I’m stuck, at work, with a request to tesselate equilateral pentagons in the simplest way possible (fewest shapes other than pentagons).
What I’ve tried so far is drawing rectangles and trapezoids around the pentagons. Not the most graceful results… anybody know if any artists have had good luck with this?
I think it is diamonds and pentagons. My former physics professor, H.C. Van Baeyer, had an essay on the difficulty of tiling pentagons in his book “The Fermi Solution.” I was vaguely remembering that this is what he wrote… but a little experiementation seems to have proved out my memory. Try it.
Oh, for this to work, the pentagons dont all face the same way. Some face “point up” while others face “point down.” Imagine 1 diamond, surrounded by 6 pentagons. One pentagon touches the diamond point to point on each end, two pentagons arranged butt-to-butt are on the left and right sides. Am I explaining this well? The result is not unattractive.
Here’s a page with some different tesselations involving regular pentagons. Hello Again’s tesselation is at the bottom of this page, and is as simple a tesselation as you’re gonna get (you can’t tesselate with just regular pentagons alone).
Just a quick point: Is the problem “regular pentagons” or “equilateral pentagons”? Because there are several equilateral pentagons (i.e., all sides equal, but not necessarily all angles) which can be tessled exactly. For instance, if you have two right angles next to each other, then you get a shape like an equilateral triangle attached to a square, which can be tiled. You also get a very elegant tiling (sort of like a tortoise shell) from equilateral pentagons with right angles separated by another angle: Unfortunately this is difficult to describe better. On second thought, just take a look at the background on my homepage.
If you need regular pentagons rather than just equilateral ones, then you still have other options just as simple as the one Cabbage linked to. For instance, there are a few tilings using those same pentagons and rhombi that have two translational symmetries, rather than rotational symmetry. Or you can slice each rhombus into two equal parallelograms, and make another tiling using those.
Thanks, everyone. Hello Again’s solution is exactly what I’m looking for. Once again, the day has been saved by the Straight Dope Message Board.
I was indeed asking about regular pentagons, but I’m glad I checked out Chronos’s equilateral but regular pentagons - very simple, but it clumps in interesting ways (hexagons and some kind of escher-type pinwheel, to name two).
Thanks, all!