Post on 23-Feb-2016
description
ITERATED TRIANGLES
Natalie WeiresKathy Lin
THE QUESTION: 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?
FAILED ATTEMPTS 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.
THE NEW ANGLES 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.
MORE SIMPLIFYING… 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]
THE PATTERN Gen 1 1/6 2/6 3/6
Gen 2 1/12 1/12 2/12 2/12 3/12 3/12 7/128/12 9/12
Gen 3 1/24 1/24 1/24 1/24 2/24 2/24 2/242/24 3/24 3/24 3/24 3/24 7/24 7/24
8/24 8/24 9/24 9/24 13/24 13/2414/24 14/24 15/24 15/24 19/24 20/2421/24
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 16 etc
As appears in this histogram…
HISTOGRAM
0 100 200 300 400 500 600 700 8000
20
40
60
80
100
120
140
Numerators
Freq
uenc
ies
PLOTTING ON A PLANE 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 ANGLES
00.1
0.20.3
0.40.5
00.1
0.20.3
0.40.5
0
0.1
0.2
0.3
0.4
0.5
FOR ALL ANGLES So if we take every point on the plane
and apply the matrices to each one, we wonder what happens…
IT LOOKS FAMILIAR…
0
0.5
1
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
1
HOW COME? 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…
SO WHAT IF… We let the angle bisectors go all the
way through and create six triangles?
USING ONE STARTING TRIANGLE…
00.2
0.40.6
0.8 1
0
0.5
1
0
0.2
0.4
0.6
0.8
1
WITH THE WHOLE PLANE…
00.5
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1