zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 {\it B. Grünbaum} and {\it 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 $v\ge 4$; 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:
05B45Tessellation and tiling problems
52C20Tilings in $2$ dimensions (discrete geometry)
52C22Tilings in $n$ dimensions (discrete geometry)
52A10Convex sets in $2$ dimensions (including convex curves)
WorldCat.org
Full Text: DOI
References:
[1] Grünbaum, B.; Shephard, G. C.: Tilings and patterns. (1987) · 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, No. 2, 37-50 (1984) · Zbl 0584.05022
[3] Heath, T.: Euclid. elements. (1947)
[4] Heath, T.: A history of Greek mathematics. (1921) · Zbl 48.0046.01
[5] Sommerville, D. M. Y.: Semi-regular networks of the plane in absolute geometry. Trans. R. Soc. edinb. 41, 725-747 (1905) · Zbl 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). · Zbl 1324.52014
[8] Kepler, J.: Harmonice mundi, lincii. (1619)
[9] German translation: M. Caspar (1939)
[10] 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) · Zbl 0584.05022
[13] 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). · 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)