Abstract:
What general regularity manifests itself in the fact
that a triangle, and in general any convex polygon, cannot be tessellated by
non-convex quadrangles? Another question: it is known that for $n>6$
the plane cannot be tessellated by convex $n$-gons
if their diameters are bounded, while the areas are
separated from zero; can this fact be generalized for non-convex
polygons? In the present paper we introduce the characteristic $\chi(M)$
of a polygon $M$. We answer the above questions in terms of $\chi(M)$
and then study tessellations of the plane by $n$-gons equivalent to $M$,
that is, with the same sequence of angles greater than and smaller than $\pi$.
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