A symmetry of an object is a motion of that object that
doesn't change its size or shape. That is, a symmetry is something you
do to an object.
Examples for objects in the plane: slide the object (officially called
translation); turn it around (officially called rotation );
flip it over, or replace it with its mirror image (officially called
reflection); flip it, then slide
(officially called glide reflection).
In fact, these 4 motions are the only types of symmetries for objects in
the plane. For more details, including pictures, see
The 4 plane symmetries.
An object or pattern has symmetry of a certain type if it looks the
same when that symmetry is done to it.
Examples: A valentine heart has reflection symmetry: reflection across
its center line switches its right and left halves, so afterwards it still
looks the same. A pinwheel has rotational symmetry: if you rotate it by
90 degrees, it looks the same.
Problem of the Day
This house shape has reflection symmetry, and no other symmetries
(except the "do nothing" symmetry).
Can you make a pattern using only copies of the house that has
rotational symmetry, but not reflection symmetry?
Extensions and related problems
Make a pattern using only copies of the house that has rotational
symmetry and reflection symmetry.
Make a pattern using only copies of the house that has translation
symmetry and no other symmetries.
Make a pattern using only copies of the house that has glide
reflection symmetry and no other symmetries.
Make a pattern using only copies of the house that has translation
symmetry and rotation symmetry.
What other combinations of symmetries can you make with copies
of the house?
Can you find a symmetric pattern of houses that tessellates the
plane (covers the whole plane with no gaps or overlaps)?
