Google showed its mathematical side last week

If you did a Google search on February 3, you may have noticed that their logo looked a bit different.  In honor of Gaston Julia's 111th birthday, Dennis Hwang, the logo artist at Google, created the logo shown to the left.  Julia fractals were added to the logo for the day.  The search engine company will often change their logo for special days.  You can visit http://www.google.com/holidaylogos.html for other examples of special Google logos.

To read more about the fascinating life of the French mathematician Julia, see http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Julia.html.  You can also visit the Julia and Mandelbrot Set Explorer at http://aleph0.clarku.edu/~djoyce/julia/explorer.html to read more about these interesting fractal sets and create some of your own!  (Or click on the Google image to the left to do an image search for Julia fractals and see some marvelous, non-interactive graphics.)

Summer opportunities for research should be investigated soon

As mentioned in the last newsletter, the Department of Mathematics has an NSF-REU Summer Research Grant.  This coming summer, professors Aaron Cinzori and Tim Pennings will be the research mentors.  Although students apply from all over the country, Hope students are given special consideration.  So if you are interested, see the web site at http://www.math.hope.edu/reu.html for more details.  If you are interested in doing summer research, but not at Hope, check out the other REU sites around the country.  A list of these can be found at http://www.maa.org/students/reustuff/pages/REU.html.  The deadline for applying to the Hope REU is February 29.  Other sites have deadlines around this time as well.  Since this date is fast approaching, apply soon if your are interested.

Famous Curves

With Valentine's Day just around the corner, we thought it appropriate to introduce a famous curve of the fortnight.  Our curve for this issue is the cardioid, whose name comes from Latin for heart. A cardioid is a curve traced by a point on the circumference of a circle rolling completely around another circle with the same radius.  An example of a cardioid in polar coordinates is the equation r = 2 - 2sin(Ө) whose graph is shown to the right.  More information about cardioids can be found at http://www-gap.dcs.st-and.ac.uk/~history/Curves/Cardioid.html.

Problem Solvers of the Fortnight

Congratulations to Nick Sumner, who devised a method for our seven paperless and penless (but not penniless) actuaries to compute their average salary without any of them having to reveal his or her salary to the rest of the group.  Nick's scheme involved an averaging of each of the digits in the salaries.  Another solution bandied about the department was based on a probability scheme where, if the actuaries had a coin available, a secret flip would either add or subtract a predetermined fixed amount to each salary.  Probably the best solution we came up with -- one that involves no external assumptions or equipment and one that Leticia Grandia also graciously provided -- was a round robin scheme where the first person adds a fixed amount to his or her salary, then whispers that amount to the second person, who adds his or her salary to the amount, and then passes that figure on to the third person, and so on.  When the sum of the salaries and the amount the first person added to his or her salary reaches the first person once again, he or she can then subtract the fixed amount by which he or she altered his or her salary to compute the sum of the salaries, and divide by seven to obtain the average salary.

Many thanks to Leticia Grandia once again for providing us with this monetary mystery and a very elegant solution!


Problem of the (200)4-tnight

Using all the digits in 2004 -- in order and exactly once -- and any operations, it is possible to write expressions that equal each of the numbers from 0 to 50 (at least).  For example, 19 = 20 + 0! - √4.  Exponents must come from the digits in 2004; for example 8 cannot be written as 2² + 0 + 0 + 4, but you may write 5 = 2⁰ + 0 + 4. 

Write 15 of the numbers from 0 to 50 using the digits in 2004 and any operations to qualify for a prize.  The blue ribbon prize will be awarded to the person who finds the most numbers from 0 to 50, with the edge in a tiebreaker given to the person who provides the greatest number of ways of finding these numbers (there may be more than one way to write the number 19, for example). 

Write your solution on an old Valentine and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, February 20.

Mathography

In this issue of Off on a Tangent we highlight Josiah Willard Gibbs, who was born on February 11, 1839 and is probably most widely recognized for his contributions to thermodynamics.  Gibbs received the first American Ph.D. in engineering from Yale University in 1863.  To read more about Gibbs, see
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Gibbs.html.

It's impossible to resist introducing two other mathematicians on this occasion: Hermann Hankel was born on Valentine's Day in 1839 and studied under Möbius, Riemann, Weierstrass and Kronecker; David Hilbert died on Valentine's Day in 1943 and is best known for his work in geometry and for posing 23 problems of fundamental importance at the dawn of the twentieth century.   To find out more about our Valentine's Day mathematicians, see
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hankel.html
and  http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Hilbert.html.

"Wir müssen wissen, wir werden wissen." (We must know, we shall know.)
David Hilbert (1862-1943)