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# Patterns in Nature

An outline for Math 107
by

## steve kangas

on 21 March 2017

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#### Transcript of Patterns in Nature

Patterns in Nature
There are patterns all around us.
Patterns in mathematics can help us understand patterns in nature.

Size
Size, and the relative size of part to whole, affects the shape of things
Branching & Fractals
Flow
Symmetry
A "symmetry " is a way of moving so it looks the same after it has moved.
Symmetries around a central point
Rotation and reflection symmetries
Fractals
Abtractly, branching
structures can be
represented by mathematical graphs
Cyclic symmetry group
Rotations but no reflections
Order 3
Order 4
Dihedral symmetry group
Rotations and reflections both
Order 6
Order 8
Symmetry groups can tell things apart
Order 4
Order 8
Symmetry groups show some things are similar
Symmetries
along a line
"frieze patterns"
Roations, reflections, translations, glide reflections
Symmetries in 3D
Symmetries in 2D
"Wallpaper groups"
Rotations, reflections, translations, glide reflections
Tiling by regular polygons
"Crystallographic groups"
3D symmetries around a point
"Platonic solids"
Symmetry
Size
Branching & Fractals
Growth
Bubbles, packing, cracking
Flow
Growth
Bubbles, Packing, Cracking
Surface to volume
Surface to volume ratio

Flattening
Whiskers
Wrinkles
Internal tubes
Branching
Power Laws
Formula
Tool
Results
V - E + F = 2
Here
V = Vertices
E = Edges
F = Faces
This is true for all polyhedra
To study the proportional changes in features of animals as the size of the animal changes, we use a power law to get an allometric formula.
By using scaling factors a and the
appropriate b, we can relate the sizes of two features of an animal.
We also have scaling factors for cities!
To find the scaling factor a
we plotted data points on
log log graph paper.
We can describe a branching structure (such as a river system) by different measures (such as Strahler number)
For actual trees
there are rules about angles between branches
Fractals are shapes which look similar at different scales
Fractals are shapes which look similar at different scales
etc.
Fractal dimension
Statistically self similar
Perfectly self similar
Fibonacci:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
Golden ratio
Bubbles in Foam
Bubbles form hexagonal arrangements at the surface,
with angles of 120 degrees.
Can't have only hexagons because of V - E + F = 2
So there may be pentagons or other polygons
In 3 dimensions, bubbles come together in fours with an angle between of 109.5 degrees.
This also happens with arrangements of atoms in some molecules, such as this methane molecule.
This can be seen when spheres pack together as closely as possible
12 spheres surround a central sphere
.
Soap films are minimal surfaces (they have the smallest area for a surface with a given boundary)
Soap films satisfy Plateau's Laws:

1) Soap films are made of entire (unbroken) smooth surfaces.

2) Soap films always meet in threes along an edge called a Plateau border, and they do so at an angle of 120°.

3) These Plateau borders meet in fours at a vertex, and they do so at an angle of ≈ 109.47° (the tetrahedral angle)
Patterns like this appear everywhere...
... dragonfly wings, cells, insect eyes...
Cracks can form 120 degree angles
Cracks can also form 90 degree angles where new cracks meet old cracks.
Fluids flowing around an obstacle can form different patterns
The type of pattern corresponds to the Reynolds number:
In 3 dimensions, fluids can form vortex rings
Waves combine to form more complex waves
Waves curl over and break as water reaches shallow depths
Circulation patterns
tie life together
Golden rectangle and spiral
Phyllotaxis
Logarithmic spirals
Exponential growth
can be described by a formula
where the base a is a number and the exponent x changes
Full transcript