This Friday, June 17, is the inaugural World Tessellation Day. I am normally skeptical of attempts to create new holidays, but I am so fond of filling up the plane with shapes that I just can’t help myself.

Emily Grosvenor is the tiling enthusiast behind the push to make Tessellation Day a thing. She recently successfully crowdfunded Tessalation, a book about a girl named Tessa who sees patterns everywhere. The book will be available for non-backers on June 17, Tessellation Day. (Disclosure: I backed the project on Kickstarter.) The date was chosen because it is the birthday of M. C. Escher, one of the most famous tessellators.

Luckily, it’s easy to celebrate Tessellation Day. Just tessellate! Appropriately enough, tessellations.org has a tessellation tutorial to get you started. John Golden also has a page of tessellation resources on his blog math hombre, and John Baez has some cool posts about tilings on his AMS blog Visual Insight. Update: Emily Grosvenor also just published a list of 23 simple ways to celebrate World Tessellation Day.

The easiest shapes to base a tessellation on are equilateral triangles, squares, and regular hexagons—the regular shapes that fill the plane all by themselves—but there are lots of other shapes that can form the foundation of your tessellation. I’m particularly fond of pentagons, and Laura Taalman has instructions for 3D printing all the tessellating pentagons if you’d like to make yourself a desk organizer or other plastic object from irregular pentagons. To break free from the repetition of those tilings, the Penrose tiling is probably everybody’s favorite aperiodic tessellation. You can learn how to knit yourself a Penrose tiling from Woolly Thoughts.

One way to make your tessellations a little more exciting is to move them to the hyperbolic plane. Escher of course made a lot of good Euclidean tessellations, but I’m partial to his Circle Limit paintings, which tile the Poincaré disc model of the hyperbolic plane. There are good articles about Escher’s use of mathematics and the Circle Limit series in particular by several authors, including Doris Schattschneider, Bill Casselman, and Doug Dunham.

To elevate your tessellations, you might try venturing into the third dimension. It’s not hard to make a 3-dimensional tessellation, or honeycomb, by taking a 2-dimensional tessellation and making it into a prism, but there are other ways of filling 3-space with repeating polyhedra. One of my favorites is the combination of truncated cubes and cubes in this experimental water bottle by Portuguese design firm Pedrita.

If you want to really get wacky, you can combine hyperbolic geometry and the third dimension and learn about how to visualize hyperbolic honeycombs from Roice Nelson and Henry Segerman.

How will you celebrate World Tessellation Day? Share with the hashtag #WorldTessellationDay or #WorldTessDay.

Thanks a lot for this awesome article. I just cannot wait to share it with my friends. Will soon start making tessellations…

Great selection of resources! Those water bottles, though…

So cool! Thank you so much for these ideas. I really like the water bottles — now that’s an idea for a tessellation party.