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Title: Through the Eyes of the Mathematician Through the Eyes of the Mathematician Through the Eyes of the Mathematician Through the Eyes of the Mathematician


1
Through the Eyes of the MathematicianThrough the
Eyes of the MathematicianThrough the Eyes of the
MathematicianThrough the Eyes of the
Mathematician
  • Cheryl Ooten, Ph.D.
  • Mathematics Professor Emerita
  • Santa Ana College
  • Bernie Russo, Ph.D.
  • Mathematics Professor Emeritus
  • University of California at Irvine

2
  • Preview
  • I. Set the StageSome Models
  • Myths
  • II. Whats It All About?
  • III. Sampling the Branches
  • IV. Stories about The Big Ones
  • Solved Unsolved

3
I.Set the StageSome Models Myths
4
The math world is like a restaurant
5
The math world is like a restaurant
with a kitchen dining room
6
Math is like gossip.
Devlin, K.
7
Math is like gossip.
Its about relationships.
Devlin, K. ?
8
Models for Managing the Mean Math
Blues
9
  • a) Fight negative math rumors.
  • 1. Some people have a math mind and some dont.
  • 2. I cant do math.
  • 3. Only smart people can do math.
  • 4. Only men can do math.
  • 5. Math is always hard.
  • 6. Mathematicians always do math problems quickly
    in
    their heads.
  • 7. If I dont understand a problem immediately, I
    never will.
  • 8. There is only one right way to work a math
    problem.
  • 9. It is bad to count on fingers.
  • 10. Negative math-experience memories never go
    away.

10
  • b) Use reframes.
  • What can a reframe do?
  • Affect attitude change feelings.
  • Neutralize negativity.
  • Change a helpless victim to an in-charge owner.
  • Remind us of the YET.

Ref Ooten Moore, Managing the Mean Math Blues
11
Cognitive Psychotherapy Model
of how people work

THOUGHTS
EMOTIONS
BEHAVIORS
BODY SENSATIONS
Ref Ooten Moore. Managing the Mean Math Blues
12
EMOTIONS I am frightened by math
THOUGHTS I cant do math.
BEHAVIORS I avoid numbers I dont practice math
BODY SENSATIONS My stomach tenses when I see
numbers
Ref Ooten Moore, Managing the Mean Math Blues,
p 154
13
EMOTIONS Relief Curiosity about what else I can
learn Joy with skills I have
THOUGHTS I can do some math. I can learn more. I
dont need to get it all right now.
BEHAVIORS Take a deep breath Write a possible
solution. Try something new. Ask questions
BODY SENSATIONS Relax Become calmer Heart rate
slows
Ref Ooten Moore, Managing the Mean Math Blues,
p 154
14
c) Check your Mindset. (Carol Dweck,
Stanford Psychology Prof)
vs
  • Growth Mindset
  • Your smartness increases with hard work.
  • Fixed Mindset
  • You can learn but cant change your basic level
    of intelligence

Ref Boaler, Mathematical Mindsets Dweck,
Mindset The New Psychology of Success
15
Mindset Model vs
  • Growth Mindset
  • Focus on effort
  • Skip judging
  • Ask
  • What can I learn?
  • How can I improve?
  • What can I do
  • differently?
  • Fixed Mindset
  • Focus on ability
  • Evaluate label Good-bad
  • Strong-weak

Choose this!
Ref Boaler, Mathematical Mindsets Dweck,
Mindset The New Psychology of Success
16
Just because some people can do something with
little or no training, it doesnt mean that
others cant do it (and sometimes do it even
better) with training. Reframe anxiety by
focusing on effort, not ability. Math skills are
learnable!
Ref Dweck
17
d) Anxiety comes from being required to
stay in an uncomfortable situation over which we
believe we have no control.
18
References for Managing the Mean Math
Blues Boaler, J. (2016). Mathematical mindsets
Unleashing students potential through
creative math, inspiring messages and
innovative teaching. Jossey-Bass San
Francisco, CA. Boaler, J. (2015). Whats math
got to do with it? Penguin Books New York,
N.Y. Dweck, C. S. (2006). Mindset The new
psychology of success. Ballantine Books,
N.Y. Ooten, C. K. Moore. (2010). Managing the
mean math blues Math study skills for
student success. Pearson Upper Saddle River,
N.J.
19
Watch for math myths everywhere!
Math Writer John Derbyshire wrote I dont
believe this topic can be explained using math
more elementary than I have used here, so if you
dont understand it after finishing my book,
you can be pretty sure you will never understand
it.
Ref Derbyshire, Prime Obsession, p. viii
?
20
UC Berkeley Mathematics Professor
Edward Frenkel says One of my teachersused
to say People think they dont understand math,
but its all about how you explain it to them.
If you ask a drunkard what number is larger,
2/3 or 3/5, he wont be able to tell you. But if
you rephrase the question what is better, 2
bottles of vodka for 3 people or 3 bottles of
vodka for 5 people, he will tell you right away
2 bottles for 3 people, of course. My goal is
to explain this stuff to you in terms that you
will understand. Ref
Frenkel, Love and Math, p 6

21
Our goal is to talk about a few math things we
think are cool that were not taught in school and
hopefully to talk about them in ways that make
sense!
II.Whats It All About? Maths beautiful,
Maths everywhere, and Maths huge!
?
22
Our goal is to talk about a few math things we
think are cool that were not taught in school and
hopefully to talk about them in ways that make
sense!
II.Whats It All About? Maths beautiful,
Maths everywhere, and Maths huge!
23
The world of mathematics is a hidden parallel
universe of beauty and elegance, intricately
intertwined with ours. Ref Frenkel,
Love and Math, p 1

These are mathematical models of fractals.
24
  • Example
  • Neuroscientist/musician Daniel Levitin says
    Music is organized sound.
  • We say
  • Math is the organizing tool.
  • Music is intricately twined with math.
  • Example
  • Dave Brubeck with Paul Desmonds
  • Take Five
  • Ref Levitin, This Is Your Brain on Music
    ?

25
How much math does
one person know?
By the late 1800s, math had passed out of the
era when really great strides could be made by a
single mind working alone. It became a
collegial enterprise in which the work of even
the most brilliant scholars was built upon, and
nourished by, that of living colleagues.
Ref Derbyshire, Prime Obsession, p 165
26
Four Branches of Mathematics
Arithmetic
Algebra
Geometry
Analysis
27
Four Branches of Mathematics
1. Arithmetic (Counting)
2. Algebra (Symbolic Manipulation)
3. Geometry (Figures,Drawing)
4. Analysis (Calculus, Limits)
28
60 Mathematics Research Specialties
(A.M.S.)
Combinatorics Number Theory Algebraic
Geometry K-Theory Topological Groups Real
Functions Potential Theory Fourier
Analysis Abstract Harmonic Analysis Integral
Equations Functional Analysis (Bernie) Operator
Theory Calculus of Variations Optimization Convex/
Discrete Geometry
Differential Geometry General Topology Algebraic
Topology Manifolds Cell Complexes Global
Analysis Probability Theory Statistics Numerical
Analysis Computer Science Fluid Mechanics Quantum
Theory Game Theory Economics Operations
Research Systems Theory Mathematical Education
(Cheryl) AND MORE
29
Each specialty is related
to one or more branches.
  • 1.Arithmetic
  • Combinatorics
  • Number Theory
  • Statistics

2.Algebra Algebraic Geometry K-Theory Group
Theory
4.Analysis Fourier Analysis Differential
Equations Functional Analysis
  • 3.Geometry
  • Convex/Discrete Geometry
  • Differential Geometry
  • General Topology

Deeper results are possible when different
research specialties are connected.
30
  • Mathematical research is either
  • Pure (theory for its own sake)
  • Applied (e.g. credit cards security)

31
III.Sampling the Branches
32
  • School math is
  • centuries old
  • tiny part of the whole field of math.
  • Lets see some cool things
    that arent always shown in school.
  • ?

33
1.Arithmetic (Counting or Number Theory)

34
Number Theory prime numbers have given us
cryptography , thus, the ability to securely use
credit cards online.
35
Prime numbers are a rich source of ideas.
36
The FUNdamental Theorem of Arithmetic
Every positive integer greater than 1 is either a
prime or it can be factored as the unique product
of prime numbers.
37
Review What is a prime?
The numbers 2, 3, 4, 5, are either Composite
or Prime
Composites 4, 6, 8, 9, 10, 12, 14, 15, 16, 18,
20, 21, 22, 24, 25, 26, 27, 28, 30
Primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
Primes cannot be factored in an interesting way.
38
Review What is a prime?
The numbers 2, 3, 4, 5, are either Composite
or Prime
Composites 4, 6, 8, 9, 10, 12, 14, 15, 16, 18,
20, 21, 22, 24, 25, 26, 27, 28, 30
Primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
Primes cannot be factored in an interesting way.
Even numbers after 2 are composite.
39
The FUNdamental Theorem of Arithmetic
  • Every positive integer greater than 1 is either a
    prime or it can be factored as the unique product
    of prime numbers.
  • e.g. 2 and 3 are prime
  • 4 22
  • 5 is prime
  • 6 is 23
  • 7 is prime ?

40
The FUNdamental Theorem of Arithmetic
Every positive integer greater than 1 is either a
prime or can be factored as the unique product of
prime numbers.
Lets prove it
Good Numbers primes or can be factored as the
unique product of prime numbers Bad Numbers
all the others
41
The FUNdamental Theorem of Arithmetic
  • Consider the smallest bad number.
  • It cant be prime (its bad, not good.)
  • That means its composite and can be factored
    into the product of smaller numbers.

BAD
Smaller number times Smaller number
42
The FUNdamental Theorem of Arithmetic
  • These smaller numbers must either be prime or
    able to be factored as primes.
  • Whoops! The BAD number isnt bad after all.

BAD
Smaller number times Smaller number
43
Since the smallest bad number couldnt be bad, we
continue up to the next smallest bad number but
the same thing happens. And on and on. That means
that there are no bad numbers and our theorem
is true.
ITS TRUE Every positive integer greater than 1
is a prime or can be factored as the unique
product of prime numbers.
44
The FUNdamental Theorem of Arithmetic
ITS TRUE Every positive integer greater than 1
is either a prime or can be factored as the
unique product of prime numbers.
45
The FUNdamental Theorem of Arithmetic
  • This proof is done by induction which is like
    setting up dominoes so each domino could push the
    next over then starting them to fall.
  • ?

46
  • Arithmetic has the peculiar characteristic that
    it is rather easy to state problems in it that
    are ferociously difficult to prove.
  • Example
  • In 1742, Christian Goldbach (age 52) made a
    conjecture.
  • It is not yet proved or disproved
  • But it is the subject of a novel called Uncle
    Petros and Goldbachs Conjecture.

Ref Derbyshire, Prime Obsession, p 90
47
  • Arithmetic has the peculiar characteristic that
    it is rather easy to state problems in it that
    are ferociously difficult to prove.
  • Example
  • In 1742, Goldbach made a conjecture.
  • It is not proved or disproved yet.
  • But its the subject of a novel called Uncle
    Petros and Goldbachs Conjecture.

Ref Derbyshire, Prime Obsession, p 90
48
Goldbachs Conjecture Every even number greater
than 2 is the sum of two primes.
422 633 835 1037
1257 14311 16511
18711 20? 22?
49
 

50
 

51
 

Are there any more???
52
There are many other interesting hypotheses about
primes. 1-3 are known to be true
  1. There are infinitely many primes.
    (Euclid 300 BC)
  2. The higher you go, the sparser the primes.
  3. Successive primes can be any distance apart. They
    can also be very close. (e.g. 1113 or 3741)
  4. There are infinitely many twin primes. (i.e. 2
    apart like 1113 or 2931)
  5. Riemann Hypothesis (More later)

53
  • 2. Algebra (Symbolic Manipulation)

54
In algebra, we like to look for general symbolic
formulas. Example The Pythagorean Theorem
55
Example The Pythagorean Theorem
56
Example The Pythagorean Theorem
15
57
The Pythagorean Theorem
may be pictured this way But why is
c²a²b²?
58
Once we find a formula, we need to convince
ourselves that it is true. Here is one of many
(112) proofs of The Pythagorean Theorem.
(Only one is needed! Were in
the kitchen now.) Step 1 Consider a right
triangle.
59
Proof of The Pythagorean Theorem Step 2 Label
its sides.
60
Proof of The Pythagorean Theorem Step 3 Make 4
triangles the same size.
a
c
b
61
Proof of The Pythagorean Theorem Step 4 Place
the 4 triangles together in two different ways to
make two squares that are the same size (ab).
b
a
a
b
a
b
b
a
62
Proof of The Pythagorean Theorem Step 5 Notice
that the extra white space forms 3 smaller
squares.
63
Proof of The Pythagorean Theorem Step 6
Notice The white square on the left has area
c². The two white squares on the right have
areas b² a².
64
Proof of The Pythagorean Theorem Step 7 Since
the entire left square is the same size as the
entire right square, removing the four triangles
gives us c²a²b² ?
65
The Pythagorean Theorem The square of the
hypotenuse Of a right triangle Is equal to the
sum of the squares Of the two adjacent sides.
66
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67
(No Transcript)
68
Are there cases of right triangles whose sides
are whole numbers? C 19.209
15
69
Yes, there are cases when the a, b, c in
c²a²b² are whole numbers. For example 5, 4,
3 or 13, 12, 5 These are called Pythagorean
Triples.
70
We could say that for Pythagorean Triples, this
theorem separates squares into the sum
of two squares. c² a² b² 5² 3² 4² 25 9
16 For now, lets just think about whole numbers
for a, b, c.
71

In 1637, Pierre de Fermat wrote On the
other hand, it is impossible to separate a cube
into two cubes, , or generally any power except
a square into two powers with the same exponent.
I have discovered a truly marvelous proof of
this, which, however, the margin is not large
enough to contain.
(Ref Aczel, p. 9)
72

 
?
73
(Fermat was a tease.)
356 years later, after thousands of
mathematicians tried to prove it, Andrew Wiles
did, in 1993. (So much for the myth that
mathematicians do problems quickly in their
heads.)
74
It is clear to us now that entire fields of math
had to be developed by hardworking mathematicians
to prove Fermats Last Theorem. So much
for Fermats truly marvelous proof.
Sophie Germain
75
3. Geometry (Figures Drawing)
76
3. Geometry (Figures Drawing) Its more than
the Euclidean geometry that we learned in high
school.
77
Geometry Makes Me Happy when it meets art.
Jen Starks kaleidoscope brings to mind geodes,
topographic maps, fractal geometry.
Ref Geometry Makes Me Happy, p 33
78

Fractals
geometry self-similarity
Fractals can come from nature or the
imagination
Ref http//fractalfoundation.org/OFC/OFC-12-2.htm
l
79

Fractals decribe the real world better than
Euclidean geometry. Engineers can perform
biomimicry.
Ref http//fractalfoundation.org/OFC/OFC-12-2.htm
l
80
Fractal patterns (from nature) are used to cool
silicon chips in computers.
Ref http//fractalfoundation.org/OFC/OFC-12-2.htm
l
81
The Sierpinski Triangle Fractal
Ref Burger Starbird
82
The Sierpinski triangle was used to design
antennas for cell phones and wifi.
Ref http//fractalfoundation.org/OFC/OFC-12-2.htm
l
83
Fractals provide ways to mix fluids carefully
when just stirring doesnt work.
Ref http//fractalfoundation.org/OFC/OFC-12-2.htm
l
84
  • 4. Analysis (Limits)
  • ?

85
4. Analysis (Limits)
 
 
 
86
4. Analysis (Limits)
 
 
87
4. Analysis (Limits)
 
 
88
Using limits, the study of Calculus
has two basic themes Differentiation
Integration
89
Isaac Newton Gottfried Leibnitz
These two lovely gentlemen came up with calculus
independently about 200 years ago. It took 150
years to prove that
their ideas were correct.
90
Calculus Idea 1 Differentiation
Differentiation is about finding weird slopes.

Example Slope is steepness of a roof
or the grade of a road or the incline
of a treadmill
91
Calculus Idea 1 Differentiation
Differentiation is about finding weird slopes.
Example Slope is steepness of a roof

What is the slope of a flat roof? What is the
slope of a curved roof like a dome?
92
Calculus Idea 1 Differentiation
Differentiation is about finding weird slopes.
In calculus, we try to find the slope of a line
at each point on a curve
Example Slope is steepness of a roof
93
Calculus Idea 2 Integration
Integration is about finding weird areas.
Not-weird area A red carpet measuring
3 feet by 18 feet has area 54 square feet
or 6 square yards.
94
Calculus Idea 2 Integration
Integration is about finding weird areas like the
area in gray in this picture.
95
We know how to find the area of rectangles. So,
we approximate the area with a few rectangles.

 
96
Then we increase the number of rectangles.

 
97
We keep increasing the rectangles.

 
98
We keep increasing the number of rectangles and
use the idea of limit to find the exact area.
99
AND, miracles of miracles, it turns out that
differentiation integration are opposite
processes of each other in a similar way
that adding subtracting are.
100
IV.Stories about The Big OnesSolved
Unsolved ?
101
Solved Problems
102
1. Arithmetic (Counting)
 
103
2. Algebra (Manipulating)
a

d
b
c
4-Color Theorem (1852-1976)
104

The computer was necessary for
part of the proof.
105
3. Geometry (Figures/Drawing)
Poincarés Conjecture
(1904-2006) Grigori Perelman proved this in 2006
but turned down 1 million in prize money, left
mathematics, and moved back to Russia with
family. (He also turned down the Fields
Medal.) Refhttps//laplacian.wordpress.com/
2010/03/13/the-poincare-conjecture/
106
4. Analysis
 
?
107
P.S. Another Algebra Theorem
Classification of Symmetries (1870-1985)
108
  • Symmetry
  • Is pervasive in nature
  • e.g. starfish, diamonds, bee hives,
  • Is about connections between different parts of
    the same object.


109
For a mathematician, a symmetry is something
active, not passive. For the mathematician, the
pattern searcher, understanding symmetry is one
of the principal themes in the quest to chart the
mathematical world. Ex Lets look at symmetries
of a round table top aRRed square table top

Ref Du
Sautoy, Symmetry
110
For a mathematician, a symmetry is something
active, not passive. For the mathematician, the
pattern searcher, understanding symmetry is one
of the principal themes in the quest to chart the
mathematical world. Ex Lets look at symmetries
of a round table top a square table top

Ref Frenkel
111
Consider all possible transformations of the two
tables which preserve their shape
position. Those transformation are called
symmetries. A round table has many symmetries. A
square table only has four. When we combine these
symmetries, we get a mathematical entity called a
group.

112
Ex Look at all possible transformations of the
two tables which preserve their shape
position. Those transformation are called
symmetries. A round table has many symmetries. A
square table only has four. When we combine these
symmetries, we get a mathematical entity called a
group.

113
Other nice examples of symmetries forming a group
can be seen with each of the Five Platonic
Solids.

48
48
24
120
120
114
  • Child Prodigy Evariste Galois (1811-1832)
  • recorded a great gift to math in a long letter to
    a friend the evening before he was killed in a
    duel at age 20.
  • His concept (the group) proved to be most
    significant, with applications in
  • physics
  • chemistry
  • engineering
  • many fields of math


Ref Frenkel, p 75
115
  • In math, discoveries using Galois idea led to
    advances in
  • Computer algorithms
  • Logic
  • Geometry
  • Number theory


116
A primary aim in any science is to identify
study basic objects from which all other
objects are constructed. In biologycells In
chemistryatoms In physicsfundamental
particles So too for mathematics In number
theoryprime numbers In group theorythe simple
groups

117
A primary aim in any science is to identify
study basic objects from which all other
objects are constructed. In biologycells In
chemistryatoms In physicsfundamental
particles So too for mathematics In number
theoryprime numbers In group theorythe simple
groups

Ref Devlin. Mathematics, the New Golden Age
118
A primary aim in any science is to identify
study basic objects from which all other
objects are constructed. In biologycells In
chemistryatoms In physicsfundamental
particles So too for fields in mathematics In
number theoryprime numbers In group theorythe
simple group

Galois contribution
119
  • Unsolved Problems
  • ?

120
Unsolved Problems
In May, 2000, in Paris, Clay Mathematical
Foundation announced 7,000,000 in prizes. One
million dollars apiece for solution of 7
mathematical problems called The
Millenium Problems.
121
i. The Riemann Hypothesis (MORE LATER) ii.
YangMills Theory the Mass Gap
HypothesisDescribe mathematically why electrons
have mass iii. The P vs. NP ProblemHow
efficiently can computers solve problems iv. The
Navier-Stokes EquationsMathematically describe
wave/fluid motion v. The Poincaré ConjectureFind
the mathematical difference between an apple a
donut vi. The Birch Swinnerton-Dyer
ConjectureKnowing when an equation cant be
solved vii. Hodge ConjectureClassifying abstract
objects
Ref Devlin, K. The Millennium Problems ?
122
100 years ago, Hilbert stated 23 problems. All
have been solved except for Riemanns Hypothesis.
Ref Derbyshire
123
The Great White Whale of the 20th Century has
been (and still is) The Riemann Hypothesis.
Ref Derbyshire, Prime Obsession pp 197-198
?
124
The Riemann Hypothesis was stated by Bernhard
Riemann in 1859
Ref Derbyshire, Prime Obsession pp 197-198
125
It is slippery to state and slippery to understand
Ref Derbyshire, Prime Obsession pp 197-198
126
 
Ref Derbyshire, Prime Obsession pp 197-198
127
20th century mathematicians have been obsessed by
this challenging problem. Mathematicians have
worked on it in different ways, according to
their mathematical inclinations. There have
been computational, algebraic, physical, and
analytic threads.
Ref Derbyshire, Prime Obsession pp 197-198
128
The Golden Key
adding
multiplying

Notice the whole numbers.
Notice the prime numbers.
The link between analysis (Riemanns zeta
function) arithmetic (prime numbers)
129
Lets end on a nice note.
Ref clip art
130
www.youtube.com/watch?voomGHjJN-RE
131
Other References
Aczel, A. D. (1996). Fermats last theorem
Unlocking the secret of an ancient mathematical
problem. Burger, E.D. Starbird, M. (2005). The
heart of mathematics An invitation to effective
thinking. Derbyshire, J. (2004). Prime obsession
Bernhard Riemann and the greatest unsolved
problem in mathematics. Devlin, K. (1999).
Mathematics The new golden age. Devlin, K.
(2002). The math gene How mathematical thinking
evolved and why numbers are like gossip. Devlin,
K. (2002). The millennium problems The seven
greatest unsolved mathematical puzzles of our
time. Du Sautoy, Marcus. (2008). Symmetry A
journey into the patterns of nature. Estrada, S.
(publ). (2013). Geometry makes me happy. Fractal
Foundation http//fractalfoundation.org/OFC/OFC-1
2-2.html Frenkel. E. (2013). Love
math. Levitin, D. J. (2007). This is your brain
on music The science of a human
obsession. Ronan, M. (2006). Symmetry and the
monster One of the greatest quests of
mathematics. Russo, B. Freshman Seminars UCI.
Prime Obsession, Millennium Problems, Love
and Math. http//www.math.uci.edu/brusso/freshw0
5.html http//www.math.uci.edu/brusso/fr
eshs15.html http//www.math.uci.edu/brus
so/freshs14.html
132
Other References continued
  • Russo, B. High School Presentation. The Prime
    Number Theorem and the
  • Riemann Hypothesis.
  • http//www.math.uci.edu/brusso/slidRHKevin0
    60811.pdf
  • Singh, S. (1998). Fermats enigma The epic quest
    to solve the worlds
  • greatest mathematical problem.
  • Sabbagh, K. (2002) The Riemann hypothesis The
    greatest unsolved problem
  • in mathematics.
  • Smith, S. (1996). Agnesi to Zeno Over 100
    vignettes from the history of
  • math.
  • Szpiro, G. G. (2008). Poincarés prize The
    hundred-year quest to solve one
  • of maths greatest puzzles.

133
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134
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135
The FUNdamental Theorem of Arithmetic
ITS TRUE Every positive integer greater than 1
is either a prime or can be factored as the
unique product of prime numbers.
Said another way Every number is either prime or
divisible exactly by a prime.
136
Theorem Every number is interesting.
137
Theorem Every number is interesting. Proof Cons
ider the smallest uninteresting number. Isnt
that interesting? QED
138
Geometry Makes Me Happy when it meets
architecture. This is an underground car park in
Sydney, Australia.
Ref Geometry Makes Me Happy, p 175
139
Geometry Makes Me Happy when it meets industrial
design.
Textile designer Elisa Strozyk gave textile
properties to wood to make this throw.
Ref Geometry Make Me Happy, p 149
140
  • Koch Triangle
  • Fractals from the imagination

Ref Burger Starbird
141
The FUNdamental Theorem of Arithmetic
Every positive integer greater than 1 is a prime
or can be factored as the unique product of prime
numbers. e.g. 24 2223
Eenie, meenie, miney, moe Catch a
number by its toe If
composite, make it pay With
prime numbers all the way. My
teacher told me to always look for primes.

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The largest known twin primes currently Known
are 37568016956852666669 1 or
37568016956852666669 1 They each have
200,700 decimal digits!
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Lets play a number game with primes.
primes product 1 prime? 2,3
23 1 7 yes 2,3,5
235 1 31 yes 2,3,5,7
2357 1 211 yes 2,3,5,7,11
23571112311 ? 2,3,,p
235p1 Q So, Q is either prime or
composite. If prime, we have found a prime
larger than p. If composite, the FUN Theorem of
Arithmetic guarantees a prime divisor. We could
show that we have found a prime larger than
2,3,,p.
145
Insert the other number game that shows that
Large strings of composites. See Notes.
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