On Canonical Substitution Tilings - Versionsgeschichte

Gbachelier am 18. November 2014 um 11:21 Uhr

2014-11-18T11:21:33Z

Gbachelier: Die Seite wurde neu angelegt: „== Reference == Edmund O. Harriss: On Canonical Substitution Tilings. PhD Thesis. Imperial College London 2004. == DOI == == Abstract == This thesis is conc…“

2014-11-18T10:55:55Z

Die Seite wurde neu angelegt: „== Reference == Edmund O. Harriss: On Canonical Substitution Tilings. PhD Thesis. Imperial College London 2004. == DOI == == Abstract == This thesis is conc…“

Neue Seite

== Reference ==
Edmund O. Harriss: On Canonical Substitution Tilings. PhD Thesis. Imperial College London 2004.

== DOI ==

== Abstract ==
This thesis is concerned with canonical substitution tilings. These are tilings
generated by the canonical projection method which admit substitution rules,
and include the famous Penrose tilings. We characterise all canonical substitution
tilings and consider the question what the set of all substitution rules is for a given
tiling. In many cases, including the Penrose tilings, we are able to characterise
all the substitution rules for the tiling. Our methods are constructive and give an
algorithm to construct the substitution rules and tilings.

== Extended Abstract ==

== Bibtex ==

== Used References ==
[AG95] F Axel and D Gratias (eds.), Beyond quasicrystals, Springer-Verlag,
Berlin, 1995, Papers from the Winter School held in Les Houches, March 7–18,
1994. MR 97e:00004 1.1.1, 7.3.2
[AGS92] R Ammann, B Grünbaum, and G C Shephard, Aperiodic tiles, Discrete
Comput. Geom. 8 (1992), no. 1, 1–25. MR 93g:52015 1.1.3
[AI01] P Arnoux and S Ito, Pisot substitutions and Rauzy fractals, Bull.
Belg. Math. Soc. Simon Stevin 8 (2001), no. 2, 181–207, Journées Montoises
d’Informatique Théorique (Marne-la-Vallée, 2000). MR 2002j:37018 1.1.4
[AS03] J-P Allouche and J Shallit, Automatic sequences, Cambridge University
Press, Cambridge, 2003, Theory, applications, generalizations. MR 1 997 038
3.2
[Bee82] F P M Beenker, 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, 1982. 1.1.3, 1.1.4, 4
[Ber66] R Berger, The undecidability of the domino problem, Mem. Amer. Math.
Soc. No. 66 (1966), 72. MR 36 #49 1.1.2
[BGS92] S Baranidharan, E S R Gopal, and V Sasisekharan, Quasiperiodic
tilings with fourfold symmetry, Acta Cryst. Sect. A 48 (1992), no. 5, 782–
784. MR 93h:52026 2.3.1
[BJK91] M Baake, D Joseph, and P Kramer, The Schur rotation as a simple
approach to the transition between quasiperiodic and periodic phases, J.
Phys. A 24 (1991), no. 17, L961–L967. MR 92k:82071 2.3.1
[BJKS90] M Baake, D Joseph, P Kramer, and M Schlottmann, Root lattices
and quasicrystals, J. Phys. A 23 (1990), no. 19, L1037–L1041. MR 91k:82065
1.1.3, 2.3.1
[BJS91]
M Baake, D Joseph, and M Schlottmann, The root lattice D4 and pla-
nar quasilattices with octagonal and dodecagonal symmetry, Internat. J.
Modern Phys. B 5 (1991), no. 11, 1927–1953. MR 92m:52035 1.1.3, 2.3.1
[BKSZ90] M Baake, P Kramer, M Schlottmann, and D Zeidler, Planar pat-
terns with fivefold symmetry as sections of periodic structures in 4-space,
155156
BIBLIOGRAPHY
Internat. J. Modern Phys. B 4 (1990), no. 15-16, 2217–2268. MR 92b:52041
1.1.4
[BM00a] M Baake and R V Moody (eds.), Directions in mathematical quasicrys-
tals, American Mathematical Society, Providence, RI, 2000. MR 2001f:52047
1.1.1, 7.3.2
[BM00b] , Self-similar measures for quasicrystals, in Directions in math-
ematical quasicrystals [BM00a], pp. 1–42. MR 1 798 987 1.1.4
[BMS98] M Baake, R V Moody, and M Schlottmann, Limit-(quasi)periodic point
sets as quasicrystals with p-adic internal spaces, J. Phys. A 31 (1998), no. 27,
5755–5765. MR 99f:82061 1.1.4
[Boh52] H Bohr, Collected Mathematical Works. Vol. I. Dirichlet series. The Rie-
mann zeta-function. Vol. II. Almost periodic functions. Vol. III. Almost
periodic functions (continued). Linear congruences. Diophantine approx-
imations. Function theory. Addition of convex curves. Other papers. En-
cyclopædia article. Supplements, Dansk Matematisk Forening, København,
1952. MR 15,276i 1.1.3
[BSJ91] M Baake, M Schlottmann, and P D Jarvis, Quasiperiodic tilings with
tenfold symmetry and equivalence with respect to local derivability, J. Phys.
A 24 (1991), no. 19, 4637–4654. MR 92k:52043 1.1.2, 2.3.2, 3.8

[BT87] E Bombieri and J E Taylor, Quasicrystals, tilings, and algebraic number theory: some preliminary connections, The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985), Amer. Math. Soc., Providence, RI, 1987, pp. 241–264. MR 89a:82031 1.1.3

[dB81a] N G de Bruijn, Algebraic theory of Penrose’s nonperiodic tilings of the plane. I, II, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 39–52, 53–66. MR 82e:05055 1.1.3, 1.1.4, 6.3

[dB81b] , Sequences of zeros and ones generated by special production rules, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 27–37. MR 82i:10074 1.1.2

[DMO89] M Duneau, R Mosseri, and C Oguey, Approximants of quasiperiodic structures generated by the inflation mapping, J. Phys. A 22 (1989), no. 21, 4549–4564. MR 91c:52024 2.3.2

[FHK02] A H Forrest, J R Hunton, and J Kellendonk, Cohomology of canonical projection tilings, Comm. Math. Phys. 226 (2002), no. 2, 289–322. MR 1 892 456 2.3.2, 5.2.3

[Fie79] J V Field, Kepler’s star polyhedra, Vistas in Astronomy 23 (1979), 109–141. 1.1.2

[FT93] A Fröhlich and M J Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993. MR 94d:11078 26

[Gar77] M Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Sci. Am. November (1977), 110–119. 1.1.2

[GK97] F Gähler and R Klitzing, The diffraction pattern of self-similar tilings, in Moody [Moo97], pp. 141–174. MR 99b:52054 1.1.3

[GKP94] R L Graham, D E Knuth, and O Patashnik, Concrete mathematics, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994, A foundation for computer science. MR 97d:68003 (document)

[GLJJ93] C Godrèche, J M Luck, A Janner, and T Janssen, Fractal atomic surfaces of self-similar quasi-periodic tilings of the plane, J. Physique I 3 (1993), no. 9, 1921–1939. 1.1.4, 2.3.2

[Gol94] S W Golomb, Polyominoes, 2nd ed., Princeton University Press, Princeton, NJ, 1994, Puzzles, patterns, problems, and packings. MR 95k:00006 1.1.2

[Goo] C Goodman-Strauss, Open questions in tilings, (preprint) http://comp.uark.edu/~cgstraus/papers. 1.1.2

[Goo98] , Matching rules and substitution tilings, Ann. of Math. (2) 147 (1998), no. 1, 181–223. MR 99m:52027 1.1.2, 1.2, 2.3.2

[Goo99] Chaim Goodman-Strauss, A small aperiodic set of planar tiles, European J. Combin. 20 (1999), no. 5, 375–384. MR 2001a:05037 1.1.2

[GR86] F Gähler and J Rhyner, Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings, J. Phys. A 19 (1986), no. 2, 267–277. MR 87e:52023 1.1.3

[GS87] B Grünbaum and G C Shephard, Tilings and patterns, W. H. Freeman and Company, New York, 1987. MR 88k:52018 1.1.2, 2.1.1, 2.1.2, 2.3.1

[Hil00] D Hilbert, Mathematische probleme, Göttinger Nachrichten (1900), 253–297. 1.1.2

[HL] E O Harriss and J S W Lamb, Canonical substitution tilings of Ammann-Beenker type, Th. Comp. Sci. (to appear). 5.1

[HNL94] D Haussler, H U Nissen, and R Lück, Dodecagonal tilings derived as duals from quasi-periodic ammann-grids, Phys. Status Solidi A-Appl. Res. 146 (1994), 425–435. 2.3.1

[Hof97] A Hof, Diffraction by aperiodic structures, in Moody [Moo97], pp. 239–268. MR 98a:52001 1.1.3

[Ing92] R Ingalls, Decagonal quasi-crystal tilings, Acta Crystallogr. Sect. A A48 (1992), 533–541. 2.3.1

[Ing93] , Octagonal quasi-crystal tilings, J. Non-Cryst. Solids 153 (1993), 177–180. 2.3.1

[Jan94] C Janot, Quasicrystals: A primer, 2nd ed., Monographs on the Physics and Chemistry of Materials, Oxford University Press, Oxford, 1994. 1.1.1

[Jan95] A Janner, Elements of a multimetrical crystallography, in Axel and Gratias [AG95], Papers from the Winter School held in Les Houches, March 7–18, 1994, pp. 33–54. MR 98a:20052 2.3.2

[JJ80] A Janner and T Janssen, Symmetry of incommensurate crystal phases. I, II, Acta Cryst. A36 (1980), 399–408, 408–415. 1, 1.1.3

[Kat95] A Katz, Matching rules and quasiperiodicity: the octagonal tilings, in Axel and Gratias [AG95], Papers from the Winter School held in Les Houches, March 7–18, 1994, pp. 141–189. MR 97k:52027 1.1.2, 2.3.1

[KD86] A Katz and M Duneau, Quasiperiodic patterns and icosahedral symmetry, J. Physique 47 (1986), no. 2, 181–196. MR 87m:52025 1.1.3, 2.3.1

[Ken90] R Kenyon, Self-similar tilings, Ph.D. thesis, Princeton University, 1990. 1.2.1

[Ken92] , Self-replicating tilings, Symbolic dynamics and its applications (New Haven, CT, 1991), Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 239–263. MR 94a:52043 1.1.2

[Ken96] , The construction of self-similar tilings, Geom. Funct. Anal. 6 (1996), no. 3, 471–488. MR 97j:52025 1.1.2

[Kep19] J Kepler, Harmonices mundi, 1619. 1, 1.1, 1.1.2, 7.3.2

[KN84] P Kramer and R Neri, On periodic and non-periodic space fillings of Emobtained by projection, Acta. Cryst. A40 (1984), 580–587. 1.1.3

[KP00] J Kellendonk and I F Putnam, Tilings, C∗ -algebras, and K-theory, in Baake and Moody [BM00a], pp. 177–206. MR 2001m:46153 1.1.2

[Lag00] J C Lagarias, Mathematical quasicrystals and the problem of diffraction, in Baake and Moody [BM00a], pp. 61–93. MR 2001f:52047 1.1.3

[Lam98] J S W Lamb, On the canonical projection method for one-dimensional quasicrystals and invertible substitution rules, J. Phys. A 31 (1998), no. 18, L331–L336. MR 99d:82075 1.2.1, 2.3.1

[Le97] T T Q Le, Local rules for quasiperiodic tilings, in Moody [Moo97], pp. 331–366. MR 98f:52028 1.1.2, 2.2, 2.3.2

[LGJJ93] J M Luck, C Godrèche, A Janner, and T Janssen, The nature of the atomic surfaces of quasiperiodic self-similar structures, J. Phys. A 26 (1993), no. 8, 1951–1999. MR 94a:82049 1.1.4, 1.2.1, 2.3.2

[LM01] J-Y Lee and R V Moody, Lattice substitution systems and model sets, Discrete Comput. Geom. 25 (2001), no. 2, 173–201. MR 2001k:52033 1.1.4

[LS84] D. Levine and P. J. Steinhardt, Quasicrystals - a new class of ordered structures, Phys. Rev. Lett. 53 (1984), 2477–80. 1, 1.1.3

[Lüc93] R Lück, Basic ideas of Ammann bar grids, Internat. J. Modern Phys. B 7 (1993), no. 6-7, 1437–1453. MR 94i:52019 2.3.1

[LW96] J C Lagarias and Y Wang, Self-affine tiles in Rn , Adv. Math. 121 (1996), no. 1, 21–49. MR 97d:52034 7.3.1

[LW03] , Substitution Delone sets, Discrete Comput. Geom. 29 (2003), no. 2, 175–209. MR 1 957 227 7.3.1

[Mey72] Y Meyer, Algebraic numbers and harmonic analysis, North-Holland Publishing Co., Amsterdam, 1972, North-Holland Mathematical Library, Vol. 2. MR 58 #5579 1.1.3

[Mey95] , Quasicrystals, diophantine approximations and algebraic numbers, in Axel and Gratias [AG95], Papers from the Winter School held in Les Houches, March 7–18, 1994, pp. 3–16. MR 97e:00004 1.1.3

[Mil71] J Milton, Paradise regained, 1671. 2

[Moo97] R V Moody (ed.), The mathematics of long-range aperiodic order, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 489, Dordrecht, Kluwer Academic Publishers Group, 1997. MR 98a:52001 1.1.1, 7.3.2

[MPP00a] Z Masáková, J Patera, and E Pelantová, Lattice-like properties of quasicrystal models with quadratic irrationalities, Quantum theory and symme-
tries (Goslar, 1999), World Sci. Publishing, River Edge, NJ, 2000, pp. 499–509. MR 1 796 094 3.2

[MPP00b] , Substitution rules for aperiodic sequences of the cut and project type, J. Phys. A 33 (2000), no. 48, 8867–8886. MR 2001h:11027 (document), 1.2.1, 1.2.2, 2.3.1, 3, 3.2, 3.7

[Pat98] J Patera (ed.), Quasicrystals and discrete geometry, Providence, RI, American Mathematical Society, 1998. MR 99d:52001 1.1.1, 7.3.2

[Pen74] R Penrose, The role of aesthetics in pure and applied mathematical research, Bull. Inst. Math. and its Appl. 10 (1974), 266–271. 1.1.2

[Ple00] P A B Pleasants, Designer quasicrystals: cut-and-project sets with preassigned properties, in Baake and Moody [BM00a], pp. 95–141. MR 1 798 990 2.3.2, 3.8, 5.2.3

[Pra02] T Pratchett, Night watch, hardback ed., Doubleday, 2002. (document)

[PS01] N Priebe and B Solomyak, Characterization of planar pseudo-selfsimilar tilings, Discrete Comput. Geom. 26 (2001), no. 3, 289–306. MR 2002j:37029 1.1.2

[Pyt02] N Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002, Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. MR 2004c:37005 2.3.1

[Que87] M Queffélec, Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, vol. 1294, Springer-Verlag, Berlin, 1987. MR 89g:54094 1.1.2

[Rad94] C Radin, The pinwheel tilings of the plane, Ann. of Math. (2) 139 (1994), no. 3, 661–702. MR 95d:52021 1.1.2, 2.1.1

[Rau77] G Rauzy, Une généralisation du développement en fraction continue, Séminaire Delange-Pisot-Poitou, 18e année: 1976/77, Théorie des nombres, Fasc. 1, Secrétariat Math., Paris, 1977, pp. Exp. No. 15, 16. MR 83e:10077 1.1.4

[Rob96] E A Robinson, Jr., The dynamical properties of Penrose tilings, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4447–4464. MR 97a:52041 1.1.2

[RS98] C Radin and L Sadun, An algebraic invariant for substitution tiling systems, Geom. Dedicata 73 (1998), no. 1, 21–37. MR 99j:52027 1.1.2

[SAI01] Y Sano, P Arnoux, and S Ito, Higher dimensional extensions of substitutions and their dual maps, J. Anal. Math. 83 (2001), 183–206. MR 2003c:37016 1.1.4, 5

[SBGC84] D Shechtman, I Blech, D Gratias, and J W Cahn, Metallic phase with long-range orientational order and no translational symmetry, Physical Review Letters 53 (1984), 1951–3. 1.1.1, 1, 1.1.3

[Sch98] M Schlottman, Cut-and-project sets in locally compact abelian groups, in Patera [Pat98], pp. 247–264. MR 99d:52001 1.1.3

[Séé98] P Séébold, On the conjugation of standard morphisms, Theoret. Comput. Sci. 195 (1998), no. 1, 91–109, Mathematical foundations of computer science (Cracow, 1996). MR 99a:68140 1.2.1

[Sen95] M Senechal, Quasicrystals and geometry, Cambridge University Press, Cambridge, 1995. MR 96c:52038 1.1.1, 1.1.3, 1.2

[SÖ87] P J Steinhardt and S Östlund (eds.), The physics of quasicrystals, World Scientific Publishing Co., Singapore, 1987. MR 90b:82003 1.1.3

[Soc89] J E S Socolar, Simple octagonal and dodecagonal quasicrystals, Phys. Rev. B (3) 39 (1989), no. 15, 519–551. MR 90h:52019 1.1.4, 2.3.1

[Sol97] B Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 695–738. MR 98f:52030 1.1.2

[SW92] T Soma and Y Watanabe, A class of patterns generated by modification of beenkers pattern, Acta Crystallogr. A48 (1992), 470–475. 2.3.1

[Thu89] W P Thurston, Groups, tilings and finite state automata, AMS Colloquium Lectures (1989). 1.1.2

[Wan61] H Wang, Proving theorems by pattern recognition, Bell Systems Tech. J. 40 (1961), 1–41. 1.1.2

[WW94] Z X Wen and Z Y Wen, Local isomorphisms of invertible substitutions, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 4, 299–304. MR 95f:20097 1.2.1, 2.3.1

== Links ==
=== Full Text ===
http://www.mathematicians.org.uk/eoh/files/Harriss_Thesis.pdf

[[intern file]]

=== Sonstige Links ===
http://www.mathematicians.org.uk/eoh/

@@ Zeile 18: / Zeile 18: @@
 == Used References ==
-[AG95] F Axel and D Gratias (eds.), Beyond quasicrystals, Springer-Verlag,
+[AG95] F Axel and D Gratias (eds.), Beyond quasicrystals, Springer-Verlag, Berlin, 1995, Papers from the Winter School held in Les Houches, March 7–18, 1994. MR 97e:00004 1.1.1, 7.3.2
-      Berlin, 1995, Papers from the Winter School held in Les Houches, March 7–18,
-. MR 97e:00004 1.1.1, 7.3.2
+[AGS92] R Ammann, B Grünbaum, and G C Shephard, Aperiodic tiles, Discrete Comput. Geom. 8 (1992), no. 1, 1–25. MR 93g:52015 1.1.3
 [AGS92] R Ammann, B Grünbaum, and G C Shephard, Aperiodic tiles, Discrete
-        Comput. Geom. 8 (1992), no. 1, 1–25. MR 93g:52015 1.1.3
+[AI01] P Arnoux and S Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), no. 2, 181–207, Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000). MR 2002j:37018 1.1.4
 [AI01] P Arnoux and S Ito, Pisot substitutions and Rauzy fractals, Bull.
-      Belg. Math. Soc. Simon Stevin 8 (2001), no. 2, 181–207, Journées Montoises
+[AS03] J-P Allouche and J Shallit, Automatic sequences, Cambridge University Press, Cambridge, 2003, Theory, applications, generalizations. MR 1 997 038 3.2
-        d’Informatique Théorique (Marne-la-Vallée, 2000). MR 2002j:37018 1.1.4
-[AS03] J-P Allouche and J Shallit, Automatic sequences, Cambridge University
+[Bee82] F P M Beenker, 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, 1982. 1.1.3, 1.1.4, 4
-      Press, Cambridge, 2003, Theory, applications, generalizations. MR 1 997 038
-.2
+[Ber66] R Berger, The undecidability of the domino problem, Mem. Amer. Math. Soc. No. 66 (1966), 72. MR 36 #49 1.1.2
 [Bee82] F P M Beenker, Algebraic theory of non-periodic tilings of the plane by
-       two simple building blocks: a square and a rhombus, TH-Report 82-WSK04,
+[BGS92] S Baranidharan, E S R Gopal, and V Sasisekharan, Quasiperiodic tilings with fourfold symmetry, Acta Cryst. Sect. A 48 (1992), no. 5, 782–784. MR 93h:52026 2.3.1
-      Eindhoven University of Technology, 1982. 1.1.3, 1.1.4, 4
-[Ber66] R Berger, The undecidability of the domino problem, Mem. Amer. Math.
+[BJK91] M Baake, D Joseph, and P Kramer, The Schur rotation as a simple approach to the transition between quasiperiodic and periodic phases, J. Phys. A 24 (1991), no. 17, L961–L967. MR 92k:82071 2.3.1
-       Soc. No. 66 (1966), 72. MR 36 #49 1.1.2
-[BGS92] S Baranidharan, E S R Gopal, and V Sasisekharan, Quasiperiodic
+[BJKS90] M Baake, D Joseph, P Kramer, and M Schlottmann, Root lattices and quasicrystals, J. Phys. A 23 (1990), no. 19, L1037–L1041. MR 91k:82065 1.1.3, 2.3.1
-       tilings with fourfold symmetry, Acta Cryst. Sect. A 48 (1992), no. 5, 782–
-. MR 93h:52026 2.3.1
+[BJS91] M Baake, D Joseph, and M Schlottmann, The root lattice D4 and planar quasilattices with octagonal and dodecagonal symmetry, Internat. J. Modern Phys. B 5 (1991), no. 11, 1927–1953. MR 92m:52035 1.1.3, 2.3.1
 [BJK91] M Baake, D Joseph, and P Kramer, The Schur rotation as a simple
-       approach to the transition between quasiperiodic and periodic phases, J.
+[BKSZ90] M Baake, P Kramer, M Schlottmann, and D Zeidler, Planar patterns with fivefold symmetry as sections of periodic structures in 4-space, Internat. J. Modern Phys. B 4 (1990), no. 15-16, 2217–2268. MR 92b:52041 1.1.4
-      Phys. A 24 (1991), no. 17, L961–L967. MR 92k:82071 2.3.1
-[BJKS90] M Baake, D Joseph, P Kramer, and M Schlottmann, Root lattices
+[BM00a] M Baake and R V Moody (eds.), Directions in mathematical quasicrystals, American Mathematical Society, Providence, RI, 2000. MR 2001f:52047 1.1.1, 7.3.2
 and quasicrystals, J. Phys. A 23 (1990), no. 19, L1037–L1041. MR 91k:82065
-.1.3, 2.3.1
+[BM00b] , Self-similar measures for quasicrystals, in Directions in mathematical quasicrystals [BM00a], pp. 1–42. MR 1 798 987 1.1.4
 [BJS91]
-M Baake, D Joseph, and M Schlottmann, The root lattice D4 and pla-
+[BMS98] M Baake, R V Moody, and M Schlottmann, Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces, J. Phys. A 31 (1998), no. 27, 5755–5765. MR 99f:82061 1.1.4
-nar quasilattices with octagonal and dodecagonal symmetry, Internat. J.
-Modern Phys. B 5 (1991), no. 11, 1927–1953. MR 92m:52035 1.1.3, 2.3.1
+[Boh52] H Bohr, Collected Mathematical Works. Vol. I. Dirichlet series. The Riemann zeta-function. Vol. II. Almost periodic functions. Vol. III. Almost periodic functions (continued). Linear congruences. Diophantine approximations. Function theory. Addition of convex curves. Other papers. Encyclopædia article. Supplements, Dansk Matematisk Forening, København, 1952. MR 15,276i 1.1.3
-[BKSZ90] M Baake, P Kramer, M Schlottmann, and D Zeidler, Planar pat-
-terns with fivefold symmetry as sections of periodic structures in 4-space,
+[BSJ91] M Baake, M Schlottmann, and P D Jarvis, Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability, J. Phys. A 24 (1991), no. 19, 4637–4654. MR 92k:52043 1.1.2, 2.3.2, 3.8
 [BT87] E Bombieri and J E Taylor, Quasicrystals, tilings, and algebraic number theory: some preliminary connections, The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985), Amer. Math. Soc., Providence, RI, 1987, pp. 241–264. MR 89a:82031 1.1.3