WOLFRAM|DEMONSTRATIONS PROJECT

Any Quadrilateral Can Tile

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midpoint parallelograms
Any quadrilateral
Q
tiles the plane. The tiling is formed by rotating
Q
by 180° about the midpoints of its sides. The same process is then applied to the four new quadrilaterals, and so on.
Another approach is to use the parallelogram
P
formed by joining the midpoints of
Q
's adjacent sides. Tile the plane with
P
in the obvious way. Translate the parts of
Q
from the copies of
P
adjacent to
P
across
P
into the opposite copy of
P
. Then copy the three different patterns in the copies of
P
to form the same tiling as before.
The tiling works whether
Q
is convex or not, simple or not, and even when
Q
is 3D.