Math 44, Spring 2009 Home page
Green sheet
Class 21, Th., June 18, 2009
We had final projects.
Here's a sample
problem list from a previous quarter. Yours will be similar.
Here's another list
of sample problems.
Class 20, Tue., June 16, 2009
Worked on a fractal dimension handout.
Also made some simple polyhedra with loops of string.
Projects are due on Thursday.
Class 19, Thu., June 11, 2009
We went over more material on fractals, this time on fractal dimension.
Class 18, Tue., June 9, 2009
We went over the cellular automata handout, also went
over more about "chaos," and what are also known as dynamical systems.
Papers due tomorrow.
Do HW through ch. 6.5, we'll finish ch. 6 tomorrow.
Class 17, Thu., June 4, 2009
We went over tesselations - turn in your pentomino tesselation on
Tues., and show all symmetries.
We went over the cellular automata - complete and hand in Tuesday.
We also went over material from section 6.2-6.4. Do homework through
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.
Your next paper is due next Thursday, see below for description.
Class 16, Tue., June 2, 2009
We received handouts on cellular automata. Exams are due Th.
Class 15, Thu., May 28,
2009
I've started a Google groups discussion group for the class. If
you are on my official registration list, you received an invitation at
the email address you listed on your registration form. If not, email
me and I'll send you the invitation.
We spent class time on the take-home exams. Class also received
a handout to complete for next class, introducing the notion of fractals.
Complete the chapter 5 homework. Begin chapter 6 and do the homework
for section 6.1.
Here is your final
project form, for the project which is due the last week of class.
I will ask you to fill out the form in the next week. Filling it out completely
is part of your project grade (15%). I will also email everyone in the
class list the form as an MS Word file, so you can more easily fill it
out. Please bring to class on Tuesday and we will discuss.
Due Tuesday: An element of the tiling assignment is to find
all the symmetries in your tiling. We observed four kinds of symmetries
in class: Translation, Rotation, Reflection, and Glide. Label each symmetry
you found in your tiling. Here is a handout
showing you how to find and display the symmetries in your tiling.
For those of you wanting more info on pentominoes, here is a standard intro, with pictures
of all 12 pentominoes. Your homework from two classes ago was to "Choose
one of the pentominoes, make a tiling of the plane, find the symmetries,
and bring to class! " 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.
Your next written report is due on Thursday, June. 11. (The take-home
exam will be due a week before that on Th. June 4t.) 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 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., June 11th.
We have gone over a variety of graph theory concepts, including
Euler Circuits and Hamiltonian Cycles (these are not in your text). An
assigned reading is this site
which has excellent explanations of all these graph concepts, with links
to good diagrams. Please do these
Euler
& Hamiltonian circuits and cycles problems for Tuesday's class.
We also looked at the "Small World Phenomenon,"
and "six
degrees of separation,"
Here are some more links:
Six degrees
of Kevin Bacon.
Oracle of Bacon, where
you can look up actor's links to other actors.
Kevin Bacon's charitable SixDegrees.org.
The Erdos-Bacon
number site.
The Facebook
Six Degrees site.
The Erdos Number project.
Class 14, Tue., May 26, 2009
We went over material related to symmetry, including how
the four planar symmetries, and how to combine them. We worked on knots,
made "human" knots, learned the knotted string trick.
Class 13, Thu., May 21, 2009
Here is take-home exam, and here is corresponding fraction/take
away game list. We also went over material from chapters 5.3 and 5.4. Do
homework up to chapter 5.4. Turn in chapter 4 homework on Thursday.
Class 12, Tue., May 19, 2009
We heard biographical reports.
Class 11, Tue., May 12, 2009
Class 10, Th., May 7, 2009
Sorry this is late.
We went over new material from chapter 5, and also more on symmetries
in the plane. Here is the four-fold
rotation "geoboard" handout. I'll delay turning in things that were
to be due tomorrow (Tuesday) since this note went up late. But we will
move ahead with the material in the text. You should still have
chapter 4 homework on Tuesday, since that was assigned last week.
Class 9, Tue., May 5, 2009
We learned about
polyhedra, and introduced chapter 5 in section 5.1.
Homework: sections 4.4 and 4.5
Be prepared to turn in chapter 4 homework on Tuesday.
Class 8, Th., Apr. 30, 2009
We worked on symmetry. Try to write your name using either
rotational or reflection symmetry and bring the result to class, we'll
look at them on Tuesday.
Here is the list of bios
for the next paper. I'll email interviews to several of you.
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.
Here are here are handouts on rotational
symmetry and reflection
symmetry, do as much as you can we will probably complete and hand
in next class.
Bring the Poinsot
Stars handout to class and we will go over it on Tuesday.
Here is the first
exam. Correct any problem you missed, not just by choosing the
correct answer, but by writing an explanation in words and symbols as
to why that answer is correct. If you do all you missed correctly, you
will get back half the points you missed on exam 1. You must turn this
in next Thursday, May 7. You must also turn in your scantron form - you'll
get it back!
Do the chapter 4.4 HW.
Class 7, Tue., Apr. 28, 2009
We had exam 1, and began to look at symmetry.
Please visit Scott Kim's
homepage and look at the many examples of inversions.
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!
Read section 4.4 and begin working
on the 4.4 homework.
Here is the list of bios for the
next paper. I'll email interviews to several of you.
Class 6, Th., Apr. 23, 2009
We went over sections 4.2 and 4.3.
Do homework from sections 4.2 and 4.3 (list at bottom).
First exam on Tuesday, goes thru sections we've covered, up
to 4.3, will be partially a scantron exam.
Here is a study
guide for the first exam, from another quarter. There may be a
few problems that we have not gone over, and thus won't be on the exam!
Bring a half page scantron (kind with 50 questions per side, brown or
green) - most questions will be multiple choice.
On Tuesday, turn in ch. 2 homework, two mod. arithmetic handouts.
Class 5, Tue., Apr. 21, 2009
We went over chapter several homework problems and some
material on modular arithmetic and patterns. We went over error-detecting
codes in chapter 2.5, and began section 2.6. We also briefly began chapter
4.1.
We will skip chapter 3 and the rest of chapter 2 after section
2.6.
Do the homework from chapter 2.5 and 4.1. I will take up chapter
2 homework on Tuesday, so you have it to study for the exam.
If you are wondering where the homework assignments from the
text are, look at the bottom of this site! The horizontal line shows how
far we've gone in the text.
I will also take up the handouts
Modular
arithmetic intro
Patterns
and Modular Arithmetic
on Thursday as quizzes.
Class 4, Th., Apr. 16, 2009
We went over chapter 2.3 and more about modular arithmetic,
which is the subject of ch. 2.4. Also went over some of the chapter
1 problems.
Do the homework from ch. 2.3, and start on ch. 2.4 homework.
Here are some links to Fibonacci sites, please take a look at
them:
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.
Class 3, Tue., Apr. 14, 2009
We went over some problems from chapter 1, and also sections
2.1 and 2.2.
Start working on problems from sections 2.1 and 2.2 (see list
at bottom of site).
Be prepared to turn in HW from chapter 1, problems 1-15.
Also, can you figure out why the number of ways to add 1s and
2s to get a particular number is always a Fibonacci number?
Please print out these
handouts, and bring them to class on Thursday, will not collect them
yet:
Modular
arithmetic intro
Patterns
and Modular Arithmetic
Class 2, Thu., Apr.
9, 2009
We worked on problems from the text, especially 1-3.
We also worked on modular arithmetic, and also the take-away game.
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 Tuesday: Math Autobiography (see page 2 of the Green
sheet for a description.)
Also, continue to work on problems 1-15 at the end of chapter
1, we start chapter 2 on Tuesday.
Please print out these handouts, start working on them, and
bring them to class on Tuesday:
Modular
arithmetic intro
Patterns
and Modular Arithmetic
Class 1, Tue., Apr. 7, 2009
(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
We 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 is an animation
of the billiard ball water pouring method for solving the water pouring
problem. You can actually download the program and run it on your computer
(it takes a special Mathematica player which you have to download and
install for free first). By the way, a puzzle like this was used in the
film Die Hard: With a Vengeance.
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