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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Tilings with congruent tiles
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by Branko Grünbaum and G. C. Shephard PDF
Bull. Amer. Math. Soc. 3 (1980), 951-973
References
  • Die Hilbertschen Probleme, Ostwalds Klassiker der Exakten Wissenschaften, Band 252, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1971 (German). Mit einem Nachdruck von D. Hilberts Vortrag “Mathematische Probleme”; Erläutert von einem Autorenkollektiv unter der Redaktion von P. S. Aleksandrov; Verantwortlicher Herausgeber: Hans Wussing; Redaktion der deutschen Übersetzung: Hannelore Bernhardt und Walter Purkert; Übersetzung der Kommentare aus dem Russischen: Hannelore Bernhardt, Manfred Burkhardt, Siegfried Gottwald, Walter Purkert, Jürgen vom Scheidt, Rainer Schimming und Volkmar Wünsch. MR 0345757
  • W.Baily, 1908, On models of space-filling solids, Math. Gazette 4 (1908), 297-302.
  • Janos Baracs, Juxtapositions, Structural Topology 1 (1979), 59–72, 73 (English, with French summary). MR 621629
  • Ludwig Baumgartner, Zerlegung des vierdimensionalen Raumes in kongruente Fünfzelle, Math.-Phys. Semesterber. 15 (1968), 76–86 (German). MR 235453
  • Ludwig Baumgartner, Zerlegung des $n$-dimensionalen Raumes in kongruente Simplexe, Math. Nachr. 48 (1971), 213–224 (German). MR 287430, DOI 10.1002/mana.19710480115
  • Felix E. Browder (ed.), Mathematical developments arising from Hilbert problems, Proceedings of Symposia in Pure Mathematics, Vol. XXVIII, American Mathematical Society, Providence, R.I., 1976. MR 0419125
  • G. O. Brunner and F. Laves, How many faces has the largest space-filling polyhedron?, Z. Krist. 147 (1978), no. 1-2, 39–43 (English, with German summary). MR 518541, DOI 10.1524/zkri.1978.147.1-2.39
  • Paul J. Cohen, Decision procedures for real and $p$-adic fields, Comm. Pure Appl. Math. 22 (1969), 131–151. MR 244025, DOI 10.1002/cpa.3160220202
  • H. S. M. Coxeter, The classification of zonohedra by means of projective diagrams, J. Math. Pures Appl. (9) 41 (1962), 137–156. MR 141004
  • H. S. M. Coxeter, 1978, Review of "Three-dimensional nets and polyhedra" by A. F. Wells, Bull. Amer. Math. Soc. 84 (1978), 466-470. K. Critchlow, 1970, Order in space, Viking Press, New York, 1970.
  • Ludwig Danzer, Zerlegbarkeit endlichdimensionaler Räume in kongruente Simplices, Math.-Phys. Semesterber. 15 (1968), 87 (German). MR 235454
  • H. E. Debrunner, 1980, Pflasterungen des euklidischen Raumes mit kongruenten Simplexen (to appear). B. N. Delone (= Delaunay), 1932, Neue Darstellung der geometrischen Kristallographie, Zeitschr. Kristallographie 84 (1932), 109-149.
  • B. N. Delone, Proof of the fundamental theorem in the theory of stereohedra, Dokl. Akad. Nauk SSSR 138 (1961), 1270–1272 (Russian). MR 0133056
  • B. N. Delone, On Hilbert’s eighteenth problem, Hilbert’s Problems (Russian), Izdat. “Nauka”, Moscow, 1969, pp. 200–203 (Russian). MR 0257241
  • B. N. Delone, N. P. Dolbilin, and M. I. Štogrin, Combinatorial and metric theory of planigons, Trudy Mat. Inst. Steklov. 148 (1978), 109–140, 275 (Russian). Algebra, number theory and their applications. MR 558946
  • B. N. Delone, R. V. Galiulin and M. I. Štogrin, 1979, The contemporary theory of regular decompositions of the Euclidean space, pages 235-260 of Fedorov [1979]. (Russian) B. Delone, N. Padurov and A. Aleksandrov, 1934, Mathematical foundations of the structural analysis of crystals, ONTI, Moscow-Leningrad, 1934. (Russian)
  • B. N. Delone and N. N. Sandakova, Theory of stereohedra, Trudy Mat. Inst. Steklov. 64 (1961), 28–51 (Russian). MR 0137029
  • Peter Engel, Zur Interpretation der Pattersonsynthese, Z. Krist. 151 (1980), no. 3-4, 217–230 (German, with English summary). MR 569428, DOI 10.1524/zkri.1980.151.3-4.217
  • E. S. Fedorov, Načala učeniya o figurah, Izdat. Akad. Nauk SSSR, Moscow, 1953 (Russian). MR 0062061
  • E. S. Fedorov, 1899, Regulare Plan-und Raumteilung, Abh. K. Bayer. Akad. Wiss. (II. Classe) 20 (1899), 465-588. Russian translation: Fedorov [1979].
  • E. S. Fedorov, Pravil′noe delenie ploskosti i prostranstva, “Nauka” Leningrad. Otdel., Leningrad, 1979 (Russian). Translated from the 1899 German original by A. V. Nardova; With a preface by I. I. Šafranovskiĭ, V. A. Frank-Kameneckiĭ and K. P. Janulov; With papers by Šafranovskiĭ, Frank-Kameneckiĭ, B. N. Delone, R. V. Galiulin and M. I. Štogrin. MR 571363
  • W. Fischer, 1979, Homogene Raumteilungen in konvexe Polyeder, Collected Abstracts, Symposium on Mathematical Crystallography, Riederalp, Wallis (Switzerland), August 1979, page 5. L. Föppl, 1914, Der Fundamentalbereich des Diamantgitters, Phys. Zeitschr. 15 (1914), 191-193. M. Gardner, 1975a, On tessellating the plane with convex polygon tiles, Scientific American, July 1975, pp. 112-117. M. Gardner, 1975b, Mathematical games, Scientific American, December 1975, pp. 116-119. M. Gardner, 1977, Extraordinary nonperiodic tiling that enriches the theory of tiles, Scientific American, January 1977, pp. 110-121.
  • Jack Giles Jr., Infinite-level replicating dissections of plane figures, J. Combin. Theory Ser. A 26 (1979), no. 3, 319–327. MR 535163, DOI 10.1016/0097-3165(79)90110-9
  • Michael Goldberg, The space-filling pentahedra, J. Combinatorial Theory Ser. A 13 (1972), 437–443. MR 307048, DOI 10.1016/0097-3165(72)90077-5
  • Michael Goldberg, Three infinite families of tetrahedral space-fillers, J. Combinatorial Theory Ser. A 16 (1974), 348–354. MR 343156, DOI 10.1016/0097-3165(74)90058-2
  • Michael Goldberg, The space-filling pentahedra. II, J. Combinatorial Theory Ser. A 17 (1974), 375–378. MR 365365, DOI 10.1016/0097-3165(74)90101-0
  • Michael Goldberg, Several new space-filling polyhedra, Geometriae Dedicata 5 (1976), no. 4, 517–523. MR 470860, DOI 10.1007/BF00150781
  • Michael Goldberg, On the space-filling hexahedra, Geometriae Dedicata 6 (1977), no. 1, 99–108. MR 448241, DOI 10.1007/BF00181585
  • Michael Goldberg, On the space-filling heptahedra, Geom. Dedicata 7 (1978), no. 2, 175–184. MR 500559, DOI 10.1007/BF00181630
  • Michael Goldberg, Convex polyhedral space-fillers of more than twelve faces, Geom. Dedicata 8 (1979), no. 4, 491–500. MR 553685, DOI 10.1007/BF00183263
  • Branko Grünbaum and G. C. Shephard, The eighty-one types of isohedral tilings in the plane, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 2, 177–196. MR 448233, DOI 10.1017/S0305004100053810
  • Branko Grünbaum and G. C. Shephard, Isohedral tilings of the plane by polygons, Comment. Math. Helv. 53 (1978), no. 4, 542–571. MR 511847, DOI 10.1007/BF02566098
  • Branko Grünbaum and G. C. Shephard, Tilings and patterns, A Series of Books in the Mathematical Sciences, W. H. Freeman and Company, New York, 1989. An introduction. MR 992195
  • Branko Grünbaum and G. C. Shephard, Spherical tilings with transitivity properties, The geometric vein, Springer, New York-Berlin, 1981, pp. 65–98. MR 661770
  • Georg Hajós, Über einfache und mehrfache Bedeckung des $n$-dimensionalen Raumes mit einem Würfelgitter, Math. Z. 47 (1941), 427–467 (German). MR 6425, DOI 10.1007/BF01180974
  • E. Heesch, H. Heesch and J. Loef, 1944, System einer Flachenteilung und seine Anwendung zum Werkstoff- und Arbeitsparen, Reichsminister für Rüstling und Kriegsproduktion, 1944. H. Heesch, 1934, Über Raumteilungen, Nachr. Gesellschaft Wiss. Göttingen, Neue Folge, 1 (1934), 35-42. H. Heesch, 1935, Aufbau der Ebene aus kongruenten Bereichen, Nachr. Gesellschaft Wiss. Göttingen, Neue Folge, 1(1935), 115-117.
  • Heinrich Heesch, Reguläres Parkettierungsproblem, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 172, Westdeutscher Verlag, Cologne, 1968 (German, with English and French summaries). MR 0230218, DOI 10.1007/978-3-322-98547-7
  • H. Heesch, 1968b, Parkettierungsprobleme, Der Mathematikunterricht, Ernst Klett Verlag, Stuttgart, 1968, No. 4, pp. 5-45. N. F. M. Henry and K. Lonsdale, 1965, International Tables for X-Ray Crystallography, Vol. 1, Kynoch Press, Birmingham, 1965. D. Hilbert, 1900, Mathematische Probleme, Göttinger Nachrichten 1900, pp. 253-297; also Arch. Math. Phys. Ser. 3, 1 (1901), pp. 44-63 and 213-237. English translation: Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), 437-479; reprinted in Browder [1976], pp. 1-34. H. Jansen, 1909, Lückenlose Ausfüllung des Rn mit gitterförmig angeordneten n-dimensionalen Quadern, Dissertation, Universität Kiel, Teubner, Leipzig 1909. I. Kaplansky, 1977, Hilbert’s Problems (Preliminary edition), Lecture Notes in Mathematics, Univ. of Chicago, 1977. O.-H. Keller, 1930, Über die lückenlose Erfüllung des Raumes mit Würfeln, J. Reine Angew. Math. 163 (1930), 231-248.
  • R. B. Kershner, On paving the plane, Amer. Math. Monthly 75 (1968), 839–844. MR 236822, DOI 10.2307/2314332
  • E. Koch, 1972, Wirkungsbereichspolyeder und Wirkungsbereichsteilungen zu kubischen Gitterkomplexen mit weniger als drei Freiheitsgraden, Ph. D. Thesis, Philipps-Universität Marburg/Lahn, 1972. E. Koch and W. Fischer, 1972, Wirkungsbereichstypen einer verzerrten Diamantkonfiguration mit Kugelpackungscharakter, Zeitschr. Kristallographie 135 (1972), 73-92. E. Lampe, 1911, Review of Baily [1908], Fortschritte Math. 39 (1911), p. 552. J. Lawrence, 1980, Tiling Rd by translates of the orthants (to appear) H. D. Löckenhoff and E. Hellner, 1971, Die Wirkungsbereiche der invarianten kubischen Gitterkomplexe, Neues Jahrbuch für Mineralogie, Monatshefte. 1971, pp. 155-174.
  • Benoit B. Mandelbrot, Fractals: form, chance, and dimension, Revised edition, W. H. Freeman and Co., San Francisco, Calif., 1977. Translated from the French. MR 0471493
  • J. Milnor, Hilbert’s problem 18: on crystallographic groups, fundamental domains, and on sphere packing, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R.I., 1976, pp. 491–506. MR 0430101
  • H. Minkowski, 1897, Allgemeine Lehrsätze über die konvexen Polyeder, Nachr. Gesellschaft Wiss. Göttingen, Math.-Phys. Kl. 1987, pp. 198-219. Reprinted in: Gesammelte Abhandlungen, vol. 2, pp. 103-121. Teubner, Leipzig 1911; reprint Chelsea, New York, 1967. W. Nowacki, 1935, Homogene Raumteilung und Kristallstruktur, Ph. D. Thesis, E. T. H. Zürich, 1935. W. Nowacki, 1976, Über allgemeine Eigenschaften von Wirkungsbereichen, Zeitschr. Kristallographie 143 (1976), 360-368.
  • R. Penrose, Pentaplexity: a class of nonperiodic tilings of the plane, Math. Intelligencer 2 (1979/80), no. 1, 32–37. MR 558670, DOI 10.1007/BF03024384
  • Oskar Perron, Über lückenlose Ausfüllung des $n$-dimensionalen Raumes durch kongruente Würfel, Math. Z. 46 (1940), 1–26 (German). MR 3041, DOI 10.1007/BF01181421
  • Oskar Perron, Modulartige lückenlose Ausfüllung des $R_n$ mit kongruenten Würfeln. I, Math. Ann. 117 (1940), 415–447 (German). MR 3042, DOI 10.1007/BF01450026
  • Oskar Perron, Modulartige lückenlose Ausfüllung des $R_n$ mit kongruenten Würfeln. II, Math. Ann. 117 (1941), 609–658 (German). MR 6068, DOI 10.1007/BF01450033
  • K. Reinhardt, 1918, Über die Zerlegung der Ebene in Polygone, Ph. D. Thesis, Univ. Frankfurt a.M. Noske, Borna-Leipzig, 1918. K. Reinhardt, 1928, Zur Zerlegung der euklidischen Räume in kongruente Polytope, S.-Ber. Preuss. Akad. Wiss. Berlin, 1928, pp. 150-155.
  • Raphael M. Robinson, Multiple tilings of $n$-dimensional space by unit cubes, Math. Z. 166 (1979), no. 3, 225–264. MR 526466, DOI 10.1007/BF01214145
  • R. M. Robinson, 1980, Can cubes avoid meeting face to facet The Mathematical Gardner, D. A. Klarner, ed. Prindle-Weber-Schmidt, Boston 1980 (to appear).
  • S. S. Ryshkov and E. P. Baranovskiĭ, $S$-tipy $n$-mernykh reshetok i pyatimernye primitivnye paralleloèdry (s prilozheniem k teorii pokrytiĭ), Izdat. “Nauka”, Moscow, 1976 (Russian). Trudy Mat. Inst. Steklov. 137 (1976). MR 0469874
  • Doris Schattschneider, Tiling the plane with congruent pentagons, Math. Mag. 51 (1978), no. 1, 29–44. MR 493766, DOI 10.2307/2689644
  • A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954), 365–374. MR 63994, DOI 10.2307/1969640
  • K. Seitz, Investigations in the Hajós-Rédei theory of finite Abelian groups, Karl Marx University of Economics, Department of Mathematics, Budapest, 1975. DM 75-6. MR 0396795
  • C. S. Smith, 1978, Structural hierarchy in science, art, and history, On Aesthetics in Science, edited by J. Wechsler. The MIT Press, Cambridge, Mass., 1978, pp. 9-53. F. W. Smith, 1965, The structure of aggregates-a class of 20-faced space-filling potyhedra, Canad. J. Physics 43 (1965), 2052-2055.
  • S. K. Stein, Algebraic tiling, Amer. Math. Monthly 81 (1974), 445–462. MR 340063, DOI 10.2307/2318582
  • M. I. Štogrin, Regular Dirichlet partitions for the second triclinic group, Dokl. Akad. Nauk SSSR 183 (1968), 793–796 (Russian). MR 0236816
  • M. I. Shtogrin, Pravil′nye razbieniya Dirikhle-Voronogo dlya vtoroĭ triklinnoĭ gruppy, Izdat. “Nauka”, Moscow, 1973 (Russian). Trudy Mat. Inst. Steklov. 123 (1973). MR 0370330
  • A. V. Šubnikov and V. A. Kopcik, 1972, Symmetry in science and art, Nauka, Moscow 1972. (Russian) English translation: A. V. Shubnikov and V. A. Koptsik, Symmetry in Science and Art., Plenum Press, New York and London, 1974.
  • Alfred Tarski, A decision method for elementary algebra and geometry, University of California Press, Berkeley-Los Angeles, Calif., 1951. 2nd ed. MR 0044472, DOI 10.1525/9780520348097
  • Robert Williams, The geometrical foundation of natural structure, Dover Publications, Inc., New York, 1979. A source book of design; Corrected reprint. MR 614293
  • A. M. Zamorzaev, On non-normal regular partitions of Euclidean space, Dokl. Akad. Nauk SSSR 161 (1965), 30–32 (Russian). MR 0175030
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Additional Information
  • Journal: Bull. Amer. Math. Soc. 3 (1980), 951-973
  • MSC (1980): Primary 51M20, 52A45; Secondary 05B45, 10E30, 52A15, 52A25, 52A43
  • DOI: https://doi.org/10.1090/S0273-0979-1980-14827-2
  • MathSciNet review: 585178