I learned about the

*versatile*, which makes it possible to create spiral tilings:

As it turns out, the versatile is a combination of two pattern blocks, so I hope to explore it further when I get home.

This led me to wonder whether a tiling pentomino spiral would be possible. Here are two attempts, one of which seemed successful.

I still need to clarify what I mean by a pentomino spiral, and to figure out some sort of proof that the one that seemed to work could be continued indefinitely. I suspect the P pentomino is the only one that will work for this, but who knows.

Pentagon tiles make challenging puzzles. I solved five of them:

And that is not all! I got new insights into the relationship between tiling and symmetry, and learned how the use of dual tilings can help unmask the underlying structure of some Escher tessellations. All and all a productive way to spend three mornings!

Many thanks to Christopher and all the participants.

-- Henri

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PS:

I've been interested in tiling and its uses in the classroom for some time. A whole section of Geometry Labs addresses this. Four labs: Tiling with Polyominoes, Tiling with Pattern Blocks, Tiling with Triangles and Quadrilaterals, Tiling with Regular Polygons. The last two are very "curricular", in that they address topics central to secondary school geometry. (The book is a free download on my Web site.) See also the final lab in the book, which uses a pattern block tiling in a lesson about "π" for regular polygons.

My Geometry Labs template is an excellent tool for exploring tiling, because it has every sort of triangle and quadrilateral, many regular polygons, and all the pattern blocks. Check it out here, purchase it here.

If you use the Notebook software for the SmartBoard, you can explore tiling with this document. (Includes clonable pentominoes, pattern blocks, triangles, quadrilaterals, and regular polygons.)

Finally, the Zome Geometry book I co-authored with George Hart has a section on tiling, including activities about dual tessellations, spiral tilings, and non-periodic tilings.