I had a great session the other day with two wonderful kids, age 5 and 7. After warming up with a quick game of Hex, we jumped into our activity for the hour: playing with pattern blocks.

Pattern blocks have been loved on here at the Math for Love blog before, and no doubt on countless other blogs about math. This is because they are awesome. Pattern blocks allow for lessons on adding and subtracting, multiplication and division, fractions, geometry, plane tilings, primes, and symmetry, to name a few. After spending a few minutes with the kids building beautiful shapes out of the blocks, we stopped to discuss what we had built. The process of building with pattern blocks draws on a person’s deep intuition for some beautiful ideas of mathematics, even if we can’t put words to why. For some reason, it’s inexplicably delightful to fit the blocks together, to find the piece that fills in a gap, to balance color and shape throughout the design. And indeed, the kids had done this.

I asked them about their shapes, and observed that one of them had built a highly symmetrical design while the other had built a highly *un*symmetrical design, and this, of course, blew the door open onto the topic for the next 45 minutes. Symmetry–one of the great ideas in mathematics that somehow we all intuitively understand but that mathematics gives us tools to describe and to explore deeply. We briefly talked about mirror symmetry but that was not to keep our focus for long. Rotational symmetry seemed so much more delicious. The kids and I spent the hour classifying the rotational symmetry of individual objects–the blocks, other things in the room, people, etcetera–and grouping them according to what degree of rotational symmetry they possessed. The square went in the four pile. The triangle in the three. The trapezoids, people, and many of the other things we looked at went in the one pile. I held up the bright red circular lid of the container of pattern blocks. “What about this one? What kind of rotational symmetry does this have?” They thought for a bit, and then one of the children smiled and suggested it had infinite rotational symmetry. Boy did it ever.

The kids then built designs using the pattern blocks that had various sizes of rotational symmetry. We worked our way up to eight–a shape which has eight angles through which it can be rotated and look the same. Ending with a couple questions (a fantastic mathematical habit of mind for all ages), we asked if it would be possible to build a pattern block shape with infinite rotational symmetry, and if not, what was the largest possible rotational symmetry value we could build? Something to ruminate on next time.

I love this line of inquiry because it’s so ripe for further exploration. Kids can classify most objects they see in the world according to what kind of rotational symmetry the object has, and this will start to seed in them a stronger sense of how rotational symmetry works. It leads naturally into ideas of multiplication and addition: objects with 4-rotational symmetry have 2-rotational symmetry, those with 6-rotational symmetry have 3-rotational symmetry and 2-rotational symmetry, and so on. It even allows kids a glimpse of the ideas of limits (as was wisely noted by the student who said that circles have infinite rotational symmetry). Primes can be explored here, as those shapes with prime rotational symmetry have no smaller values of rotational symmetry. And of course it touches on the deep mathematical ideas of isometries and symmetric groups, sophisticated and elegant ‘high-level’ mathematics that is built on the humblest of all things, a pattern block with rotational symmetry.