Tuesday, March 28, 2006
20:37
One perquisite of this job is that I have access to Adobe Illustrator. I discovered recently that it can make regular polygons on demand, and this led me to renew my interest in tesselations. One thing the program won't do is make said polygons with sides of a specified length; instead it makes them with a specified radius, which is the distance from the center point to a vertex. So in order to make different polygons with sides the same length, I had to do a little bit of trig first to approximate the radii.
I am specifically interested in tesselations of the plane using equilateral triangles, squares, and regular hexagons. Each of these can tesselate the plane if used alone, but there are a lot of interesting tesselations using combinations of them. There are infinitely many such tesselations, but I've been trying to find a way to catalogue the repeating ones. My current approach is to start with certain "base shapes" formed by taking a polygon, adding another polygon to each of its sides, and also adding any polygons that are definitely determined by these. For instance, if you have a 90 degree angle, the only thing that can go there is a square. I believe I have drawn all of these, and there are 10 with triangles in the center, 15 with squares and 28 with hexagons. (I'm not counting reflections and rotations.) It surprised me that only 6 of these 53 shapes have no lines of symmetry. There were some possibilities that I didn't draw because they wouldn't allow for eventual tesselation; namely, a hexagon and a square cannot be placed on adjacent edges of a hexagon, because the angle between them would be 30 degrees.
The next step will be to see which of these basic shapes can tesselate by themselves.
I am specifically interested in tesselations of the plane using equilateral triangles, squares, and regular hexagons. Each of these can tesselate the plane if used alone, but there are a lot of interesting tesselations using combinations of them. There are infinitely many such tesselations, but I've been trying to find a way to catalogue the repeating ones. My current approach is to start with certain "base shapes" formed by taking a polygon, adding another polygon to each of its sides, and also adding any polygons that are definitely determined by these. For instance, if you have a 90 degree angle, the only thing that can go there is a square. I believe I have drawn all of these, and there are 10 with triangles in the center, 15 with squares and 28 with hexagons. (I'm not counting reflections and rotations.) It surprised me that only 6 of these 53 shapes have no lines of symmetry. There were some possibilities that I didn't draw because they wouldn't allow for eventual tesselation; namely, a hexagon and a square cannot be placed on adjacent edges of a hexagon, because the angle between them would be 30 degrees.
The next step will be to see which of these basic shapes can tesselate by themselves.
# posted by Fibonacci @ 28.3.06 
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You ought to post some pictures of these.
Question: Are you only using regular polygons because that's what the program produces, or are those the only "legal" ones for tesselation? Like, why not fill in your 30 degree gap with a non regular triangle?
Question: Are you only using regular polygons because that's what the program produces, or are those the only "legal" ones for tesselation? Like, why not fill in your 30 degree gap with a non regular triangle?
Well, if I open up to the possibility of other shapes, the problem just gets too big. I'm tackling a specific subset that interests me.
Here's a description of my current favorite:
Draw a square. Attach a hexagon to one edge. Attach a triangle to an adjacent edge. Attach two more squares to the two remaining edges. Given that only equilateral triangles, squares and regular hexagons are allowed, and given the assumption that every square has that assortment of shapes attached to it grouped in that way (though possibly with reverse orientation), you can construct the tesselation.
Here's a description of my current favorite:
Draw a square. Attach a hexagon to one edge. Attach a triangle to an adjacent edge. Attach two more squares to the two remaining edges. Given that only equilateral triangles, squares and regular hexagons are allowed, and given the assumption that every square has that assortment of shapes attached to it grouped in that way (though possibly with reverse orientation), you can construct the tesselation.
I looked up tesselation, and found that in general it seems to be grouped by the number of vertices that come together at a single point (assuming I understood their notation).They had some really wild ones. :^)
That doesn't seem like sufficient classification to me. The tesselation I described in my last post has four vertices coming together at every point, but it is much different from a grid of all squares.
From Wikipedia:
"Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex."
Okay, I oversimplified.
"Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex."
Okay, I oversimplified.
Ah, but with uniform tiling every vertex is the same. Most of the tilings I'm looking at are non-uniform.
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