Conjugated coupling for two Penrose tiles pair and their elementary unit cell , for translation tiles

Chung Yuan Kung , retired Professor , Chung Hsing University, 250, Kuo Kuang rd. 402,Taichung, Taiwan, R.O.C

Key words: decagon, Penrose tiling, Pentagon shape , translational periodic Penrose tiles, unit cell

Abstract: We constructed two different types of pentagonal Penrose tiles, And using them to make a translational periodic Penrose tiles (crystals)

The Penrose tiling (1) is one of the most famous and complex tessellations and has been drawn and represented in many different ways, most of them are in artistic form. This five-fold symmetric Penrose tiling diagram helps to explain the five-fold nature of quasicrystals ( 2) and has sparked many studies in this field ( 3,4 ).                                                                                                                      In this paper, the authors only focus on strictly defined defect-free Pentose tiles and how to systematically and orderly extend tiles to infinity. The following is one of the most systematic procedures for drawing defect-free Penrose tiles, the most important of which is to ensure that the tile is defect-free under strict definitions, with a perceivable and repeatable internal structure, and can be extended to infinity.

Two rhombuses with acute angle 36 and 72 degree can make six different decagons Fig 1. In 1974 almost fifty years ago Penrose (1) using these two rhombic created a famous Penrose-tile which can be extended unlimitedly, as shown in Fig 1. According to the original pattern configuration found by Penrose, several rhombusl (bricks) were added outwards, and a regular pentagonal pattern with five decagonal on each side was produced, as shown in Figure 2a. The pattern contains three decagons of a, type-b, and type-c (or could be only type-a and type-b two decagons). The pentagon shape in fig 2a is little bit bloated. At the same time, a counterpart to a bulky Penrose tile was found, with a slightly slimmer pentagonal shape.as shown in fig 2 b

All the vertices of each pentagonal Penrose tiles are designated as a-type decagons . ( Here, it must be noted that these five vertices can be replaced by other types of decagons, which may produce extensions in different way). For type-a decagons in the dilated pentagonal tile (fig 2 a) are specially marked with red as red stars and for type-a decagon on the slim pentagonal (FIG 2b) one are marked green color , so as to lay the groundwork of inner defect revision for future coupling

alignment. The central regions for both bloated (dilated) and slim Pentagons are marked in dark blue for beneficial of coupling alignment.

Several examples of    pair coupling are shown below as in the fig 3   and fig 4 and fig 5    , Some of them still retain the remaining mismatched regions to show the difficulty of making defect-free Penrose tiles. The different coupling mentioned above may result defects in the overlapping seam area,   Most cases, these coupling defects can be easily removed to accomplish an even larger zero-defect Penrose tiles

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The details of these Bloated-slim types of coupling are demonstrated as shown in fig 3 , the examples of bottom-to-bottom opposite direction coupling type, and  Figure 4 the examples vertex - to-vertex opposite direction coupling type and fig 5 bottom-vertex same direction type . Fig 3a shows a simple skin coupling, only the decades on outer layer are involved and perfectly coupled, no revision needed. Fig 3b demonstrated a deeper coupling, with outer three layers involved. Fig 3b1 is coupling before revision still retain the remaining mismatched regions and fig 3b2 is after revision, and the new type-e ingenerated (*created) to meet defect free  requirements , Fig 3c shows a even deeper coupling, covers almost all the region slim type tile , as shown in fig 3c1 and able to make an unbelievable defect free revision as shown in Fig 3c2. Fig 3d demonstrated a completed coupling before and after  revision of these two pair, Fig3d2, revealed a circular Penrose tile. Fig 3e f demonstrated a slanted shift coupling of these Penrose -tile pairs and again ingenerate a type-e decagon for meeting defect free requirement. Fig 4a1 is the  result of a deep coupling before revision, Fig 4 a2 is the same coupling after revision,

fig 5 shows the coupling results of these two Penrose tiles on the same direction.

bottom-vertex      coupling type . 5a1, before revision 5a2 after revision, (5b) a deeper coupling after revision ,( 5c) slant shift coupling           after revision As described in fig3 and fig 4, most defects on overlapped coupling can be removed with some but not straight forward.. The defects are left for reader to revise them and will not be discussed here. In fact, If not count the defect ingenerated, all kind of coupling can be performed easily for these two types of Penrose tiles, as shown in the case in fig 5 . Since these two Penrose tiles are pretty much coupling possibility with one another at many different coupling condition, they can be defined as mutual

conjugate. The different coupling mentioned above may result defects in the

overlapping seam area, in most cases, these coupling defects can be easily removed to accomplish a larger zero-defect Penrose tiles.*( be noted here that the traditional Penrose tiling allowed the existence of defects)

All these coupled pairs can be translational (or slanted shift) coupled by their self as shown in fig 6 a ,b., to form a translational aperiodic Penrose tiles. In this paper, only fig 3c2 is employed to demonstrate self-coupled to infinity (as shown in fig 7 a)

and from there on extracted the elementary unit cell as shown in fig 7b. And this elementary unit cell can be tessellated to construct a translational periodic Penrose tiling.

Much like rolling up different types of carbon nanotubes, four different Penrose tubes can be fabricated by rolling up a translating Penrose tiles along the x-direction or along the y-direction or along an 18-degree or 54-degree tilt angle.. One may do rolling on an unit cell , a three dimensional Penrose unit ball may be constructed. All these have been in another paper. ( 5)

Summary:       In this short article, two different types Pentagon shape of Penrose tiles are constructed based in original Penrose tiling pattern and it is found that these two types of Pentagonal shapes can be well coupled in different depth and defect free.    An elementary unit cell can be extracted from repeated coupling results, and this unit cell can be recombined (tessellated, mosaiced) to build translational periodic Penrose tiles. And a model of Penrose multiple concentric nanotube is        established to facility a further 3-D Penrose-tiling studies.

Figure captions;

Figure      1)    Original Penrose tile

Figure      2)     a regular pentagonal pattern with five decagonal on each side, (a) Bloated , (b)   slim

Figure      3)     the examples of bottom-to-bottom opposite direction coupling type

, ( 3a )    shows a simple skin perfectly coupling    (3b) deeper coupling, with outer three layers involved  (3b1)                  coupling before revision remaining mismatched regions and fig ( 3b2) is after revision, ( 3c) shows an even deeper coupling, covers almost all the region slim type tile , (3c1) before revision, ( 3c2 ) after revision                  ( 3d) completed coupling (3d1)before and(3d2) after revision            revealed a circular Penrose tile. Fig 3e        a slanted shift coupling .

Figure 4 the examples vertex - to-vertex opposite direction coupling type Fig 4a result of a deep coupling before revision, Fig 4 b                                           after revision,

fig 5    bottom-vertex         coupling in the same direction type . 5a1, before revision 5a2 after revision, (5b) a deeper coupling                after revision ,( 5c) slant shift coupling after revision

fig 6)    coupled pairs can be translational (or slanted shift) coupled by their self as shown in fig 6 a ,b.,

fig 7) fig 7 a)Tile in fig 3c2 self-coupled to infinity,  fig 7b anunit cell    extracted from tile in fig ta   fig 8)unit cell can be tessellated to construct a translational

periodic Penrose tiling.

fig 4 a 1

fig 4 a 2

i

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1)     Penrose, R. (1974) The role of aesthetics in pure and applird mathematical research. Bulletin of the Institute of Mathematics and Its Applications, 10, 266-271.

2. Quasicrystals.I definition and structure, D. Levine, and Paul J. Steinhardt

Phys. Rev Reference:

3. . B 34, 617596 – Published 15 July 1986