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# Area tiling

This topic contains 3 replies, has 2 voices, and was last updated by  Aritra 3 months, 2 weeks ago.

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• #28915

Participant

Suppose we have to cover the xy plane with identical tiles such that no two tiles overlap and no gap is left between the tiles. Suppose that we can chose the tiles of the following shapes: equilateral triangle, square, regular pentagon, regular hexagon.  Then the tilings can be done with the tiles of

A) all four shapes

B) exactly three of the four shapes

C) exactly two of the four shapes

D) exactly one of the four shapes

#29006

Aritra
Moderator

if you tile the plane by congruent regular polygons, there must be nn polygons meeting at each vertex. Thus the interior angles of each polygon must be 2π/n, for some positive integer n .

now see that for n≤6 the angle is greater than π/3 .

but for , n>6 the polygons would need to have angles less than π/3, which is impossible.

but when n= 5 then internal angle is 108 which doesnot divide 360 so pentagon is not possible   ,  option a is correct

• This reply was modified 3 months, 2 weeks ago by  Aritra.
• This reply was modified 3 months, 2 weeks ago by  Aritra.
• This reply was modified 3 months, 2 weeks ago by  Aritra.
#29062

Participant

Ok… so a pentagon is not possible and the other shapes are possible right .

? Thank you…

#29064

Aritra
Moderator

yes, even except those three no tiling is possible

• This reply was modified 3 months, 2 weeks ago by  Aritra.
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