Fractals and Biology By: May and Phyllis May 1, 2013 Period 9 How do you measure life? In straight lines? In circles? Or in squares? You don't! Life isn't measured in simple shapes. It's measured in FRACTALS! What is a fractal? Fractal - an irregular geometric object that displays self-similarity at all scales Fractals can look like this: Or this: They're identical! Or like this: The first "fractal" Cantor Set: developed by Georg Cantor Helge von Koch Koch Snowflake First instance of self-similarity Gaston Julia Julia Set Studied iterative functions and attractors/repellors Benoit Mandelbrot Mandelbrot Set: self-similar and contains Julia sets Invented the field of fractal geometry,

popularized applications of fractals in other scientific fields Now some history... how do we measure them? Since fractals are so IRREGULAR, 4 methods Modified Pixel Dilation

Perimeter-Area Method

Ruler-counting Method

Box-counting Method Box-Counting Method As the grid gets smaller, the number of squares occupied gets bigger exponentially! Let's try measuring a plant root s= 1/7.4 s=1/3.8 s= 1/1.5 N(s)= 4 N(s)= 7 N(s)= 17 x-axis of the grid is s where s=1/(width of the grid).

N(s)= number of boxes the fractal touches

Resize and Repeat

Plot the values on a graph

x-axis is the log(s)

y-axis is the log[N(s)]

Draw in the line of best fit

The box-counting dimension measure= slope of the line of best fit s= 1/1.0 x= log 1 N(s)= 32 Data Table x= logS = log(1/width of the graph)

y= logN(s)= log(number of blocks the picture touches) Once calculated the plotted points look like this: The line of best fit looks like this: Once calculated, the equation of the line of best fit is y=1.039x+log32 The dimension measure of the plant is 1.039 Fractals occur inside our bodies! But how does this relate to people? So...where are the fractals? They can be found in: attractors -

points in space which attract other points to them repellors -

points in space that repel other points Box-Counting Method What first led us to believe we contain fractals? Inconsistencies between measurements of cell membranes of liver cells between laboratories

6 sq. meters vs. 11 sq. meters

Surface density appeared to increase as resolution increased

Similar to trying to measure coastlines - "Coast of Britain" effect Cell membranes and organelles

Respiratory system - lungs

Nervous system - neurons

Digestive system - small intestine

Circulatory system - cardiac tissue

Brain tissue

Tumor cells Cell Membranes and Organelles Found in folded or indented membranes, like: Inner mitochondrial membrane

Rough endoplasmic reticulum (RER)

Surface density increased as magnification increased with rough membranes Changes of surface density estimates from outer and inner mitochondrial membranes Cardiac System Fractal-like network of veins and arteries transport blood to and from the heart

Fractal-like "canopy" of connective tissues within the heart

chordae tendinae - connects valves to muscles

Branching in the His-Purkinje system

Conducts electrical signals from atria to ventricles Large vessels branch into smaller vessels, which branch even further into smaller vessels. Sierpinski Triangle Self-similarity Recursion iteration -

a phase of recursion (repetition) Coastlines are fractals too! x= log1/7.4 y= log4 Fractal Dimension- degree of complexity by evaluating how quickly our measurements increase or decrease as our scale becomes larger or smaller coastlines - impossible to measure accurately Respiratory System Lungs branch off:

Trachea

Left and right bronchi

Bronchioles

Alveoli

Displays self-similarity on all levels Nervous System Similar to respiratory system

Dendrites on neurons branch off

3 iterations

Displays self-similarity x= log1/3.8 y= log7 Digestive System Small intestine: Lined with finger-like projections called villi

Increases absorbency of the tissue

Displays self-similarity x= log1/1.5 y= log 17 y= log32 FRACTAL Mandelbrot set Tumor: a mass of cells that no longer recognize the growth limits of a normal cell. Tumor Cells A normal cell would stops growing when it contacts other cells or experiences a change in growth regulating genes BUT, A tumor cell doesn't! Formation of a Koch snowflake When a tumor first begins to grow, it is supplied nutrients and oxygen via diffusion from nearby blood vessels TUMOR ANGIOGENESIS As the tumor enlarges, it releases substances that stimulate branching of the blood vessels. TUMOR ANGIOGENESIS Gradually, more branches are added and the tumor is able to enlarge. TUMOR ANGIOGENESIS TUMOR ANGIOGENESIS "angiogenesis"--process of blood vessel growth and branching This grid - extracellular matrix of a tumor mass.

The dark spaces- resistant to blood vessel invasion Once tumor has commandeered blood vessels, tumor cells may break off and spread metastasize shortest path- minimum path of vascular tissue Box-counting!! Better diagnose the presence of tumors Researchers can use this information to: Present new options for the delivery of chemotherapy Suggest new ways of fighting tumors So what???? "starving" the tumor But...why fractals? Amplify surface area to: Absorb

Small intestine - absorbs nutrients

Distribute

Blood vessels - distribute blood to entire body

Bile ducts - distribute bile to digest food

Bronchial tubes - distribute oxygen and to entire body

Process information

Neurons - process information from the environment Irregularity and redundancy resistant to injury

Operate under wide range of conditions Adaptable and flexible are EVERYWHERE!!!! Open Your EYES!! Fractals are here: Fractals are there: You just have to look really closely... ...to see the BIG PICTURE Activity Time! Remember the Sierpinski Triangle... Using your spaghetti, create as many iterations of the Sierpinski Triangle Fractal as you can in three minutes. Ready? Set? GO!!!!

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# Fractals and Biology: A Geometry Project

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