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MF97: The Stern-Brocot tree, matrices and wedges

We discuss further manifestations of the Stern-Brocot tree, which we discussed in the previous lecture. Here we introduce the matrix form of the tree, which sheds new light on how to move smoothly down the tree. To do this we introduce some basic definitions on 2x2 matrices and their determinants, including the operations of addition and multiplication, and the three special matrices I,L and K.Then we exhibit a somewhat novel planar, geometrical representation of the Stern-Brocot tree, identifying fractions with visible integral points in the positive quadrant. This gives an explanation of the curious role of the pseudo-fraction 1/0 which we used in the last lecture to generate the tree. Finally we introduce a new geometrical structure that results from combining the matrix and geometrical forms of the Stern-Brocot tree; the theory of wedges. A wedge is a particular triangle associated to a fraction, or to a matrix in the matrix form of the S-B tree, and these provide a curious tesselation of the visible positive quadrant, rather challenging to visualize!
Length: 34:14


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