Seeing Symmetry All Around Us

In the classroom, use familiar objects, corporate logos, or symbols seen in everyday life to observe symmetry and to generate discussion:

1. Have your students handle a baseball - it is spherical in shape. Notice that the outer surface is composed of two identical pieces of leather (but for the writing on them -- asymmetry). The two pieces of leather are the same size and shape, sewn together. This composition exhibits rotational symmetry. See if you can figure out where the point of rotation may be located. (Hint: think three-dimensionally).

2. Look at, think of, or draw each letter of the alphabet and identify the symmetry. (Hints: S, Z, N use rotation, order two; F,G, R, P and others have no symmetry; A and E have different axes of reflection; X and H have reflection along both horizontal and vertical axes).

3. Locate a joker from a pack of playing cards - there's lots of anti-symmetry here (symmetry with a significant property reversal). How many examples of anti-symmetry can you find? (Hints: color, contrast, profile/full face view, open eye/closed eye, suits, and more).

4. Take a pack of playing cards and look at the Queen of Clubs, Queen of Diamonds, King of Hearts, etc. Jacks, Queens, and Kings typically exhibit rotational symmetry (order two, around a central point). If you turn the card upside down, it looks the same as it looks right side up.

5. Yin-yang also exhibits rotational symmetry, order two. It has anti-symmetry because of the white/black interchange. The comma-shaped form is a classic asymmetrical shape. Combined, it forms a circle, with the comma-shape rotated around the central point. Now think of this two-dimensional motif (plane symmetry) in three dimensions. Can you relate it to the baseball?

6. Take a walk around the block and observe automobile hubcaps. How many different kinds of symmetry can you find? (Hint: they are all rotational symmetries around a central point, with each hubcap being a circle divided into two, three, four, five, six, or more sections, each of which is the same, rotated that many times around.

7. Have your students look through the junk mail they receive at home, or advertisements in local papers, to gather company logos. Then have them analyze the symmetries. Many banks and insurance companies use rotational symmetry in their logos. The logos of many companies and organizations also use color alternation or color change. Very few play with asymmetry or symmetry-breaking, but Apple does, probably for the very reason that the bite out of the apple suggests a human dimension. The four-handed logo for Teaching Tolerance uses rotational symmetry, order four. Each hand must be repeated four times to complete its circuit around the central point to arrive at its identity position. Color change makes the rotational symmetry of form less immediately apparent.

8. Washington, DC, is a city of reflections. Consider the reflecting pools along the mall. What happens to the Washington Monument? Or the Jefferson Memorial reflected in the Tidal Basin? Other cities also utilize reflecting pools to emphasize public monuments. A horizontal axis of symmetry is created in each instance.

9. Military formations and many dances utilize different symmetry operations. Can the students offer suggestions for what symmetry operations are used in familiar situations?

10. A teleidoscope has a lens that refracts rays of light. This creates a pattern of repetition based upon the symmetry operation of translation. Each repeated image is the same, just moved along an axis sideways or up and down. A kaleidoscope, on the other hand, utilizes mirrors to create reflections.

11. Modern dance often plays with symmetry, asymmetry, and symmetry-breaking. There is also dissymmetry (a little bit of asymmetry or symmetry-breaking) and anti-symmetry (symmetry with a property reversal). Male/female is a significant property reversal (gender).

12. Crystals-- the study and analysis of symmetry of the plane (two dimensions) evolved from the study of crystal growth patterns. Crystals exhibit three-dimensional symmetry, which is based upon the same four basic symmetries of translation, reflection, glide reflection, and rotation.