OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
March 8, 2006 Vol. 4, No. 10
http://www.math.hope.edu/newsletter.html


You can take a look at dynamical systems in tomorrow's colloquium
In tomorrow's colloquium, Anne McCarthy from Northwestern University will speak on "Group Actions on the Circle: A Dynamical Perspective."  She will give an introduction to both group actions and the study of dynamical systems.  She will then discuss how the methods of dynamical systems can be used to classify actions of discrete groups on compact spaces by looking at an example on the circle.




  Tea at 3:30 in VWF 222 (Reading Room) on Thursdays before colloquia

As part of our colloquium series, the mathematics department will host a "tea time" in the Reading Room (VWF 222) at 3:30.  If tea isn't really your cup of tea, have no fear -- we'll provide some other beverages and snacks, too.  So please join us for a little food and fellowship before you go to the colloquia.  It'll be a great time to chat with the speaker, your professors and other students. 


The 30th Lower Michigan Mathematics Competition will be held soon

The 30th Annual Lower Michigan Mathematics Competition will be held at Hope this year on Saturday, April 1---no foolin'.  Students from colleges and universities in Michigan will gather here to challenge themselves on 10 interesting problems, working together in teams of up to three people. The competition runs from 9:30 a.m. to 12:30 p.m. After the problem session in the morning, there will be a break for lunch followed by a solutions session in the afternoon. Registration is free and students in Math 131 and up are encouraged to participate. Interested students may sign up individually or in teams. The deadline for registering is Thursday, March 16. 

Hope has a history of strong showings at the LMMC, including several championships, and we'd like to regain the title this year and bring the Klein Bottle Trophy back to Hope!


From the Margins of My Textbook
A Math Major’s Musings 

By JENNICA SKOUG | Essayist

“Pure mathematics is, in its way, the poetry of logical ideas.” –Albert Einstein, a reflection on the death of mathematician Emmy Noether

During my sophomore year of high school, the math teachers were occupied in designing and implementing a new portion of the math curriculum.  It was to be an advanced course for students who weren’t particularly fond of math.  “Math for poets,” they called it.  It seemed to imply that math and poetry are oil and water disciplines: they don’t mix.  It took me a long time to get over this.  When I did, it was somewhat of a sluggish revelation, and I believe it grew from the “Mathematical Poetry” contest that was held my freshman year.  (Refer to Dr. Cinzori. He took second place.)  This somewhat strange event grew to an understanding of how mathematics is in fact equivalent to poetry.  They emulate each other in both creation and presentation. 

The purpose of both disciplines is, at a fundamental level, to strip the observable world (existent or imaginary) not just to nakedness, but to its very bones, in hopes of exposing the most essential parts of its structure.  Furthermore, each seeks to manipulate this structure, to see it bend and twist in a way no one has yet imagined.  The only difference is that, as building material, mathematicians prefer numbers (and their various substitutes), while poets prefer words.  During the first week of topology this semester, one of the other students in class said this: “The thing I love about math is that in any definition or proof, you are always striving to include just enough information, but never too much.  It’s very exact.”  She was right, I think.  Math as well as poetry, says author K.C. Cole, is “a way of taking a big idea and condensing and honing it until it communicates exactly the right information.”  In a poem, as in a proof, each word or turn of logic is vitally important.  The world contains many complicated problems, be they intuitive or scientific.   These problems double or triple in complexity (and often in delusion) when coupled with all the unnecessary baggage we tend to load on.  Math and poetry are wonderful distillers of such luggage – they move information from quagmire to quintessence, and leave you with only a carry-on.  The trick is that you fill it only with the very best, that which is paramount to your understanding.  

Sometimes, poetry comes in a smooth, rhyming mold, and leads the reader directly to a not unexpected ending.  Yet some of the best poetry does not always give you what you might expect.  If it did, you would write it yourself.  The trick is to leave the reader with some new idea they weren’t anticipating when they came.  It must be read several times before its real meaning becomes clear.  Pieces of logic are smattered across the page and come together at the end in a whirlwind of conclusion.  The reader presses through the beginning in good faith, and once at the end, goes back to re-puzzle the beginning; that is, once they have gotten over the inevitable whiplash.  But upon leaving the poem, the reader commands it with an audacity that suggests they might have produced it themselves, if only they had had the time. 

So it is with math.  Ironically, “math – that most logical of sciences,” says Cole, “shows us that the truth can be highly counterintuitive and that sense is hardly common.”   Infinite series come to mind. Here, it is possible to add up billions upon billions (etc.) of numbers and find that the total value of your sum is, for example, one third.  Or perhaps “imaginary” numbers make you tick.  If this is not a highly poetic idea, which the casual observer might dismiss with derision but the devoted student digests with fortitude (and a little extra stomach acid), then I am not sure what is.  An elegant and powerful proof, like a poem of its same kind, has a certain grace and flow which, although it may have taken weeks or years to formulate, meets the reader as if it were a casual, Sunday afternoon entertainment.  Plutarch describes Archimedes’s determination of circular area in just this way: “No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required” (Dunham).  Certainly, Archimedes was no John Doe of mathematics.  But he did not stand alone.  Mathematicians throughout history have taken this path of unexpected beauty – “the poetry of logical ideas.”

The beauty comes in the understanding, and it startles you. 


Problem Solvers of the Fortnight

Congratulations to James Daly, Mary DeYoung, Matt Paarlberg and Megan Patnott for wending their way through, or rather across the surface of, the box problem.  The number of ways to get from one corner of the 2x3x4 box to a corner diagonally opposite is 372.  There are many ways to think about this problem, but one interesting tack involves cutting the box, laying it flat and counting the number of ways to get to each vertex.  Following this approach, you'll find Pascal's triangle -- or at least a portion of it.  Hmmmm!


Problem of the Fortnight

Three 1x1 squares are joined to make a figure -- call it L.


The figure L

For which positive integers m and n is it possible to tile an mxn rectangle with copies of L so that the copies do not overlap or extend beyond the rectangle?

Write your solution on an L-shaped piece of paper and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office by 3 p.m. on Thursday, March 16.



Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu

Dear Friends,

Next week Tuesday (3/14), math fans from around the world will be celebrating Pi Day.  I don't think we get classes off for that day, but hopefully you can find something to do that will make the day more special (like working a few extra homework problems or trying to tackle the problem of the fortnight.)  

If doing math problems is not your idea of celebrating, you can send a loved one a Pi Day electronic greeting card.  Just go online to http://www.123greetings.com/events/pi/ and chose one of their “interesting” (but not entirely mathematically correct) cards.  It you want to show our love for Pi Day for all to see, you can purchase temporary tattoos that celebrate Pi Day from students at Pepperdine.  You can view their designs at www.math.pepperdine.edu/~kkillpat/MathEvents/mathevents.html.  If you enjoy singing, you can find some lyrics to Pi Day songs at http://www.winternet.com/~mchristi/piday.html.

How ever you choose to celebrate Pi Day, I hope you have a great time, but do so responsibly.  For beware of the ides of March!

I didn't have any questions for this fortnight.  I hope you have some soon, or I will have to stop coming to work everyday!  You can email questions to me at elvis@hope.edu.




Everything should be made as simple as possible, but not simpler.
Albert Einstein (1879-1955)