We previously looked at the following strip:
Now we will divide up the enneagon slightly differently and look at the tessellations that result. The curved edge represents one that we can modify in any way we like.
Note how the two rows of anchor points move relative to one another. If we can overlay a T strip so that its anchor points coincide with the anchor points of both rows then we will get an efficient dissection. Unfortunately this does not work for the square, but does work for the pentagon, hexagon and heptagon.
We will demonstrate the next stage using the pentagon. But first we will modify our tessellation by dissecting the enneagon slightly differently. Note that we can now vary the tessellation over a smaller range.
The next overlay diagram shows how a strip can be overlaid. The strip varies in width by a small amount as the tessellation varies. We could have positioned the strip in a number of different ways by changing which anchor points we chose to align it to.
Fortunately the pentagon’s strip has a width that fits the tessellation so we can now overlay this tessellation over a pentagon strip to get the following overlay. We also align the anchor points of the tessellation and the strip as this is a T strip.
One last trick gives us our final dissection. We adapt the curved edges to exactly match an edge of the pentagon strip saving one more piece. Just ten pieces for this dissection is an impressive result: this is one of my favourite dissections.
All the above tricks also work for the hexagon and for the heptagon. The first of these also requires further tricks to save an extra piece.
Next we will look at the {12/2} dodecagram. This was not an easy tessellation to find.
What is so useful is that we can overlay strips of varying width.
So to dissect to a heptagon we select the tessellation with just the right width to fit the heptagon strip. Here is the result.
Much the same method also works for the square, heptagon, octagon, enneagon, decagon, dodecagon, pentagram, {8/2} octagram, greek cross and latin cross. All of these use P strips except the last which uses a T strip.