Introduction
Knowing techniques for tessellating any triangle, we proceed to
consider quadrilaterals. Quadrilaterals, four-sided polygons,
have one more side than triangles do, making them the next logical
step. Let's consider the arrangement of polygons around a single
vertex, such as we did earlier with regular polygons and triangles.
Is there a way to arrange the quadrilateral around a point so
as to perfectly fill 360 degrees?
Technique: Rotations Around Midpoints
Recall that the sum of the interior angles of any quadrilateral
is 360 degrees. If we managed to fit one of each of the four angles
of a quadrilateral around a point, then the 360 degrees would
be filled exactly.
There are many different ways to arrange the angles of a quadrilateral
around a point to fill 360 degrees. Note that around each vertex,
there is one copy of each angle. (The angles of a quadrilateral
total 360 degrees.)
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As you can see from the examples, most of the arrangements will
not tessellate, mainly because the sides of the triangles do not
line up. Let us try to keep sides lined up.
Rotating the quadrilateral around the midpoints of its sides produces
an arrangement that will tessellate.
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This technique can be used to make a tessellation out of any quadrilateral.
Additional examples:
An application of the technique. The shaded area represents the
pattern produced by repeated rotations.
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An application of the technique. The shaded area represents the
pattern produced by repeated rotations.
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An application of the technique. The shaded area represents the
pattern produced by repeated rotations.
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In summary, any quadrilateral will tessellate using the technique
of rotations about the midpoints of the sides.
  
A real example of a tessellation
of quadrilaterals obtained
by the above technique:
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Now, experiment yourself! (No template required.) Draw any quadrilateral
and use the technique described above to create a tessellation.
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