## Fractals Inside Us

Whenever your body has to send something from place to place...

Signals propagating along a nerve

Hormones, blood, oxygen, or anything that needs to be carried throughout your body to service your cells.

Those things are transported on a system of

Heart --> Aorta --> arteries --> Arteorles -->Capillaries

Branch around 15 times

Pulmonary Arteries carry dark blood

Place blood in close proximity to the air you inhale.

Carbon dioxide... Blood and Gas Exchange

Bronchial Tree is the airways from oxygen to come in.

Veins of a leaf... Also a fractal network to

Photosynthesized food to rest of plant.

Plumbing system is the best way.

Signals propagating along a nerve

Hormones, blood, oxygen, or anything that needs to be carried throughout your body to service your cells.

Those things are transported on a system of

**fractal branching networks.**Heart --> Aorta --> arteries --> Arteorles -->Capillaries

Branch around 15 times

Pulmonary Arteries carry dark blood

Place blood in close proximity to the air you inhale.

Carbon dioxide... Blood and Gas Exchange

Bronchial Tree is the airways from oxygen to come in.

Veins of a leaf... Also a fractal network to

Photosynthesized food to rest of plant.

**Why has evolution favored Fractal Branching Networks.****1)**Fractal Networks are a very efficient way to provide resources or communication links throughout a 3-D solid body, using only 1-D tubes. Reach every point in a 3-D body, to every point or cell... Make a treePlumbing system is the best way.

**2)**Fractal Structure creates an enormous amount of surface area in a confined space, crucial for processes like gas exchange in lungs or absorption of food in intestines.**3)**Easy to program -- grow a tube, split it, and repeat...## Fractals May Hold a Key to Explaining Lifes Greatest Mysteries

1) Why do we live 80 years? Why 10's of years?

Why does a mouse live a year or two... Same genes and cells at molecular level.

Molecular Mechanisms of Aging...

Look at life through a physicists Life....

Broad Unifying Principles... Ignore minor details...

1997 - Geoffrey West - Particle Physicist.

Mathematical Regularities

Smallest microbes to biggest animals

Size of a creature, metabolism and lifespan

How many calories each day?

What are our metabolic needs...

Heat, Oxygen, Calories...

Metabolic rates for diverse mammals...

Metabolic rates fall neatly on a curve when they are graphed as a function of the body mass of the animal.

Economy of scale to being big

Shrew eats its body weight a day.

Metabolic rate grows in relation to body mass x^3/4

3/4 power law

Thought it would be 2/3 power... same tissue and density and euclidian shapes, the volume of a thing is proportional to its volume or width CUBED and its surface area is only a SQUARED

SURFACE to VOLUME ratio... Effects surface dominated. Heat loss surfaces.... X^2/3

Why three quarters not 2/3.

3/4 power extends some 27 orders of magnitude

Lifetime... M^1/4

Heartbeats, breathes, M^1/4

Physiological time is somehow related to MASS^1/4

Any animal from a mouse to us, can expect to live the same number of heartbeats.

No matter how big.

We all live about 1.5 billion heartbeats.

300,000,000 breath cycles.

At ends of the network

Terminal units... Cappilaries same size for all mammals.

Invariant terminal units.

3/4 power comes out of that...

List of 20 or 30 scaling laws.

Respiratoty enzymes are end of the line in the mitochondria.

Mammals cannot small... There is a limit... a few grams...

Pulsitile flow impossible smaller than a shrew... oozing or steady flow.

Optimality assumption... Natural Selection does not need to be optimal.

OPtimality still a natural benchmark...

Drastic simplications to see broad trends... Miss all kinds of details...

Human lives longer.....

Deeper chemical or physical principles....

Take body temperature into account for cold blooded...

Biology should explain...

Why does a mouse live a year or two... Same genes and cells at molecular level.

Molecular Mechanisms of Aging...

Look at life through a physicists Life....

Broad Unifying Principles... Ignore minor details...

1997 - Geoffrey West - Particle Physicist.

Mathematical Regularities

Smallest microbes to biggest animals

Size of a creature, metabolism and lifespan

How many calories each day?

What are our metabolic needs...

Heat, Oxygen, Calories...

Metabolic rates for diverse mammals...

Metabolic rates fall neatly on a curve when they are graphed as a function of the body mass of the animal.

Economy of scale to being big

Shrew eats its body weight a day.

Metabolic rate grows in relation to body mass x^3/4

3/4 power law

Thought it would be 2/3 power... same tissue and density and euclidian shapes, the volume of a thing is proportional to its volume or width CUBED and its surface area is only a SQUARED

SURFACE to VOLUME ratio... Effects surface dominated. Heat loss surfaces.... X^2/3

Why three quarters not 2/3.

3/4 power extends some 27 orders of magnitude

Lifetime... M^1/4

Heartbeats, breathes, M^1/4

Physiological time is somehow related to MASS^1/4

Any animal from a mouse to us, can expect to live the same number of heartbeats.

No matter how big.

We all live about 1.5 billion heartbeats.

300,000,000 breath cycles.

**Every point in 3D and has to be space filling.**

Fast as possible with minimal energy...

Using these criteria and laws fluid mechanics

Network had to have a self-similar, branching and fractal structure.Fast as possible with minimal energy...

Using these criteria and laws fluid mechanics

Network had to have a self-similar, branching and fractal structure.

At ends of the network

Terminal units... Cappilaries same size for all mammals.

Invariant terminal units.

3/4 power comes out of that...

List of 20 or 30 scaling laws.

**Networks inside the cells... Cytoskeleton, or chemical network.**

There is a network in the cell, then another network in the mitochondria.There is a network in the cell, then another network in the mitochondria.

Respiratoty enzymes are end of the line in the mitochondria.

Mammals cannot small... There is a limit... a few grams...

Pulsitile flow impossible smaller than a shrew... oozing or steady flow.

**Cells taken from animal in culture... SAME metabolic rate in vitro.**

But inside big animals there is less capillary flow per cell, in elephant, so metabolic rate has to slow down.

In culture not applicable, because no network in culture, every cell getting nutrients.But inside big animals there is less capillary flow per cell, in elephant, so metabolic rate has to slow down.

In culture not applicable, because no network in culture, every cell getting nutrients.

Optimality assumption... Natural Selection does not need to be optimal.

OPtimality still a natural benchmark...

Drastic simplications to see broad trends... Miss all kinds of details...

Human lives longer.....

Deeper chemical or physical principles....

Take body temperature into account for cold blooded...

Biology should explain...

## Mandelbrot Set

What is the math behind it. What does it mean?

Logistic map - iterated map to visualize the transition to chaos in biology

Chemical oscillations, fluid mechanics.

Analogous to logistic map... He had the idea to study iterated maps on a plane instead of a line.

Complex numbers... A complex of two things stuck together... real and imaginery welded together.

Instead of a number on a number, it is a generalization to a plane.

Very natural in higher math, electricity, magnetism, waves, oscillators, quantum mechanics.

Sequence of complex numbers hopping around complex plane.

Is it bounded or does it hop out to infinitely.

Other colors, how fast do you hop to infinity.

Subtly self similar - not too orderly or too random , like good art.

Pollock said.. I AM NATURE.

Used Chaos in Nature.

He moved around

Systematic increase in fractal dimension.

Chaotic pendulum like pollock,,

Logistic map - iterated map to visualize the transition to chaos in biology

Chemical oscillations, fluid mechanics.

Analogous to logistic map... He had the idea to study iterated maps on a plane instead of a line.

Complex numbers... A complex of two things stuck together... real and imaginery welded together.

Instead of a number on a number, it is a generalization to a plane.

Very natural in higher math, electricity, magnetism, waves, oscillators, quantum mechanics.

**Mandelbrot Sets encodes information about complex-number maps, much like the orbit diagram did for the logistic map.**

Orbit diagram is like a family portrait of logistic map showing what would happen if we changed a parameter in it... LIKE TURNING A KNOB, only adjusting a parameter.

Growth rate in population model...

Mandelbrot Set

Complex knob is like 2 knobs...

Yellow, blue

Colors describe

Choose a complex number c

Define new number = (old number)^2 + c

0 --> c (zero maps to c)

c ---> c^2 + cOrbit diagram is like a family portrait of logistic map showing what would happen if we changed a parameter in it... LIKE TURNING A KNOB, only adjusting a parameter.

Growth rate in population model...

Mandelbrot Set

Complex knob is like 2 knobs...

Yellow, blue

Colors describe

Choose a complex number c

Define new number = (old number)^2 + c

0 --> c (zero maps to c)

c ---> c^2 + c

Sequence of complex numbers hopping around complex plane.

Is it bounded or does it hop out to infinitely.

Other colors, how fast do you hop to infinity.

Subtly self similar - not too orderly or too random , like good art.

Pollock said.. I AM NATURE.

Used Chaos in Nature.

He moved around

Systematic increase in fractal dimension.

Chaotic pendulum like pollock,,