1. Natalie Weires Kathy Lin

2.  We take an arbitrary triangle and bisect its angles to create three new triangles. If we repeat this process on each generated triangle, what kinds of pattern can we find when looking at the angles?

3.  We first tried to use degrees and tried to find patterns in each generation, but the numbers became too complicated, so we switched to radians and factored out the π for simplicity.  For the remainder of the presentation, all angles will be presented in terms of radians with π omitted.

4.  When the angles of a triangle (A, B, C) are bisected to form three smaller triangles, the new triangles have angles  A/2, B/2, C + A/2 + B/2  A/2,C/2, B + A/2 + C/2  B/2,C/2, A + B/2 + C/2  because all the angles in a triangle must add up to one, using our modified angle measurements.

5.  Each triangle can be represented by a vector and the three new triangles can be generated from an original triangle by multiplying its vector by three matrices:  [ 1 0.5 0.5 ] [ 0.5 0 0 ] [ 0 0.5 0 ] [ 0.5 1 0.5 ] [ 0 0 0.5 ][ 0 0 0.5 ] [ 0.5 0 0 ] [ 0 0.5 0 ] [ 0.5 0.5 1]

6.  Gen 11/62/63/6  Gen 21/121/122/122/123/123/127/12 8/129/12  Gen 31/241/241/241/242/242/242/24 2/243/243/243/243/247/247/24 8/248/249/249/2413/2413/2414/24 14/2415/2415/2419/2420/2421/24  The numbers appear in triplets that skip by the starting denominator  The numerators appear in frequencies that follow Dress’s Sequence:  1  1 2  1 2 2 4  1 2 2 4 2 4 4 8  1 2 2 4 2 4 4 8 2 4 4 8 4 8 8 16etc  As appears in this histogram…

7. Numerators Frequencies

8.  For any triangle, A + B + C = 1.  This defines a plane in space.  Given a starting triangle, the three new triangles will appear on the plane as three points.

10.  So if we take every point on the plane and apply the matrices to each one, we wonder what happens…

12.  Two of the three dimensions are always divided by two, so the points always map to halfway in each direction.  We will attempt to show this on the board…

13.  We let the angle bisectors go all the way through and create six triangles?