Wednesday, February 10, 2010

Topologically Equivalent Surfaces in Felt for the Math Nerd in You

This is a set of 4 topologically equivalent surfaces that I just listed. Each surface has two faces shown in the two colors of felt, and each surface also has exactly two edges, shown with button hole stitching in green and black.

All of these surfaces can be THEORETICALLY* deformed into any of the other three, but you might have to allow the surface to intersect itself to do it. In mathematical terms, each of the four surfaces is homeomorphic to the other three since they all have Euler characteristic -2. Secondly, three of the surfaces are isotopic to each other, but one of the surfaces is not isotopic to the other three.  The one with the three-fold spiral is the odd man out (rightmost piece in the first photo).  You can see that this one is weird since the black edge is a trefoil knot.   That's how you know that you would have to intersect the suface to deform it into the other surfaces.  Each of the other edges on all four pieces can be deformed to a circle without crossing itself.

*Note, you can't physically deform these felt models to make the other surfaces.  You have to imagine or visualize how to do this.  

The Euler characteristic is calculated by drawing a map on the surface and counting the number of faces, edges and vertices as V + F - E.  For 3D polyhedra, the Euler characteristic is always positive 2.  For example, the cube gives 8 + 6 - 12 = 2.  The tetrahedron gives 4 + 4 - 6 = 2.

I made these by wet felting pure wool over a cotton base, and when they were dry, I hand stitched the edges with cotton yarn. These little sculptures are stiff enough to hold their shape, but flexible enough to be folded. Three of them can be arranged to sit in at least two different configurations.  It's hard to measure how big these are exactly, but they range from about 2 to 4 inches.

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