OFF ON A TANGENT |

A Fortnightly Electronic Newsletter from the Hope
College Department of Mathematics |

October 5, 2005 |
Vol. 4, No. 3 |

http://www.math.hope.edu/newsletter.html |

Tomorrow's colloquium to be on the Poincaré Conjecture

- Thursday, October 6 at 4:00 p.m.
- 104 VWF

A space that at small scales resembles Euclidean 3-space is called a 3-dimensional manifold. Such spaces can have complicated and interesting large-scale structure. Recent breakthrough work by the Russian mathematician Grisha Perelman appears to have resolved one of the major questions regarding 3-manifolds — the century-old Poincaré Conjecture. Perelman's solution is most surprising in that he establishes a much more sweeping result, known as Thurston's Geometrization Conjecture, that relate the geometry and topology of 3-manifolds.

In this talk Prof. Venema will attempt to explain what Perelman has proved about 3-manifolds by analogy with 2-dimensional results. In particular he will explain the difference between the topology and geometry of a space and what it means for a space to have a natural geometry of constant curvature. If time permits he will conclude with some speculation about implications for the shape of the physical universe.

Don't forget that tea time will precede the colloquium at 3:30 p.m. in VWF 222.

No colloquium scheduled for next week

You all know that next week Wednesday (when you return from fall break) is declared to be a Monday by the registrar's office. The mathematics department (for colloquium purposes only) is declaring next week Thursday to be a Tuesday. Since we have colloquia on Thursdays, and there will not be a Thursday next week, we will not have a colloquium. We thought this method of changing days around would be the most logical for all. So remember next week will go Monday, Tuesday, Monday, Tuesday, Friday. (I think the SAC movie next week should be Groundhog Day!) We would like to change Friday to a Saturday, but that is out of our hands.

Just to make things a bit more difficult. Next week October 12 is Columbus Day. However, somebody at the Post Office, I think, likes to have Monday holidays. This means that next week Monday (which is a Monday) will be October 12. So the dates for next week Monday through Friday go 12th-11th-10th-13th-14th. Got it? Good!

Let's play Mathematical Jeopardy!

The mathematics department will conduct a Mathematical Jeopardy competition on Thursday, October 20 at 4:00 p.m. (tentatively in VWF 102). Entry is free and there will be prizes for the winning team! Sign up on Professor Stephenson’s door (VWF 210) by Monday, October 17 or email him at stephenson@hope.edu if you would like to be a contestant. Here are some of the particulars.

- You may sign up as a complete team (4 students or less). Include the names of all team members and a team name. (You must make up a team name, or we will choose one for you.)
- You may also sign up individually, and we will put you on a team.
- This contest is open to everyone – students at all mathematical levels are encouraged to participate. Questions will be chosen from a broad range of topics, including films and history.
- Refreshments will be served following the competition.

Statistics Career Day next month at GVSU

On October 14 the Southwest Michigan Chapter of the American Statistical Association (ASA) and the Grand Valley State University Department of Statistics will present a Statistics Career Day on their Allendale Campus. Exhibits on employment in statistics from governmental agencies and private industry will be available as well as those from graduate schools. There will also be a number of talks given about employment as well as a keynote address from ASA president Fritz J. Scheuren.

For more information about this event you may contact either Prof. Bekmetjev or Prof. Tintle. (Information can also be found at Statistics Career Day link in the preceding paragraph.)

It is time to think about competing in the fall mathematics competitions

Two mathematics competitions that take place each fall at Hope College are the MATH Challenge and the Putnam Exam. Information about each of these follows.

The MATH Challenge

The 2005 Michigan Autumn Take Home Challenge (or MATH Challenge) will take place on the morning of Saturday, October 29 this year. 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 Wednesday, October 12, 2005 at 5:00 p.m. Interested students can sign up by sending Prof. 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 encourage 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/.

The Putnam Exam

The William Lowell Putnam Mathematical Competition, administered by the Mathematical Association of America, is the most prestigious mathematical competition for undergraduates in the nation. If you are interested in taking the 66th Annual Wm. Lowell Putnam Exam, you must sign up by this Friday, Oct. 7 with Prof. Stoughton. The date of the exam is Saturday, December 3, 2005. There is both a morning and an afternoon session of this exam; lunch will be provided by the mathematics department during the break. For more information about the Putnam Exam visit http://math.scu.edu/putnam/. (As of today, this official Putnam site hasn't been updated since last year, but still gives some useful information.) For questions and solutions from past exams visit http://www.kalva.demon.co.uk/putnam.html.

The Michigan Undergraduate Mathematics Conference will be held this month

The Department of Mathematics at the University of Michigan at Flint will be hosting the eighth annual Michigan Undergraduate Mathematics Conference (MUMC) on Saturday, October 22, 2005. Hope College will be taking a group of students and faculty. They will leave early in the day and return in the evening.

Undergraduate students will be giving 20-minute oral presentations
on many areas of mathematics, statistics or related discipline. Such
areas include undergraduate research projects, interesting class
projects, history of mathematics, or expository talks on interesting
mathematics. Students are also encourage just to attend as there
will be presentations on careers in mathematics, information about
mathematics graduate programs and REU programs.

For students interested in attending need to sign up with Prof.
Darin Stephenson by October 12 (he has a sign-up sheet outside his
office door, VWF 210). Visit the MUMC web page at http://www.umflint.edu/departments/math/mumc2005/index.html
for more information about the conference.

Problem of the Fortnight

A little problem about big numbers. . . .

Find the smallest N, or show that none exists, for which the decimal representation of ends in exactly 2005 zeros.

Write your solution on the back of two 2005 American League Championship Series tickets, or on the back of a $2005 bill, and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 Friday, October 14.

Problem Solvers of the Fortnight

We received many interesting solutions to the problem in the last issue, including two written on styrofoam egg cartons. Only nine people, though, found the best strategy. Congratulations to Amanda Allen, Benjamin Crumpler, James Daly, Erica Dickinson, Derek Duncan, Nathan Makowski, Jeff Mastin, Jon Moerdyk, and Megan Patnott for handling this egg problem with such delicate care and hatching elegant solutions!

The least number of drops that are needed to determine with certainty, even in the worst case scenario, is eight. The idea is to keep the total number of drops the same. First drop the egg from the eighth floor. If it breaks, the only thing to do is go to floor one and proceed upward; in the worst case, you need to go to the seventh floor, in which case eight drops were needed to determine this. If, however, the egg survives a drop from the eighth floor, we want to keep the total number of drops the same, and so having already made one drop, we can only go up seven floors. So drop it from the 15th floor. If it doesn't survive, go to the ninth floor and proceed upwards; in the worst case, you need to go to the 14th floor, in which case a total of eight drops (one from floor eight and seven from floors 9 - 14) were required. Proceed to floors 21, 26, 30, 33, and 35, as necessary. If an egg dropped from the 35th floor survives, we will have made 7 drops already, and a final drop from the 36th floor will determine exactly where the egg breaks.

The answer to this problem is beautiful. It involves triangular numbers, which are numbers obtained by adding all positive integers less than or equal to a given integer. For example, 10 is a triangular number: 10 = 1 + 2 + 3 + 4. They are called triangular numbers because the integers involved in the sum may be arranged to form a triangle. For example, the integers 1, 2, 3, 4 may be arranged like pins on a bowling alley to form a triangle.

For more on triangular numbers, visit http://mathworld.wolfram.com/TriangularNumber.html.

So . . . what can I do with a degree in mathematics?

On Monday President Bush nominated Harriet Miers to fill the vacancy on the Supreme Court left by the retirement of Sandra Day O'Connor. In the newsroom at "Off on a Tangent" our ears perked up when we heard the radio news report that she had obtained a B.S. in mathematics from SMU. So if you're thinking about majoring or minoring in mathematics but are worried that your choice of studies may not further your career goal of being associate justice of the Supreme Court of the United States, you need not worry. All joking aside, many law schools look favorably on applicants who majored in mathematics because of their ability to construct tightly reasoned, logical arguments.

Political pundits have been speculating that diversity of the Court would be a primary consideration for President Bush in choosing a nominee. The current justices, along with their majors and undergraduate institutions, are:

- Sandra Day O'Connor – Economics, Stanford
- John Paul Stevens - English Literature, University of Chicago
- Stephen Breyer - Philosophy, Stanford
- Ruth Bader Ginsburg - Government, Cornell
- Anthony M. Kennedy - Political Science, Stanford
- Antonin Scalia - History, Georgetown
- Clarence Thomas - English, Holy Cross
- David Souter - Philosophy, Harvard
- John Roberts - History, Harvard

Got a Math Question? Ask Elvis ... ... email him at elvis@hope.edu |

Dear Friends,

Did you hear that Akira Haraguchi, from Japan, recited the first 83,431 digits of pi from memory this past summer? He finished this astonishing feat in 13 hours. He did, however, have some trouble along the way. He lost his place after reciting the first 16,000 digits and was forced to start over. (Don't you hate it when that happens?) After starting over he managed to break the old record of 42,195 digits set by a 21-year-old Japanese student in 1995.

I don't know about you, but I don't like to spend too much time memorizing digits. I do, however, have committed to memory the smells of 83,431 different food and non food odors. For example, I can detect the subtle differences between the smell of beet soup and moldy shoe leather. I find this to be a much more rewarding talent.

I do have one letter to answer today. Don't forget to keep those questions coming in. It is always nice to get non-spam email.

3.1415926535897932384626433832795028841971693993751058209749445923078164062

Dear Elvis,

I have heard (from slashdot.com) that a new way of doing trigonometry has been developed. This new way doesn't involve using any classical trigonometric functions. If you follow this link, you will find a PDF of the first chapter of the book that it is revealed in. It seems fairly simple and the author, Dr. Norman Wildberger, of the South Wales University in Australia, does a good job of explaining it. However I was wondering if you could share your thoughts on this new technique.

Best Itches (uh...I mean wishes),

Karl

Dear Karl,

You know what they say. It is hard to teach an old dog new tricks. That is the way I feel about rational trigonometry. I know all the angles on classical trigonometry. I know the difference between a cosine and a coconut, as well as the difference between a tangent and a tangerine. Therefore, I am comfortable with the old ways. However, you young whippersnappers are certainly welcomed to explore new ways of doing mathematics. For that is the way new discoveries are made!

Elvis

The right to swing my fist ends where the other man's nose begins.

Oliver Wendell Holmes, Jr., Associate Justice of the U.S. Supreme Court from 1902-1932