Penrose tiling (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Penrose tiling" in English language version.

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ams.org

Radin 1996. Radin, Charles (April 1996). "Book Review: Quasicrystals and geometry" (PDF). Notices of the American Mathematical Society. 43 (4): 416–421.
Austin 2005a Austin, David (2005a). "Penrose Tiles Talk Across Miles". Providence: American Mathematical Society..
"However, as will be explained momentarily, differently colored pentagons will be considered to be different types of tiles." Austin 2005a; Grünbaum & Shephard 1987, figure 10.3.1, shows the edge modifications needed to yield an aperiodic set of prototiles. Austin, David (2005a). "Penrose Tiles Talk Across Miles". Providence: American Mathematical Society.. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
Austin 2005b Austin, David (2005b). "Penrose Tilings Tied up in Ribbons". Providence: American Mathematical Society..
"... any finite patch that we choose in a tiling will lie inside a single inflated tile if we continue moving far enough up in the inflation hierarchy. This means that anywhere that tile occurs at that level in the hierarchy, our original patch must also occur in the original tiling. Therefore, the patch will occur infinitely often in the original tiling and, in fact, in every other tiling as well." Austin 2005a Austin, David (2005a). "Penrose Tiles Talk Across Miles". Providence: American Mathematical Society..

mathscinet.ams.org

Gummelt 1996 Gummelt, Petra (1996). "Penrose tilings as coverings of congruent decagons". Geometriae Dedicata. 62 (1): 1–17. doi:10.1007/BF00239998. MR 1400977. Zbl 0893.52011.

archim.org.uk (Global: low place; English: low place)

Penrose 1978, p. 32 Penrose, Roger (1978). "Pentaplexity" (PDF). Eureka. Vol. 39. pp. 16–22. Archived from the original (PDF) on 9 August 2022. Retrieved 2 February 2021.. (Page numbers cited here are from the reproduction as Penrose, R. (1979–80). "Pentaplexity: A class of non-periodic tilings of the plane". The Mathematical Intelligencer. 2: 32–37. doi:10.1007/BF03024384. S2CID 120305260..)

archive.org (Global: 6^th place; English: 6^th place)

General references for this article include Gardner 1997, pp. 1–30, Grünbaum & Shephard 1987, pp. 520–548 &amp, 558–579, and Senechal 1996, pp. 170–206. Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
- Grünbaum & Shephard 1987, p. 520 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- Grünbaum & Shephard 1987, p. 584 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- Grünbaum & Shephard 1987, p. 525 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- Grünbaum & Shephard 1987, section 2.5 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- The P1–P3 notation is taken from Grünbaum & Shephard 1987, section 10.3 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- Grünbaum & Shephard 1987, section 10.3 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- "However, as will be explained momentarily, differently colored pentagons will be considered to be different types of tiles." Austin 2005a; Grünbaum & Shephard 1987, figure 10.3.1, shows the edge modifications needed to yield an aperiodic set of prototiles. Austin, David (2005a). "Penrose Tiles Talk Across Miles". Providence: American Mathematical Society.. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- Grünbaum & Shephard 1987, pp. 537–547 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- Grünbaum & Shephard 1987, p. 543 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- In Grünbaum & Shephard 1987, the term "inflation" is used where other authors would use "deflation" (followed by rescaling). The terms "composition" and "decomposition", which many authors also use, are less ambiguous. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..
- Grünbaum & Shephard 1987, p. 546 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3..

archnet.org (Global: 5,343^rd place; English: 3,881^st place)

"Indian Institute of Information Technology, Allahabad". ArchNet.

books.google.com (Global: 3^rd place; English: 3^rd place)

Berger 1966 Berger, R. (1966). The undecidability of the domino problem. Memoirs of the American Mathematical Society. Vol. 66. ISBN 978-0-8218-1266-2..
Kepler, Johannes (1997). The harmony of the world. American Philosophical Society. p. 108. ISBN 0-87169-209-0.
Zaslavskiĭ et al. 1988; Makovicky 1992 Zaslavskiĭ, G.M.; Sagdeev, Roal'd Z.; Usikov, D.A.; Chernikov, A.A. (1988). "Minimal chaos, stochastic web and structures of quasicrystal symmetry". Soviet Physics Uspekhi. 31 (10): 887–915. Bibcode:1988SvPhU..31..887Z. doi:10.1070/PU1988v031n10ABEH005632.. Makovicky, E. (1992). "800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired". In I. Hargittai (ed.). Fivefold Symmetry. Singapore–London: World Scientific. pp. 67–86. ISBN 978-981-02-0600-0..

cam.ac.uk (Global: 670^th place; English: 480^th place)

repository.cam.ac.uk

Colbrook; Roman; Hansen (2019). "How to Compute Spectra with Error Control". Physical Review Letters. 122 (25) 250201. Bibcode:2019PhRvL.122y0201C. doi:10.1103/PhysRevLett.122.250201. PMID 31347861. S2CID 198463498.

cea.fr (Global: low place; English: low place)

inac.cea.fr

Lançon & Billard 1988 Lançon, Frédéric; Billard, Luc (1988). "Two-dimensional system with a quasi-crystalline ground state" (PDF). Journal de Physique. 49 (2): 249–256. CiteSeerX 10.1.1.700.3611. doi:10.1051/jphys:01988004902024900. Archived from the original (PDF) on 20 July 2011. Retrieved 28 November 2009..

curbed.com (Global: 2,477^th place; English: 1,402^nd place)

sf.curbed.com

Kuchar, Sally (11 July 2013). "Check Out the Proposed Skin for the Transbay Transit Center". Curbed.

denverpost.com (Global: 770^th place; English: 472^nd place)

Rinaldi, Ray Mark (12 July 2019). "Two dueling Clark Richert exhibitions explore the career of one of Colorado's biggest painters". The Denver Post. Denver, Colorado. Retrieved 27 March 2025.

doi.org (Global: 2^nd place; English: 2^nd place)

General references for this article include Gardner 1997, pp. 1–30, Grünbaum & Shephard 1987, pp. 520–548 &amp, 558–579, and Senechal 1996, pp. 170–206. Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
- Gardner 1997, pp. 20, 23 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
  - Culik & Kari 1997 Culik, Karel; Kari, Jarkko (1997). "On aperiodic sets of Wang tiles". Foundations of Computer Science. Lecture Notes in Computer Science. Vol. 1337. pp. 153–162. doi:10.1007/BFb0052084. ISBN 978-3-540-63746-2.
  - Wang 1961 Wang, H. (1961). "Proving theorems by pattern recognition II". Bell System Technical Journal. 40: 1–42. doi:10.1002/j.1538-7305.1961.tb03975.x..
  - Gardner 1997, p. 5 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
    - Robinson 1971 Robinson, R.M. (1971). "Undecidability and non-periodicity for tilings of the plane". Inventiones Mathematicae. 12 (3): 177–190. Bibcode:1971InMat..12..177R. doi:10.1007/BF01418780. S2CID 14259496..
    - Luck 2000 Luck, R. (2000). "Dürer-Kepler-Penrose: the development of pentagonal tilings". Materials Science and Engineering. 294 (6): 263–267. doi:10.1016/S0921-5093(00)01302-2..
    - Gardner 1997, p. 6 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
      - Gardner 1997, p. 19 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, chapter 1 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Baake, M.; Kramer, P.; Schlottmann, M.; Zeidler, D. (December 1990). "Planar Patterns with Fivefold Symmetry as Sections of Periodic Structures in 4-Space". International Journal of Modern Physics B. 04 (15n16): 2217–2268. Bibcode:1990IJMPB...4.2217B. doi:10.1142/S0217979290001054.
        Penrose 1978, p. 32 Penrose, Roger (1978). "Pentaplexity" (PDF). Eureka. Vol. 39. pp. 16–22. Archived from the original (PDF) on 9 August 2022. Retrieved 2 February 2021.. (Page numbers cited here are from the reproduction as Penrose, R. (1979–80). "Pentaplexity: A class of non-periodic tilings of the plane". The Mathematical Intelligencer. 2: 32–37. doi:10.1007/BF03024384. S2CID 120305260..)
        "The rhombus of course tiles periodically, but we are not allowed to join the pieces in this manner." Gardner 1997, pp. 6–7 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 8 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, pp. 10–11 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 12 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 9 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 27 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 7 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Lord & Ranganathan 2001 Lord, E.A.; Ranganathan, S. (2001). "The Gummelt decagon as a 'quasi unit cell'" (PDF). Acta Crystallographica. A57 (5): 531–539. CiteSeerX 10.1.1.614.3786. doi:10.1107/S0108767301007504. PMID 11526302. Archived from the original (PDF) on 21 July 2011. Retrieved 13 December 2009.
        Gummelt 1996 Gummelt, Petra (1996). "Penrose tilings as coverings of congruent decagons". Geometriae Dedicata. 62 (1): 1–17. doi:10.1007/BF00239998. MR 1400977. Zbl 0893.52011.
        Steinhardt & Jeong 1996; see also Steinhardt, Paul J. "A New Paradigm for the Structure of Quasicrystals". Archived from the original on 18 October 2016. Retrieved 29 November 2009. Steinhardt, Paul J.; Jeong, Hyeong-Chai (1996). "A simpler approach to Penrose tiling with implications for quasicrystal formation". Nature. 382 (1 August): 431–433. Bibcode:1996Natur.382..431S. doi:10.1038/382431a0. S2CID 4354819..
        Colbrook; Roman; Hansen (2019). "How to Compute Spectra with Error Control". Physical Review Letters. 122 (25) 250201. Bibcode:2019PhRvL.122y0201C. doi:10.1103/PhysRevLett.122.250201. PMID 31347861. S2CID 198463498.
        Luck, R (1990). "Penrose Sublattices". Journal of Non-Crystalline Solids. 117–8 (90): 832–5. Bibcode:1990JNCS..117..832L. doi:10.1016/0022-3093(90)90657-8.
        Lançon & Billard 1988 Lançon, Frédéric; Billard, Luc (1988). "Two-dimensional system with a quasi-crystalline ground state" (PDF). Journal de Physique. 49 (2): 249–256. CiteSeerX 10.1.1.700.3611. doi:10.1051/jphys:01988004902024900. Archived from the original (PDF) on 20 July 2011. Retrieved 28 November 2009..
        Godrèche & Lançon 1992; see also Dirk Frettlöh; F. Gähler & Edmund Harriss. "Binary". Tilings Encyclopedia. Department of Mathematics, University of Bielefeld. Godrèche, C; Lançon, F. (1992). "A simple example of a non-Pisot tiling with five-fold symmetry". Journal de Physique I. 2 (2): 207–220. Bibcode:1992JPhy1...2..207G. doi:10.1051/jp1:1992134. S2CID 120168483..
        Zaslavskiĭ et al. 1988; Makovicky 1992 Zaslavskiĭ, G.M.; Sagdeev, Roal'd Z.; Usikov, D.A.; Chernikov, A.A. (1988). "Minimal chaos, stochastic web and structures of quasicrystal symmetry". Soviet Physics Uspekhi. 31 (10): 887–915. Bibcode:1988SvPhU..31..887Z. doi:10.1070/PU1988v031n10ABEH005632.. Makovicky, E. (1992). "800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired". In I. Hargittai (ed.). Fivefold Symmetry. Singapore–London: World Scientific. pp. 67–86. ISBN 978-981-02-0600-0..
        Lu & Steinhardt 2007 Lu, Peter J.; Steinhardt, Paul J. (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture" (PDF). Science. 315 (5815): 1106–1110. Bibcode:2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056. S2CID 10374218..
        Kemp 2005 Kemp, Martin (2005). "Science in culture: A trick of the tiles". Nature. 436 (7049): 332. Bibcode:2005Natur.436..332K. doi:10.1038/436332a..

ernet.in (Global: 4,058^th place; English: 2,544^th place)

iisc.ernet.in

Ramachandrarao, P (2000). "On the fractal nature of Penrose tiling" (PDF). Current Science. 79: 364.

materials.iisc.ernet.in

Lord & Ranganathan 2001 Lord, E.A.; Ranganathan, S. (2001). "The Gummelt decagon as a 'quasi unit cell'" (PDF). Acta Crystallographica. A57 (5): 531–539. CiteSeerX 10.1.1.614.3786. doi:10.1107/S0108767301007504. PMID 11526302. Archived from the original (PDF) on 21 July 2011. Retrieved 13 December 2009.

hal.science (Global: low place; English: low place)

Godrèche & Lançon 1992; see also Dirk Frettlöh; F. Gähler & Edmund Harriss. "Binary". Tilings Encyclopedia. Department of Mathematics, University of Bielefeld. Godrèche, C; Lançon, F. (1992). "A simple example of a non-Pisot tiling with five-fold symmetry". Journal de Physique I. 2 (2): 207–220. Bibcode:1992JPhy1...2..207G. doi:10.1051/jp1:1992134. S2CID 120168483..

harvard.edu (Global: 18^th place; English: 17^th place)

ui.adsabs.harvard.edu

General references for this article include Gardner 1997, pp. 1–30, Grünbaum & Shephard 1987, pp. 520–548 &amp, 558–579, and Senechal 1996, pp. 170–206. Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
- Gardner 1997, pp. 20, 23 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
  - Gardner 1997, p. 5 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
    - Robinson 1971 Robinson, R.M. (1971). "Undecidability and non-periodicity for tilings of the plane". Inventiones Mathematicae. 12 (3): 177–190. Bibcode:1971InMat..12..177R. doi:10.1007/BF01418780. S2CID 14259496..
    - Gardner 1997, p. 6 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
      - Gardner 1997, p. 19 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, chapter 1 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Baake, M.; Kramer, P.; Schlottmann, M.; Zeidler, D. (December 1990). "Planar Patterns with Fivefold Symmetry as Sections of Periodic Structures in 4-Space". International Journal of Modern Physics B. 04 (15n16): 2217–2268. Bibcode:1990IJMPB...4.2217B. doi:10.1142/S0217979290001054.
        "The rhombus of course tiles periodically, but we are not allowed to join the pieces in this manner." Gardner 1997, pp. 6–7 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 8 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, pp. 10–11 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 12 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 9 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 27 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Gardner 1997, p. 7 Gardner, Martin (1997). Penrose Tiles to Trapdoor Ciphers. Cambridge University Press. ISBN 978-0-88385-521-8.. (First published by W. H. Freeman, New York (1989), ISBN 978-0-7167-1986-1.)
        Chapter 1 (pp. 1–18) is a reprint of Gardner, Martin (January 1977). "Extraordinary non-periodic tiling that enriches the theory of tiles". Scientific American. Vol. 236, no. 1. pp. 110–121. Bibcode:1977SciAm.236a.110G. doi:10.1038/scientificamerican0177-110..
        Steinhardt & Jeong 1996; see also Steinhardt, Paul J. "A New Paradigm for the Structure of Quasicrystals". Archived from the original on 18 October 2016. Retrieved 29 November 2009. Steinhardt, Paul J.; Jeong, Hyeong-Chai (1996). "A simpler approach to Penrose tiling with implications for quasicrystal formation". Nature. 382 (1 August): 431–433. Bibcode:1996Natur.382..431S. doi:10.1038/382431a0. S2CID 4354819..
        Colbrook; Roman; Hansen (2019). "How to Compute Spectra with Error Control". Physical Review Letters. 122 (25) 250201. Bibcode:2019PhRvL.122y0201C. doi:10.1103/PhysRevLett.122.250201. PMID 31347861. S2CID 198463498.
        Luck, R (1990). "Penrose Sublattices". Journal of Non-Crystalline Solids. 117–8 (90): 832–5. Bibcode:1990JNCS..117..832L. doi:10.1016/0022-3093(90)90657-8.
        Godrèche & Lançon 1992; see also Dirk Frettlöh; F. Gähler & Edmund Harriss. "Binary". Tilings Encyclopedia. Department of Mathematics, University of Bielefeld. Godrèche, C; Lançon, F. (1992). "A simple example of a non-Pisot tiling with five-fold symmetry". Journal de Physique I. 2 (2): 207–220. Bibcode:1992JPhy1...2..207G. doi:10.1051/jp1:1992134. S2CID 120168483..
        Zaslavskiĭ et al. 1988; Makovicky 1992 Zaslavskiĭ, G.M.; Sagdeev, Roal'd Z.; Usikov, D.A.; Chernikov, A.A. (1988). "Minimal chaos, stochastic web and structures of quasicrystal symmetry". Soviet Physics Uspekhi. 31 (10): 887–915. Bibcode:1988SvPhU..31..887Z. doi:10.1070/PU1988v031n10ABEH005632.. Makovicky, E. (1992). "800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired". In I. Hargittai (ed.). Fivefold Symmetry. Singapore–London: World Scientific. pp. 67–86. ISBN 978-981-02-0600-0..
        Lu & Steinhardt 2007 Lu, Peter J.; Steinhardt, Paul J. (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture" (PDF). Science. 315 (5815): 1106–1110. Bibcode:2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056. S2CID 10374218..
        Kemp 2005 Kemp, Martin (2005). "Science in culture: A trick of the tiles". Nature. 436 (7049): 332. Bibcode:2005Natur.436..332K. doi:10.1038/436332a..

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Colbrook; Roman; Hansen (2019). "How to Compute Spectra with Error Control". Physical Review Letters. 122 (25) 250201. Bibcode:2019PhRvL.122y0201C. doi:10.1103/PhysRevLett.122.250201. PMID 31347861. S2CID 198463498.
Lu & Steinhardt 2007 Lu, Peter J.; Steinhardt, Paul J. (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture" (PDF). Science. 315 (5815): 1106–1110. Bibcode:2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056. S2CID 10374218..

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Steinhardt & Jeong 1996; see also Steinhardt, Paul J. "A New Paradigm for the Structure of Quasicrystals". Archived from the original on 18 October 2016. Retrieved 29 November 2009. Steinhardt, Paul J.; Jeong, Hyeong-Chai (1996). "A simpler approach to Penrose tiling with implications for quasicrystal formation". Nature. 382 (1 August): 431–433. Bibcode:1996Natur.382..431S. doi:10.1038/382431a0. S2CID 4354819..

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Lord & Ranganathan 2001 Lord, E.A.; Ranganathan, S. (2001). "The Gummelt decagon as a 'quasi unit cell'" (PDF). Acta Crystallographica. A57 (5): 531–539. CiteSeerX 10.1.1.614.3786. doi:10.1107/S0108767301007504. PMID 11526302. Archived from the original (PDF) on 21 July 2011. Retrieved 13 December 2009.
Lançon & Billard 1988 Lançon, Frédéric; Billard, Luc (1988). "Two-dimensional system with a quasi-crystalline ground state" (PDF). Journal de Physique. 49 (2): 249–256. CiteSeerX 10.1.1.700.3611. doi:10.1051/jphys:01988004902024900. Archived from the original (PDF) on 20 July 2011. Retrieved 28 November 2009..

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Prange, Sebastian R.; Peter J. Lu (1 September 2009). "The Tiles of Infinity". Saudi Aramco World. Aramco Services Company. pp. 24–31. Archived from the original on 13 January 2010. Retrieved 22 February 2010.

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Penrose 1978, p. 32 Penrose, Roger (1978). "Pentaplexity" (PDF). Eureka. Vol. 39. pp. 16–22. Archived from the original (PDF) on 9 August 2022. Retrieved 2 February 2021.. (Page numbers cited here are from the reproduction as Penrose, R. (1979–80). "Pentaplexity: A class of non-periodic tilings of the plane". The Mathematical Intelligencer. 2: 32–37. doi:10.1007/BF03024384. S2CID 120305260..)
Steinhardt & Jeong 1996; see also Steinhardt, Paul J. "A New Paradigm for the Structure of Quasicrystals". Archived from the original on 18 October 2016. Retrieved 29 November 2009. Steinhardt, Paul J.; Jeong, Hyeong-Chai (1996). "A simpler approach to Penrose tiling with implications for quasicrystal formation". Nature. 382 (1 August): 431–433. Bibcode:1996Natur.382..431S. doi:10.1038/382431a0. S2CID 4354819..
Colbrook; Roman; Hansen (2019). "How to Compute Spectra with Error Control". Physical Review Letters. 122 (25) 250201. Bibcode:2019PhRvL.122y0201C. doi:10.1103/PhysRevLett.122.250201. PMID 31347861. S2CID 198463498.
Godrèche & Lançon 1992; see also Dirk Frettlöh; F. Gähler & Edmund Harriss. "Binary". Tilings Encyclopedia. Department of Mathematics, University of Bielefeld. Godrèche, C; Lançon, F. (1992). "A simple example of a non-Pisot tiling with five-fold symmetry". Journal de Physique I. 2 (2): 207–220. Bibcode:1992JPhy1...2..207G. doi:10.1051/jp1:1992134. S2CID 120168483..
Lu & Steinhardt 2007 Lu, Peter J.; Steinhardt, Paul J. (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture" (PDF). Science. 315 (5815): 1106–1110. Bibcode:2007Sci...315.1106L. doi:10.1126/science.1135491. PMID 17322056. S2CID 10374218..

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Penrose 1978, p. 32 Penrose, Roger (1978). "Pentaplexity" (PDF). Eureka. Vol. 39. pp. 16–22. Archived from the original (PDF) on 9 August 2022. Retrieved 2 February 2021.. (Page numbers cited here are from the reproduction as Penrose, R. (1979–80). "Pentaplexity: A class of non-periodic tilings of the plane". The Mathematical Intelligencer. 2: 32–37. doi:10.1007/BF03024384. S2CID 120305260..)
Lord & Ranganathan 2001 Lord, E.A.; Ranganathan, S. (2001). "The Gummelt decagon as a 'quasi unit cell'" (PDF). Acta Crystallographica. A57 (5): 531–539. CiteSeerX 10.1.1.614.3786. doi:10.1107/S0108767301007504. PMID 11526302. Archived from the original (PDF) on 21 July 2011. Retrieved 13 December 2009.
Steinhardt & Jeong 1996; see also Steinhardt, Paul J. "A New Paradigm for the Structure of Quasicrystals". Archived from the original on 18 October 2016. Retrieved 29 November 2009. Steinhardt, Paul J.; Jeong, Hyeong-Chai (1996). "A simpler approach to Penrose tiling with implications for quasicrystal formation". Nature. 382 (1 August): 431–433. Bibcode:1996Natur.382..431S. doi:10.1038/382431a0. S2CID 4354819..
Lançon & Billard 1988 Lançon, Frédéric; Billard, Luc (1988). "Two-dimensional system with a quasi-crystalline ground state" (PDF). Journal de Physique. 49 (2): 249–256. CiteSeerX 10.1.1.700.3611. doi:10.1051/jphys:01988004902024900. Archived from the original (PDF) on 20 July 2011. Retrieved 28 November 2009..
Prange, Sebastian R.; Peter J. Lu (1 September 2009). "The Tiles of Infinity". Saudi Aramco World. Aramco Services Company. pp. 24–31. Archived from the original on 13 January 2010. Retrieved 22 February 2010.
The Penrose Tiling at Miami University Archived 14 August 2017 at the Wayback Machine by David Kullman, Presented at the Mathematical Association of America Ohio Section Meeting Shawnee State University, 24 October 1997
"New Building Project". Archived from the original on 22 November 2012. Retrieved 30 November 2013.

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Veritasium (30 September 2020). The Infinite Pattern That Never Repeats. Retrieved 22 October 2025 – via YouTube.

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Gummelt 1996 Gummelt, Petra (1996). "Penrose tilings as coverings of congruent decagons". Geometriae Dedicata. 62 (1): 1–17. doi:10.1007/BF00239998. MR 1400977. Zbl 0893.52011.

Penrose tiling (English Wikipedia)

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