In my search for a method of tiling with a pentagon I came across a very cool way to tile. I started by slicing up a pentagon into smaller shapes.
By using the diagonals of a pentagon two consistent properties are found in all of these shapes. First, all of the internal angles are whole multiples of 36 degrees. Second, all of the sides are whole multiples of phi, which is also known as the golden mean ~1.618.
The blue and red triangles are special for a variety of reasons, and I view them as inversions of each other. They are each combined with a copy to form the two famous tiles for aperiodic tiling known as Penrose tiles.
What I find particularly interesting is that since all of the shapes involve whole multiple lengths of phi, we can scale them indefinitely, either larger or smaller, and they will always fit within each other. This opens up all kinds of tiling possibilities.
From this starting point, any shape of these three shapes can be completely tiled by any of the others. For instance, the golden triangle will fill itself, the golden gnomon, or the pentagon, given a decomposition based on an infinite scaling.
Of course this pattern, or any like it can be expanded and repeated indefinitely.
Pentagons are surprisingly easy to make at any scale, and with a virtually infinite number of patterns.
However, the trick works in reverse as well. We can use the pentagon to decompose the shapes with infinite scaling also.
So now we can imagine that any surface can be filled by pentagons, albeit not pentagons of the same size, but pentagons that are scaled by whole multiples of phi.
These are fun to play with and think about. I have put together an Adobe Illustrator file with the three shapes at these five scales with all of the possible rotations. It is easy to use Alt+Left mouse to copy and drag as many tiles as you need to make a big mosaic without having to rotate or size any of the individual tiles.
Download tiling2.zip Illustrator
10.0 (160 kb)
Download tiling3.zip Illustrator 4.0 (8 kb)
Here is a gallery of some mosaics. E-mail us some of your creations and perhaps we will add them.
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