Discussion
Which one of the following shapes can be used to tile (completely cover by repeating) a flat plane, extending to infinity in all directions, without leaving any empty spaces in between them? The copies of the shape used to tile are identical and are not allowed to overlap.
Answer: D
A rhombus tiles the plane.
Copies of any parallelogram, a rhombus included, can be laid in rows by translation: each row fits edge to edge, and the next row sits on top of it, so the plane is covered with no gaps or overlaps. At each vertex the angles meet as two acute and two obtuse angles, which sum to 360°.
Circles always leave curved gaps between them. A regular polygon tiles the plane on its own only if its interior angle divides 360° exactly: the regular pentagon’s 108° and the regular octagon’s 135° do not, so both leave gaps (octagons need extra squares to fill the space between them). Only the equilateral triangle, the square and the regular hexagon tile among regular polygons.
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