We know that octagons do not tessellate the plane, but on the other hand there is a nice semiregular tessllation involving octagons and squares with a common side-length. Geoffrey Bennett probably used this fact when he discovered a beautiful dissection of an octagon to a square.

Problem O1. Find a 5-piece dissection of an octagon into a square.

We have already seen tessellations involving squares of differenrt sizes. Perhaps we might take such a tessellation with the smaller squares the same size as the small squares illustrated at right, and the larger squares having the same area as the octagons. Drag this square tessellation so that a big square is centred on a yellow square. What do you notice about the octagons now? Can you describe the dissection which solves the problem?

Another technique which has been useful is using the natural subdivisions of our polygons.

Problem O2. Find an 8-piece dissection of 2 octagons to 1 octagon.

Using the area argument, we observe that the larger octagon must be scaled by a factor of 2. Perhaps you can use some of the lines in the pretty flower pattern at left to dissect the large octagon into pieces which exactly cover the two smaller octagons? Remembering the scaling, which of the flower segments must match the edges of the smaller octagons? If you need some help, click here. Will this give the required number of pieces? Can you rearrange the pieces to fit in the smaller octagons? And here is the solution.

If you are interested in checking out dissections of octagons to various other polygons, see Gavin Theobald’s site.