Geodesic
Extension Theorem in the Theory of Isohedral Tilings and Its Applications
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 146-169

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A detailed proof is given for one of the basic theorems in the theory of isohedral tilings, the extension theorem, which describes necessary and sufficient conditions under which a given polyhedron admits an isohedral tiling of a space of constant curvature. 
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     author = {N. P. Dolbilin and V. S. Makarov},
     title = {Extension {Theorem} in the {Theory} of {Isohedral} {Tilings} and {Its} {Applications}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     publisher = {mathdoc},
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     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2002_239_a9/}
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N. P. Dolbilin; V. S. Makarov. Extension Theorem in the Theory of Isohedral Tilings and Its Applications. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Discrete geometry and geometry of numbers, Tome 239 (2002), pp. 146-169. http://geodesic.mathdoc.fr/item/TM_2002_239_a9/