Mosaics and their isometric embeddings
Izvestiya. Mathematics , Tome 66 (2002) no. 3, pp. 443-462.

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The main result of this paper is an enumeration (given in the Table) of all mosaics $T$ among the 165 mosaics presented in Chavey's catalogue [2] (which includes the most important classifications of mosaics) such that either $T$ or the dual mosaic $T^*$ is embeddable in $Z_n$ isometrically or with scale $\lambda=2$. We also study some interesting infinite-dimensional hypermetric spaces.
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M. Deza; M. I. Shtogrin. Mosaics and their isometric embeddings. Izvestiya. Mathematics , Tome 66 (2002) no. 3, pp. 443-462. http://geodesic.mathdoc.fr/item/IM2_2002_66_3_a0/

[1] Chavey D., Periodic tilings and tilings by regular polygons, Ph. D. Thesis, Univ. of Wisconsin-Madison, 1984

[2] Chavey D., “Tilings by regular polygons. II: A catalog of tilings”, Computers Math. Applic., 17 (1989), 147–165 | DOI | MR | Zbl

[3] Chepoi V., Deza M., Grishukhin V. P., “Clin d'oeil on $l_1$-embeddable planar graphs”, Discrete Applied Math., 80 (1997), 3–19 | DOI | MR | Zbl

[4] Debroey I., Landuyt F., “Equitransitive edge-to-edge tilings by regular convex polygons”, Geom. Dedicata, 11 (1981), 47–60 | DOI | MR | Zbl

[5] Tylkin M. E., “O geometrii Khemminga edinichnykh kubov”, DAN, 134:5 (1960), 1037–1040 | MR

[6] Deza M. M., Loran M., Geometriya razrezov i metrik, Izd-vo MTsNMO, M., 2001

[7] Deza M., Shpectorov S., “Recognition of $l_{1}$-graphs with complexity $O(nm)$, or Football in a Hypercube”, European J. of Combinatorics, 17:2,3 (1996), 279–289 | DOI | MR | Zbl

[8] Deza M., Shtogrin M. I., “Izometricheskie vlozheniya polupravilnykh mnogogrannikov, razbienii i im dualnykh v giperkuby i kubicheskie reshetki”, UMN, 51:6 (1996), 199–200 | MR | Zbl

[9] Deza M., Shtogrin M. I., “Vlozhenie grafov v giperkuby i kubicheskie reshetki”, UMN, 52:6 (1997), 155–156 | MR | Zbl

[10] Deza M., Shtogrin M. I., “Uniform Partitions of $3$-space, their Relatives and Embedding”, European J. of Combinatorics, 21 (2000), 807–814 | DOI | MR | Zbl

[11] Deza M., Shtogrin M. I., “Vlozhenie khimicheskikh grafov v giperkuby”, Matem. zametki, 68:3 (2000), 339–352 | MR | Zbl

[12] Deza M., Shtogrin M. I., “Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices”, Advanced Studies in Pure Mathematics, 27, Kinokuniya company LTD, Tokyo, 2000, 73–92 | MR | Zbl

[13] Grünbaum B., Shephard G. C., Tilings and patterns, W. H. Freeman and Company, N. Y., 1987 | MR | Zbl

[14] Krötenheerdt O., “Die homogenen Mosaike $n$-ter Ordnung in der euklidischen Ebene, 1”, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-natur. Reihe, 18 (1969), 273–290 | MR

[15] Krötenheerdt O., “Die homogenen Mosaike $n$-ter Ordnung in der euklidischen Ebene, 2”, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-natur. Reihe, 19 (1970), 19–38 | MR | Zbl

[16] Krötenheerdt O., “Die homogenen Mosaike $n$-ter Ordnung in der euklidischen Ebene, 3”, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-natur. Reihe, 19 (1970), 97–122 | MR | Zbl

[17] Wells D., The Penguin Dictionary of Curious and Interesting Geometry, Penguin Books, London – N. Y., 1991 | MR | Zbl

[18] Novikov S. P., “Topologiya”, Topologiya – I., Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 12, VINITI, M., 1986, 5–252 | MR

[19] Dolbilin N. P., Shtanko M. A., Shtogrin M. I., “Kubicheskie mnogoobraziya v reshetkakh”, Izv. RAN. Ser. matem., 58:2 (1994), 93–107 | MR | Zbl