Academic Journal of Engineering and Technology Science, 2023, 6(7); doi: 10.25236/AJETS.2023.060706.
Du Peiwen
School of Electrical and Computer Engineering, Guangzhou Southern College, Guangzhou, Guangdong, 510900, China
Penrose tiling is an example of the two-dimensional quasicrystal tessellation model, which British mathematician Roger Penrose first proposed in 1974. In this paper, we investigate the construction rules of the P3 Penrose tiling in the two-dimensional quasicrystal theoretical model. We successfully generate the first seven generations of complete P3 Penrose tiling using the self-similar transformation method.
quasicrystal; mosaic model; penrose tiling; self-similar transformation method; quasi-periodic structure
Du Peiwen. Construction Rules of P3 Penrose Tiling and Its MATLAB Implementation. Academic Journal of Engineering and Technology Science (2023) Vol. 6, Issue 7: 30-34. https://doi.org/10.25236/AJETS.2023.060706.
[1] Shechtmen D. S., Blech I., Gratias D., et al. Metallic Phase with Long- Range Orientational Order and No Translational Symmetry. Phys[J]. Rev. Lett, 1984, 53(20):1951-1953.
[2] R. Penrose. A Class of Non-Periodic Tilings of the Plane. 1979, Oxford, MI. U. K. (32-37).
[3] He L. X., Li X. Z., Zhang Z., et al. One-dimensional quasicrystal in rapidly solidified alloys[J]. Phys. Rev. Lett, 1988(61):1116-1118.
[4] A. V. Shutov, A. V. Maleev. Study of Penrose Tiling Using Parameterization Method [J]. Crystallography Reports, 2019, 64(3):376-385.
[5] Collins Laura C; Witte Thomas G; Silverman Rochelle; Green David B; Gomes Kenjiro K . Imaging quasiperiodic electronic states in a synthetic Penrose tiling[J]. Nature Communications, 2017, 8(1): 15961.
[6] Guo Xin. 8th and 12th order symmetries and related quasicrystal discoveries [J]. Physics, 1990, 20(1):11-14.
[7] Youyan Liu, Xiujun Fu, Xiuqing Huang. Physical properties of one-dimensional quasicrystals [J]. Advances in Physics, 1997(17):1-23.
[8] Fu Xiujun, Cheng Bolin, Zheng Dafang, Liu Youyan. Two-dimensional Fibonacci quasicrystal electron energy spectra[J]. Advances in Physics, 1991(40):1667-76.
[9] Liao Longguang. Structural properties of eight and twelve times symmetric quasicrystal covering models [Z]. Master's thesis, South China University of Technology Man, 2008.
[10] Peng Benyi. Conformations and transformation properties of Penrose puzzles[Z]. Master's thesis, South China University of Technology, 2015.
[11] Guo Xin. Quasi-crystals [J]. Science Bulletin, 1990(22):1691-1695.
[12] Peng Benyi, Fu Xiujun. Configurations of the Penrose Tiling beyond Nearest Neighbors[J]. Chinese Physics Letters, 2015, 32(5)056101-1-5.
[13] Liao Longguang, Fu Hong, Fu Xiujun. Self-similar transformation and quasi-cell construction of quadratic symmetric quasi-periodic structures[J]. Journal of Physics, 2009, 58(10):7088-93.
[14] Peng Caixia. Statistical properties of complex networks based on Penrose puzzles[Z]. Master's thesis, South China University of Technology Man, 2016.
[15] N. G. de Bruijn. Sequences of zeros and ones generated by special production rules[Z]. MathematicS, Proceedings A 84(1), March 20, 1981:27-37.
[16] Fu Xiujun, Zhang Xiaowei and Hou Zhilin. Band structure and localization of electronic states in a fivefold symmetric quasicrystal model [J]. Journal of Non-Crystalline Solids, 2008(354):1740-1743.
[17] Guo Xin. Five times symmetry and quasicrystalline states[J]. Physics, 1985, 14(8):449-451.
[18] Jeong H, Tombor B, Albert R, et al. The large-scale organization of metabolic networks[J]. Nature, 2000(407):651-654.
[19] R. Penrose. Introducing to mathematics of quasicrystals. Boston: Academic Press [M], 1989.