Tomorrow
in History . . .
On April
12, 1852 Carl Louis Ferdinand von Lindemann was born in Hannover,
Germany. Lindemann is famous for proving that π is
transcendental
-- i.e. that π is not the root of any polynomial with rational
coefficients. His proof is based on the fact that
e is transcendental and Euler's
celebrated result
e
iπ + 1 = 0.
Before Lindemann,
it had been proved that if π is transcendental, then the ancient
problem of squaring the circle by ruler and compass constructions
cannot be accomplished. Lindemann received his Ph.D. under the
direction of Felix Klein at the University of Erlangen, and while he
was a professor at the University of Königsburg, he supervised the
doctoral theses of more than 60 students, including David Hilbert and
Hermann Minkowski. To read more about Lindemann, please visit
http://www-history.mcs.st-andrews.ac.uk/Biographies/Lindemann.html.
Problem
Solvers of the Fortnight
The Problem of the Fortnight in the last issue was: Let {a
n}
be a (possibly
infinite) sequence of positive integers. A creature like
is called a
continued fraction
and is sometimes denoted by [a
0, a
1,
a
2, a
3, . . .
].
The Problem of the Fortnight in the last issue was: Find the exact
value of the continued fraction [1, 2, 3, 1, 2, 3, 1, 2,
3, . . . ].
Congratulations to Dirk Van Bruggen, Benjamin Crumpler, Ashten Wallace,
Steven Barbachyn, Jon Moerdyk and Luc Leavenworth for determining that
[1, 2, 3, 1, 2, 3,
. . . ] = (4 + √37)/7.
Erratum: The editors regret omitting Steven Barbachyn from the
list of Problem Solvers of the Fortnight for the problem posed in Vol.
5 No. 12.
Problem of the
Fortnight 
We've saved the best for last! The final Problem of the Fortnight
for the year involves a little geometry and some ideas from Calculus 1
-- but nothing more! -- and so everyone should be able to take a crack
at it. It's a great problem, and we hope you enjoy working on it!
Suppose that circles of equal diameter are packed tightly in
n rows
inside an equilateral triangle. (The figure at left illustrates
the case
n = 3.)
If
A is the area of the
triangle and
An is the
total area occupied by the
n
rows of circles, find the limit of the ratio
An / A as
n goes to infinity; i.e. find
lim n →∞ An / A
Write your solution -- showing all your work, please! -- on the back of
your
favorite picture of your favorite mathematician and
drop it by the Problem of the Fortnight slot outside Dr. Pearson's
office (VWF 212) by
3:00 p.m. on Friday,
April 20. As always, be sure to include your name, you
math class(es) and the name(s) of your professor(s) -- e.g. Dave
Hilbert,
Professor Lindemann, Calculus 1 with Early Transcendentals -- on your
solution.
Hungary
for Math?
Martha Precup, a
junior math major from Boyne City, MI, is currently whetting her
appetite for further study of mathematics in Hungary, where she is
studying abroad in the Budapest Semesters in Mathematics Program.
Recently she wrote us to tell us about her experiences there.
Here's what she had to say:
"Budapest is great! I love the city, and the math classes are
very challenging, but I'm learning so much.
"Studying abroad has become one of the staples of the 'college
experience,' and if you're interested in studying math, then Budapest
is the place to be. The city is beautiful, clean, and a safe
environment. My apartment is very nice and located just a few
block from Hero's Square, City Park, and the famous Hungarian Opera
House. There are so many attractions: traditional Hungarian folk
music in the local bars, operas, ballets, art, and hundreds of
restaurants to choose from! Not to mention that Hungary is a
great location from which to travel to other areas of Europe.
Transportation to cities like Vienna and Prague is readily available
and very cheap.
"Another reason to come to Budapest is the math. Never have I
been so challenged in my academics. The program has a wide
variety of classes available from introductory courses to those of a
graduate school level. The first three weeks allow you to take as
many classes as you'd like and then choose which you enjoy the
most. This semester I'm taking: complex functions, topology,
introductory number theory, and advanced abstract algebra. In
addition to mathematics courses there are a number of language and
humanities classes offered as well. You need to work hard, but it
is worth it. If you are at all interested in studying math beyond
the undergraduate level, then you should consider this program
seriously."
