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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:

d1a d1a

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:

d2 d3

d4 d5

d6 d7

d8 d9

d10 d11

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:

jg1

From Mallorca, Spain, Daniel Ruiz Aguilera:

dra1 dra2

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.

dra3

From Toulouse, France, Simon Gregg:

sg1sg2

sg3sg4

sg5

And these tripled dodecagons:

sg6sg7

You too, and your students, can play this game!
Download the dodecagon outlines, and get started!

Discussion

  1. Look for smaller dodecagons in the larger ones. In some cases, they overlap. Can you find a symmetric, but not regular dodecagon?
  2. If the area of the square is 1, and the area of the triangle is T, what is the area of the dodecagon with side 1? side 2? side 3?
  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)`?
  4. What combinations of blocks will work for a side-1 dodecagon? (Do not try to find all the combinations, but try to put in words, or in formulas, what must be true of the numbers of each type of blocks.)
  5. What symmetries are possible for a dodecagon? Are all of them represented on this page?
  6. Can you create pattern block dodecagons of your own for each type of symmetry? (Side 1? Side 2?)
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