Middle School Math Circle Lesson

                    GEOMETRIC SYMMETRY

This could be done in two meetings, or shortened for one meeting.

The idea of a SYMMETRY is: a way to move an object onto itself, without stretching or distorting.

I. Plane symmetry. 

Materials: paper, pencils, scissors, blackboard.

1. Fold a blank piece of paper in half, trace and cut a shape along the folded edge. Unfold to get a shape with two-fold reflection symmetry. Namely, there are two ways to move the shape onto itself: either don't move at all (identity symmetry), or reflect across the fold. Doing the reflection twice brings each point back to itself, giving the identity symmetry.

2. Do the same with a paper folded in half, then again in half at right angles to make a folded corner. Unfolded shape has the four-fold symmetry of a rectangle: identity symmetry, or reflect across one of the folds, and what is the last symmetry? Ans: 180 deg rotation, which is the result of doing the two reflections one after another. 

The set of all symmetries, with the composition operation (doing one symmetry after another) is called the SYMMETRY GROUP of the object.

3. One more time, folding in half three times: two at right angles, then one more at 45 deg angle to make a wedge. Unfolded shape has symmetry of a square (dihedral group of 8 symmetries). Class contributes to list and sketch all 8 symmetries on the board. If you get stuck, produce new symmetries by composing known ones: rotation and reflection compose to give a new reflection.

4. Try to draw a plane shape with rotation symmetry, but no reflection symmetry. Ans: a propeller with any number of blades can be rotated onto itself, but reflection makes it turn the other way, not moved onto itself. 

II. Spatial symmetry

Materials: large paper cube for teacher, lump of clay for each student, wire or knife to cut clay.

1. Each student makes his or her own cube out of clay, about 2 cm. Use opposing fingers and flat table to get it even, square on all sides. Now consider only ROTATION symmetries (not mirror reflections, which we cannot do in physical space). Teacher can illustrate with a larger, paper cube.

The cube has three types of symmetry (other than the identity).

a) 90 deg rotations around an axis through midpoints of two opposite faces: these have order 4, meaning doing it four times brings the cube back to its original position.

b) amazingly, 120-deg rotations around an axis through two opposite faces: these have order 3. Look at the cube along a corner axis to see the three 120 deg lines around it.

c) The last type is really surprising: 180 deg rotations around an axis through midpoints of two opposite edges. This is hard to picture, must see it on the model.

2. To illustrate the order 3 symmetries of the clay cube, slice along a plane perpendicular to the corner-to-corner axis. This plane makes a hexagonal "equator" passing through the midpoints of the 6 edges which are not adjacent to the axis corners, cutting an isosceles right triangle off EVERY face. Scratch these lines on one face after another to get the plane, then cut through with a knife or thin wire(teacher can do this?). Each half reveals a regular hexagon along the slice, and has the symmetries of an equilateral triangle: a surprising shape.

Peter Magyar
Dept of Math, Michigan State University
magyar@math.msu.edu