Mathematical symmetry is commonly found in tessellations.

A pattern is symmetric if there is at least one symmetry
(translation, rotation, glide reflection, reflection)
that leaves the pattern unchanged.

Translational Symmetry

Rotational Symmetry

Glide Reflection Symmetry

Reflection Symmetry

Symmetries create patterns that help us organize our world conceptually.
Symmetric patterns occur in nature, and are invented by artists,
 craftspeople, musicians, choreographers, and mathematicians.

Note: Tessellations extend to infinity; the diagrams shown below are finite portions of infinite tessellations.

Translational, rotational, and glide refection symmetry are the three mathematical symmetries most commonly found in tessellations.

1. Translational Symmetry - a tessellation has translational symmetry if it can be translated by some vector and remain unchanged. Any tessellation with this property has inifinitely many different translation vectors due to the infinite extent of tessellations. The tessellation below has translational symmetry; two possible vectors are shown.

2. Rotational Symmetry - a tessellation has rotational symmetry if it can be rotated by some angle about some point and remain unchanged. A tessellation which can be rotated by 1/n of a full revolution and remain unchanged is said to posses n-fold rotational symmetry. In the example below, point A is a point of 3-fold rotational symmetry, while point B is a point of 2-fold rotational symmetry.

3. Glide Reflection Symmetry - a tessellation has glide reflection symmetry if it can be translated by some vector and then reflected about that vector and remain unchanged. A special case of glide rereflection symmetry is simple reflection or mirror symmetry, where the vector has a value of zero. The example below illustrates glide reflection. Try to find some lines of simple reflection symmetry for the first tessellation above.

How many tessellation tilings can you find in your house? How many can you find in your classroom, in nature or outdoors? Locate examples of real-world tessellations, scan or photograph them, and describe tessellations found in the real world (at three different tessellations per person should be included in your group presentation). Name and describe the examples that you find. Turn in printed copies of the examples that you found.

What is symmetry? (Even though we all understand and recognize symmetry intuitively, it is a little harder to say just what it is.)

Our Tessellation WebQuest developed by Cynthia R. Parker
7th Grade Mathematics Instructor
Alice Drive Middle School
Sumter School District #17
Sumter, South Carolina
July 2004