List of aperiodic sets of tiles

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A periodic tiling with a fundamental unit (triangle) and a primitive cell (hexagon) highlighted. A tiling of the entire plane can be generated by fitting copies of these triangular patches together. In order to do this, the basic triangle needs to be rotated 180 degrees in order to fit it edge-to-edge to a neighboring triangle. Thus a triangular tiling of fundamental units will be generated that is mutually locally derivable from the tiling by the colored tiles. The other figure drawn onto the tiling, the white hexagon, represents a primitive cell of the tiling. Copies of the corresponding patch of coloured tiles can be translated to form an infinite tiling of the plane. It is not necessary to rotate this patch in order to achieve this.

In geometry, a tiling is a partition of the plane (or any other geometric setting) into closed sets (called tiles), without gaps or overlaps (other than the boundaries of the tiles).[1] A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is composed of a single fundamental unit or primitive cell which repeats endlessly and regularly in two independent directions.[2] An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic.[3] The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".)

The first table explains the abbreviations used in the second table. The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. This list of tiles is still incomplete.

Explanations[edit]

Abbreviation Meaning Explanation
E2 Euclidean plane normal flat plane
H2 hyperbolic plane plane, where the parallel postulate does not hold
E3 Euclidean 3 space space defined by three perpendicular coordinate axes
MLD Mutually locally derivable two tilings are said to be mutually locally derivable from each other, if one tiling can be obtained from the other by a simple local rule (such as deleting or inserting an edge)

List[edit]

Image Name Number of tiles Space Publication Date Refs. Comments
Trilobite and cross.svg
Trilobite and cross tiles 2 E2 1999 [4] Tilings MLD from the chair tilings
Penrose P1.svg
Penrose P1 tiles 6 E2 1974[5] [6] Tilings MLD from the tilings by P2 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Kite Dart.svg
Penrose P2 tiles 2 E2 1977[7] [8] Tilings MLD from the tilings by P1 and P3, Robinson triangles, and "Starfish, ivy leaf, hex"
Penrose P3 arcs.svg
Penrose P3 tiles 2 E2 1978[9] [10] Tilings MLD from the tilings by P1 and P2, Robinson triangles, and "Starfish, ivy leaf, hex"
Binary tiling arcs.svg
Binary tiles 2 E2 1988 [11][12] Although similar in shape to the P3 tiles, the tilings are not MLD from each other. Developed in an attempt to model the atomic arrangement in binary alloys
Robinson tiles.svg
Robinson tiles 6 E2 1971[13] [14] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
No image Ammann A1 tiles 6 E2 1977[15] [16] Tiles enforce aperiodicity by forming an infinite hierarchal binary tree.
Ammann A2.svg
Ammann A2 tiles 2 E2 1986[17] [18]
Ammann A3.svg
Ammann A3 tiles 3 E2 1986[17] [18]
Ammann A4.svg
Ammann A4 tiles 2 E2 1986[17] [18][19] Tilings MLD with Ammann A5.
Ammann A5.svg
Ammann A5 tiles 2 E2 1982[20] [21][22] Tilings MLD with Ammann A4.
No image Penrose Hexagon-Triangle tiles 2 E2 1997[23] [23][24]
No image Golden Triangle tiles 10 E2 2001[25] [26] date is for discovery of matching rules. Dual to Ammann A2
Socolar.svg
Socolar tiles 3 E2 1989[27] [28][29] Tilings MLD from the tilings by the Shield tiles
Shield.svg
Shield tiles 4 E2 1988[30] [31][32] Tilings MLD from the tilings by the Socolar tiles
Square triangle tiles.svg
Square triangle tiles 5 E2 1986[33] [34]
Self-replication of sphynx hexidiamonds.svg
Sphinx tiling 91 E2 [35]
Starfish ivyleaf hex.svg
Starfish, ivy leaf and hex tiles 3 E2 [36][37][38] Tiling is MLD to Penrose P1, P2, P3, and Robinson triangles
Robinson triangle decompositions.svg
Robinson triangle 4 E2 [17] Tiling is MLD to Penrose P1, P2, P3, and "Starfish, ivy leaf, hex".
Danzer triangles.svg
Danzer triangles 6 E2 1996[39] [40]
Pinwheel 1.svg
Pinwheel tiles E2 1994[41][42] [43][44] Date is for publication of matching rules.
Socolar-Taylor tile.svg
Socolar–Taylor tile 1 E2 2010 [45][46] Not a connected set. Aperiodic hierarchical tiling.
No image Wang tiles 20426 E2 1966 [47]
No image Wang tiles 104 E2 2008 [48]
No image Wang tiles 52 E2 1971[13] [49] Tiles enforce aperiodicity by forming an infinite hierarchy of square lattices
Wang 32 tiles.svg
Wang tiles 32 E2 1986 [50] Locally derivable from the Penrose tiles.
No image Wang tiles 24 E2 1986 [50] Locally derivable from the A2 tiling
Wang 16 tiles.svg
Wang tiles 16 E2 1986 [17][51] Derived from tiling A2 and its Ammann bars
Wang 14 tiles.svg
Wang tiles 14 E2 1996 [52][53]
Wang 13 tiles.svg
Wang tiles 13 E2 1996 [54][55]
No image Decagonal Sponge tile 1 E2 2002 [56][57] Porous tile consisting of non-overlapping point sets
No image Goodman-Strauss strongly aperiodic tiles 85 H2 2005 [58]
No image Goodman-Strauss strongly aperiodic tiles 26 H2 2005 [59]
Goodman-Strauss hyperbolic tile.svg
Böröczky hyperbolic tile 1 Hn 1974[60][61] [59][62] Only weakly aperiodic
No image Schmitt tile 1 E3 1988 [63] Screw-periodic
SCD tile.svg
Schmitt–Conway–Danzer tile 1 E3 [63] Screw-periodic and convex
Socolar Taylor 3D.svg
Socolar-Taylor tile 1 E3 2010 [45][46] Periodic in third dimension
No image Penrose rhombohedra 2 E3 1981[64] [65][66][67][68][69][70][71]
Nets for icosahedral aperiodic tile set.svg
Mackay-Amman rhombohedra 4 E3 1981 [36] Icosahedral symmetry. These are decorated Penrose rhombohedra with a matching rule that force aperiodicity.
No image Wang cubes 21 E3 1996 [72]
No image Wang cubes 18 E3 1999 [73]
No image Danzer tetrahedra 4 E3 1989[74] [75]
I and L tiles.png
I and L tiles 2 En for all n ≥ 3 1999 [76]

References[edit]

  1. ^ Grünbaum, Branko; Shephard, Geoffrey C. (1977), "Tilings by Regular Polygons", Math. Mag., 50 (5): 227–247, doi:10.2307/2689529, JSTOR 2689529 
  2. ^ Edwards, Steve, "Fundamental Regions and Primitive cells", Tiling Plane & Fancy, Kennesaw State University, archived from the original on 2010-09-16, retrieved 2017-01-11 
  3. ^ Wagon, Steve (2010), Mathematica in Action (3rd ed.), Springer Science & Business Media, p. 268, ISBN 9780387754772 
  4. ^ Goodman-Strauss, Chaim (1999), "A Small Aperiodic Set of Planar Tiles", European J. Combin., 20 (5): 375–384, doi:10.1006/eujc.1998.0281  (preprint available)
  5. ^ Penrose, Roger (1974), "The role of Aesthetics in Pure and Applied Mathematical Research", Bull. Inst. Math. and its Appl., 10 (2): 266–271 
  6. ^ Mikhael, Jules (2010), Colloidal Monolayers On Quasiperiodic Laser Fields (PDF) (Dr. rer. nat thesis), p. 23, archived (PDF) from the original on 2010-09-28 
  7. ^ Gardner, Martin (January 1977), "Mathematical Games: Extraordinary nonperiodic tiling that enriches the theory of tiles", Scientific American, 236 (1): 110–121, Bibcode:1977SciAm.236a.110G, doi:10.1038/scientificamerican0177-110 
  8. ^ Gardner, Martin (1997), Penrose Tiles to Trapdoor Ciphers (Revised ed.), The Mathematical Association of America, p. 86, ISBN 9780883855218 
  9. ^ Penrose, Roger (1978), "Pentaplexity", Eureka, 39: 16–22 
  10. ^ Penrose, Roger (1979), "Pentaplexity", Math. Intell., 2 (1): 32–37, doi:10.1007/bf03024384, archived from the original on 2010-09-23 
  11. ^ Lançon, F.; Billard, L. (1988), "Two-dimensional system with a quasi-crystalline ground state" (PDF), Journal de Physique, 49 (2): 249–256, doi:10.1051/jphys:01988004902024900, archived (PDF) from the original on 2010-09-29 
  12. ^ Godrèche, C.; Lançon, F. (1992), "A simple example of a non-Pisot tiling with five-fold symmetry" (PDF), J. Phys. I France, 2 (2): 207–220, Bibcode:1992JPhy1...2..207G, doi:10.1051/jp1:1992134, archived (PDF) from the original on 2010-09-29 
  13. ^ a b Robinson, Raphael M. (1971), "Undecidability and nonperiodicity of tilings in the plane", Inventiones Mathematicae, 12 (3): 177–209, Bibcode:1971InMat..12..177R, doi:10.1007/BF01418780 
  14. ^ Goodman-Strauss, Chaim (1999), Sadoc, J. F.; Rivier, N., eds., "Aperiodic Hierarchical tilings", NATO ASI Series, Series E: Applied Sciences, 354 (Foams and Emulsions): 481–496, doi:10.1007/978-94-015-9157-7_28, ISBN 978-90-481-5180-6 
  15. ^ Gardner, Martin (2001), The Colossal Book of Mathematics, W. W. Norton & Company, p. 76, ISBN 978-0393020236 
  16. ^ Grünbaum, Branko & Shephard, Geoffrey C. (1986), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1194-X , according to Dutch, Steven (2003), Aperiodic Tilings, University of Wisconsin - Green Bay ; cf. Savard, John J. G., Aperiodic Tilings Within Conventional Lattices 
  17. ^ a b c d e Grünbaum, Branko & Shephard, Geoffrey C. (1986), Tilings and Patterns, New York: W. H. Freeman, ISBN 0-7167-1194-X 
  18. ^ a b c Ammann, Robert; Grünbaum, Branko; Shephard, Geoffrey C. (July 1992), "Aperiodic tiles", Discrete & Computational Geometry, 8 (1): 1–25, doi:10.1007/BF02293033 
  19. ^ Harriss, Edmund; Frettlöh, Dirk, "Ammann A4", Tilings Encyclopedia, Bielefeld University 
  20. ^ Beenker, F. P. M. (1982), Algebraic theory of non-periodic tilings of the plane by two simple building blocks: a square and a rhombus, TH Report, 82-WSK04, Eindhoven University of Technology 
  21. ^ Komatsu, Kazushi; Nomakuchi, Kentaro; Sakamoto, Kuniko; Tokitou, Takashi (2004), "Representation of Ammann-Beenker tilings by an automaton", Nihonkai Math. J., 15 (2): 109–118, archived from the original on 2010-09-29 
  22. ^ Harriss, Edmund; Frettlöh, Dirk, "Ammann-Beenker", Tilings Encyclopedia, Bielefeld University 
  23. ^ a b Penrose, R. (1997), "Remarks on tiling: Details of a (1+ε+ε2) aperiodic set.", NATO ASI Series, Series C: Mathematical and Physical Sciences, 489 (The Mathematics of Long-Range Aperiodic Order): 467–497, doi:10.1007/978-94-015-8784-6_18, ISBN 978-0-7923-4506-0 
  24. ^ Goodman-Strauss, Chaim (2003), An aperiodic pair of tiles (PDF), University of Arkansas 
  25. ^ Danzer, Ludwig; van Ophuysen, Gerrit (2001), "A species of planar triangular tilings with inflation factor ", Res. Bull. Panjab Univ. Sci., 50 (1–4): 137–175, MR 1914493 
  26. ^ Gelbrich, G (1997), "Fractal Penrose tiles II. Tiles with fractal boundary as duals of Penrose triangles", Aequationes Mathematicae, 54: 108–116, doi:10.1007/bf02755450, MR 1466298 
  27. ^ Socolar, Joshua E. S. (1989), "Simple octagonal and dodecagonal quasicrystals", Physical Review B, 39 (15): 10519–51, Bibcode:1989PhRvB..3910519S, doi:10.1103/PhysRevB.39.10519 
  28. ^ Gähler, Franz; Lück, Reinhard; Ben-Abraham, Shelomo I.; Gummelt, Petra (2001), "Dodecagonal tilings as maximal cluster coverings", Ferroelectrics, 250 (1): 335–338, doi:10.1080/00150190108225095 
  29. ^ Savard, John J. G., The Socolar tiling 
  30. ^ Gähler, Franz (1988), "Crystallography of dodecagonal quasicrystals"" (PDF), in Janot, Christian, Quasicrystalline materials: Proceedings of the I.L.L. / Codest Workshop, Grenoble, 21–25 March 1988, Singapore: World Scientific, pp. 272–284 
  31. ^ Gähler, Franz; Frettlöh, Dirk, "Shield", Tilings Encyclopedia, Bielefeld University 
  32. ^ Gähler, Franz (1993), "Matching rules for quasicrystals: the composition-decomposition method" (PDF), Journal of Non-crystalline Solids, 153–154 (Proceddings of the Fourth International Conference on Quasicrystals): 160–164, Bibcode:1993JNCS..153..160G, doi:10.1016/0022-3093(93)90335-u, archived (PDF) from the original on 2010-10-01 
  33. ^ Stampfli, P. (1986), "A Dodecagonal Quasiperiodic Lattice in Two Dimensions", Helv. Phys. Acta, 59: 1260–1263 
  34. ^ Hermisson, Joachim; Richard, Christoph; Baake, Michael (1997), "A Guide to the Symmetry Structure of Quasiperiodic Tiling Classes", J. Phys. I France, 7 (8): 1003–1018, Bibcode:1997JPhy1...7.1003H, CiteSeerX 10.1.1.46.5796Freely accessible, doi:10.1051/jp1:1997200 
  35. ^ Goodman-Strauss, Chaim (2009), "Aperiodic Tilings" (PDF), Quasi periodic tile and language theory, Kyoto, 8–10 June, 2009, p. 74 
  36. ^ a b Lord, Eric. A. (1991), "Quasicrystals and Penrose patterns" (PDF), Current Science, 61 (5): 313–319, archived (PDF) from the original on September 27, 2010 
  37. ^ Olamy, Z.; Kléman, M. (1989), "A two dimensional aperiodic dense tiling" (PDF), J. Phys. France, 50 (1): 19–33, doi:10.1051/jphys:0198900500101900, archived (PDF) from the original on 2010-11-01 
  38. ^ Mihalkovič, M.; Henley, C. L.; Widom, M. (2004), "Combined energy-diffraction data refinement of decagonal AlNiCo", Journal of Non-Crystalline Solids, 334–335 (8th International Conference on Quasicrystals): 177–183, arXiv:cond-mat/0311613Freely accessible, Bibcode:2004JNCS..334..177M, doi:10.1016/j.jnoncrysol.2003.11.034 
  39. ^ Nischke, K.-P.; Danzer, L. (1996), "A construction of inflation rules based on n-fold symmetry", Discrete & Computational Geometry, 15 (2): 221–236, doi:10.1007/bf02717732 
  40. ^ Hayashi, Hiroko; Kawachi, Yuu; Komatsu, Kazushi; Konda, Aya; Kurozoe, Miho; Nakano, Fumihiko; Odawara, Naomi; Onda, Rika; Sugio, Akinobu; Yamauchi, Masatetsu (2009), "Abstract: Notes on vertex atlas of planar Danzer tiling" (PDF), Japan Conference on Computational Geometry and Graphs, Kanazawa, November 11–13, 2009 
  41. ^ Radin, Charles (1994), "The pinwheel tilings of the plane", Annals of Mathematics, Second Series, 139 (3): 661–702, CiteSeerX 10.1.1.44.9723Freely accessible, doi:10.2307/2118575, JSTOR 2118575, MR 1283873, retrieved 2013-09-25 
  42. ^ Radin, Charles (1993), "Symmetry of Tilings of the Plane", Bull. Amer. Math. Soc., 29: 213–217, CiteSeerX 10.1.1.45.5319Freely accessible, doi:10.1090/s0273-0979-1993-00425-7 
  43. ^ Radin, Charles; Wolff, Mayhew (1992), "Space tilings and local isomorphism", Geom. Dedicata, 42 (3): 355–360, doi:10.1007/bf02414073, MR 1164542 
  44. ^ Radin, C (1997), "Aperiodic tilings, ergodic theory, and rotations", NATO ASI Series, Series C: Mathematical and Physical Sciences, Kluwer Acad. Publ., Dordrecht, 489 (The mathematics of long-range aperiodic order), MR 1460035 
  45. ^ a b Socolar, Joshua E. S.; Taylor, Joan M. (2011), "An aperiodic hexagonal tile", Journal of Combinatorial Theory, Series A, 118 (8): 2207–2231, arXiv:1003.4279v1Freely accessible, doi:10.1016/j.jcta.2011.05.001 
  46. ^ a b Socolar, Joshua E. S.; Taylor, Joan M. (2011), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419v1Freely accessible, doi:10.1007/s00283-011-9255-y 
  47. ^ Burger, Robert (1966), "The Undecidability of the Domino Problem", Memoirs of the American Mathematical Society, 66, doi:10.1090/memo/0066, ISBN 978-0-8218-1266-2 
  48. ^ Ollinger, Nicolas (2008), Two-by-two Substitution Systems and the Undecidability of the Domino Problem (PDF), Lecture Notes in Computer Science, 5028, Springer, pp. 476–485, doi:10.1007/978-3-540-69407-6_51 
  49. ^ Kari, J.; Papasoglu, P. (1999), "Deterministic Aperiodic Tile Sets", Geometric and Functional Analysis, 9 (2): 353–369, doi:10.1007/s000390050090 
  50. ^ a b Lagae, Ares; Kari, Jarkko; Dutré, Phillip (2006), Aperiodic Sets of Square Tiles with Colored Corners, Report CW, 460, KU Leuven, p. 12, CiteSeerX 10.1.1.89.1294Freely accessible, archived from the original on 2010-10-02 
  51. ^ Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000), Pattern Formation in Biology, Vision and Dynamics, Singapore: World Scientific, ISBN 981-02-3792-8 
  52. ^ Kari, Jarkko (1996), "A small aperiodic set of Wang tiles", Discrete Mathematics, 160 (1–3): 259–264, doi:10.1016/0012-365X(95)00120-L 
  53. ^ Lagae, Ares (2007), Tile Based Methods in Computer Graphics (PDF) (PhD thesis), KU Leuven, p. 149, ISBN 978-90-5682-789-2, archived from the original (PDF) on 2010-10-06 
  54. ^ Culik, Karel; Kari, Jarkko (1997), On aperiodic sets of Wang tiles, Lecture Notes in Computer Science, 1337, pp. 153–162, doi:10.1007/BFb0052084 
  55. ^ Culik, Karel (1996), "An aperiodic set of 13 Wang tiles", Discrete Mathematics, 160 (1–3): 245–251, CiteSeerX 10.1.1.53.5421Freely accessible, doi:10.1016/S0012-365X(96)00118-5 
  56. ^ Zhu, Feng (2002), The Search for a Universal Tile (PDF) (BA thesis), Williams College 
  57. ^ Bailey, Duane A.; Zhu, Feng (2001), A Sponge-Like (Almost) Universal Tile (PDF), CiteSeerX 10.1.1.103.3739Freely accessible 
  58. ^ Goodman-Strauss, Chaim (2010), "A hierarchical strongly aperiodic set of tiles in the hyperbolic plane" (PDF), Theoretical Computer Science, 411 (7–9): 1085–1093, doi:10.1016/j.tcs.2009.11.018 
  59. ^ a b Goodman-Strauss, Chaim (2005), "A strongly aperiodic set of tiles in the hyperbolic plane", Invent. Math., 159: 130–132, Bibcode:2004InMat.159..119G, doi:10.1007/s00222-004-0384-1 
  60. ^ Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben I", Matematikai Lapok, 25: 265–306 
  61. ^ Böröczky, K. (1974), "Gömbkitöltések állandó görbületü terekben II", Matematikai Lapok, 26: 67–90 
  62. ^ Dolbilin, Nikkolai; Frettlöh, Dirk (2010), "Properties of Böröczky tilings in high dimensional hyperbolic spaces" (PDF), European J. Combin., 31 (4): 1181–1195, arXiv:0705.0291Freely accessible, doi:10.1016/j.ejc.2009.11.016 
  63. ^ a b Radin, Charles (1995), "Aperiodic tilings in higher dimensions" (pdf), Proceedings of the American Mathematical Society, American Mathematical Society, 123 (11): 3543–3548, doi:10.2307/2161105, JSTOR 2161105, retrieved 2013-09-25 
  64. ^ Mackay, Alan L. (1981), "De Nive Quinquangula: On the pentagonal snowflake" (PDF), Sov. Phys. Crystallogr., 26 (5): 517–522, archived (PDF) from the original on 2010-10-06 
  65. ^ Meisterernst, Götz, Experimente zur Wachstumskinetik Dekagonaler Quasikristalle (PDF) (Dissertation), Ludwig Maximilian University of Munich, pp. 18–19, archived (PDF) from the original on 2010-10-08 
  66. ^ Jirong, Sun (1993), "Structure Transition of the Three-Dimensional Penrose Tiling Under Phason Strain Field", Chinese Phys. Lett., 10 (8): 449–452, Bibcode:1993ChPhL..10..449S, doi:10.1088/0256-307x/10/8/001 
  67. ^ Inchbald, Guy (2002), A 3-D Quasicrystal Structure 
  68. ^ Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2001), "Quasicrystals: tiling versus clustering" (PDF), Philosophical Magazine A, 81 (11): 2645–2651, doi:10.1080/01418610108216660, archived (PDF) from the original on 2010-10-06 
  69. ^ Rudhart, Christoph Paul (June 1999), Zur numerischen Simulation des Bruchs von Quasikristallen (Thesis), University of Stuttgart, p. 11, doi:10.18419/opus-4639 
  70. ^ Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000), "Tilings, coverings, clusters and quasicrystals" (PDF), Current Science, 78 (1): 64–72, archived (PDF) from the original on 2010-11-01 
  71. ^ Katz, A. (1988), "Theory of Matching Rules for the 3-Dimensional Penrose Tilings", Communications in Mathematical Physics, 118 (2): 263–288, Bibcode:1988CMaPh.118..263K, doi:10.1007/BF01218580 
  72. ^ Culik, Karel; Kari, Jarkko (1995), "An aperiodic set of Wang cubes", The Journal of Universal Computer Science, 1 (10), CiteSeerX 10.1.1.54.5897Freely accessible, doi:10.3217/jucs-001-10-0675 
  73. ^ Walther. Gerd; Selter, Christoph, eds. (1999), Mathematikdidaktik als design science : Festschrift für Erich Christian Wittmann, Leipzig: Ernst Klett Grundschulverlag, ISBN 3-12-200060-1 
  74. ^ Danzer, L. (1989), "Three-Dimensional Analogs of the Planar Penrose Tilings and Quasicrystals", Discrete Mathematics, 76 (1): 1–7, doi:10.1016/0012-365X(89)90282-3 
  75. ^ Zerhusen, Aaron (1997), Danzer's three dimensional tiling, University of Kentucky 
  76. ^ Goodman-Strauss, Chaim (1999), "An Aperiodic Pair of Tiles in En for all n ≥ 3", European J. Combin., 20 (5): 385–395, doi:10.1006/eujc.1998.0282  (preprint available)

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