# Problem of the Day #1

### Definitions

• 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)?

To: Math Forum: ICME 8 || California Math Show at Cal State San Bernardino