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1 PERIODIC TILINGS AND TILINGS BY REGULAR POLYGONS by DARRAH PERRY CRAVEY A thesis submitted in partia1 fu1fi11ment of the requirements for the degree of Doctor of Philosophy (Mathematics) at the UNIVERSITY OF WISCONSIN - MADISON 1984

3 PERIODIC TILINGS AND TILINGS BY REGULAR POLYGONS Darrah Perry Chavey Under the supervision of Professor Donald W. Crowe Abstract: We assume a tiling has, under its symmetry group, v orbits of vertices; e orbits of edges; and t orbits of tiles. Inequalities are established relating these parameters, both for arbitrary tilings and for tilings by regular polygons, and we show that some of these inequalities are sharp. In the case of tilings by regular polygons, we classify those tilings with v ~ 3, e ~ 3, or t~ 2, and show that the number of tilings with some fixed number of orbits of vertices [or edges; or tiles] is finite. The edge figures which can occur in a tiling by regular polygons are Classified, as are tilings which contain at most three different types of these edge figures. Progress is made towards classifying those tilings by regular polygons which contain at most two different types of vertex figures. with respect to tilings by regular polygons which contain only two types of tiles (two congruence classes of polygons), the number of possible orbits of each polygon is determined. Tilings by regular polygons in which any two congruent tiles are equivalent under the symmetries of the tiling are classified, as are tilings which satisfy a similar condition on the edges.

4 ii "We're all in it - we're all tiled, here." Olga. The Grand Duke, by Gilbert and Sullivan. "He's got 'em on the list - he's got 'em on the list; And they'll none of them be missed - they'll none of them be missed." Chorus of Men, The Mikado, by Gilbert and Sullivan. Dedicated to the two women I love Peggy and Eunice Chavey.

5 iii Acknowledgements Now that it's almost over, it seems amazing to me that my friends and my thesis committee (which are not exclusive) have managed to put up with me for the last month or so. They are among the many people I wish to thank for helping to make this thesis possibl.e. None of this work woul.d have been possible without the excellent survey of the subject by GrUnbaum and Shephard, and I wish to thank them for making their advance copy avail.abl.eto us. Professors Donald Crowe and Michael Bl.eicher deserve thanks for their efforts in creating and sustaining a seminar covering this work, and it was from this seminar that most of these results developed. Much of the work in this thesis owes a great deal, in ways that are difficult to pin down, to conversations with Don Crowe and Mike Bleicher; but some of the work can be more directly attributed to my colleagues. Mary Leland discovered one class of tilirlgs used in the proof of theorem 2.3 (as mentioned there), and this class helps to extend the known range of realizable parameters in tilings. The nice proof of fact 1 in section 1.3 is a drastic improvement of my original, and this proof was pointed out by John Rosenberg. Elsa Gunter vol.unteered to draw most of the til.ings in figures on a Carnegie-Mellon laser printer, and these figures (one of prettier aspects of the thesis) would have been impossible without her help.

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17 188 References A. Badoureau 1881 Memoire sur les figures isosceles. J. tcole Poly technique, 49(1881), M. Breen 1983 A Characterization Theorem for Tilings having Countably Many Singular Points. J. of Geometry, 21(1983), D. Chavey 1984 Periodic Tilings and Tilings by Regular Polygons I: Bounds on the Number of Orbits of Vertices, Edges, and Tiles. Mitteilungen aus dem Mathem. Seminar Giessen, 164(2), 1984, pp J. P. Conlan 1976 Derived Tilings. J. Combinatorial Theory (A) 20(1976), E. S. Fedorov 1891 Symmetry in the Plane. (Russian). zapiski Rus. Mineralog. Obscestva, Sere 2, 28(1891), I plates. n

18 189 n B. Grunbaum and G. C. Shephard 1977a Tilings by Regular Polygons. " Math. Magazine, 50(1977), b ~~he Eighty-one Types of Isohedral Tilings in the Plane. Math. Proc. Cambridge Philos. Soc. 82(1977), a Isohedral Tilings of the Plane by Polygons. Comment Math. Helv., 53(1978), b Isotoxal Tilings. Pacific J. Math., 76(1978), c The Ninety-one Types of Isogonal Tilings in the Plane. Trans. Amer. Math. Soc., 242(1978), a Incidence Symbols and their Applications. Relations between combinatorics and other parts of mathematics. Proc. Sympos. Pure Math., XXXIV, Amer. Math. soe,, b Erratum to "The Ninety-one ~pes of Isogonal Tilings in the Plane". Trans. Amer. Math. Soc. 249(1979), a Tilings with Regular Polygons. Models in the Plane from the Days of Kepler to Today, with Recent'Results and Unsolved Problems. (Italian) Archimede 32(1980), b Errata corrige: "Tilings with Regular Polygons". (Italian) Archimede 32(1980), Tilings and patterjs, preliminary edition, 1983.

19 190 T. Heath 1921 A History of Greek Mathematics, Vol. II, Euclid, The Elements, Vol. II, J. Kepl.er 1619 Harmonice Mundi. Lincii, German translation: Weltharmonik, M. Caspar, O. Krotenheerdt 1969 Die homogenen Mosaike n-ter Ordnung in der euklidischen 1970a Ebene. I, II, III. 1970b Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.- Natur. Reihe 18(1969), , 19(2) (1970), and 19(6) (1970), Sur les pavages a l'aide de polygones reguliers. Bull. de la Societe Philomatique de Paris, (8) 3(1891), Question 262. I Intermed. Math. 1(1894), p. 147 and 7(1900), p. 153.

20 191 P. Niggli / 1926 Die regelmassige Punktverteilung langs einer Geraden in einer Ebene. (Symmetrie yon Bordurmuster.) z, Krist. 63(1926), D. M. Y. Sommerville 1905 Semi-regul.ar Networks of the Pl.ane in Absol.u1;eGeometry. Trans. Roy. Soc. Edinburgh 41(1905), and 12 plates. A. Val.ette 1981 Tilings of the Plane by Topolgical Disks. Geometriae Dedicata, 11(1981),

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