Algebraic theory of non-periodic tilings of the plane by two simple building blocks : a square and a rhombus Citation for published version (APA): 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 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected]providing details and we will investigate your claim. Download date: 15. Jul. 2020
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Algebraic theory of non-periodic tilings of the plane by twosimple building blocks : a square and a rhombusCitation for published version (APA):Beenker, F. P. M. (1982). Algebraic theory of non-periodic tilings of the plane by two simple building blocks : asquare 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
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
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. 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) 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 :::: Y
3, 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 + Re«zn)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
-31-
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
-32-
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
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)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.
-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"12). 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)
-42-
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
-43-
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
<-l)j(o. - ~}nj ....
=-E. = -$0' J J
where
0 = -Yo + Y1 - Y + 1 , 0 3
°1 = Yo - Yl + Y2 (6.9)
°2 = Yl - Y + Y3 . 2 03
== -Yo + Y2 - Y3 + 1.
The o's obtained from the above consi~erations will now be used for the
description of the tetragrids which·will be shown to underly the GR-pattern
and its inflation.
Definition 6.1. Let ~ be the GR-pattern generated by a regular tetragrid with Y
parameters Yo"'" Y3 and let ~o be the GR-pattern generated by a regular
tetragrid with parameters 00"", 03 as given in (6.9). Then the inflation
X of ~ is given by Y
(6.10) x = p~ . 0
where P~o is the pattern obtained from the points and lines of ~o by
multiplication with p (in the sense of multiplication in the complex plane) .
Theorem 6.1. If the tetragrid with parameters Yo"'" Y3 is singular then
the tetragrid with parameters 00"", 0 3 ' given by (6.9)' is also singular,
and conversely.
Proof. According to theorem 5.1 the tetragrid with parameters °0"", 03
•
-44-
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
-45-
p$o - pattern) is also a vertex of the original pattern (the $y - pattern) •
Theorem 6.3. Ej
kjnj is a vertex offue p~o - pattern if and only if
(6.15)
. -1 . Proof. E. k.n J is a vertex of the p$~ - pattern if and only if E. k. P nJ
J J \J J J is a vertex of the $ <5 - pattern. According to theorem 3. 1, (6'.6) and (6. 7)
this means that Ej
kjnj is a vertex of the P$8 : pattern if and only if
(6.16)
By a complete elaboration of (6.16) and by using relation (6.8) and
transformation (6.9) one easily finds that (6.16) is equivalent with
(6.17) o
Since V contains _p~1.v, (6.15) yields that E. k.n j is a vertex of the 1> -J J y
pattern.
From this theorem we deduce that the question whether a point Ejkjnj
is a vertex of the inflation pattern can be answered by means of a corres--1~ .
ponding point in the set -p V, (cf. theorem 3.1). In figure 6.1 we have ~ -1 .....
drawn the sets V and -p V.
-1 ...... Fig. 6.1. The set -p V contained in the set v.
-46-
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
-47-
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
-48-
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
-49-
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).
'---<f----" (A)
Fig. 6.4. The square and rhombus provided with arrows.
This arrow condition can be extended to the deflation in the way as
is shown in figure 6.5.
Fig. 6.5. The deflation of the arrowed square and rhombus.
Hence, the arrow-property is "deflation-invariant".
By using the arrowed square and rhombus, we are now able to give a
geometrical definition of the inflation of a GR-pattern. If we consider a
GR-pattern as a pattern built up by squares and rhombuses provided with
arrows then, by using the transition-relations of table 6.1, it is easy
to describe the inflation of a GR-pattern in a geometrical way:
Omit all the vertices which have a side with arrow directed outwards.
The remaining vertices are the vertices of the inflation pattern X. We
connect two vertices of X if their distance is 1 + 12 times the edge-length
-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.
-51-
Fig. 6.6. An inflation-deflation pattern.
Forcing of
a vertex of
type 2.
Forcing of a
vertex of type
-52-
-53-
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
-54-
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). •
-55-
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 72
0 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
-56-
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
-57-
(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.
-58-
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, k1
» 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) •
-60-
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
-61-
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 <P be an AR-pattern with the stated property. Then its deflation
has the form p-1<jl(1), where p = 1 + 12. We have noticed bef~re that q,(1) (-1 )
is again an AR-pattern. Similarly, the inflation of <jl has the form p<jl ,
where, according to the properties of <P, <jl(-1) is also an AR-pattern. We
d f ' ,;, ( 2 ) ,;, ( 3 ) b '" (n+ 1 ) (,;, (n) ) (1) d" 1 1 ,( - 2) ,;, ( - 3 ) e l.ne 'I' ''I' , • • • Y 'I' = 'I' I an Sl.ml. ar Y <p , • 'I' , •••
by <p (-n+ 1) = (<P (-n) ) (1) •
Consider two AR-patterns <p and ~. Assume that <p and ~ have a vertex z o in common, and that the set of neighbors of z in <p is the same as in ~. o At the same time we assume that the common edges have the same arrows in the
same direction. Let K be the union of the closed interiors of the rhombuses
of <p and ~ that meet in z . From the definition of deflation of an AR-pattern o
we easily obtain that the deflations of <p and ~ coincide at l,east inside
K. We conclude that q,(n) and ~(n) coincide inside pnKI for n = 1,2, .... .. Assertion. For any R > 0 and for any AR-pattern <p which has the stated
properties there exists a regular GR-pattern ~ such that q, and ~
coincide in the region given by Izi < R.
•
Proof. Take n € N such that pntan(~/8) > 2R and consider. <p(-(n+2)~ In (-n)
<p the point 0 belongs to a closed square or rhombus. Let z be the o vertex in <jl(-n} which is closest to O. By checking the second deflation
(";;(n+2» of all the possible vertices of the AR-pattern cf> one finds
that z and all the neighbors of z are vertices of the type 1, ... , 6. o 0
Again let K denote the union of the closed squares and rhombuses
meeting at z . The distance of 0 to the boundary of K is at least o
~tan(w/8), cf. figure 8.2.
Fig. 8.2. The distance ~tan(~/8).
-62-
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 ) + Y
J.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
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