Hi everyone,
I'm working on "direct regular tilings of the plane", that is, for E a euclidian plane, a couple (G,P) with G a subgroup of Is+(E) (direct isometries of E) and P a compact set with non-empty interior such that :
- the union of g(P) for g in G is E ;
- if g(Int(P)) intersects g'(Int(P)), then g(P)=g'(P) ;
I want to show that G must contain a translation (other than the identity). I supposed G doesn't contain one. So it only contains rotations. If there's only one center for these rotations, then it can't cover the plane. So there are at least two centers C1 and C2. Consider two rotations R1 and R2 whose centers are C1 and C2. R1•R2 would be a translation if the sum of the angles t1+t2 was 0 (mod 2pi). So the two angles won't sum up to a multiple of 2pi. R1•R2 is another rotation, whose center can be found with the angles, and the centers C1 and C2 and whose angle is t1+t2.
I can't find any contradiction... If someone could give me a hint that would be wonderful !
Thanks a lot and merry Christmas.
I'm working on "direct regular tilings of the plane", that is, for E a euclidian plane, a couple (G,P) with G a subgroup of Is+(E) (direct isometries of E) and P a compact set with non-empty interior such that :
- the union of g(P) for g in G is E ;
- if g(Int(P)) intersects g'(Int(P)), then g(P)=g'(P) ;
I want to show that G must contain a translation (other than the identity). I supposed G doesn't contain one. So it only contains rotations. If there's only one center for these rotations, then it can't cover the plane. So there are at least two centers C1 and C2. Consider two rotations R1 and R2 whose centers are C1 and C2. R1•R2 would be a translation if the sum of the angles t1+t2 was 0 (mod 2pi). So the two angles won't sum up to a multiple of 2pi. R1•R2 is another rotation, whose center can be found with the angles, and the centers C1 and C2 and whose angle is t1+t2.
I can't find any contradiction... If someone could give me a hint that would be wonderful !
Thanks a lot and merry Christmas.