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Aperiodic Sets of Prototiles Extracted from the Penrose Rhomb Tiling


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We present aperiodic sets of prototiles whose shapes are based on the well-known Penrose rhomb tiling. Some decorated prototiles lead to an exact Penrose rhomb tiling without any matching rules. We also give an approximate solution to an aperiodic monotile that tessellates the plane (including five types of gaps) only in a nonperiodic way.
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Aperiodic Sets of Prototiles Extracted
From the Penrose Rhomb Tiling
Mike Winkler*
at f¨
ur Mathematik, Ruhr-Universit¨
at Bochum, Germany,
We present aperiodic sets of prototiles whose shapes are based
on the well-known Penrose rhomb tiling. Some decorated pro-
totiles lead to an exact Penrose rhomb tiling without any match-
ing rules. We also give an approximate solution to an aperiodic
monotile that tessellates the plane (including five types of gaps)
only in a nonperiodic way.
1 Introduction
The Penrose P2 (kite and dart) and P3 (rhomb) tilings are certainly the
most popular examples of nonperiodic tilings. Both tilings are strongly
related and generate the same mld-class1. For more details we refer the
reader to [1][2][4]. In this article we will take a closer look at cutouts
from the P3 tiling, which themselves form aperiodic sets of two prototiles
(decorated or undecorated) and which have not yet been published or
mentioned. The smallest structures of edge to edge connected rhombs
we have found are given in Figure 1, whereby T1is not a connected tile,
and T3represents a connected pair of T1and T2. The aperiodic sets of
T1, T2or T2, T3inevitably lead to an exact P3 tiling without matching
rules. A slight modification of T1and T2will give us an aperiodic set of
two undecorated tiles in the shape of a snake and a dog (Fig. 4).
1Two tilings are called mld (mutually locally derivable), if one is obtained from the
other in a unique way by local rules, and vice versa [1].
arXiv:2106.08155v3 [math.GM] 11 Aug 2021
Figure 1: Cutouts of the P3 tiling.
Tilings like P2 and P3 have a scaling self-similarity as a fractal. This
is due to the substitution rules that allow each tile to be decomposed into
smaller tiles of the same shape as those used in the tiling. Thus allow
larger tiles to be composed of smaller ones. Penrose tilings are strongly
related with Fibonacci numbers2. For instance, the ratio between the
number of different prototiles which can tessellate a larger tile is always
given by such numbers [1][2]. This Fibonacci ratio also applies to the
prototiles in Figure 1. T1consists of five thin and eight thick rhombs,
T2of eight thin and thirteen thick rhombs, and T3of thirteen thin and 21
thick rhombs.
Figures 2 and 3 show the arrangements for T1, T2respectively T2, T3
based on the superordinate kite and dart shape. The use of the substitution
rules for the P2 tiling then lead to a nonperiodic tiling (Fig. 10). Please
note that the right sided arrangements of Figures 2 and 3 are completely
part of the left sided ones.
Figure 2: Arrangements for T1, T2and T2, T3.
Figure 3: Tiles of Figure 2 with decoration.
The dashed kite and dart shapes in Figure 3 can be completely deco-
rated or filled by rhombs, including half rhombs with their diagonals on
the dashed edges. The arrangements in Figures 2 and 3 can also be inter-
preted as modifications of the kite and dart shape, as mentioned in [2] on
page 539, where the long and short sides of kite and dart are replaced by
two J-curves.
2 Variants of the Prototiles
From T1and T2we can get two new undecorated prototiles, nicknamed
Snake and Dog, by shifting two outside half thin rhombs to the corre-
sponding gaps (Fig. 4)3.
T1Snake T2Dog
Figure 4: Building the snake and dog tiles.
Again the snake and dog tiles can be transformed into further and
smoother shapes given by a hexagon and a pentagon, both concave and
3The black dots within the tiles should represent the animals’ eyes only.
irregular. Figures 5 and 6 show this process from the left to the right by
shifting three or four half rhombs as shown. T8and T9, as well as Snake
and Dog, are balanced tiles with the same acreage as T1and T2.
Snake (b) (c)T8
Figure 5: Building T8from the Snake.
Dog (b) (c)T9
Figure 6: Building T9from the Dog.
Figure 7: T8and T9with Robinson triangles.
As well as the Penrose tiles, T8and T9can be decomposed into Robin-
son triangles. T8can be decomposed into two golden gnomons and two
golden triangles, T9into four golden gnomons and two golden triangles
(Fig. 7).
The decorated snake and dog tiles (Fig. 5band 6b) lead to a P3 tiling,
whereas a tiling of the decorated T8and T9tiles will always contain small
errors due to incomplete edges. These incorrect areas are related to the
decoration in the top corner of T8and the lower left corner of T9. Figure
8 shows an aperiodic set of two decorated prototiles (based on T8and T9),
which allow an exact P3 tiling again (Fig. 12). We leave it to the reader
to locate the three areas with the rectified edge-errors.
T10 T11
Figure 8: Aperiodic set for a P3 tiling.
Without substitution rules the undecorated T8,T9, and T11 tiles also
allow periodic tilings (Fig. 9). A tessellation of the plane only with T8or
T10 tiles is not possible.
Figures 11 and 12 show larger portions with the snake and dog tiles
and the undecorated T10, T11 set. Note the special feature in the last one,
where the T11 tiles never touch each other.
(a) (b)
Figure 9: Matching rules for periodic tilings with T8,T9and T11 .
The substitution rules for all aperiodic sets in this article are the same
as for the Penrose kite and dart, due to their same mld-class. Figure 10
shows these rules using T8and T9as an example.
Figure 10: Substitution rules for T8and T9.
Figure 11: Snake and Dog tiling.
Figure 12: Tiling with the undecorated T10 , T11 set.
3 Aperiodic Monotiles
One of the most exciting open problems in plane geometry is the exis-
tence of an aperiodic monotile. It asks about a single connected prototile
that by itself forms a strongly aperiodic set. Such a tile can tessellate
the Euclidean plane only in a nonperiodic way without matching rules.
The smallest aperiodic sets known to date consist of two prototiles, like
the Penrose tiles. Note that the P2 and P3 sets must be modified to get
strongly aperiodic sets without matching rules. Examples can be seen in
[2] on page 539 (kite, dart) and page 544 (rhombs).4
The currently best approximations to an aperiodic monotile were given
by Petra Gummelt in 1996, and Joshua Socolar and Joan Taylor in 2010.
4Further examples: tiling#Rhombus tiling (P3)
Gummelt constructed a decorated decagonal tile and showed that when
two kinds of overlaps between pairs of tiles are allowed, these tiles can
cover the plane only nonperiodically [3]. Socolar and Taylor presented
an undecorated, but not connected aperiodic monotile that is based on a
regular hexagon [5].
Figure 13 shows an approximation to an aperiodic monotile from the
Penrose tiles, a shape that represents both kite and dart as well as possible.
Please note that it is not possible to create a truly aperiodic monotile with
a 5-fold rotational symmetry from the Penrose prototiles because they do
not have the same acreage. But other shapes as given by T12 are possible.
It is also possible to modify edges (e.g. by identically shaped bumps and
notches) to enforce the tiling rules.
T12, an irregular concave dodecagon, is based on a connected pair of
T10, T11 tiles (Fig. 13a), and can be subdivided into a rhombus (a, b, c, p),
two congruent kites (c, d, r, p)and (g, h, i, r), two non-congruent trape-
zoids (d, e, f, r)and (i, j, k, r), which we will denote by T13 and T14, and
an irregular quadrilateral (l, n, q, r)(Fig. 13b).
Figure 13: The approximated aperiodic monotile T12 .
Several properties and common features of the Penrose tilings involve
the golden ratio ϕ= (1 + 5)/21.618 [2]. The edge lengths in Figure
13b, in relation to the unit length edges of the decorated rhombs, are
|pa|=|pc|=|qr|=|fr|=ϕ+ 1,|dr|=|gr|=|hj|=|ir|=|lr|=
|pr|= 2ϕ+ 1, and |de|=|kl|=|lm|= 2.
AT12 tiling contains always five types of gaps. An irregular triangle
(l, m, n), an irregular quadrilateral (n, o, p, q), and three types of rotors
(concave pentadecagons) that can be subdivided into T13 and T14 tiles
(Fig. 13b and 15). Note that a connected pair of T13 and T14 could build a
rhombus with edge lengths of 2ϕ+ 1. The decorated T12 tiles (Fig. 13a)
also allow a P3 tiling with the mentioned gaps. The substitution rules for
T12 are given in Figure 16.
Figure 14: A T12 tiling with gaps.
The possibility of an exact P3 tiling by the decorated tile sets men-
tioned in this article can be proved with the seven possible vertex figures
in a P2 tiling. Details are given in [2] on page 561 or at Wikipedia5.
5 tiling#Kite and dart tiling (P2)
T14 T13
T14 T14
Figure 15: The three types of rotor gaps in a T12 tiling.
Figure 16: Substitution rules for T12 .
[1] D. Frettl¨
oh, E. Harriss, F. G¨
ahler: Tilings encyclopedia,
[2] Gr¨
unbaum, Branko and Shephard, G. C. (1987), Tilings and Patterns,
New York: W. H. Freeman, pp. 519–548, ISBN 978-0-7167-1193-3.
[3] Gummelt, Petra (1996), Penrose tilings as coverings of congruent
decagons, Geometriae Dedicata, 62 (1), doi:10.1007/BF00239998.
[4] Penrose, Roger (1979–80), Pentaplexity: A class of nonperiodic
tilings of the plane, The Mathematical Intelligencer, 2: pp. 32–37,
doi:10.1007/BF03024384, S2CID 120305260.
[5] Socolar, Joshua E. S. and Taylor, Joan M. (2011), An Aperiodic
Hexagonal Tile, Journal of Combinatorial Theory, Series A. 118 (8):
2207–2231, doi:10.1016/j.jcta.2011.05.001.
ResearchGate has not been able to resolve any citations for this publication.
The open problem of tiling theory whether there is a single aperiodic two-dimensional prototile with corresponding matching rules, is answered for coverings instead of tilings. We introduce admissible overlaps for the regular decagon determining only nonperiodic coverings of the Euclidean plane which are equivalent to tilings by Robinson triangles. Our work is motivated by structural properties of quasicrystals.
We show that a single prototile can fill space uniformly but not admit a periodic tiling. A two-dimensional, hexagonal prototile with markings that enforce local matching rules is proven to be aperiodic by two independent methods. The space--filling tiling that can be built from copies of the prototile has the structure of a union of honeycombs with lattice constants of $2^n a$, where $a$ sets the scale of the most dense lattice and $n$ takes all positive integer values. There are two local isomorphism classes consistent with the matching rules and there is a nontrivial relation between these tilings and a previous construction by Penrose. Alternative forms of the prototile enforce the local matching rules by shape alone, one using a prototile that is not a connected region and the other using a three--dimensional prototile.