Chief Edito, titled: “The definition of unlimited, pentagonal shape Penrose tile that is defect-free.”
I believe that the article will significantly contribute to the body of literature as it tends to modify the original Penrose tile into a pentagon. It was found that the original penrose tile is composed of three types of decagons: type-a, type-b and type-c, and that Pentagonal penrose tiles may not always be five –fold or mirror symmetry since they consist of any combination of six types of decagons.
Given that Pentagonal penrose tiles should consisted of any combination of six types of decagons, for further investigations need to be carried out in order to determine that their coupling to form infinite Penrose tiles with invisible boundaries is defect free.
This manuscript has not been partially or fully published or presented elsewhere, and it is not being considered by another journal. I have read and understood your journal’s policies, and we believe that neither the manuscript nor the study violates any of these. There are no conflicts of interest to declare
defiition of unlimited, pentagonal shape Penrose tile that is defect-free
ABSTRACT
Penrose tiles has been reported to creat patterns with irregular repetitions. They exhibit "quasiperiodicity," which indicates that they lack translational symmetry despite having long-range order. In this study, the very original Penrose tile was modified into a pentagon simply by adding some tiles out ward and found that the original penrose tile is composed of three types of decagons: type-a, type-b and type-c. Also, since the Pentagonal penrose tiles should consisted of any combination of six types of decagons, they may not always be five –fold or mirror symmetry.
Key words: Decagon, Penrose tiling,defect, pentagon shape.
INTRODUCTION
It is cumbersome to design Penrose tiles that are defect-free as a result of thier aperiodic nature and the complexity of the rules governing their arrangement (Penrose, 1974). Socolar and Steinhardt (1986) derive a prescription for constructing the special LI class corresponding to the three-dimensional (3D) icosahedral analogue of the original Penrose tilings (configurations which can be generated by local matching and deflation rules). In a study, Levine and Steinhardt (1986) use the case of icosahedral orientational symmetry to illustrate how to compute the diffraction pattern of a quasilattice. Moreover, in resolving the structure of the decagonal quasicrystal Al72Ni20Co8, Steinhardt et al. (1998) present experimental evidence for the quasi-unit-cell model.
It was a stroke of luck that, in this study, the very original Penrose tile (Figure 1) was modified into a pentagon (Figure 2) simply by adding some tiles out ward and found that the original penrose tile is composed of three types of decagons: type-a, type-b and type-c. These three types of decagons belong to one of the the six different decagons consisting of two rhombuses with acute angles of 36 and 72 degrees (Figure 4). The purpose of using these five-fold Pentagonal shapes as standard examples for Penrose tiles was to demonstrate that they can be extended to infinity. When Penrose tiles were being made to infinity in this case, one rigorous newly specified "defect" was added.
Since the Pentagonal penrose tiles should consisted of any combination of six types of decagons, they may not always be five –fold or mirror symmetry. Thus, it can be proven that no potential defects are created when coupling them together to form infinite Penrose tiles with invisible boundaries.
For ease of future follow-up extension work, it is specifically stated here for pentagonal Penrose tile that “all single bricks must belong to any one of the six decagons, and all decagons in the pentagonal penrose tiling must not miss any brick”. This minor adjustment causes the Penrose tiles to extend in a manner that is logical, predictable, and completely visible when they reach infinity. It provides the most reliable assurance that Penrose tiles are defect-free.
With conventional Penrose tiles, defects are not a problem (they need not be taken into account because it only fixes the fitting of two rhombuses).
Here is an illustration of a defect-free Penrose tile that uses the recently discovered standard pentagonal tile to show how easy to reverse the defects generated in the process of coupling. The unique five-fold symmetry of the standard pentagonal Penrose tile with 5 decagons on each side shows that all decagons behave internally the same after a 72-degree rotation.Therefore, we can properly cut the edge along the center of the top rightmost red star, as shown in Figure 2a. The tile in Figure 2a should then be rotated 72 degrees to the right and cut through the center of the leftmost red star. The cutting results of the first round of pre-coupled Penrose tiles are shown in Figure 2b.
Then the Penrose tile in Figure 3 is used as the unit of the 72 degree rotational tessellation (mosaic). Figure 4 shows the tessellation (splicing) results of the first 72-degree rotation and the second 144-degree rotation, with the dark grey colour representing the defects generated during the tessellation process. A larger pentagon with ten decagonal Penrose tiles on each side is constructed following the fifth round of splicing (that is, after the first round of coupling). Using the same procedure, defect-free Penrose tiles with 20 decagons on each side can be obtained after the second round of coupling, as shown in Figure 4. Absolutely, these coupling defects (unwanted decagons) created in the seam area are all transformed into a proper decagon. Also, by using the same procedure, an infinitely number of defect-free Penrose tiles can be obtained.
It has been reported (ref 2) that the expansion technique can be applied to different types of Penrose tiles. The results of coupling two Penrose tiles with two distinct inner structures are shown in Figure 5. Sometimes it requires a lot of effort to resolve the unanticipated (unwanted) defects in the tessellation seam area. To reduce the difficulty of the tiling work (for the later larger tiles), it is necessary to identify the potential "build-in defects" in the pentagonal Penrose tiles at earliest stage. The majority of Pentagonal-shaped Penrose tiles do not always have a five-fold symmetry, so it often takes more time than necessary to resolve the defects in the seam region.
Summary
In this short communication, The simplest and easiest-to-understand coupling scheme is demonstrated through the use of a pentagonal Penrose tile as an illustration. Here, a concept of defect is introduced for the coupling of different types of Pentagonal Penrose tiles with a strict condition. Finally, the defect free Penrose tile of unlimited size is conceiveable.
Figure 1. Two fat and thin rhombuses with angles of 36 degrees and 72 degrees forming six different basic configurations of decagons. Penrose tiling under the strict definition requires that all fat and thin rhombuses (bricks) must be contained in any regular decagon
Figure 2. (A) Original Penrose tile, (b) superposition coincidence of the original penrose tile and a pentagonal shape Penrose tile.
Figure 3. The pentagonal shape penrose tile with five-fold symmetry and mirror symmetry.
Figure 4. (a) The pentagonal shape penrose tile with proper cut on the far end right, (b) the right side cutted pentagonal shape penrose tile with properlyappropriate cut on the far end left side.
Figure 5. The configuration of splicing of the properly appropriate cut (pentagonal shape) penrose tile with first 72 degree rotation and 144 degrees rotation. defect region are marked as dark grey, can be replaced by a type-b decagon. Properly cut (pentagonal) Penrose tiles are first rotated 72 degrees for the splicing of two tiles with each other, then rotated 144 degrees to add a splicing of third tile, marked as dark gray tile defect areas can be replaced with a decagon b-type.
Figure 6. First round coupling resulting a pentagonal shape Penrose tile with ten decagons per side.
Figure 7. Second round coupling resulting a pentagonal shape Penrose tile with twenty decagons per side.
Figure 8. The result of coupling with different coupling schemes.
REFERENCES
Penrose R (1974). The role of aesthetics in pure and applied mathematical research. Bulletin of the Institute of Mathematics and Its Applications, 10:266-271.
Levine D, Steinhardt PJ (1986). Quasicrystals.I definition and structure. Phys. Rev. B 34: 617596 – Published 15 July 1986.
Socolar JES, Steinhardt PJ (1986). Quasicrystals. II. Unit-cell configurations. Phys. Rev. B 34:617 – Published 15 July 1986.
Steinhardt P, Jeong H-C, Tsai A (1998). Experimental verification of the quasi-unit-cell model of quasicrystal structure. Published 5 November 1998
5 ) Kun CY. Unpublished
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