Governor’s School for the Sciences

MathematicsMathematicsDay 9

MOTD: Sofia Kovalevskaya

• 1850 to 1891 • Worked on

differential equations

• Considered the first woman mathematician

Self-Similarity

• Typical feature of a fractal is self-similarity, i.e. where parts look like the whole:

How to create self-similarity

• Multiple Reduction Copy Machine: a copy machine with K lenses each lens makes a copy of the image reduces it and places it on the copy

• Substitution Rules: rule for each geometric object (GO) replace a GO with other GOs

Multiple Reduction Copy Machine (MRCM)

• 3 lens example• Take the output and run it back

through the copier

How to create a MRCM• Each ‘lens’ is a geometric

transformation which maps the whole to a part

• Identify the tranformations by marking the parts in the whole

T1 =

0

@12 0 00 1

2 00 0 1

1

A

T2 =

0

@12 0 1

20 1

2 00 0 1

1

A

T3 =

0

@12 0 1

40 1

212

0 0 1

1

A

Using a MRCM

• Start with a figure• Apply all the transformations to

create a new figure• Apply all the tranformations to this

new figure to get another new figure

• Repeat ad infinitum• Can also do randomly (Tuesday)

Results

• If all the transformations are contractions (i.e. move points closer together) then the original figure will be reduced to an unidentifiable dot and thus the final figure does not depend on the original figure!

• If the original figures has N points and the MRCM has k lenses then after M copies on the MRCM the figure has kM N points.

Twin Christmas Tree

Sierpinski Carpet

3-fold Dragon

Cantor Maze

Substitution Rules (SR)

• For each geometric object in a figure, have a rule that replaces it by a collection of other geometric objects

• By doing this over and over with the same rules, you get a fractal image

• Following are some examples

Given any line segment, replace it by a series of (connected) line segments starting and ending at the same endpoints as the original line, for example:

Given a triangle, replace by three triangles by connecting the side midpoints and discarding the middle

Building a SR

• Just like in the MRCM, the maps from the original object to the new pieces in the substitution rule are geometric transformations

• Main difference from the MRCM is that the original figure must be the correct original geometric object that defines the rule

Short Break

• Demo of Geometer’s Sketch Pad by Laura

• How to construct transformations• Lab time

Building Transformations

• Three types: line to line, triangle to triangle, square to parallelogram

• Process is to either solve in general or build from basic transformations

• Final form always looks like: | a b c | T = | d e f | | 0 0 1 |

Work it out:

• Construct a transformation that takes the line segment [(0,0)->(1,0)] to the line segment [(1,2)->(2,3)]

• Construct a transformation that takes the unit square (LL at (0,0)) to a square ¼ the size with LL at (½, ½)

Line to Line

• Segment: (0,0)->(1,0) to (a,b)->(c,d)

• Transformation: | c-a b-d a | T = | d-b c-a b | | 0 0 1 |

• Built as a rotation and a scaling followed by a translation

Triangle to Triangle• Start with the general form

| a b c | T = | d e f | | 0 0 1 |

• Original: (xi,yi), Target: (ui,vi) i=1,2,3• Solve the system for a,…,f:

a*xi + b*yi + c = ui d*xi + e*yi + f = vi

• Six equations for 6 unknowns\• Can also build from basic transforms,

esp. if original is a ‘standard’ triangle

Square to Quadrilateral

• If restricting to linear transformations only, then just use matching 3 points from each figure and use triangle technique

• If allowing more general transformations, then use bilinear form and solve for the coefficients

Lab Preview

• Enter the data for your name as a list of pairs of points

• For each segment construct a transformation

• Draw your name, but for each segment, replace it with your name by applying all the transformations to the segment

• Challenge: go one level deeper, i.e. replace the segments in the replacement with your name