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# Fractals

Math project about fractals, fractal trees, and chaos theory.

Lauren Kronenberg

Pre Calculus G block

December 17, 2010 Fractal Trees What is a fractal? A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is a smaller copy of the whole: self-similarity

Fractals are complex shapes based upon repeating the patterns of the same shape Some Different Fractals Sierpinski Triangle Koch Snowflake Mandelbrot Romanesco broccoli Barnsley's Fern Lichtenberg figure Fractal Trees One common type of fractal is the fractal tree.

Using fractal trees can make beautiful patterns and pictures,

as well as simulate real life-like trees.

The main idea in creating fractal trees or plants is to have a base object and to then create smaller, similar objects protruding from that initial object.

Each of the branches is a smaller version of the main "tree trunk"

The angle, length and other features of these "children" can be randomized for a more realistic look.

This method is a recursive method, meaning that it continues for each "child" branch down to a finite number of steps.

•A binary fractal tree is defined by symmetric binary branching. - The trunk of length L splits into two branches of length r

each making an angle X (a variable) with the direction of the trunk

- Both of these branches divides into two branches of length r^2

each making an angle X (the same angle as earlier variable) with the direction of its parent branch

- Continuing in this way for infinitely many branchings, the tree is the set of branches, together with their limit points, called branch tips Many Kinds of Fractal Trees Fractal trees with very large angles Fractal trees with small angles Asymmetrical fractal trees Sierpinski Cube Bifurcation Bifurcation is very important in fractal trees

Bifurcation means the splitting of a main body into two parts Fractal dimension Dimension= log N

log a -The fractal dimensions (D) of two types of trees were measured with high-tech 3D measurments.

-The results showed that both trees have fractal dimension 2.

-This is a minimal value (fractal dimension) which allows for the plants to absorb sunlight effectively (maximum surface area).

-This small value also allows for sufficient air flow to cool down the leaves for the plant.

-This structure also contributes to tree's ability to transfer variables: such as moisture. Works Sited The Chaos Game -Place 3 dots in the shape of a triangle

Color top vertex red, lower left green, and right blue

Take a die: color 2 faces each of red, green, and blue

-Pick an arbitrary starting point in the plane

Starting at this point, ALGORITHM: Roll the die, then move your point half the distance toward the colored vertex (which was determined by the dice roll)

(move half way from where you started, to the point rolled on dice) Starting point How this relates to the real world The resulting picture is one of the most famous fractals, the Sierpinski triangle Keep on rolling the dice and recording the points: Elements which Effect the Look of Fractal Trees -Number of branches protruding from each "parent" branch

-Angle of protruding branches

-Length of protruding branches

-Symmetry of protruding branches (makes overal look of fractal tree symmetrical or not) http://www.math.union.edu/research/fractaltrees/

http://www.zonalandeducation.com/mmts/geometrySection/fractals/tree/treeFractal.html

http://www.isprs.org/proceedings/XXXVII/congress/4_pdf/97.pdf

http://dl.begellhouse.com/journals/1bb331655c289a0a,79a1a64252fdbc4b,1f52189e4083f72f.html

http://archives.math.utk.edu/topics/fractals.html

http://math.bu.edu/DYSYS/chaos-game/chaos-game.html

http://www.math.umass.edu/~mconnors/fractal/fractal.html

http://math.bu.edu/DYSYS/FRACGEOM/FRACGEOM.html What do you notice about a fractal tree with asymmetrically protruding branches? How does the angle of the protruding branches effect the fractal tree? Observation Questions: D= log (# self-simlar pieces)

log (magnification factor) Website with a great fractal tree maker activity:

http://www.zonalandeducation.com/mmts/geometrySection/fractals/tree/treeFractal.html Refer to piece of paper which further explains Fractal Dimension Formula for Understanding and Making Fractal Trees Also notice that a small change early on in the beginning (like the repeating branch) has a huge effect on the overall look of a fractal tree. scroll over video to play--> <-- scroll over video to play This figure shows the moves associated to rolling red, green, blue, and blue in order. Although, a fractal tree does not have an exact formula, one could use this information to better understand how fractal trees are formed This chaos game and chaos theory relate to fractal trees because using a different version of this chaos game, produces a fractal fern: which are similar to fractal trees. Thank you!Full transcript

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Tweet## Lauren Kronenberg

on 17 December 2010#### Transcript of Fractals

Lauren Kronenberg

Pre Calculus G block

December 17, 2010 Fractal Trees What is a fractal? A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is a smaller copy of the whole: self-similarity

Fractals are complex shapes based upon repeating the patterns of the same shape Some Different Fractals Sierpinski Triangle Koch Snowflake Mandelbrot Romanesco broccoli Barnsley's Fern Lichtenberg figure Fractal Trees One common type of fractal is the fractal tree.

Using fractal trees can make beautiful patterns and pictures,

as well as simulate real life-like trees.

The main idea in creating fractal trees or plants is to have a base object and to then create smaller, similar objects protruding from that initial object.

Each of the branches is a smaller version of the main "tree trunk"

The angle, length and other features of these "children" can be randomized for a more realistic look.

This method is a recursive method, meaning that it continues for each "child" branch down to a finite number of steps.

•A binary fractal tree is defined by symmetric binary branching. - The trunk of length L splits into two branches of length r

each making an angle X (a variable) with the direction of the trunk

- Both of these branches divides into two branches of length r^2

each making an angle X (the same angle as earlier variable) with the direction of its parent branch

- Continuing in this way for infinitely many branchings, the tree is the set of branches, together with their limit points, called branch tips Many Kinds of Fractal Trees Fractal trees with very large angles Fractal trees with small angles Asymmetrical fractal trees Sierpinski Cube Bifurcation Bifurcation is very important in fractal trees

Bifurcation means the splitting of a main body into two parts Fractal dimension Dimension= log N

log a -The fractal dimensions (D) of two types of trees were measured with high-tech 3D measurments.

-The results showed that both trees have fractal dimension 2.

-This is a minimal value (fractal dimension) which allows for the plants to absorb sunlight effectively (maximum surface area).

-This small value also allows for sufficient air flow to cool down the leaves for the plant.

-This structure also contributes to tree's ability to transfer variables: such as moisture. Works Sited The Chaos Game -Place 3 dots in the shape of a triangle

Color top vertex red, lower left green, and right blue

Take a die: color 2 faces each of red, green, and blue

-Pick an arbitrary starting point in the plane

Starting at this point, ALGORITHM: Roll the die, then move your point half the distance toward the colored vertex (which was determined by the dice roll)

(move half way from where you started, to the point rolled on dice) Starting point How this relates to the real world The resulting picture is one of the most famous fractals, the Sierpinski triangle Keep on rolling the dice and recording the points: Elements which Effect the Look of Fractal Trees -Number of branches protruding from each "parent" branch

-Angle of protruding branches

-Length of protruding branches

-Symmetry of protruding branches (makes overal look of fractal tree symmetrical or not) http://www.math.union.edu/research/fractaltrees/

http://www.zonalandeducation.com/mmts/geometrySection/fractals/tree/treeFractal.html

http://www.isprs.org/proceedings/XXXVII/congress/4_pdf/97.pdf

http://dl.begellhouse.com/journals/1bb331655c289a0a,79a1a64252fdbc4b,1f52189e4083f72f.html

http://archives.math.utk.edu/topics/fractals.html

http://math.bu.edu/DYSYS/chaos-game/chaos-game.html

http://www.math.umass.edu/~mconnors/fractal/fractal.html

http://math.bu.edu/DYSYS/FRACGEOM/FRACGEOM.html What do you notice about a fractal tree with asymmetrically protruding branches? How does the angle of the protruding branches effect the fractal tree? Observation Questions: D= log (# self-simlar pieces)

log (magnification factor) Website with a great fractal tree maker activity:

http://www.zonalandeducation.com/mmts/geometrySection/fractals/tree/treeFractal.html Refer to piece of paper which further explains Fractal Dimension Formula for Understanding and Making Fractal Trees Also notice that a small change early on in the beginning (like the repeating branch) has a huge effect on the overall look of a fractal tree. scroll over video to play--> <-- scroll over video to play This figure shows the moves associated to rolling red, green, blue, and blue in order. Although, a fractal tree does not have an exact formula, one could use this information to better understand how fractal trees are formed This chaos game and chaos theory relate to fractal trees because using a different version of this chaos game, produces a fractal fern: which are similar to fractal trees. Thank you!