Well, I was off work this entire week. I should have a post for this coming Monday… but I still have to write a summary for it. I should even have a post for the following Monday… but in that case I actually have to write almost all of the connecting narrative. The images have already been uploaded and inserted, but I have yet to explain what they are and what I’m doing.

I’ve continued playing with fractional linear transformations, categories, circuit theory, and linear programming. In particular, I might be right on the edge of understanding one last detail in what I want to show you about linear programming.

And, in a blinding flash of insight, it is clear to me why barycentric coordinates are not unique unless we use triangles. I asked that question back in February 2009, toyed with it for a little while, put it down and moved on. And I really did move on, thinking about the question only when something reminded me of it.

Simply put… take a rectangle, cut it into 2 triangles with one of the diagonals, and then draw a point inside one of the triangles. Well, using barycentric coordinates based on the triangle which contains the point, all of its coordinates are between 0 and 1… but using barycentric coordinates based on the other triangle, one of its coordinates is greater than 1 because it’s outside the diagonal edge of that triangle.

In other words, if we want to use rectangles to cut up something (say for finite elements), and if we want to use barycentric coordinates, then within each rectangle we would have to define a triangle to select one specific set of coordinates. We might as well use triangles in the 1st place.

Mathematics, however, is not the only thing I do in this week between Christmas and New Year’s Day. This is when I write an end of the year letter to serve as a Christmas card. I’m still working on it… I don’t have much time for mathematics for the rest of this weekend… possibly not for much of this coming week. That’s one reason I have 2 posts already in stage V… so I don’t have to do very much other than my letter for a while.

And so, for a change, this post closes with: I’m off to do some non-mathematical writing.

Happy New Year.