Math 44
Fall 2009 Home Page
Green Sheet
Class 20, Tue., Dec. 1, 2009
We heard/saw projects. Thursday's class is for review, bring
questions!
Class 19, Tue., Nov. 24, 2009
We went over several handouts - please finish the handout
on "web" diagrams and chaos.
Projects are due Tuesday. Projects should include a written report (600
words), one report per project. In the report you must describe - in
detail - what each participant did, what course mathematics was
involved and how it was used, what steps you took to create the
project, and must include any references. Refer to the final
project
form for more information about the projects.
Class 18, Thu., Nov. 19, 2009
Here is the link to the Math 44 google
group.
You are all members already under the email address which I have for
you. But you must still register with Google. Go to the group site and
they will ask you to register, if you have not already. Please use it
to work on the take-home exam.
We did a "body" symmetry activity in which we constructed the table of
"compositions" of four planar symmetries:
T for Translation (or slide),
G for Glide (also called glide reflection),
M for Mirror (or reflection), and
R for Rotation (180 degree turn).
Here's the table we constructed:
|
T
|
G
|
M
|
R
|
T
|
T
|
G
|
M
|
R
|
G
|
G
|
T
|
R
|
M
|
M
|
M
|
R
|
T
|
G
|
R
|
R
|
M
|
G
|
T
|
We also worked on the take-home exam, and did a fractal dimension
handout.
Class 17, Tue., Nov. 17, 2009
We went over fractal dimension and some polyhedra calculations.
If you did not turn in your project form Tuesday, you MUST turn it in
Thursday. If I returned it to you for corrections, please make them and
turn in again on Thursday.
Do homework through section 6.5.
Turn in chapter 5 homework on Thursday.
Class 16, Th., Nov. 12, 2009
We went over more material on fractals.
We also worked on final projects. Please turn in your final
project
form, on Tuesday.
I emailed your take-home exam to your email address. Let me know if
you did not receive it.
Complete section 6.4 homework.
Class 15, Tue., Nov. 10, 2009
We went over material on fractals and cellular automata.
Please complete handouts given out in class and bring back to class.
We also began talking about complex numbers.
Do homework for sections 6.3 and begin homework from section 6.4.
Here is your final
project
form, for the project which is due the last week of class.
Fill out the form and turn in Tuesday.
Class 14, Thu., Nov. 5, 2009
We heard reports, saw more of the Martin Gardner video, and began
chapter 6.
Do homework for sections 6.1 and 6.2.
Class 13, Tue., Nov. 3, 2009
We went over graph theory material.
Your papers are due on Thursday; be prepared to give a short oral
summary of what you reported on.
Class 12, Thu., Oct. 29
W worked on the Poinsot
Stars
handout, due Tuesday.
We also saw some video of the Gathering For Gardner, learned how to tie
a knot without letting go of the ends, and made some human knots.
Please do the rest of the homework from section 5.4.
Your papers are due next Thursday, Nov. 5, see below.
Class 11, Tue., Oct. 27, 2009
We went over the modular arithmetic handout. Complete the Poinsot
Stars
handout and turn in Thursday.
We experimented with paper Mobious strips.
We also began section 5.3 on graph theory. Do homework for section 5.3.
Class 10, Thu., Oct. 22, 2009
We went over Patterns
and
Modular
Arithmetic, which is to be turned in on Tuesday.
Bring the Poinsot
Stars
handout on Tuesday, we will start that as well.
We also built many polyhedra, and just began chapter 5. Please begin
the problems for sections 5.1 and 5.2.
I will take up Chapter 2 homework on Tuesday, which is long over due! I
will also take up chapter 4 homework next Thursday (we did not cover
all
sections of chapter 4).
Your pentomino tiling, with symmetries identified, is due on Tuesday
also.
Use this handout
showing
you
how
to
find
the
symmetries
in your tiling to understand
how to complete it.
Your next paper is due Thursday, Nov. 5. For your next written report,
you are to report on a mathematical subject connected to this course,
that has caught your interest. I suggest that you read a chapter or two
from one of Martin Gardner's books on mathematics, he covers a wide
variety of topics. Here is a list of a few of the 70 books by Martin
Gardner. Choose a chapter (or two) that is of interest to you and
report on it:
The Colossal Book of Mathematics
Mathematical Circus
Mathematical Magic Show
The Magic Numbers of Dr. Matrix
Mathematical Carnival
Knotted Doughnuts and Other
Mathematical Entertainments
Wheels, Life, and other Mathematical
Amusements
Time Travel and Other Mathematical
Bewilderments
You might also want to look at books by Ian Stewart, Keith Devlin, or
Ivars Peterson.
Please check out one of the books and sign up to do a report on one
chapter. You can email me which chapter you want to do a report on, if
you have any question. This will be a 600 word paper, as before. The
main criteria for how you select which chapter to report on is that it
should be of interest to you. You will also give a brief 2-3 minute
oral report on what you learned to the class, if we have time, on Th.,
Nov. 6.
Class 9, Tue., Oct. 20, 2009
We heard the mathematician biography reports. Here is an article
published in yesterday's New York Times about Martin
Gardner, in honor of his 95th birthday.
We also learned a little more about polyhedra. Please do the homework
problems in section 4.5.
Also, prepare to turn in the pentomino tiling. You should show the
symmetries in your tiling. Use this handout
showing
you
how
to
find
the
symmetries
in your tiling!
Class 8, Thu., Oct. 15, 2009
We went over symmetry handouts. Please complete
Rotational
symmetry
handout.
4-Fold
rotational
symmetry
handout.
and turn in on Tuesday.
Your mathematician bio is due on Tuesday. Here is the list showing who is assigned which bio.
The reference books, Mathematical
People and More Mathematical
People is on reserve in the campus library. The report will be
600
words, and will also involve a short 2 to 3 minute oral report. Include
in your paper some response by yourself to the person - is this someone
you might like to have a conversation with, or is it someone who does
not
seem very interesting to you? Your main reference must be a printed
source,
not a web site, and you must cite any sources you use.
Please visit Scott
Kim's homepage and look at the many examples of Inversions,
some from his book by the same name, some animated. Recently other
artists have produced these letterform images that exhibit symmetry,
for example here is a site
for
ambigram
tattoos by Mark Palmer. The artist John Langdon has also been
creating Ambigrams for many years, and produced those used by Dan Brown
in his
book Angels and Demons. See Langdon's site for examples of
his work, or look for his book Wordplay.
Here's
a
site
by
the
mathematician
and
artist
Burkard
Polster with lots of
playful examples.
Symmetry leads us to the field of "tilings" or "tessellations." There
is an enormous amount of recent mathematics and art on this
subject. Take a look at David
Eppstein's enormous collection of links on tilings. He calls his
collection of such links the Geometry Junkyard;
it's
very
entertaining,
in
a
mathematical
and
arts sort
of way!
Here's a simple
example
of
how
the
RSA
code
works.
You have an assignment to choose a pentomino and use it
to create a tiling of the plane, then find the symmetries in your
tiling.
You might also color the tiling in an interesting way! Indicate
on
the
tiling
exactly
where
the
symmetries
are. Here's a handout
showing
you
how
to
find
the
symmetries
in your tiling! Here are
hundreds of links
to some amazing tiling sites. Pentominoes each have
five identical unit squares, linked complete edge to edge. For example,
we might try to solve the same problem for the "hexomino:" there are 34
hexominoes, each having six squares. Here is a diagram
indicating
briefly
how
all
34
hexominoes
tile
the plane. These are
not shown in complete detail; your penomino tiling should extend
further
than these, and indicate how the pattern you found really could extend
in all directions throughout the plane.
Class 7, Tue., Oct. 13, 2009
We went over material on symmetry and you took some handouts home -
please complete them and bring back to class. For those not in class
print out and bring to class:
Rotational
symmetry
handout.
4-Fold
rotational
symmetry
handout.
Please visit Scott
Kim's
homepage and look at the many examples of inversions.
Bring the Poinsot
Stars handout, we'll do some work on it in class.
Here's a simple
example
of
how
the
RSA
code
works.
Here's a list
of
"distributed"
mathematical
projects now being conducted
throughout
the internet.
Here's the "World
Community
Grid" site listing many online projects in which you (and your
computer)
can participate!
Here's another site for "Volunteer
Computer
Grids."
Here's an online article by
Ivars
Peterson on drivers license codes.
Here's Joseph
Gallian's
site on check digits in codes.
Read sections 4.3 And 4.4 and begin homework from those sections.
Class 6, Thu., Oct. 8, 2009
We went over sections 4.1 and 4.2. Please do homework for these
sections. Sorry this update is late.
Class 5, Tue., Oct. 6, 2009
We have first exam on Thursday.
We went over commercial codes that use modular arithmetic, and some
homework about prime numbers.
Here is the list of mathematician bios,
showing
who
is
assigned
to
whom.
Due
Tuesday, Oct. 20.
Work on homework from chapter 2.5 also.
Class 4, Thu., Oct. 1, 2009
We went over Fibonacci numbers, and explored a pattern
demonstrating that the sum of the squares of two consecutive Fibonacci
numbers is another Fibonacci number. We completed and turned in the Modular
Arithmetic
Intro
handout.
We also looked at prime numbers. Here is a 2 minute slightly nutty but
mathematically accurate music video
(professionally produced by Nova) of SJSU's Dan Goldston's discoveries
about the Twin Prime Conjecture.
Here are some links to Fibonacci sites, it is required that you (at
least) take a look at them. The second and third have lots of pictures!
A
great
site
about
Fibonacci
numbers.
Here's another Fibonacci
site
with
lots
of
pictures
and
interactive
applets.
Here's an interactive site
that helps explain phyllotaxis, which is the pattern of spirals in
many plants. Within the site is a short film clip that is part of the
explanation as to why the Fibonacci numbers appear in plants.
Here is a study
guide
for
the
first
exam
(next Thursday) from a previous quarter.
We haven't covered all topics yet, and we've covered some that are not
here, but this will give you a start. First exam is individual, open
book, open notes, calculator but not computer allowed.
Here is a list
of
the
contemporary
mathematicians you will report on Oct. 20.
Everyone will be assigned one mathematician on Tuesday, so you might
want to look up some info on them first.
Homework:
Work problems in section 2.3 and 2.4. Section 2.4 is about modular
arithmetic, which we've already studied, and its application to codes.
The list of homework problems is at the bottom of this page.
Here is the Fibonacci
number
assignment, turn in Tuesday as a "quiz." The first half of
the quiz was your participation in counting the number of spirals in
the pine cones; everyone in class received full credit. Make sure that
the plant you choose exhibits spiral patterns, not just one of the
Fibonacci numbers.
Please print out this handout, and bring them to class on Tuesday:
Patterns and Modular Arithmetic
Class 3, Tue., Sep. 29, 2009
We went over new material from the text on the pigeonhole
principle,
and also the Fibonacci numbers (sections 2.1 and 2.2).
We also worked on modular arithmetic and the take-away game.
In the pigeonhole principle "magic trick," I asked you to choose seven
numbers
from the list 1,2,3,...,12. The properties each of your lists had were:
(1) A pair of your numbers had a sum of 13.
(2) A pair of your numbers had a difference of 6.
(3) A pair of your numbers had a difference of 3.
(4) A pair of your numbers had the property that their only common
factor
was 1.
(5) A pair of your numbers had the property that one divided the other
equally.
We saw how the pigeonhole principle explained why properties 1,2, and 3
are
true. For Thursday, can you use the pigeonhole principle to explain
property
4? Remember, it's all in how you label to (six) pigeonholes!
Property 5 is more difficult to explain; in this case the six
pigeonholes
have different numbers of numbers assigned to them!
We also went over the "water pouring" problem, and saw how to solve it
using
the billiard ball technique. Here's a site
where you can get free software for implementing the billiard ball
approach
to the water pouring problem. You
have
to
download
the
"Mathematica
Player"
(also
free) first.
Due Thursday: Homework problems 1-15 from chapter 1, to be turned in.
Also due Thursday: work all problems in this Modular
Arithmetic
Intro
handout, to turn in.
For Thursday, also work on problems from chapters 2.1 and 2.2 (see list
at
bottom of this page).
Please print out this handout, and bring them to class on Thursday:
Patterns
and
Modular
Arithmetic
Class 2, Thu., Sep. 24 2009
We went over several of the problems at the end of chapter 1. (See
the list of homework from the text for the whole quarter at the bottom
of
this page.) Complete for homework by Tuesday, but will take up next
Thursday
(you'll have other assignments to complete by Thursday also, so we need
to
be done with them by Tuesday.
Also due Tuesday is your Math Autobiography. See page 2 of the green
sheet for a complete description.
We also began to discuss modular arithmetic.
Class 1, Tue., Sep. 22, 2009
Had a substitute, Hassan Bourgoub, who played the pattern game and
did the frogs on a log problem.
Assignment: Read chapter 1 and work problems 1-15 at the end of chapter
1.
(1) Get your textbook! (See green sheet.)
(2) Read Ch. 1 and begin working on problems at the end, though I won't
collect them yet. You should do them
by the end of week 2: #1-15, pg 28-32.
(3) We worked on the pattern game and associated problems.
(4) We also worked on the "Frogs on a log" problem, and found that with
1 frog per side it took a minimum of 3 moves to exchange places
2 frogs per side it took a minimum of 8 moves to exchange places
3 frogs per side it took a minimum of 15 moves to exchange places
You might have guessed that 4 frogs per side would require 24 moves, 5
frogs per side would require 35 moves.
Figure out how to do these exchanges also, in the minimum number of
moves. (See below for more explanation.)
Three round frogs (O's) and three crossed frogs (X's)
are sitting on a log with seven spaces, and want to take each other's
places. A frog can move one step to a vacant square, or jump over one
neighbor to a vacant square. In class we learned that it can be done in
15 moves.
Two frogs on a three space log can take each other's places
in three moves:
Four frogs on a five space log can change places in 8
moves, as we also discovered in class.
Here's a list of all the problems from the text book for the quarter.
There might be some revisions, and there will be other additions.
Ch. 1 #1-15, pg 28-32.
Ch. 2.1 I: 4,6,9,10-12,14-18,21
Ch. 2.2: 1-7,16,18,21,24,26
Ch. 2.3: 1-3,6,7,13,14,19,20,22,37,39
Ch. 2.4 #3,4,5,7,9,19,28,29,30,32
Ch. 2.5: 3,6,7,8,11,15
Ch. 4.1 #2,11,12,13,15,18,19
Ch. 4.2 #5,7,9,11,21
Ch. 4.3 # 1-8
Ch. 4.4 # 1-5
Ch. 4.5: 11,12,15,16,21
Ch. 5.1: 4-6, 12,14,16,22,25,26,36
Ch. 5.2: Do any three of 8,9,10,13,14. Also do 18,19,22, any one of
26-28, 37
Ch. 5.3: 2,4,5,7-10
Ch. 5.4: 1-5,13,16,19,20
Ch. 6.1 1,2,3,5,7,8
Ch. 6.2 3,12-23
Ch. 6.3 #1,2,4,5,9,13,21,23
Ch.
6.4 #2,3,6,8,12-15
Ch.
6.5 1-10,
26-28,38-40
Ch. 6.6 #1-6,10-12,15