Math 44, Fall 2008 Home page
Green Sheet
Class 20, Th., Dec. 4
We reviewed, especially Poinsot stars, polyhedra, Euler circuits, by means
of string and finger geometry!
Watch this space for review sheet.
For more on the math/dance stuff I do:
Here's info on the performance
we're doing Sunday.
Listen to the interview with of Sept. 12, 2008, on The Math Factor, the podcast by the
Univ. of Arkansas' Chaim Goodman-Strauss.
Discoveries and Breakthroughs in Science
May 2008 story + video clip on Erik Stern and my work integrating math and
dance (click on "Do the Math Dance," under the mathematics heading).
Class 19, Tue., Dec. 2
We saw some wonderful projects!! Thank you!
We also went over the exams. I am going to post some further homework problems
shortly.
Thursday will be review.
Class 18, Tue., Nov. 25
We worked on final projects, which are due on Tuesday. Thursday is for review
for final.
We also went over the calculation of fractal dimension.
Here's a good explanation
of fractal dimension by Robert Devaney. Here's another.
Final text homework:
Ch. 6.6 #1-6,10-12,15
Class 17, Thu., Nov. 20
EXAM UPDATES:
Here's an update
to one of the problems, may help you understand it...
Here are
take-home problems 4-6.
NOTES: all the take-home problems, 1-6, are due this Tuesday!
Also, thanks Hannah for reminding me, the final exam date on the green sheet
is incorrect, the correct date and time for the final exam is Wed., Dec. 10
from 1:45 to 3:45 PM.
Your projects are due the week after next, which is why the project forms
are due on Tuesday!
We worked on take-home problems, due on Tuesday.
Please also bring final
project form on Tuesday, we will discuss more.
We went over more about fractals and chaos, from chapter 6.
Some new handouts, but they're not due till after Thanksgiving.
New homework, due after Thanksgiving:
Ch. 6.4 #2,3,6,8,12-15
Ch. 6.5 1-10, 26-28,38-40
Class 16, Tue., Nov. 18
We went over Conway's game of life, and some material on fractals.
Here's some material on the game of life:
Executable Java applet that will run the game of life on the web; you can
also download a free executable program: http://www.bitstorm.org/gameoflife/
Easy to understand background and explanation of the game of life:
http://www.math.com/students/wonders/life/life.html
We went over a variety of concepts related to chaos, fractals (see
this link to fractal galleries), and
"dynamical systems" (systems that change over time - here is a site with some neat animations).
Here is another site devoted to
fractals.
Here is a site with interactive
software for creating fractals and learning about chaos.
Here's a site for looking at
the Mandelbrot set.
Here is your final
project form, which is due the last week of class. I will ask you to fill
it out 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 Thursday
and we will discuss.
Homework:
Ch. 6.2 3,12-23
Class 15, Tue., Nov. 11, 2008
We worked on knots, made "human" knots, learned the knotted string trick,
and started learning about fractals.
Papers are due on Thursday.
Here are your first
three take-home problems. More soon. Here is a list
that goes with them!
Bring to class on Thursday, you will have time to work together on the problems.
New homework:
Ch. 6.1 1,2,3,5,7,8
Ch. 6.2 (We'll go over this section next week.)
Ch. 6.3 #1,2,4,5,9,13,21,23
Class 14, Thu., Nov. 6, 2008
We went over a variety of graph theory concepts, including Euler Circuits,
Hamiltonian Cycles, the so-called "Chinese Postman" Problem, and the Travelling
Salesperson Problem (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.
We also looked at the "Small World Phenomenon,"
and "six
degrees of separation," and began looking at knots.
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.
Homework from text:
Ch. 5.4: 1-5,13,16,19,20
Continued from last class: homework
problems on voting, and Benin
handout.
Do these homework problems also on Mobius
strips and Euler
& Hamiltonian circuits and cycles.
I also asked you, as homework, to find two "simple" graphs (undirected, and
no loops or multiple edges allowed) that are not the same, but that have the
same numbers of vertices, the same numbers of edges, and the same set of
degrees. This can be done with six or fewer vertices in a number of ways.
On Tuesday we will do more with knots, and begin the chapter on fractals.
I will also hand out the take-home exam, with due date updated to give you
two weeks to work on it, though I may post it here before that.
Class 13, Tue., Nov. 4, 2008
We went over electoral mathematics, in honor of today being election day!
Here are the homework
problems on voting; we did number 1 in class, please work out the problem
on the 1980 New York Senate race (and the other two problems) for Thursday.
The site I referred you to is FiveThirtyEight.
Here is a site
that explains the various voting methods we talked about today. Especially
look at the 3rd section of the site.
We also went over planar symmetries, so please complete the Benin
handout for Thursday.
Also, complete the Poinsot
Stars handout for Thursday, we went over it in class today.
Here is the link for Jeffrey Weeks'
site, he has lots of free downloadable software for creating tilings,
etc. Please try out at least two of his programs and be prepared to tell us
what you thought!
Class 12, Thu., Oct. 30, 2008
We heard the last two biographies, learned about four
symmetries in the plane (Translation, Reflection, Rotation, Glide), and
practiced them, then saw how they combine in a line. We also learned about
graphs of vertices and edges.
Due Tuesday: We also worked a little on the
four-fold geoboard rotation problem, please complete it and bring to class
to turn in on Tuesday.
Due Tuesday: We also looked at the symmetries of frieze patterns,
here are the frieze
symmetry slides I showed you, and here is the Benin
handout in which I would like you to determine which of the 7 symmetries
is shown in each pattern (some appear more than once, some do not appear.)
Please hand this in on Tuesday also.
For those of you who missed them in class, here are the handouts on rotational
symmetry, reflection
symmetry, and polyhedra;
for those who turned them in, I'll return them on Tuesday.
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, color it, 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.
Due Tuesday: A further element of this assignment is now 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.
Your next written report is due on Thursday, Nov. 12. (The take-home exam
will be due after that - it will be distributed soon.) 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
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 on Th. Nov. 13.
Bring the Poinsot
Stars handout to class and we WILL go over it on Tuesday.
New homework from text:
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
Class 11, Tue., Oct. 28, 2008
We spent most of the class on the bios - we'll finish on Thursday. Bring
handout
showing you how to find the symmetries in your tiling! Bring your pentomino
tiling. Bring Poinsot
Stars handout to class on Thursday.
The link to the handout
showing you how to find the symmetries in your tiling is now fixed!
Your report on a contemporary mathematician is due Tues., Oct.
28.
Here's the list of mathematicians
along with who is reporting on whom.
The reference books, Mathematical
People and More Mathematical People
should be 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.
Class 10, Th., Oct. 23, 2008
We went over the exam, counted the number of pentominoes (12), and
looked at how to tile the plane.
Homework: Choose one of the pentominoes, make a tiling of the plane, color
it, and bring to class! You might try to find all the symmetries
in the tiling, and indicate on it exactly where they 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.
Ch. 4.5: 11,12,15,16,21
Ch. 5.1: 4-6, 12,14,16,22,25,26,36
Also due Tuesday is your mathematician biography.
We also introduced topology (chapter 5), so read ahead into chapter
5.2. some of which we went over today.
By the way, here's a fascinating article on John Conway.
Just in case you also want to learn more about the person Grace is going to
report on!
Class 9, Tue., Oct. 21, 2008
We worked on polyhedra, and did some symmetry work also. Please bring
the two symmetry handouts to class on Thursday completed.
Please visit Scott Kim's homepage
and look at the many examples of inversions.
Also looked at slides of polyhedra and related mathematical
objects, including "DNA Origami". Here is Paul Rothemund's home page,
you can find much more on this subject there. You really should check
out some of the amazing web sites about polyhedra, for example George Hart's
Pavilion of Polyhedrality
(you may need to update your software to see everything.) Hart seems
to be channeling some kind of alien polyhedral knowledge.
The Math Forum site http://mathforum.org/alejandre/applet.polyhedra.html
has a nifty Java Applet that allows you to choose and rotate various polyhedra.
New homework:
Ch. 4.5: 11,12,15,16,21
Class 8, Thu., Oct. 16, 2008
We went over sections 4.2 and 4.3, and introduced symmetry. See the many
links at the bottom of this entry.
Do the homework for sections 4.3 and 4.4; also finish the homework for 4.1
and 4.2 if you have not yet done that.
Print out and bring the Poinsot
Stars handout, and try the problems. We'll do some work on it in class
also.
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 list of all the problems from the text book for the quarter. There
might be some revisions, and there will be other additions. New homework
is assigned up to ch. 4.4 for Tuesday. Some of 4.4 we still need to go over
Ch. 1 #1-15, pg 28-32.
Ch. 8.4: #1, 3, 5, 7, 10-15
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
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.
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.
Class 7, Tue., Oct. 14, 2008
We went over some of the newer material, or some things that had been left
out, for example, the Fibonacci number notation and patterns, the RSA code
(section 2.5, including "Fermat's Little Theorem"), the billiard ball solution
method to the pouring problems, and the Art Gallery Theorem (section 4.2).
Here is an animation of
the billiard ball water pouring method. 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.
Please have homework for chapters 1 and 2 completed for me to check on Thursday.
Remember that we skipped sections 2.6 and 2.7.
Also remember that we covered section 8.4, so you will have a voting question.
Here are some homework
problems on voting; we did number 1 in class, but work on 2,3, and 4.
Here are solution to the patterns
and modular arithmetic handout.
New HW:
HW Ch. 2.5: 3,6,7 (we did 6 and 7 in class!),8,11,15
HW: Ch. 4.1 #2,11,12,13,15,18,19
HW: Ch. 4.2 #5,7,9,11,21
I don't insist that you have HW for 4.1 and 4.2 done by Thursday, but I will
probably ask a question from 4.1 and one from 4.2 on the exam, like on the
practice test.
Class 6, Thu., Oct. 9, 2008
We went over the modular arithmetic homework. We also learned about
Fibonacci numbers. Do the Fibonacci
assignment to turn in on Tuesday. We will briefly go over the material
in ch. 2.5, then skip ch. 2.6. We will then go on to chapter 4. You might
want to read ahead to chapters 4.1 and 4.2.
Read the Where's Fido? handout and
work the other two problems in it.
Here is a study
guide for the first exam, from another quarter. There are a few problems
that we have not gone over, but we will on Tuesday.
Several more homework problems will appear here due Tuesday, check again
on Friday.
Class 5, Tue., Oct. 7, 2008
We went over voting homework from ch. 8.4, also chapter 2.1 to 2.3.
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
I'll check homework thru section 2.2 on Thursday.
Also be prepared to turn in the handouts
Modular
arithmetic intro
Patterns
and Modular Arithmetic
Here's the link to the site for the "Mersenne
primes" and perfect numbers search.
You can download the software and join the search.
Here's the SETI site, if you
want to help search for extraterrestrial intelligence!
Class 4, Thu., Oct. 2, 2008
We went over more chapter 1 problems, also the pigeonhole principle, an intro
to the Fibonacci numbers, and more about modular arithmetic.
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.
Finsh HW from 8.4, work HW problems for sections 2.1 and 2.2:
Ch. 2.1 I: 4,6,9,10-12,14-18,21
Ch. 2.2: 1-7,16,18,21,24,26
Also work the problems on these two handouts:
Modular
arithmetic intro
Patterns
and Modular Arithmetic
Here's that modular arithmetic powerpoint:
Modular Arithmetic Powerpoint slides
Be prepared to turn in the handouts on Tuesday.
Class 3, Tue., Sep. 30, 2008
We went over voting math problems, also some of the problems at the end of
chapter 1. Please do some more work for Thursday on the chapter one problems.
Problem #15 is very difficult, it's better known as the counterfeit coin
problem.
Here are more references to counterfeit coin problems:
http://www.cut-the-knot.org/blue/weight3.shtml
http://www.maa.org/mathland/mathtrek_2_16_98.html
http://www.cut-the-knot.com/blue/weight1.shtml
New homework:
Ch. 8.4: #1, 3, 5, 7, 10-15
Class 2, Thu., Sep. 25, 2008
We went over the pattern problems, also the "Frog on a Log" problem, and
did the Electoral College quiz.
Homework: Turn in your math autobiography on Tuesday (see Green Sheet). Also begin work on the
problems at the end of chapter 1, but I won't collect them on Tuesday.
I'll place more background here shortly on voting, check in again in a day
or two.
Class 1, Tue., Sep. 23, 2008
We played the "pattern game." We also worked on the "Frogs on a log" problem.
Homework:
(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) During class I had you create a 6 by 6 pattern, with columns numbered
0,1,2,3,4,5, and rows numbered 0,1,2,3,4,5. Figure out what will be in boxes
(50,0), (50,50), (100,0), and (100,100), and write an explanation of why
you think your pattern extends in the way you say. I plan to collect these.
(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.)
(5) If you want to read ahead, also read Ch. 8.4 on election methods, which
we'll talk about on Thursday.
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 showed how 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.