Conway in http://olympiads.mccme.ru/lktg/2009/4/articles/conway.pdf provided some elegant techniques for identifying tiling of simply connected regions. Are there similar techniques for regions that are not simply connected such as torus in higher dimensions (in the sense that the tiles can be conformal shaped based on their position on the torus)?
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The short answer is "no, not really". In general, there are virtually no positive 3 (and higher) dimensional results on tilings, see my old survey, section 8 on a few sporadic results. Instead, most results are negative, proving hardness of tilebility (see e.g. our recent 3dim domino paper). 

