References
A. Andreini (1907) Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative, Mem. Soc. Ital. delie Scienze, Ser. 3, 14(1907), 75–129.
D. Barnette, E. Jucovič and M. Trenkler (1971) Toroidal maps with prescribed types of vertices and faces, Mathematika 18(1971), 82–90.
W. Barrett (1939) A note on some networks of polygons, Math. Notes Edinburgh Math. Soc. 31(1939), 23–24.
B. N. Delone (1959) Theory of planigons, [In Russian]. Izv. Akad. Nauk SSSR, Ser. Matem. 23(1959), 365–386.
V. Eberhard (1891) Zur Morphologie der Polyeder, Teubner, Leipzig 1891.
E. S. Fedorov (1916) 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 (1969) Scheibenpackungen konstanter Nachbarnzahl, Acta Math. Acad. Sci. Hungar. 20(1969), 375–381.
B. Grünbaum (1967) Convex Polytopes, Interscience (Wiley), London 1967.
B. Grünbaum and G. C. Shephard (1981) Tilings and Patterns, W. H. Freeman and Co., San Francisco (to appear).
B. Grünbaum and J. Zaks (1974) The existence of certain planar maps, Discrete Math. 10(1974), 93–115.
F. Haag (1932) Strukturformeln für Ebenenteilungen, Zeitschr. für Kristallographie 83(1932), 301–307.
H. Heesch (1933) Über topologisch gleichwertige Kristallbindungen, Zeitschr. für Kristallographie 86(1933), 399–407.
S. Jendrol and E. Jucovič (1977) Generalization of a theorem by V. Eberhard, Math. Slovaca 27(1977), 383–407.
E. Jucovič (1976) On face-vectors and vertex-vectors of cell-decompositions of orientable 2-manifolds, Math. Nachr. 73(1976), 285–295.
E. Jucovič and M. Trenkler (1973) A theorem on the structure of cell-decompositions of orientable 2-manifolds, Mathematika 20(1973), 63–82.
J. Kraeft (1977) Über 3-realisierbare Folgen mit beliebigen Sechseckzahlen, J. Geometry 10(1977), 32–44.
F. Laves (1931) Ebenenteilung und Koordinationszahl, Zeitschr. für Kristallographie 78(1931), 208–241.
A. L. Loeb (1976) Space Structures, Their Harmony and Counterpoint, Addison-Wesley, Reading, Mass. 1976.
P. Mani-Levitska, B. Guigas and W.E. Klee (1979) Rectifiable n-periodic maps, Geom. Dedicata 8(1979), 127–137.
W. Meretz and B. L. van der Waerden (1966) Statistische Theorie der aequalen Zellteilung, Naturwissenschaften 53(1966), 8–11.
I. Niven (1978) Convex polygons that cannot tile the plane, Amer. Math. Monthly 85(1978), 785–792.
K. Reinhardt (1918) Über die Zerlegung der Ebene in Polygone, Inaugural-Dissertation, Univ. Frankfurt a.M., R. Noske, Borna-Leipzig, 1918.
C. S. Smith (1954) The shape of things, Scientific American 190 (January 1954), 58–64.
C. S. Smith (1978) Structural hierarchy in science, art, and history. In On Aesthetics in Science, J. Wechsler ed., MIT Press, Cambridge, Mass. 1978, pp. 9–53.
A. Šubnikov (1916) On the question of the structure of crystals, [in Russian]. Bull. Acad. Imp. Sci., Ser. 6, Vol. 10(1916), 755–779.
T. R. S. Walsh (1972) Characterizing the vertex neighborhoods of semi-regular polyhedra, Geom. Dedicata 1(1972), 117–123.
Author information
Authors and Affiliations
Additional information
Research supported in part by The National Science Foundation, Grant No. McS77-01629 A01
Rights and permissions
About this article
Cite this article
Grünbaum, B., Shephard, G.C. The theorems of Euler and Eberhard for tilings of the plane. Results. Math. 5, 19–44 (1982). https://doi.org/10.1007/BF03323298
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03323298