Nature of Mathematics:
The
Final Research Paper
–
The Topics
These topics are
(strongly recommended)
suggestions.
I think each
one would work
just fine. If
you have another idea, suggest it to me and if I approve it, then you may do that topic.
This Fi
nal Research Paper takes the place of the Final Exam.
Requirements are three things:
(1)
Group
Class P
resentation
–
not long, maybe 1/2 hour, containing some history
/context
and
the statements of some Theorems.
Maybe a tiny proof, or indication of a pr
oof, of something.
This will be the last day of class.
I
n all likelihood your final paper
will not be done by then,
but this is fine, for the hard part
of your paper
, the part that will take you the most time, is the
preparation of the
mathematics compon
ent
, and you are not really going
to present this to the
class.
(2)
Group
Paper:
Paper should contain both M
athematical and
Other (
historical/contextual
)
content. It should be 2000 to 2500 words (this is approximately 8 to 10 pages, double spaced,
1.25"
left/right margins, 1" top/bottom margins, size 12 Times New Roman

excluding
diagrams [if you have any]).
This is due by the day the Final Exam is scheduled.
(3)
Individual
Oral Presentation just to me (during the Final Exam
Period
–
or earlier if you
are
ready.
)
This is just the mathematics of your Paper, and will, in all likelihood will "simply" be
the presentation to me of a particular proof.
NOTE:
To be clear: b
oth the Class Presentation and the Paper are group projects. The Oral
Presentation wil
l be done singly.
You are
strongly
encouraged to show me a draft of your paper and/or to meet with me
about it to
discuss its
mathematical content.
"Mathematical Content" means the paper must contain the very careful statements of some
substantial t
heore
ms and then some arguments
: full
proofs
, heuristic
proofs
,
or partial proofs (
i.e.
,
a special case is proven or a piece of a theorem is proven
or some examples given
)
of one or more
theorems.
I've indicated those I
very
strongly
really
thi
nk ought to be i
nclude in
a given topic by
(!) or (!!) or (!!!) or (!!!
!). [You ar
e welcome to ask me if some other theorem/proof
you have
found in your research would be appropriate.] The number of !'s indicate
the difficulty. You
should
certainly include those with (
!) and, when possible, those marked (!!)
–
the better papers o
f
course would include (!!)'s.
It's
icing on the cake to in
clude those with more !'s than two of them
.
Icing is good, of course.
OR
–
if you do a "home run" on a (!!) or (!!!), then you don'
t have to do
a (!).
Check with me what you plan to do.
"Other Content" means history
(of people and/or cultures)
,
"stories" about some theorems (like
who proved them and when and why),
commentary, reflections,
connections w
it
h other areas of
mathematics.
Each paper ought to contain a definition/vocabulary list
and
a bibliography, properly
formatted
. These are
in addition
to the 8
to 10
required pages.
Some comments about these topics. First, they are all
important, both as part of the subject of
mat
hematics but also in their effect on
other disciplinary thinking in, for example, biology,
physics, engineering
,
and computer science.
Further, their
scope ranges
from ideas that can be taught in high school to
those from
fantastically
advanced mathematic
s, so yo
u'll have to insert yourself in
to the appropriate level
. Remember,
some
you will be able to understand and include some proofs of
the basic
theorems
but you will
only
be able to state, without proof,
the more advanced ones. But these latter theor
em
s
, which are
really at the edge of our knowledge
,
need to be included/stated.
I'm going to give some references below.
"
Math Univ
erse
"
= our text.
"
Journey
"
= Journey
through Genius: The Great Theorems of Mathematics
, by William Dunha
m
, Penguin Books
(1990). This is readily available (in our library, I'm sure, in various book stores, and, you may
borrow mine to duplicate the few pages you might need).
(1) All about
the number π
Give a bit of the history of π, both old and new stuff.
STATE:
Formulas using π (like area or surface area or volume), with
arguments, when
possible.
[Any calculus book has these formulas, often in a table inside the front of back
jacket.]
STATE:
Formulas
/expressions
for that that give the value of π.
There are lots of them.
[These, too, can be found in any Calculus text and are readily available elsewhere as well.]
They look like fancy sums or products, for example. Some involve trigono
metry.
Discuss some p
eople
who have contributed to our understanding of π
:
Archimedes
,
Lambert, Hermit,
(and discuss the Greek problem of "squaring the circle").
Some words
you might include
: Irrational, Algebraic, Transcendental, Series
(!) Use the fact t
hat π is not
al
gebraic, show the circle can't be squared.
[Skim
a bit of
"Journey
"
–
pages 11

13 and then look at
middle of page 24 to top of page 26.
]
(!!)
Derive
:
Archimedes
'
estimation of π
.
[See "Math Univ
erse
"
–
start in the middle of
page 28 and co
ntinue until the bottom of page 31.]
(2) A Focus on Archimedes
The story of his life and times.
The story (and explanation) of the "crown
that was not
solid
gold
."
State/discuss his Theorem on the Cylinder and the Sphere (it's on his tombstone)
, for bot
h
volume
and surface area
.
(!) Derive the Theorem on the Cylinder and the Sphere from the formulas (which you may
assume
)
for the volume
and surface area
of a cylinder and that of a s
p
here
.
[See "Math
Univ
erse
"
–
page 235, top
. Read from the top but disc
uss, for sure, from the paragraph
towards the bottom starting "So, this is the great
theorem. from….
"
]
(!!) Derive: Archimedes' estimation of π [See "Math Univ
erse
"
–
start in the middle of page
28 and continue until the bottom of page 31.]
(3
)
Fibonacc
i
Numbers (named for
Leonardo Pisano Fibonacci
)
[There are just tons of references
for this. The web is a great
re
source.]
Connection with Art, Architecture, Biology, and Music
Golden Rectangle and the number the golden ratio Ф (phi).
Formula
s
for the n
t
h
Fibonacci
number
Formulas involving Ф
Relations among Fibonacci numbers
A discussion of recursion relationships
(!) Derive a formula for Ф using the quadratic formula
.
(!!)
Prove ratios of consecutive Fibonacci numbers approach Ф.
(!!!) Derive a formul
a for the n
th
Fibonacci number.
[The answer involves the square root
of 5
. You can just quote the relevant theorem on recursive relations. You'll see the words:
"characteristics equation
or characteristics polynomial
."
]
(4
)
Logic: The Foundations of G
ood Thinking
[Any elementary book on Mathematical Logic is a
good source.]
Truth Tables
: "and," "or," "not," "implies"
Q
uantification (uses of the phrases
"
for
all" and "
there
exists"
)
Axiomatic Systems
People:
Aristotle (who created logic),
Boole, Turing
, Russell, Godel
Direct proofs, indirect proofs, contrapositive, converse, tautology,
syllogism
.
Describe Russell's Paradox
Describe Godel's Incompleteness Theorem.
Describe the rule of inference called
Monus Ponens,
(!) Use truth tables to prove that
(A→
B)
↔
(not B → not A) is a tautology
and d
o the same
for [(R and P) →Q]
↔ (R→(P→Q)
(!!)
Discuss p
roofs via Mathematical Induction
and give an
example
.
(5
)
Pythagoras
and the Pyth
agoreans
–
Version A
The Pythagoreans as mystics: philosophy, music, and sci
ence
Maybe start with a bit on Zeno's Paradoxes
–
connect with the Pythagoreans.
(!) Prove t
he Pythagorean Theorem two specific
"e
asy ways"
See
http://www.cut

the

knot.org/pythagoras/index.
shtml
and do proofs #
2 and number and # 4 (which is the same as the proof in Mathematical
Universe, pages 91/92)/
(!!)
Prove:
Pythagorean Theorem
via
Euclid's original proof
(in many geometry texts)
[This
is a wonderful proof.
If you choose this, ver
ify with me that you've found
Eu
c
lid's
proof.
It’s here
http://www.jimloy.com/geometry/pythag.htm
for example, as proof #4.
]
(6
) Pythagoras and the Pythagoreans
–
Version B
The Pythagoreans as
mystics: philosophy, music, and science
.
Maybe start with a bit on Zeno's Paradoxes
–
connect with the Pythagoreans.
(!) Find
a formula which generates
(produces) Pythagorean Triplets,
that
is triplets of whole numbers (a, b, c) such that a
2
+b
2
=c
2
.
Here
are two such
examples:
(3,
4,
5) and (5, 12, 13) and then show that this formula indeed always produces
Pythagorean Triplets.
[Most elementary Number Theory books contain this.]
(!!) Show
that every Pythagorean Triplet
is of the form described by your
for
mula, above.
That is, show that if one is give
n a Pythagorean Triplet,
it is produced from your formula,
above.
[This is also in most elementary Number Theory books.]
(7
) Fermat's Last Theorem
and his Method of Descent
Histo
ry of the Fermat's Last The
orem
, including the history of it
s final solution.
(!!!)
Prove:
There
is no solution to a
4
+b
4
=c
4
using Fermat's
Method of Descent
.
[Actually,
many sources prove that a
4
+b
4
=c
2
has not solution,
so that the sum of two fourth powers
cannot even be a square
,
let alone fourth power.
[Y
ou can and should assume the
Theorem
on Pythagorean Triplets
(
numbers
x, y, z
that satisfy x
2
+y
2
= z
2
.
)
You can assume the
formula that characterizes Pythagorean
Triplets
. This is in most elementary Number Theory
books.
]
(8
)
The Square Root of 2 is Irrational, using
F
ermat's Method of Descent
A bit about Fermat (including, maybe, a statement and example of Fermat's Little Theorem)
and some other work he did.
)
(!!) Prove: The square root of 2 is irrational, using Fermat's
Method of Descent
. [See "Math
Univ
erse
" pages 206 to 208.]
(9
) Solving
Polynomial
Equations
with one unknown
: The Theory of Algebraic Equations
It's
one of the great mathematical problems of Greek Antiquity
.
A bit about t
he situation for
d
egree 1, 2, 3
,
4 and then degree 5
Contributions by different cultures: Greek, Western, Arab, Indian
The History: Babylonians, Euclid, Arabs,
Omar Khayyam
,
Al

Kashi
,
Niccolo Fontana
(Tartaglia)
,
Girolamo Cardan
o
,
Francois Viete
,
Lagrange
,
Gauss
[Fundamental Theorem of
A
lgebra],
Niels Henrik Abel
[a major player]
,
Evariste Galois
[the
"
winner
"
].
(!) Derive the solution
for degree 2.
[This is in many high school algebra books.]
(!
!
) State the solution for degree 3.
[A piece of this [one of the roots] is g
iven in "Math
Un
iv
erse
" chapter Z.
You could discuss this chapter a bit, if you wish.]
(!!!
!
) Derive the solution for degree 3
(10
) The Mathematics
of Motions and Cosmology:
Kepler
and Conics
A brief history of Cosmology: Ptolemy, Copernicus, Brahe, Kepler
The Conic S
ections: Parabola, Ellipse (and the special case of circle), and Hyperbola
.
Tell how an ellipse is made with two pins, a piece of string, and a pencil.
Discuss Kepler's Three Laws of Motion.
Give the algebraic equations of the parabola, ellipse and hyperbo
la.
Describe the reflective properties of the Parabola
, and give an application
(so that you need
to discuss the role of the "focus" of a parabola).
(!) Graph the ellipse x
2
/(3)
2
+ y
2
/(4)
2
= 1.
[Just to be clear, this is the same equation as
x
2
/9 + y
2
/1
6 = 1. When you research this problem, you'll see why I wrote 9 as 3
2
and 16
and 4
2
.
(!!) Prove the reflective property of the ellipse (there is a nice
, geometric, proof that does
not
use calculus.
This proof is hard to find and I can show it to you if y
ou need to see it.
Here
is a copy of the argument. It’s sort of terse, so you can ask me about it.
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