Pattern Block Dodecagons

Henri Picciotto

(Dodecagon outlines)

A classic activity is to cover a 1-inch-side dodecagon with pattern blocks. This provides a great context to discuss symmetry (see Geometry Labs 5.6.) Here are two ways to do it:

A close look at these figures should help you find the area of the tan block. (The square has area 1 square inch.)

Pattern blocks can be divided into two families: on the one hand, the triangle and its multiples (blue, red, and yellow); and on the other hand orange and tan blocks. What makes the dodecagon tilings so beautiful and so fun to discover is that they combine blocks from the two families. You and your students should explore this! (It is easiest to do this using a pre-drawn outline, such as the one in Geometry Labs 5.6)

I've been a fan of the pattern block dodecagon for about forty years. Just a few weeks ago, while thinking about something else, I stumbled upon a double-size, quadruple-area dodecagon. That launched me on a search for interesting tilings of that figure. Here are ten solutions I found:

I shared some of these on Twitter (where my handle is @hpicciotto), and it was not long before other pattern block buffs contributed some designs.

From Allendale, Michigan, John Golden:

From Mallorca, Spain, Daniel Ruiz Aguilera:

About the one on the right, Daniel asks: how many cubes do you see?

Notice that this next one has the same pattern as one of the dodecagons at the top of the page, with each of the blocks doubled. Since all the pattern blocks except the hexagon are rep-tiles, this method can be applied to any 1-inch-side dodecagons to create scaled versions of it, as long as it contains no yellow blocks. After doing this, one can do some substitutions to get aesthetically pleasing designs.

From Toulouse, France, Simon Gregg:

And these tripled dodecagons:

You too, and your students, can play this game!
3. If the area of the square is 1, what is T in terms of sqrt(3)? What are the dodecagon areas in terms of sqrt(3)?