(Originally published May 4, 2010)

Our story so far . . .

- There are four operations of symmetry (translation, rotation, reflection, and glide reflection);
- There are seven linear symmetry groups and seventeen planar symmetry groups as codified by crystallographers and chemists;
- Crystallographers and chemists are not necessarily quilters;
- The symmetry groups do not allow for “sideways” tiles (but for one exception);
- I have a tile, a square with lines drawn through it, whose lines always connect with lines in adjacent tiles to outline shapes unique to each arrangement of tiles.

The tile rotates in 90-degree increments, creating four distinct tiles with four different orientations.

The tile reflects, creating a mirror-image tile. The mirror-image tile rotates in 90-degree increments, creating four distinct mirror-image tiles with four different orientations.

I actually have eight tiles, not just one.

I group four tiles to create a block. There’s no reason the four tiles cannot rotate or reflect independently of the others (regardless of what the crystallographers and chemists say).

With eight possible tiles in four possible positions, there are 4,096 (8 x 8 x 8 x 8 = 4,096) possible 4-tile blocks.

I propose a systematic exploration of the possible 4-tile blocks, patterns, and tessellations, just by rotating and reflecting each tile of the block, one at a time.

Would you care to join me?

I did that same sum when I tried to work out how many variations of Drunkard’s Path there might be. If I can find the scrap of paper I will check my answer!

Then all those 4,096 can be combined to create … a heap more!

I came across a similar scrap of paper while cleaning up my studio . . .

You’re absolutely correct . . . “a heap more!!”

Your maths are as good as mine!