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Simple Geometry, Incredible Graphs Math and Science Matter Saturday, March 3, 2001

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A graph is called self-similar if the graph can be covered completely and exactly with miniature copies of itself. The “wave” on the cover of your program is a classic example, as is Serpinski’s Triangle:

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If you zoom-in on any part of the triangle, you continue to see copies of the same basic object:

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Likewise, you can compress the triangle onto a copy of itself:

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With a computer to help, it’s easy to make Serpinski’s triangle, or the wave, or any number of exotic graphs. All we need are some simple ideas from basic geometry.

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The location of every point in the plane is specified by two numbers: (3,1) the x-coordinate is 3 the y-coordinate is 1 3 1

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If we multiply the x-coordinate by 1/2, we slide the point halfway towards the vertical axis: (3,1)(1.5,1)

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Likewise, if we multiply the y-coordinate by 1/2, we slide the point halfway towards the horizontal axis: (3,1) (3,0.5)

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If we multiply both coordinates by 1/2, we slide the point halfway towards the origin: (3,1) (1.5,0.5)

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If we multiply every x-coordinate in an entire figure by 1/2, we shrink the entire figure halfway towards the vertical axis: Note that the size of the figure changes.

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Likewise, if we multiply every y-coordinate in an entire figure by 1/2, we shrink the entire figure halfway towards the horizontal axis: Multiplication changes the size of the figure.

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If we add 1 to the x-coordinate, we shift the point 1 unit away from the vertical axis: (3,1)(4,1)

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…and if we add 1 to the y-coordinate, we shift the point one unit away from the horizontal axis: (3,1) (3,2)

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If we add 1 to every x-coordinate in an entire figure, we shift the entire figure one unit away from the vertical axis: Note that the size of the figure does not change.

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Addition does not change the size of the figure. Likewise, if we add 1 to every y-coordinate in an entire figure, we shift the entire figure one unit away from the horizontal axis:

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The first one is easy: we need to shrink the entire triangle down onto the lower-left corner. To create Serpinski’s Triangle, we need to find formulas to shrink it onto the three major subsets of itself. Just multiply every x-coordinate by 1/2 and then multiply every y-coordinate by 1/2.

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Multiply x-coordinates by 1/2...

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…then multiply y-coordinates by 1/2.

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First formula:multiply x-coordinates by 1/2 multiply y-coordinates by 1/2

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We can think of this as sliding the triangle in the lower-left corner onto the triangle in the lower-right corner. The second formula is also pretty easy - we need to shrink the entire triangle down onto the triangle in the lower-right corner. If we set the length of the base to be 1 then, after multiplying all coordinates by 1/2, we should add one-half to each x-coordinate.

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Multiply x-coordinates by 1/2...

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…then multiply y-coordinates by 1/2...

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…then add half the base length to each x-coordinate.

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First formula: multiply x-coordinates by 1/2 multiply y-coordinates by 1/2 Second formula: multiply x-coordinates by 1/2 multiply y-coordinates by 1/2 add 1/2 to each x-coordinate

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We can do this by sliding the triangle in the lower-left corner onto the triangle in the upper-middle vertex. The third formula is also pretty easy - we need to shrink the entire triangle up onto the triangle in the upper-middle vertex. As it turns out, the height of this triangle is about so, after multiplying all coordinates by 1/2, we should add 0.25 to each x-coordinate and add to each y-coordinate.

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Multiply x-coordinates by 1/2...

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…then multiply y-coordinates by 1/2...

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…then add one-quarter the base length to each x-coordinate...

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…and add half the height to each y-coordinate.

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First formula: multiply x-coordinates by 1/2 multiply y-coordinates by 1/2 Second formula: multiply x-coordinates by 1/2 multiply y-coordinates by 1/2 add 0.5 to each x-coordinate Third formula: multiply x-coordinates by 1/2 multiply y-coordinates by 1/2 add 0.25 to each x-coordinate add to each y-coordinate

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Start with the point x = 0 and y = 0 (the lower-left vertex) and plot it. Randomly pick one of the three formulas and use it to transform the point (0,0) into a new point. Plot the new point. Randomly pick one of the three formulas and use it to transform the most recently plotted point into a new point. Repeat the last two steps... Finally, how do we create the entire figure? 100,000 times.

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Plotting 10 points on Serpinski’s Triangle

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100 points 500 points 2500 points10000 points points

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Although shrinking and sliding are enough to create Serpinski’s Triangle, if we add done last effect, we can create stunning graphics: Rotation.

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We can rotate a point around the origin by any angle. (2,1) (1.67,1.48) A 15 degree rotation has this effect: Think of the point as being on the circumference of a circle centered at the origin.

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Rotating an entire graph by 15 degrees has this effect: Rotation does not change the size of a figure.

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Using shrinking, shifting and rotations about the origin we can make the following self-similar objects.

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Using other techniques which we won’t discuss today, we can make self-similar graphs that look quite life-like.

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Wes Ostertag Dutchess Community College

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