Hinged Tessellations

by

D. Hembree

This page displays some common regular and semi-regular tessellations, but with a twist. The polygons are hinged at one or more vertices so that the appearance of the tessellation may be changed dynamically. All implementations are done in Geometer's Sketchpad and can be explored by clicking on the figure. This could be a fun extension of the study of tessellations in middle or secondary mathematics.

As a first example, consider a tessellation of squares as shown:

Each blue square is hinged at its vertices so that the tessellation changes as any blue square is rotated and my look like one of the following:

The blue squares have only rotated, but the yellow squares have become rhombi. Watching the transformation is fascinating and I suggest you take the time to watch it in GSP. Clicking on any of the figures will take you there (if you have GSP on your computer).

Here's my aborted attemp to make a hinged tessellation with triangles:

As you can see, it's no longer a tessellation (and some of the triangles were incorrectly placed), but you can still open the GSP file and see it rotate.

I had better luck with hexagons, as the next figure shows:

Only the green hexagons are "stable". Try it and see!

Here are hexagons again, but with triangles and squares:

Again, only the green hexagons are "stable".
This looks like a traditional quilt pattern, but it "morphs" into other patterns while maintaining a beautiful symmetry.

This traditional tile pattern is composed of octagons and squares, but the squares transform into beautiful stars and pinwheels.

The octagons rotate, adding triangles to the squares and then collapsing into a single octagon.

The final hinged tessellation here is composed of dodecagons,  hexagons, and squares. Compare the changing patterns of the squares to the tessellation above.

Compare the above tesselation to this one, which appears the same, but has "stable" squares which rotate with the dodecagons.

My final attempt was not a tessellation, but a hinged dissection puzzle of an equilateral triangle into a square by Dudeney.

Click to open GSP and try the puzzle