Parallelogon: Difference between revisions

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A parallelogon is a convex polygon such that images of the polygon under translations only tile the plane when fitted together along entire sides.^[1]

A parallelogon must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious restriction is that a parallelogon can only have four or six sides;^[1] a four-sided parallelogon is a parallelogram. A regular octagon, for example, is not a parallelogon because it does not tile the plane.

References

^ ^a ^b Aleksandr Danilovich Aleksandrov Convex Polyhedra p351

[aleksandrov-1] Aleksandr Danilovich Aleksandrov Convex Polyhedra p351

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 A '''parallelogon''' is a [[convex]] [[polygon]] such that images of the polygon under translations only tile the plane when fitted together along entire sides.<ref name=aleksandrov>Aleksandr Danilovich Aleksandrov ''Convex Polyhedra'' [http://books.google.co.uk/books?id=R9vPatr5aqYC&pg=PA351#v=onepage&q=&f=false p351]</ref>
-It is obvious that a parallelogon must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious restriction is that a parallelogon can only have four or six sides;<ref name=aleksandrov/> a four-sided parallelogon is a [[parallelogram]]. A regular [[octagon]], for example, is not a parallelogon because it does not tile the plane.
+A parallelogon must have an even number of sides and opposite sides must be equal in length and parallel (hence the name). A less obvious restriction is that a parallelogon can only have four or six sides;<ref name=aleksandrov/> a four-sided parallelogon is a [[parallelogram]]. A regular [[octagon]], for example, is not a parallelogon because it does not tile the plane.
 ==References==