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[1] SHECHTMAN, D., BLECH, I., GLATIAS, D. and CAHN, J. W. (1984): A metallic phase with long-ranged orientational order and no translational symmetry, Phys. Rev. Lett., 53, 1951-1953.

[2] Husimi, K. (1969): Moyo no kagaku (Science of patterns) XXV Midare (Disorder, irregularity or aperiodicity), Sugaku Seminar, 8, June p. 40-46 (in Japanese).

[3] PENROSE, R. (1974): The role of Aesthetics in pure and applied mathematical research, Bull. Inst. Math. & its Applns., 10, 266-271.

[4] GARDNER, M. (1977): Extraordinary nonperiodic tiling that enriches the theory of tiles, ScL Amer., Jan. p. 110-121.

[5] PENROSE, R. (1979/1980): Pentaplexity, Math. Intell., 2, 32-37.

[6] DE BRUIJN, N. G. (1981): Algebraic theorry of Penrose's non-periodic tilings of the plane, Kon. NederL Akad. Wetensch. Proc. Ser., A84, 39-52 and 53-66.

[7] MACKAY, A. L. (1981): De nive quinquangula - On the pentagonal snowflake, Sov. Phys. Crystallogr., 26, 517-522.

[8] MACKAY, A. L. (1982): Crystallography and the Penrose pattern, Physica, 114A, 609-613.

[9] LEVINE, D. and STEINHARDT, P. J. (1984): Quasicrystals: A new class of ordered solids, Phys. Rev. Lett., 53, 2477-2480.

[10] As the reviews, MACKAY, A. L. (1985): Some answers but more questions, Nature, 316,  17-18. 

[10a] NELSON, D. R. and HALPERIN, B. J. (1985): Pentagonal and icosahedral order in rapidly cooled metals, Science, 229, 233-238. See references therein.

[11] OGAWA, T. (1985): On the structure of a quasicrystal-Three-dimensional Penrose transformation, J. Phys. Soc. Jpn., 54, 3205-3208.

[12] MIYAZAKI, K. (1985): An Adventure in Multidimensional Space - The Art and Geometry of Polygons, Polyhedra, and Polytopes, Wiley, (Original Japanese edition; Katachi to ktikan, Asakura Publ. Co., Tokyo, 1983).

[13] HIRAGA, K., HIRABAYASHI, M., INOUE, A. and MASUMOTO, T. (1985): lcosahedral quasicrystals of a melt-quenched AI-MN alloy observed by high-resolution electron microscopy, Sci. Rep. Res. Inst. Tohoku Univ., A32, 309-314.

[14] BURSIL, L. A. and LIN, P. J. (1985): Penrose tiling observed in a quasi-crystal, Nature, 316, 50-51.

[15] MACKAY, A. L., private communication.

[16] PORTIER, M., SHECHTMAN, D., GLATIAS, D. and CAHN, J. W.: Crystallographic aspects and high resolution electron microscopy of the Al₆Mn quasicrystal, French J. Electron Microscopy, IOA, 30.

[17] KIMURA, K., HASHIMOTO, T., SUZUKI, K., NAGAYAMA, K., INO, H. and TAKEUCHI, S. (1985): Stoichiometry of quasicrystalline AI-Mn alloys, J. Phys. Soc. Jpn., 54, 3217-3219.

[18] HIRAGA, K., HIRABAYASHI, M., INOUE, A. and MASUMOTO, T.: Structure of AI-MN quasicrystal studied by high resolution electron microscopy, J. Phys. Soc. Jpn., 4080.

[19] KIMURA, K., HASHIMOTO, T., SUZUKI, K., NAGAYAMA, K., INO, H. and TAKEUCHI, S. (1986): Structure and stability of quasicrystalline AI-Mn alloys, J. Phys. Soc. Jpn., 55, 534-543.

[20] OGAWA, T. (1986): Three-dimensional Penrose transformation and the ideal quasicrystal, in Science on Form, Eds. Y. Kato et al., KTK Scientific Publishers, Tokyo, pp. 479-489.

[21] For example, HENLEY, C. L. (1986): Spherical packings and local environments in Penrose tilings, Phys. Rev. B, 34, 797-816.

[22] OGAWA, T. (1988): Symmetry of three-dimensional quasierystals, to be published in the Proceedings of the International Workshop on Quasicrystals (Aug. 30-Sept. 5, I987, in Beijing), Ed. K. H. Kuo, Trans Tech Publications Ltd., Switzerland.
 
 

Note added in proof

Two numerical values 104 appearing in Section 4 and 160 appearing in Section 4 and Appendix B (3) are the corrected ones respectively from the original incorrect 164 and 400, which were reported in the talk at the Seminar and written in Ref. [11].

In the text, the Penrose transformation is described as an expansion or structuration of two rhombohedra A₆ and O₆ so that the logical structure map be clearly seen. The obtained structure may be more clearly seen by noting that a point is transformed into a flower dodecahedron Fl₆₀ consisting of 20 A₆'s (exactly speaking, a concave 60-hedron which looks like a dodecahedron each of whose 12 faces is a five-petalous flower) [20]. In this point of view, the tiling elements should not be regarded as A₆ and O₆ but as Fl₆₀, F₂₀, C = 2K₃₀-O₆ and O₆.

Though the concept of self-similarity is useful, it may be better regarding it as a result of some local rule. Self-similarity, as a basic principle of a growing process, is a little curious: The situation may remind one a horoscope. If self-similarity is really the basic rule, an atom may have to know the position of its own star in deciding its position to attach. Therefore, it is expected that the basic rule is replaced by some local rule which leads the equivalent result.

In the discussion, Bennema gave a comment about the higher-dimensional crystallography. After that a very similar three-dimensional structure to the present model has been obtained as the projection of certain selected points of a six-dimensional simple cubic lattice to the three dimensional space [21]. The relation between the two models have not been known until the author figured it out [22].

Some two-dimensional Penrose-like tilings are obtained by similar hierarchical transformations. Some of them have pentagonal symmetry and some others have octahedral symmetry [20].