OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
November 17, 2004 Vol. 3, No. 6

Tomorrow's colloquium will be a "Dutch Treat?"

Tomorrow's colloquium, presented by Prof. Mark Pearson, is titled "Windmills and Wreaths."  Contrary to what the title of this talk may suggest, he will not be discussing Christmas in Holland.  Rather, he will define what a wreath product is and present a geometric model for certain wreath products.  This geometric model describes the rather complicated structure of these wreath products in a surprisingly simple and pictorial way as a kind of a windmill.  The colloquium is scheduled for Thursday, November 18 at 3:30 p.m. in VWF 104.  Don't forget that tea time (tea, soft drinks, and other goodies) will precede the talk at 3:00 p.m. in VWF 222.


The ice cream social was a big hit!

Attracted by ice cream, the opportunity to meet mathematics majors, and some easy colloquium credit, approximately 80 people crowded into VWF 222 on November 5 for an ice cream social.  In addition to eating ice cream in close proximity to other students, a drawing was held for a door prize.  Jeff Ambrose's name was drawn and he received the game Set.






MATH Challenge

Eight Hope students competed on three teams in the Michigan Autumn Take-Home (MATH) Challenge on Saturday, November 6.  Liz Adenegan, Aimin Walsh, Andrew Wells, Stefan Coltisor, Daniela Banu, Nick Sumner, Henry Gould, and Petya Dodova represented Hope College in this team event.  In groups of two or three, these students spent the morning working on ten interesting mathematical problems.

 Last year, the team of  Daniela Banu, Stefan Coltisor, and Heidi Libner from Hope College won the event.  We will be looking forward to finding out this year's results in the near future.


Hope students presented their research at a recent conference

Three Hope mathematics students presented their summer research at a recent undergraduate research symposium.  The symposium was sponsored by the Pew Midstates Science and Mathematics Consortium and was held November 5 - 7 at Washington University in St. Louis.  Michael Cortez  presented "A Mathematical Model of Tri-Trophic Interactions,"  Ryan Weaver  presented "Barging Ahead: Optimizing a Trip Up the River," and Kyle Williams presented "Why is the Learning Curve S-Shaped? A Probabilistic Model of Neural Connections."


Research opportunities available at Oak Ridge

As you make plans for next fall semester (or the fall after that), remember that there are internship opportunities to work with a scientist at Oak Ridge National Laboratory.  This can be done through the Oak Ridge Science Semester Program which is sponsored by an association of colleges of which Hope is a member. 

The program is held in the fall semester each year and can provide research internships in Biology, Chemistry, Computer Science, Mathematics, and Physics/Engineering. Applicants must have a 3.0 or better GPA overall and in their majors. The program is primarily designed for seniors, but juniors are are eligible to apply.

Student are awarded 16 credits for the semester consisting 4 credits for an Interdisciplinary Seminar, 4 credits for an advanced course, and 8 credits for research. Students generally pay regular tuition to their home institution for the semester. Students are paid a research stipend of $6,400 (pending funding approval by DOE) and their housing costs in Oak Ridge are also paid by the program.

For more information about the program, visit their website at http://www.orss.denison.edu or contact Prof. William Mungall of the Chemistry Department (mungall@hope.edu).


Problem Solvers of the Fortnight

Congratulations to Sommer Amundsen, Daniela Banu, James Boerkoel, Kim Harrison, Matt Paarlberg, Justin Shaler, Ashley Waples, and Emily Wondell, many of whom submitted their solutions on the back of discarded campaign posters and all of whom correctly determined that the ratio of the area of the largest rectangle inscribed in an ellipse whose major axis is twice its minor axis is pi/2.

This problem can be handled straightforwardly with integral calculus, but there's an easier and more ingenious way.  There is a linear transformation that maps the ellipse to the circle and a rectangle inscribed in the ellipse to a rectangle inscribed in the circle.  Of course, the linear transformation will change the area of the ellipse (by a factor of the determinant of the linear transformation), but it will change the area of the rectangle by the same amount, thus leaving the ratio fixed, and so it isn't even necessary to know what the linear transformation is.  The problem then becomes one of finding the largest rectangle in a circle, and the answer is obvious: it's a square!  A simple calculation (without any calculus) then reveals that the ratio is pi/2.

This clever solution to the problem is akin to an elegant solution to last issue's problem: the problem of finding the area of the annular merry-go-round with a single, straight-line measurement.  If we shrink the annulus so that the interior circle becomes a point, the straight-line measurement that was tangent to the inner circle then becomes a diameter of the shrunken circle with a hole poked in the middle, and since the measurement was known to be 30 feet, the area of the shrunken circle (and hence the area of the original annulus) is 225 pi square feet. 

Problem solvers are invited to stop by Dr. Pearson's office (VWF 212) to claim their sweet rewards! 

Editor's note: Daniela Banu and Aimin Walsh were among the problem solvers who correctly determined that the area of the merry-go-round deck was 225 pi square feet.  Aimin hit a home run by providing her solution on the back of pair 1968 World Series tickets that (if real) would have granted the bearer a glimpse of a Detroit Tigers victory over the St. Louis Cardinals in seven games.  We regret omitting their names in the last issue as much as we regret not having been able to use the tickets.




Problem of the Fortnight

What's the most efficient way to bisect a triangle?  Well, as it stands the question doesn't quite make sense.  What do we mean by "efficient" and "bisect"?  By "bisect" we mean bisect the area, and one bisection is more "efficient" than another if the length of the curve it uses it shorter.  For instance, the figure below shows four ways to bisect an isosceles triangle, and of these, the one on the left is clearly the most efficient.


The problem this fortnight is:  What is the most efficient way to bisect an equilateral triangle?  That is, what is the shortest curve that will bisect your equilateral piece of pumpkin pie this Thanksgiving?

Write your solution in whipped cream on top of a pumpkin pie and drop it off at Dr. Pearson's office (VWF 212) by 3:00 p.m. on Wednesday, November 24.  Happy Thanksgiving!



 Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu


As you can see by my picture, I am taking a break this week.  This is because nobody sent in any questions.  Feel free to write in with questions.  Who knows, with final exams on the horizon, maybe I can help! 

I think I am pretty smart for an Elvis dog.  In fact, check out another "Ask Elvis" on the Web at http://members.aol.com/jasonandelvis/elvisstash/ askelvis.html.  I think you will agree that my answers are a bit more intellectual than his.


  



Surfing the Web: MathDL

The Mathematical Association of America (MAA) has recently unveiled the new site for the Mathematical Sciences Digital Library (MathDL). The site includes: JOMA (the Journal of Online Mathematics and its Applications), DCR (Digital Classroom Resources), Convergence (an online magazine devoted to the use of the history of mathematics in teaching mathematics ), and OSLETS (Open Source Mathlets).  More resources will be added to this site in the future.  You can check it out at http://www.mathdl.org.


If you don't know where you are going, any road will take you there.
from Alice in Wonderland by Lewis Carroll