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Fractals and Biology: A Geometry Project
Transcript of Fractals and Biology: A Geometry Project
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
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
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:
Left and right bronchi
Displays self-similarity on all levels Nervous System Similar to respiratory system
Dendrites on neurons branch off
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
Blood vessels - distribute blood to entire body
Bile ducts - distribute bile to digest food
Bronchial tubes - distribute oxygen and to entire body
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!!!!