The Mathematical Way of Understanding the World

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The Mathematical Way of Understanding the World

• Clocks and Chaos
• Nov. 12, 2002
• Berkeley, CA

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An old view
Take away number in all things and all things
perish. Take calculation from the world and all
is enveloped in dark ignorance, nor can he who
does not know the way to reckon be distinguished
from the rest of animals. St. Isadore of
Seville, 7th century CE
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My plan
1st lecture Clocks and Chaos
linear versus nonlinear models a)
measuring time, space, money was the very thing
in 1300! Clocks work because of
linearity this turned out to be
the key which unlocked modern technology.
b) nonlinear phenomena are much harder it can
create chaos and pearls and fiber
optics.
2nd lecture Models of Thought
logical versus statistical models a)
Logic is too tight a straight-jacket b)
Statistics can be tamed, with difficulty.
c) But can it be done in our brains?
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Around 1300, big things began to happen(Crosby,
The Measure of Reality)

• Time measures in music and monumental town clocks

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The mechanical relaxation oscillator in the 1392
Wells Cathedral clock
Accurate to about 15 minutes/day or
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Around 1300, big things began to happen(Crosby,
The Measure of Reality)

• Time measures in music and monumental town
  clocks
• Space Giottos perspective and portolan charts

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A detail of the 1466 Roselli portolan chart of
the Mediterranean
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Around 1300, big things began to happen(Crosby,
The Measure of Reality)

• Time measures in music and monumental town
  clocks
• Space Giottos perspective and portolan charts
• Money money of account and double entry
  book-keeping

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Everything can and should be counted making
an inventory after Pacioli (1493)

• Begin as follows In the name of God, on the 8th
  day of November, 1493, in Venice. The following
  is an inventory of myself, of Venice, Street of
  the Holy Apostles.
• Continue listing the contents of your home cash,
  jewels and gold, designating each item by weight
  silverware with weight and alloy then linens
  bedsheets, tableclothes and such, and featherbeds
  and so on.
• Next go to your warehouse and record in precise
  weight, number and measure everything there
  spices, dyewood, pelts and so on.
• Then ones real estate and money on deposit,
  credit situation with an attempt at assessment .

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Around 1300, big things began to happen(Crosby,
The Measure of Reality)

• Time measures in music and monumental town
  clocks
• Space Giottos perspective and portolan charts
• Money money of account and double entry
  book-keeping
• Graphs Oresme made pictures of physical
  processes
• and much much more light, heat, color,
  certitude, virtue and grace are measurable!
• (Merton College scholastics).

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Nicole Oresme graphing functions of time
(University of Paris, c.1361)

• The idea of a function of
• time
• The idea of its rate of
• change and acceleration
• The idea of graphing
• x-axis called longitude
• y-axis called latitude
• Uniform accel. from v1
• to v2 has the same result
• as constant speed
• (v1v2)/2

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In 1582, Galileo watched a lamp swinging in the
Pisa Cathedral and discovered the concept of
simple harmonic motion
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The magic of simple harmonic oscillators
Blue verge-and-foliot clock, relaxation
oscillator, bang-bang, many parameters hard to
control
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All (well, almost all) modern technology stems
from this

• It predicts the future using only where you are
  now and your rate of change
• It is linear if f(t) solves it, so does c.f(t)
  if f(t) and g(t) solve it, so does f(t)g(t)

Linearity in action
i) at cocktails parties and concerts, you can
hear all the separate voices ii) the government
can sell the electro-magnetic spectrum (if we can
own land, why not own a frequency too!?) iii)
Schrödingers cat can be both dead and alive
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Linearity in waves (seen from a quiet harbor off
Mt. Desert)
Just like sound waves in air and electro-magnetic
waves in the aether, small water waves are a
superposition of sinusoidal wave trains and can
be decomposed into their components.
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Something new was seen in 1834
I was observing the motion of a boat which was
drawn rapidly along a narrow channel, when the
boat suddenly stopped. Not so the mass of water
in the channel which it had put into motion.
Assuming the form of a large solitary elevation,
a rounded smooth and well-defined heap of water,
which continued its course along the channel
apparently without change of form or diminution
of speed. I followed it on horseback and
overtook it still rolling along at 8 or 9 m.p.h.
John Scott Russell
Used in fiber-optics today!
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Nonlinear effects come in many forms
Pattern and Form in Earth Dynamics, Torino 2002
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The bakers transformation making filo pastry
Large iterations of nonlinear maps do the oddest
things! The butterfly effect on the weather,
a.k.a. Liapounov exponents
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Scaling a new organizing principle

• Some things look simple under the microscope
• they have good linear approximations.
• But some things retain their complexity
• no matter how closely you look, they
• have more and more structure.
• Simplest case this small scale structure is
  more
• or less similar to the big structure
• the object is self-similar or fractal

Self-similarity II in Buddhism Each object
in the world is not merely itself but involves
every other object and, in fact, is everything
else. (Charles Eliot)
Self-similarity I limericks So, Natralists
observe, a flea Hath smaller Fleas that on him
prey And these hath smaller Fleas to bite
em And so proceed, ad infinitum. (Jonathan
Swift)
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Indras Pearls (Caroline Series, Dave Wright and
DM)
In the heaven of the great god Indra is said to
be a vast and shimmering net, finer than a
spiders web, stretching to the outermost reaches
of space. Strung at each intersection of its
diaphanous threads is a reflecting pearl. In the
glistening surface of each pearl are reflected
all the other pearls, even those in the furthest
reaches of heaven. In each reflection, again are
reflected all the infinitely many other pearls,
so that reflections of reflections continue
without end.
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The key ingredient a spiral self-similarity
Klein symmetry is repeating the same shape,
but same neednt be taken literally.
You boil it in sawdust, you salt it in glue, You
condense it with locusts and tape, Still keeping
one principal object in view- To preserve its
symmetrical shape. (Lewis Carroll)
All figures/mpegs created by D. Wright
2 parameters speed and amount of spiraling can
be varied
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2 dances at once X S(X) T(X)
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S and T keep each other at arms length
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S and T kiss and a fractal curve emerges
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Both S and Tspiralthe dance getsinterestingan
d the result is truly fractal
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Apolloniuss Gasket
A filigree with a new kind of symmetry
parabolic
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More and more baroque on the edge of chaos
We call this an accidental parabolic group.
Like Apollonius gasket, its a simple curve for
which both the inside and the outside have
collapsed into circles.
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Plane filling Peano curve, a geometric form of
turbulence
For a long time, no one believed you could draw a
continuous curve in the plane touching every
point! The colors mark which branch of the tree
you in. The gaps are only due to the computers
limited power.
Whoops dont click. This crashes PowerPoint and
WindowsMedia. Must use QuickTime!
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A fly-by
Watch the limit set as the 2 parameters in each
of the generating symmetries are varied.
Indras Pearls, By C.Series, D.Wright, and DM,
Cambridge Univ. Press, 2002