OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
October 4, 2006 Vol. 5, No. 3
http://www.math.hope.edu/newsletter.html


Upcoming Colloquia 

It has been established that dogs - at least Elvis - knows calculus. That is, Elvis can find the optimal - fastest -  route to a ball thrown down the beach and in the water. But what happens when Elvis is positioned in the water and retrieves a ball that is also in the water? When should he swim the entire distance to the ball, and when should he swim in to the shore, run along the shore, and then swim back out to the ball? What is the bifurcation point for the change in optimal strategy? Does Elvis bifurcate? Does his fur bicate?  Does Elvis buy Kate fur?  Elvis will be available for follow-up questions.

  Join us for Tea Time on Thursdays before colloquia

As part of our colloquium series this year, the mathematics department will host a "tea time" in the Reading Room (VWF 222) at 3:45 pm.  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.

Upcoming Events at Hope 

The 2006 Michigan Autumn Take Home Challenge (or MATH Challenge) will take place on the morning of Saturday, October 28.  Teams of two or three students take a three-hour exam consisting of ten interesting problems dealing with topics and concepts found in the undergraduate mathematics curriculum.  Each team takes the exam at their home campus under the supervision of a faculty advisor.

The department pays the registration fee for each team and will provide lunch to participants afterwards. The sign-up deadline is Thursday, October 12, 2006 at 5:00 pm.  Interested students can sign up by sending Professor Cinzori an email at cinzori@hope.edu or by signing up on the list on his office door (VWF 216).

A group of students may sign up as a team.  Individual students are also encouraged to sign up; they will be assigned to a team on the day of the competition.  For more information about this competition and to view copies of old exams visit http://www.mcs.alma.edu/mathchallenge/.

Hope College won the competition in 1997 and 2003.  Let's do it again!

Problem Solvers of the Fortnight 

Congratulations to the anonymous mathematician with no class, Ben Johnson, Beth Heisel, Brad Lininger, Bryan McMahon, Dan Forro, Daniel Gruben, Evan Van Heukelom, James Daly, Jeff Mulder, Jill Immink, Joel Mulder, Katie Johnson, Laura Shears, Laura Smallegan, Matt Paarlberg, Megan Patnott, Sarah Dix, Stephen Goodrich, Tiffany Day, Tim Wahmhoff, and Tracy Albus for getting the Hamilton family across Broom Bridge in 17 minutes!  Evan Van Heukelom offered the following creative solution:

First William and Helen go over the bridge, taking 2 minutes.  Once on the other side, William turns around and walks over the bridge in 1 minute, making the total time 3 minutes.  With quick instruction, William sends Edwin and Archibald back over with the lamp, bringing to total time to 13 minutes.  Once the boys are across, Mom takes off with the lamp at the beginning of minute 13, arriving on the side William is standing at minute 15.  Mom and Dad then return to their kids on the other side of the bridge.  On their romantic talk alone, they discuss the thought of not having any more kids because it is beginning to take too long to get to church.  So 17 minutes after beginning their bridge excursion, the trip has been concluded.  And it was now time to go to church.  While sitting in church, William was trying to figure out a faster way home because he really had to use the bathroom.  He concluded that there was no quicker way, but in his attempt to distract himself from his bladder, he thought of a way to generalize the complex numbers to higher dimensions.  And rather than writing the equations in his hymnal, he wrote them on the bridge while waiting for his children to cross on the way home.  He then went home, had dinner and rested his legs after an exhausting walk to church.

Problem of the Fortnight 

Consider the polynomial

p(x) = x4 - 18x3 + kx2 + 200x - 1984

Given that p(a) = 0 = p(b) and ab = -32, find k.

Once you've found the value for k, graph the polynomial p(x) and write your solution (not just the value of k, but how you determined it) on the back of your graph and drop it in the Problem of the Fortnight slot outside Professor Pearson's office (VWF 212) by 3 pm on Friday, October 13.  Please be sure to include your name, your math class(es) and the name(s) of your professor(s) -- e.g.  Ima Student, Math 131, Professor Isaac Newton -- on your solution.


Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu

Dear Friends,

I have a couple of questions to answer this week about soccer.  Though one is really about mathematics.  If you have any questions about math or the sport of your interest, send me an email at elvis@hope.edu.  I hope to hear from you!




Dear Elvis,

I have seen you at a number of Hope College soccer games, so I have a question for you about soccer/math.  The Hope team uses the traditional soccer ball made up of hexagons and pentagons.  However, in the World Cup this year, I noticed that the ball was different.  There weren't any hexagons nor pentagons to be found on the entire ball.  What do you think of this new less mathematical ball?

Bucky

Dear Bucky,

I don't know about the new ball being less mathematical.  It is just different mathematically.

A mathematician might call the traditional soccer ball a truncated icosahedron.  It consists of 32 polygonal regions---20 hexagons and 12 pentagons.  This, of course,  is a polyhedron with flat sides.  The folks that make soccer balls, however, fill them up with air and it make an object that is pretty close to a sphere.  (By the way, did you know that in 1996, the Nobel Prize in Chemistry was awarded to three chemists, one a Hope alumnus, for their discovery of a carbon molecule that has the shape of a soccer ball?  I wonder what shapes they are now looking for.  Golf balls?  Kibble?  Cats?  That would be interesting!)

The new soccer ball made by Adidas does not consist of any polygons, but is constructed with two different types of curved panels.  Since these panels are no longer polygons, one might think that this ball is less mathematical.   However, in the branch of mathematics called topology, mathematical shapes can be preserved  even when they are stretched and twisted.  In other words, we can take a circle and smoosh it like a piece of dough and have it become a pentagon, but topologically, the shape hasn't changed.  We change shape by poking a hole in it and frying it up into a doughnut.  Looking at the new soccer ball topologically, we can make it a truncated octahedron made up of eight hexagons and six squares.  Pretty cool!

Therefore, I wouldn't say that it is less mathematical, but just has different mathematics.  I still prefer the more traditional soccer ball, however.  After all, you know what they say about old dogs and new soccer balls!

There is a nice article about the new soccer ball in Science News located at http://www.sciencenews.org/articles/20060708/mathtrek.asp.  This is also the online journal that once featured an article about me.

Elvis

Dear Elvis,

I play a lot of soccer and while I have shin guards and shoes to protect my feet, I have nothing to protect my head.  (I tend to use my head a lot and some people think I use it a bit too much!)  Don't you think that there ought to be some kind of head protection for soccer players?

Z.Z.

Dear Z.Z.

While I have never seen one, there are different head protection devices made for soccer players.  You can see a picture of one here. 

I have seen soccer players heads run into a variety of objects, so it would make sense that one's head could use a little protection.  However, Z.Z., if you do wear head protection playing soccer, I would suggest that you not run your head into anything that you should not run your head into!  I think you know what I mean.

Elvis


It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. 
~ Carl Friedrich Gauss