References
B. Bollobás,Filling the plane with congruent convex hexagons without overlapping. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.6 (1963), 117–123.
F. E. Browder (editor),Mathematical developments arising from Hilbert problems. Proc. Sympos. Pure Math., vol.28. Amer. Math. Soc., Providence RI, 1976.
B. N. Delone,Theory of planigons. [In Russian] Izv. Akad. Nauk SSSR Ser. Mat.23 (1959), 365–386.
J. A. Dunn,Tessellations with pentagons. Math. Gazette55 (1971), 366–369. See alsoibid. Math. Gazette56 (1972), 332–335.
E. S. Fedorov,Elements of the theory of figures. [In Russian], Imp. Akad. Nauk, St. Peterburg 1885. Annotated new edition, Akad. Nauk SSSR, 1953.
E. S. Fedorov,Theorie der Krystallstruktur. I. Mögliche Structurarten.25 (1895), 113–224. Russian translations in:Symmetry of crystals. Akad. Nauk SSSR, 1949. English translation in:Symmetry of crystals. Amer. Crystallog. Assoc. Monograph No. 7, 1971.
E. S. Fedorov,Reguläre Plan-und Raumtheilung. Abh. Bayer. Akad. Wiss. Math. Phys. Cl. vol.20, part 2 (1900), 465–588.
E. S. Fedorov,Systems of planigons as typical isohedra in the plane. [In Russian] Bull. Acad. Imp. Sci., Ser. 6, vol.10 (1916), 1523–1534.
L. Fejes Tóth,Reguläre Figuren, Akadémiai Kiadó Budapest 1965. English translation:Regular Figures, Pergamon, New York 1964.
M. Gardner,On tessellating the plane with convex polygon tiles. Scientific American, July 1975 112–117. Related materialibid. August 1975, pp. 112–115, and Sept. 1975, pp. 174–180.
M. Gardner,Mathematical Games. Scientific American, December 1975, pp. 116–119.
M. Gardner,Extraordinary nonperiodic tiling that enriches the theory of tiles. Scientific American, January 1977, pp. 110–121.
M. Goldberg,Central tesselations. Scripta Math.21 (1955), 253–260.
B. Grünbaum,Problem No. 15. Bull. London Math. Soc.8 (1976), 31.
B. Grünbaum andG. C. Shephard,The eighty-one types of isohedral tilings in the plane. Math. Proc. Cambridge Philos. Soc.82 (1977), 177–196.
B. Grünbaum andG. C. Shephard,The theorems of Euler and Eberhard for tilings of the plane. (To appear).
R. Guy,The Penrose pieces. Bull. London Math. Soc.8 (1976), 9–10.
F. Haag,Die regelmässigen Planteilungen. Z. Kryst. Min.49 (1911), 360–369.
F. Haag,Die regelmässigen Planteilungen und Punktsysteme. Z. Krist.58 (1923), 478–489.
F. Haag,Die Planigone von Fedorow, Z. Krist.63 (1926), 179–186.
T. Hayashi,Path of a particle moving within a polygon. [In Japanese] Tôhoku Math. J.11 (1917), 211–228.
E. Heesch, H. Heesch andJ. Loef,System einer Flächenteilung und seiner Anwendung zum Werkstoff- und Arbeitsparen. Reichsminister für Rüstung und Kriegsproduktion, 1944.
H. Heesch,Aufbau der Ebene aus kongruenten Bereichen. Nachr. Ges. Wiss. Göttingen, New Ser.,1 (1935), 115–117.
H. Heesch,Reguläres Parkettierungsproblem. Westdeutscher Verlag, Köln-Opladen 1968.
H. Heesch andO. Kienzle,Flächenschluss. Springer-Verlag, Berlin-Göttingen-Heidelberg 1963.
D. Hilbert,Mathematische Probleme. Göttinger Nachr. 1900, pp. 253–297; also Arch. Math. Phys. Ser. 3.,1 (1901), 44–63 and 213–237. English translation:Mathematical Problems. Bull. Amer. Math. Soc.8 (1902), 437–479; reprinted in [2]F. E. Browder (editor),Mathematical developments arising from Hilbert problems. Proc. Sympos. Pure Math., vol.28. Amer. Math. Soc., Providence RI, 1976, pp. 1–34.
J. Horváth,Bemerkungen zur Theorie der Planigone. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.8 (1965), 147–153.
R. B. Kershner,On paving the plane. Amer. Math. Monthly,75 (1968), 839–844.
F. Laves,Ebenenteilung und Koordinationszahl, Z. Krist.78 (1931), 208–241.
A. L. Loeb,Space structures. Their harmony and counterpoint. Addison-Wesley, Reading, Mass. 1976.
P. A. MacMahon,New mathematical pastimes. Cambridge University Press, London 1921.
P. A. MacMahon,The design of repeating patterns for decorative work. J. Roy. Soc. Arts,70 (1922), 567–578. Related discussion,ibid. J. Roy. Soc. Arts, pp. 578–582.
P. A. MacMahon andW. P. D. MacMahon,The design of repeating patterns. Proc. Roy. Soc. London,101 (1922), 80–94.
W. P. D. MacMahon,The theory of closed repeating polygons in Euclidean space of two dimensions. Proc. London Math. Soc. (2) vol.23 (1925), 75–93.
J. Milnor,Hilbert's Problem 18:On crystallographic groups, fundamental domains, and on sphere packing. Pp. 491–506 in [2]
K. Reinhardt,Über die Zerlegung der Ebene in Polygone. Dissertation, Univ. Frankfurt a. M. Noske, Borna-Leipzig 1918.
K. Reinhardt,Zur Zerlegung der euklidischen Räume in kongruengte Polytope. S.-Ber. Preuss. Akad. Wiss. Berlin, 1928, pp. 150–155.
K. Reinhardt,Über die Zerlegung der euklidischen Ebene in kongruente Bereiche. J.-Ber. Deutsch. Math.-Verein.38 (1929), p. 12 ital.
D. Schattschneider,Tiling the plane with congruent pentagons. Math. Magazine.
A. V. Šubnikov andV. A. Kopcik,Symmetry in science and art. [in Russian] Nauka Press, Moscow 1972). English translation: A. V. Shubnikov and V. A. Koptsik,Symmetry in science and art. Plenum Press, New York and London, 1974.
H. Voderberg,Zur Zerlegung der Umgebung eines Bereiches in kongruente. J.-Ber. Deutsch. Math.-Verein.46 (1936), 229–231.
H. Voderberg,Zur Zerlegung der Ebene in kongruente Bereiche in Form einer Spirale. J.-Ber. Deutsch. Math.-Verein.47 (1937), 159–160.
T. R. S. Walsh,Characterizing the vertex neighborhoods of semiregular polyhedra. Geom. Dedicata 1 (1972), 117–123.
Author information
Authors and Affiliations
Additional information
Dedicated to Hugo Hadwiger on his seventieth birthday
Research supported by the National Science Foundation Grant MPS74-07547 A01.
Rights and permissions
About this article
Cite this article
Grünbaum, B., Shephard, G.C. Isohedral tilings of the plane by polygons. Commentarii Mathematici Helvetici 53, 542–571 (1978). https://doi.org/10.1007/BF02566098
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02566098