U somewhere. We sh~ft U in such a way that these two parts U(o) coincide. By repeating this process infini-

(k) tely often we construct a sequence (U ) k E.IN of finite sets of squares and rhombuses which satisfy the arrow condition. Notice that the shift, applied after the deflations, is always the same. It is eaSily seen that this process converges to a covering U of the whole plane, where U is an AR-pattern. Furthermore, U has the property that its blown-4P deflation pattern is shift-equivalent with U. U is called a PR-pattern (PR stands for Penrose rhombus).

It remains to prove that a FR-pattern is also a GR-pattern. We have seen that a FR-pattern is also an AR-pattern. Furthermore we know that a FR-pattern has infinitely many deflations which are, when blown-up, shift-equivalent with each other. From this it immediately follows that a FR-pattern has also infinitely many inflations. If we blow up the inflation

-1 pattern with a factor p this pattern is shift-equivalent with the original pattern. Hence, every inflation pattern is again an AR-pattern. From theorem 8.1 we now may conclude that a FR-pattern is also a GR-pattern. That the converse need not to be true is easily seen from the property that the blown-up deflation pattern of a PR-pattern is shift-equivalent with the original pattern (cf. theorem 6.2). From theorem 6.2 we infer that the number of PR-patterns is countable, while the. number of GR-patterns is non-denumerable. In short notation, in this chapter we have found

(8.1) PR c GR c AR,

where all the .inclusions are strict.

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References

[1] de Bruijn, N.G. - Sequences of zeros and ones generated by special production rules, Kon. Nederl. Akad. Wetensch. Proc. Ser. A, 84 (= lndag. Math. 43 , 27 - 37, (1981).

[2] de Bruijn, N.G. - Algebraic theory of Penrose's non-periodic tilings of the plane, Kon. Nederl. Akad. Wetensch. Proc. Ser. A, 84 (= lndag. Math. ~), 39 - 66, (1981).

f 3] Gardner, M. - Scientific American, 236, (1) I (jan. 1977), 110 - 121. [4] Hardy, G. and Wright, E. - An introduction to the theory of numbers,

fifth edition, Oxford Univ. Press, 1979. [5] Kuipers, L. and Niederreiter, H. - Uniform distribution of sequences,

John Wiley & Sons, 1974. [ 6] Penrose, R. - The role of aesthetics in pure and applied

mathematical research. Bull. lnst. Math. Appl. 10, 266 - 271, (1974) . [ 7] Penrose, R. - Pentaplexity, Math. lntelligencer, vol. 3., 32 - 37,

(1979)

VoorbladContentsChapter 1. Introduction and notationChapter 2. Tetragrids and GR-patternsChapter 3. The set VChapter 4. A new parameter for the tetragridChapter 5. On singularity and symmetryChapter 6. Deflation and inflationChapter 7. The question of the existence of local joining conditionsChapter 8. PR-, GR- and AR-patternsReferences