Symmetry Cards

The Symmetry Cards introduce players to the mathematical concepts of symmetry groups which are highly visual and accessible to players of all ages and backgrounds.

What is Symmetry?

An object has symmetry if it looks the same after you move it in a certain way. The most well-known type of symmetry is mirror symmetry (or reflection symmetry). An object has mirror symmetry if it looks the same after reflecting it across a line (called a mirror line).

If you look through the cards, you’ll see that the characters ANA and BOB are interacting with a variety of different objects which are repeated in different colors in the corners of each card. Find the card where BOB is playing the double bass. Notice that the double bass has a vertical mirror line.

Now find the card where ANA is making a waffle. Notice that the waffle has 4 mirror lines: vertical, horizontal, and two diagonals.

A second type of symmetry is called rotation symmetry. An object has rotation symmetry if it looks the same after being rotated. Find the card where ANA is holding a pinwheel. Notice that the pinwheel does not have any mirror lines, but it looks the same after you rotate it a quarter turn or 90° which happens 4 times within 360° or a full circle. When you rotate an object, the number of times you can rotate it and it looks the same within 360° is called the order. So, the pinwheel has rotation symmetry of order 4. Now, look again at the card where ANA is making a waffle. Notice that if you rotate the waffle, it also looks the same 4 times within 360°. So, the waffle has rotation symmetry of order 4 in addition to 4 mirror lines, for a total of 8 symmetries in all!

Each card in the deck can be classified according to the symmetry group of the object in its corners. A symmetry group is the collection of all symmetries of an object. It turns out that all two-dimensional objects can be divided into just two types of symmetry groups: (1) The cyclic group which contains objects that have rotation symmetry only, like the pinwheel and (2) the dihedral group which contains objects that have both rotation and mirror symmetry, like the waffle. We use the letter C to represent cyclic groups and the letter D to represent dihedral groups. Furthermore, we place a subscript next to the letters C and D to indicate the order of the rotation symmetry for the objects in that group. For objects in D, the order of rotation is always equal to the number of mirror lines.

Find the card where ANA is playing football. Notice that the football has 2 mirror lines (vertical and horizontal) and rotation symmetry of order 2 because it looks the same after you rotate it 180° (which happens 2 times within 360°). Since it has both symmetries, we classify it as being a member of the symmetry group D2.

Find the card where BOB is cutting a flower. The flower has rotation symmetry of order 6 and no mirror lines . Because it only has rotation symmetry, we classify it as being a member of the symmetry group C6.

When two objects have the same symmetries, we describe them as being members of the same symmetry group. Look back at the waffle. It is a member of the symmetry group D4. Can you find a card with an object belonging to the same symmetry group as the waffle? Look back at the pinwheel. It is a member of the symmetry group C4. Can you find a card with an object belonging to the same symmetry group as the pinwheel?

Now you might be asking yourself, are there objects that don’t have any symmetry? Let’s see. Find the card where BOB is holding a coffee mug. The coffee mug doesn’t have any mirror lines, but it looks the same when rotated 360° (figure 6). Therefore, the coffee mug has rotation symmetry of order 1 and is in the symmetry group C1. This is the least symmetric an object can be because all objects look the same when rotated 360°.

On the other side of the spectrum, there are objects that have infinite symmetry. Find the card where ANA and BOB are playing darts (figure 7). The dartboard has infinitely many mirror lines and looks the same when rotated any amount! Can you guess what symmetry group it’s in? If you guessed D∞, you’re right!

Pick a few cards and try classifying them according to their symmetry group. The main thing to keep in mind is whether or not the object has any mirror lines. If it has mirror lines, then it must be in D. If it doesn’t have any mirror lines, then it will be in C. Once you decide between D or C you just have to figure out the order of rotation.

Are you getting the hang of it?

Now you’ll also notice that there are two wild cards in the deck. These cards represent any card you choose. Use them wisely!

Once you’ve got the basic gist of the deck, start playing! The cards are very similar to standard playing cards, so you can use them to play many of your favorite card games. Just treat the colors as suits, and the symmetry groups C1-C6, D1-D6, & D∞ as the 13 ranks.