1 / 15

Iterated Triangles

Natalie Weires Kathy Lin. Iterated Triangles. 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?. The Question:.

nerys
Download Presentation

Iterated Triangles

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Natalie Weires Kathy Lin Iterated Triangles

  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? The Question:

  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. Failed attempts

  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. The New angles

  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] More simplifying…

  6. Gen 1 1/6 2/6 3/6 • Gen 2 1/12 1/12 2/12 2/12 3/12 3/12 7/12 8/12 9/12 • Gen 3 1/24 1/24 1/24 1/24 2/24 2/24 2/24 2/24 3/24 3/24 3/24 3/24 7/24 7/24 8/24 8/24 9/24 9/24 13/24 13/24 14/24 14/24 15/24 15/24 19/24 20/24 21/24 • 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 16 etc • As appears in this histogram… The pattern

  7. Histogram Frequencies Numerators

  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. Plotting on a Plane

  9. Plotting angles

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

  11. It looks familiar…

  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… How come?

  13. We let the angle bisectors go all the way through and create six triangles? So what if…

  14. Using one starting triangle…

  15. With the whole Plane…

More Related