Slides Week 9 - James Franck Institute - University of Chicago

The 77 th Compton lecture series
Frustrating geometry:
Geometry and incompatibility shaping the physical world
Efi Efrati
Simons postdoctoral fellow
James Franck Institute
The University of Chicago
Lecture 9:
Fractal geometry
1

Outline:
• What are fractals ?
• Brief history of fractals
• The Mandelbrot set
• Fractals in nature and a practical fractal
2

What are fractals ?
3
Properties of fractals:
• Features in multiple scales
• Self-similar
• Fractional dimension
• Irregularity

Brief history of fractals
Koch curve
4

9a=A=8a
a=0
Brief history of fractals
Sierpinski carpet
5

Pierre Fatou
Brief history of fractals
6
Gaston Julia
Multiple iterations of a function F(z):
F(z), F(F(z)), F(F(F(z))), F(F(F(F(z)))), F(F(F(F(F(z)))))
If all the infinite points generated iteratively from a point P are
contained in a bounded domain, then P is in the Julia set of F.

Pierre Fatou
Brief history of fractals
7
Gaston Julia
Multiple iterations of a function F(z):
F(z), F(F(z)), F(F(F(z))), F(F(F(F(z)))), F(F(F(F(F(z)))))
If all the infinite points generated iteratively from a point P are
contained in a bounded domain, then P is in the Julia set of F.

Brief history of fractals
Hausdorff Dimension
N(r) = # of balls of size r needed to cover the object ~ 1/r D
8

hich completely annihilates all linear birefringent an
Brief history of fractals
chroic effects. Last we note that the maximal circ
r polarization effects Hausdorff per-scatterer Dimension read only 2/3 of th
lue for a single scatterer. Therefore the result of this a
N(r) = # of balls of size r needed to cover the object ~ 1/r D
ngement should be interpreted as selective attenuatio
undesired effects rather than an increase in response
rcular polarization. As this is the result of a geometr
inciple it is independent of frequency.
r → r/2 ⇒ N → 2N
D=1
9

Brief history of fractals
Hausdorff Dimension
N(r) = # of balls of size r needed to cover the object ~ 1/r D
10

ich completely annihilates all linear birefringent a
Brief history of fractals
chroic effects. Last we note that the maximal cir
Hausdorff Dimension
r polarization effects per-scatterer read only 2/3 of t
N(r) = # of balls of size r needed to cover the object ~ 1/r D
lue for a single scatterer. Therefore the result of this
ngement should be interpreted as selective attenuat
undesired effects rather than an increase in response
cular polarization. As this is the result of a geomet
inciple it is independent of frequency.
r → r/2 ⇒ N → 4N
D=2
11

Lewis Fry Richardson
Brief history of fractals
12
Coast-line paradox
Total length increases without bound with resolution

Brief history of fractals
Hausdorff Dimension
N(r) = # of balls of size r needed to cover the object ~ 1/r D
D= 1.25
13

Brief history of fractals
Hausdorff Dimension
N(r) = # of balls of size r needed to cover the object ~ 1/r D
D= 1.26 D= 1.1
14

Benoit Mandelbrot
Brief history of fractals
Worked for IBM research for 35 years (until 1987)
Tenured at Yale at the age of 75 (1999)
15
F(z)=z 2 +c
X(x,y)=x 2 -y 2 +cx
Y(x,y)= 2xy+cy

Benoit Mandelbrot
The Mandelbrot Set
16

Benoit Mandelbrot
The Mandelbrot Set
17

The Mandelbrot Set
First computer image of the Mandelbrot (Matelski,1978)
18

The Mandelbrot Set
19

The Mandelbrot Set
20

The Mandelbrot Set
21

Self similar
Exhibit features at all scales
Infinitely refinable
Its boundary has
fractal dimension 2
The Mandelbrot Set
22

principle it is independent of frequency.
The Mandelbrot Set
F (z) =z 2 + c
23

Fractals in nature
24

Fractals in nature
Dendritic crystals
25
The edge of a torn plastic sheet

Diffusion limited aggregation
Tom Witten and Len Sander
26
D~1.7

Electric discharge: Lichtenberg figure
27
D~1.7

Electric discharge: Lichtenberg figure
28
D~1.7

Lung surface area
D= 2.97
29

Cell phone broad band antenna
Cell phones need to communicate over a wide range of frequencies
4G frequency ~ 2.7 GHz, 1.9 GHz
GSM frequency ~ 1.9 GHz, 850 MHz
WiFi frequency ~ 20 MHz, 40MHz
30

Summary
Fractals are objects of exotic geometry
They arise naturally in many iterative processes, and scale free processes
Analyzing such structures calls for specialized tools
31

The 77 th Compton lecture series
Frustrating geometry:
Geometry and incompatibility shaping the physical world
Geometry and frustration
Efi Efrati
Simons postdoctoral fellow
James Franck Institute
The University of Chicago
32

Lecture series overview:
Geometry and frustration
• Geometric shaping principles govern the form and
function of much of our surrounding world.
• Geometric frustration (local property) leads to rich and
exotic (global) response.
33

Lecture series overview:
Geometry and frustration
• Euclid’s postulates
• Elimination of the fifth postulate: Non Euclidean geometry
• Generalized the notion of straight lines
34
Lecture 1

Lecture series overview:
Geometry and frustration
• Riemannian geometry
The metric, Parallel transport, Riemannian curvature
• Realizing non-Euclidean geometries:
paper model, crochet, Poincare disc
ds 2 = dx i dx j gij
Elliptic geometry cutout:
Cut the strips. Tape together the appropriate
edges (noted by the same number), red to
blue. Note that the turning rate decreases to
zero and even changes sign.
1
1
instability. I followed the notation
1
of Chernikova,
2
2
3
3
4
5
4 5
ding a similar logic). There’s
Hyperbolic ageometry paper
cutout:
by her (Sov.
Cut the strips. Tape together the
appropriate edges red to blue.
Note that the turning rate is constant
6) in which I believe she works things out in detail.
et it from the library.
sense of it is by adapting the electron density to
35
1
1
1
2
1
2
2
3
2
3
3
4
3
4
4
4
5
5
5
5
Lecture 2

Lecture series overview:
Geometry and frustration
• Relating intrinsic geometry and shape
• Thin sheet elasticity and isometries
• Shaping by metric prescription
neralize conclusions to more
plex systems.
Front Side
36
Plastic
Violet
Beet
Lettuce
Is there a connection?
Lecture 3

! Differential growth
Lecture series overview:
Geometry and frustration
• Residual Geometric stress frustration in plants and residual stress
• Visualizing residual stress
• Properties of residually stressed materials
Tuesday, November 17, 2009
Rhubarb: Rheum rhabarbarum
Tissue tension Brucke 1848
Green grows faster than red.
So red is in tension, green is in compression
This stress is called residual stress.
37
Lecture 4

Lecture series overview:
Geometry and frustration
• Theory of crystals, wallpaper symmetries
• Crystallographic restriction
• Penrose tiling
• Quasicrystals
L! L! L!
S! S! L! S! L!
38
Lecture 5

Lecture series overview:
Geometry and frustration
• Knot theroy
• Vortex knots
• Coherence from topology
40
Saturday, May 18, 2013
Peter Tait’s Table of Knots
Lecture 7

Lecture series overview:
Geometry and frustration
tion. In this work we study the simultaneous
confinement and incompatibility.
a + b a
= = Φ
• Exotic mathematical a properties b
• Phylotactic patterns
• Fibonacci spirals and golden rectangles
Φ = 1+√ 5
2
=1.618033988...
. DEFINITION OF THE PROBLEM
nsider a thin elastic disc to which a wedge of
was added (or removed). This constitutes a non-
41
Lecture 8
tu
s

Lecture series overview:
Geometry and frustration
• Fractals: A different kind of geometry
42
Lecture 9

The 77 th Compton lecture series
Frustrating geometry:
Geometry and incompatibility shaping the physical world
Thank you
Efi Efrati
Simons postdoctoral fellow
James Franck Institute
The University of Chicago
43