OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
April 14, 2004 Vol. 2, No. 13


Brandon Alleman to study in Budapest in the fall
Brandon Alleman, a Hope College junior majoring in mathematics and physics, has been selected to attend the Budapest Semesters in Mathematics program for the Fall 2004 term.  Through this program, mathematics majors in their junior or senior year may spend one or two semesters in Budapest and study with eminent Hungarian scholar-teachers. 

Brandon hopes to take number theory and algebra as well as one additional mathematics course in Budapest.  He will also take an intensive two-week language course before the actual semester starts and then a Hungarian history course during the semester.  While he is looking forward to learning mathematics at a deeper level, he is also looking forward to traveling.  "I have heard that Budapest is one of the most beautiful cities in Europe," Brandon stated.  "I am looking forward to traveling within Hungary and neighboring countries as well as experiencing another culture first-hand."   To make sure he gets the full experience of another culture, Brandon has applied to live with a host family.  "You have the option of living in either an apartment or with a host family," he said.  "I believe that, if there is a family that will take me, this will be an invaluable experience." 

Michael Rininger, a Hope junior majoring in mathematics, is currently studying in Budapest.  About living in Budapest he says,  "Budapest as a whole is a nice city and there is more to do here than in Holland."  He also said, "The Liszt Academy of Music is not far from me and has wonderful performances by groups from all over the world."   Michael has found the study of mathematics in Budapest to be challenging.  He has been studying with some excellent students and has found that the classes move along at a quicker pace than at Hope.  He also says, "The lectures here are quite good.  The professors stress the big picture while presenting proofs and whenever possible present the most beautiful proofs known."

If you want more information about the Budapest Semesters in Mathematics visit  http://www.stolaf.edu/depts/math/budapest/.


Congratulations go out to our graduating seniors

Fourteen mathematics majors are graduating this year.  A belated congratulations goes out to four of them that graduated last December.  They are Ben Freeburn, B.J. Schoettle, Mark Thelen, and Chris VanSlooten.  The remaining ten that are graduating this semester are Leslie Colburn, Martha Graham, Sarah McEvoy, Jason Mejeur, Ben Onken, Jonathan Sedon, Justin Shorb, Sara Tatge, Andrew Kiel, and Nathan Kooistra.  Our best wishes go out to all of them as they leave Hope College to pursue employment or graduate school.


Sixteen Hope students participate in the LMMC

Hope College had a fantastic showing at this year's Lower Michigan Mathematics Competition held on Saturday, April 3, 2004 at Kalamazoo College.  Hope was represented by 6 teams consisting of 16 students -- the largest group of students from any college.  While one of Hope's teams finished first (in terms of time), the results of the competition are not know yet.  Congratulations go out to all of the students participating.  They are Daniela Banu, Stefan Coltisar, Heidi Libner, Brandon Alleman, Andrew Wells, Henry Gould, Natalie Hoogeveen, Shova KC, Utsab Khadka, Jennica Skoug, Mike Cortez, Brad Houzenga, Brandon Hazen, Luke Olinyk, Ben Bradley, and Jen Kasten.


April is Mathematics Awareness Month: Networks

. . . And has been since 1986 (in case you weren't aware) when President Reagan proclaimed April as the month when we should remind ourselves "of the importance of this basic branch of science to our daily lives."  The theme this year is the mathematics of networks, a timely focus given the rising importance of computer networks, neural networks, and even social networks in our day-to-day lives.  Visit http://mathforum.org/mam/ to become more aware of this fascinating area of mathematics!  Also, check out the neat display about Mathematics Awareness Month on the bulletin board in the hallway on the second floor of VanderWerf!


More summer research in mathematics

In the last newsletter we mentioned a number of students that will be doing research in mathematics this summer at Hope.  There are three more that we can add to that list.  Megan Scholten, Tara Baase, and Erica Pagorek will be doing research with Prof. Mary DeYoung on trying to find the most effective way to learn the mathematics needed to become an elementary school teacher.
 

It is spring and the Greek letters are in bloom

As the end of the academic year draws closer and closer, a number of students will be receiving awards in the next couple of weeks.  Next Monday, April 19, Chris VanSlooten will receive the Sigma Xi award given to science and mathematics students that maintain a high GPA and do scientific research.  Next Sunday, April 18, mathematics majors Jon Sedon and B.J. Schoettle will join the Zeta of Michigan chapter of Phi Beta Kappa, the nation's oldest scholastic honorary society. 

Many students were recently invited to join the Michigan Delta chapter of Pi Mu Epsilon.  The purpose of this national mathematics society is to promote scholarly activity in mathematics among the students in academic institutions.  An induction ceremony will be held tomorrow, Thursday, April 15.  Those that will be joining are Brandon Alleman, Michael Cortez, W. Nicholas Sumner, Peter VanWylen, Martha Graham, Megan Noll, Troy Schrock, Sara Jongekryg, R. Ross Coleman, Carol Guess, Eric Jongekryg, Chad Kettner, Heather Nentzer, John Siehling, Lindsay Townsend, Elizabeth Jury, Erin McNary, Megan Praamsma, Kendra Slotten, Lisa Wisniewski, Elizabeth Zylstra, Jill Pinter, Heidi Dykema, Jennifer Folkert, and David Pridmore.


Problem Solvers of the Fortnight

Twelve of you rolled around in your minds our problem of inscribing six circles in an annulus of outer radius one and unknown inner radius.  Ryan Weaver wrote his solution inside computer-drawn circles, and Utsab Khadka solved the more general problem of inscribing n circles in such an annulus.  Michael Cortez and Mike Nelsen used a hexagon formed by joining the centers of the six circles to help them solve the problem.  Lindy Babcock, Mark Dandero, Kristen Greshel, Brett Jager, Daniela Banu, and  Joe Kooyers (Hope class of 1965) employed a variety of geometric arguments to get their solutions.  Jennica Skoug wrote her solution on an annulus of paper, as did Kim Harrison, who also noted the similarity to hexagonal crystal lattices.  A round of applause for all our problem solvers!  Problem solvers are invited to stop by Dr. Catalano's office (VWF 209) to claim their certificates for one of our favorite circles -- a cookie from Java Joe's!  Our blue ribbon prize winner, drawn at random from among all the solutions we received, is Jennica Skoug, who gets a certificate for another of our favorite circles -- a pizza from Pizza Hut!  Congratulations to all!


Problem of the Fortnight

The equations 5 = 2 + 3, 10 = 1 + 2 + 3 + 4, and 12 = 3 + 4 + 5 show that each of the numbers 5, 10, and 12 can be written as a sum of two or more consecutive positive integers.  As a warm-up to our problem of the fortnight, consider the following questions: Is 18 equal to such a sum?  What about 16?  Now try your hand at our last problem of the fortnight for the year:

Which positive integers are sums of two or more consecutive positive integers?

Drop your solution in the box outside Dr. Pearson's office by 1 + 2 = 3 p.m. on Friday, April 23 = 11 + 12. Prize winners will be notified by email.


Mathography: Huygens and Noether

Christiaan Huygens, the famous Dutch astronomer, was born on April 14, 1629 and is one of the two mathematicians we highlight in this issue.  Perhaps best known for his work in astronomy, including the invention of a telescope, Huygens' interest in mathematics permeated his life's work.  At the ripe age of 22 he debunked Gregory of Saint-Vincent's attempt to square the circle, and his study of the compound pendulum was the first dynamical system ever studied.  Huygens is responsible for defining the involute of a circle, which we feature in this issue's Graphography.  See http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html to read more about Christiaan Huygens. 

Emmy Noether is a seminal figure in modern algebra and contributed much to the development of ring theory and ideal theory.  Her work on algebraic invariants informed several of the concepts involved in Einstein's theory of general relativity, a basic result of which is due to Noether's proof of a relationship between symmetries in physics and conservation principles.  Said Einstien, "In the judgment of most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.  In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians."  After the Nazis took over Germany, Noether accepted a position at Bryn Mawr College and lectured at the Institute for Advanced Study in Princeton.  She died suddenly on April 14, 1935, after an operation.  See http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Emmy.html to read more about Emmy Noether.


Graphography: Involute of a circle

As time ticks down on the remainder of the semester, we thought we'd feature a curve that was originally studied for its relation to pendulum clocks.  The involute of a circle is the path traced by a point on a straight line as it rolls around the circle.  Huygens first studied involutes when he was considering pendulums for clocks that might be used on ships at sea, and he used the idea of an involute in designing his first pendulum clock.  The parametric Cartesian equations for the involute shown in the figure are:
x(t) = cos(t) + t sin(t)
y(t) = sin(t) - t cos(t)
Check out the MathWorld article on involutes at http://mathworld.wolfram.com/CircleInvolute.html.





"How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?"
-- Einstein, Albert (1879-1955)