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