Tilings with congruent edge coronae. (English) Zbl 1472.52026
Cunningham, Gabriel (ed.) et al., Polytopes and discrete geometry. AMS special session, Northeastern University, Boston, MA, USA, April 21–22, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 764, 251-272 (2021).
A normal tiling of the Euclidean plane is a cover of the plane by non-overlapping closed topological discs such that their diameters are bounded from above and their inradii are bounded from below by positive constants, and such that the intersection of any two tiles is either empty or a singleton or an arc. Such arcs are called edges of the tiling. A centred edge corona is composed of the centre of an edge and of all tiles having a non-empty intersection with that edge. A tiling is called edge-transitive or isotoxal if its symmetry group acts transitively on the set of all its edges. The authors show that every normal tiling with pairwise congruent centred edge coronae is isotoxal, and they classify such tilings.
For the entire collection see [Zbl 1467.52001].
For the entire collection see [Zbl 1467.52001].
Reviewer: Christian Richter (Jena)
MSC:
| 52C20 | Tilings in \(2\) dimensions (aspects of discrete geometry) |
| 05B45 | Combinatorial aspects of tessellation and tiling problems |
| 58D19 | Group actions and symmetry properties |
Keywords:
Euclidean plane; normal tiling; isotoxal tiling; edge corona; symmetry group; edge-transitiveReferences:
| [1] | Dolbilin, Nikolai; Schattschneider, Doris, The local theorem for tilings. Quasicrystals and discrete geometry, Toronto, ON, 1995, Fields Inst. Monogr. 10, 193-199 (1998), Amer. Math. Soc., Providence, RI · Zbl 0918.52007 |
| [2] | Gr\"{u}nbaum, Branko; Shephard, G. C., Isotoxal tilings, Pacific J. Math., 76, 2, 407-430 (1978) · Zbl 0393.51010 |
| [3] | Gr\"{u}nbaum, Branko; Shephard, G. C., Tilings and patterns, xii+700 pp. (1987), W. H. Freeman and Company, New York · Zbl 0601.05001 |
| [4] | Schattschneider, Doris; Dolbilin, Nikolai, One corona is enough for the Euclidean plane. Quasicrystals and discrete geometry, Toronto, ON, 1995, Fields Inst. Monogr. 10, 207-246 (1998), Amer. Math. Soc., Providence, RI · Zbl 0918.52006 |
| [5] | E. Schulte, private comunications, (2017). |
| [6] | E. Schulte, private comunications, (2017). |
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