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Tilings by regular polygons. II: A catalog of tilings. (English) Zbl 0704.05010

It is assumed that an (edge-to-edge) tiling by regular polygons has, under its symmetry group, v orbits of vertices, t orbits of tiles and e orbits of edges. In the terminology by B. Grünbaum and G. C. Shephard [Tilings and patterns (1987; Zbl 0601.05001)] such a tiling would be called “v-isogonal”, “t-isohedral” and “e-isotoxal” respectively. In connection with a classification of tilings in his part I [Mitt. Math. Semin. Gießen 164, 37-50 (1984; Zbl 0584.05022)] the author gives (on 15 pages) drawings of these tilings: the three “Platonic” tilings \((t=1)\); the eight “Archimedean” tilings \((v=1\), \(t>1)\); the 20 2-isogonal tilings \((v=2)\); the 39 3-isogonal tilings \((v=3)\) which are vertex-homogeneous; the 22 3-isogonal tilings \((v=3)\) which are not vertex-homogeneous; 65 vertex-homogeneous tilings with \(v\geq 4\); the unique tile-homogeneous tiling which is not vertex- homogeneous; the seven tilings with \(t=3\) which are not included in the earlier cases.
Reviewer: E.Quaisser

MSC:

05B45 Combinatorial aspects of tessellation and tiling problems
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
52A10 Convex sets in \(2\) dimensions (including convex curves)
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Full Text: DOI

Online Encyclopedia of Integer Sequences:

Number of n-isohedral edge-to-edge tilings of regular polygons.

References:

[1] Grünbaum, B.; Shephard, G. C., Tilings and Patterns (1987), Freeman: Freeman San Francisco, Calif · Zbl 0601.05001
[2] Chavey, D., Periodic tilings and tilings by regular polygons I: Bounds on the number of orbits of vertices, edges and tiles, Mitt. math. Semin. Giessen, 164, 2, 37-50 (1984) · Zbl 0584.05022
[3] Heath, T., Euclid. Elements, Vol. II (1947)
[4] Heath, T., (A History of Greek Mathematics, Vol. II (1921), Clarendon Press: Clarendon Press Oxford), commentary on Book IV, Prop. 10
[5] Sommerville, D. M.Y., Semi-regular networks of the plane in absolute geometry, Trans. R. Soc. Edinb., 41, 725-747 (1905), + 12 plates · JFM 36.0527.06
[6] DeBroey, I.; Landuyt, F., Equitransitive edge-to-edge tilings by regular convex polygons, Geom. Dedicata, 47-60 (1981) · Zbl 0458.52007
[7] D. Chavey, Tilings by regular polygons VII: Tile regularity (in press).; D. Chavey, Tilings by regular polygons VII: Tile regularity (in press). · Zbl 1324.52014
[8] Kepler, J., Harmonice Mundi, Lincii (1619)
[9] German translation: M. Caspar (1939); German translation: M. Caspar (1939)
[10] Also Johannes Kepler Gesammelte Werke. (Ed. M. Caspar), Band VI. Beck. Munich, (1940).; Also Johannes Kepler Gesammelte Werke. (Ed. M. Caspar), Band VI. Beck. Munich, (1940).
[11] Krötenheerdt, O., Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene. I, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-natur. Reihe, 18, 273-290 (1969) · Zbl 0208.50502
[12] Chavey, D., Periodic tilings and tilings by regular polygons, (Ph.D. Thesis (1984), Univ. of Wisconsin-Madison) · Zbl 0704.05010
[13] D. Chavey, Tilings by regular polygons V: Vertex regularity (in press).; D. Chavey, Tilings by regular polygons V: Vertex regularity (in press). · Zbl 1324.52014
[14] Grünbaum, B.; Shephard, G. C., Isotoxal tilings, Pacif. J. Math., 76, 407-430 (1978) · Zbl 0393.51010
[15] D. Chavey, Tilings by regular polygons VI: Edge regularity (in press).; D. Chavey, Tilings by regular polygons VI: Edge regularity (in press). · Zbl 1324.52014
[16] Krötenheerdt, O., Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene. II, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-natur. Reihe, 19, 19-38 (1970)
[17] Krötenheerdt, O., Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene. II, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-natur. Reihe, 19, 97-122 (1970)
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