This
Day in Mathematics
History . . .
On
March 28, 1809, Gauss completed his work on computing the orbits of
planets using the method of least squares. Gauss was drawn to
this work in 1801 when Piazzi discovered the dwarf planet Ceres.
Piazzi was able to observe its location only 22 times in a span of 41
days before it disappeared behind the glare of the sun. Months
later, when Ceres should have reappeared, Piazzi could not locate
it. Gauss, then 23 years old, heard about the problem and dove
into it. After three months of intensive work, he predicted a
location for Ceres in December 1801, and his prediction was within half
a degree of it actual location. Because Gauss did not disclose
his methods at the time, a mystique grew about Gauss' uncanny
mathematical abilities, and his work on Ceres contributed to his
appointment in 1807 as the director of the Göttingen Observartory,
a
position he retained until his death on February 23, 1855. To
read more about Gauss, please visit the
Wikipedia
page on Gauss.
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, . . .
]. A fact that is well known by those who know it well is that π
can be represented as the continued fraction [3, 7, 15, 1, 292, 1, 1,
1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, . . . ] The first few
convergents
are 3, 22/7, 333/106, and 355/113. The very large term 292 means
that the convergent [3, 7, 15, 1] = 355/113 is a very good
approximation to π (accurate to 6 decimal places), a fact first
discovered by astronomer Tsu Ch'ung-Chih in the fifth century
A.D.
Find the exact value (no decimal approximations
allowed) of the continued fraction [1, 1, 1, 1, . . . ].
The solution is: Let x = [1, 1, 1, 1, . . . ]. Writing this
as a continued fraction, we see that x = 1 + 1/x, which gives the
quadratic equation x
2 - x - 1 = 0. The solutions of
this equation are given by the quadratic formula: x = (1 ±
√5)/2. Since x is obviously positive, we can rule out the
solution x = (1 - √5)/2, and so x = [1, 1, 1, 1, . . . ] = (1 +
√5)/2. This number is known as the golden ratio and is commonly
denoted φ. The golden ratio has a long history not only in
mathematics, but also in art, architecture, music and nature, to
mention only a few of the areas where the golden ratio appears.
Please visit the
Wikipedia
page on the golden ratio to learn more about this interesting
number φ.
Congratulations to Jon Moerdyk, Jeff Shriner and Dirk VanBruggen for
solving the
last problem of the
fortnight! That was some φine work, φellas!
Problem of the
Fortnight
Find the exact value of the continued fraction [1, 2, 3, 1, 2, 3, 1, 2,
3, . . . ].
Write your answer on the back of your
favorite picture of Gauss and
drop it by the Problem of the Fortnight slot outside Dr. Pearson's
office (VWF 212) by
3:00 p.m. on
Thursday,
April 5. As always, be sure to include your name, you
math class(es) and the name(s) of your professor(s) -- e.g. B. Riemann,
Professor Gauss, Calculus 1 -- on your solution.
Ask Elvis . . . 
Got
a math question? Email Evis at elvis@hope.edu
Many of you have noticed that I have not written a column for the
newsletter in quite a while. The reason for my lentitude of
late has nothing to do with my ACL injury, which is scheduled for a
surgical repair soon. Rather, I've been working on a new
article. My best pal Tim tells the story like this. . . .
It all started early in the 21st
century when Elvis, my Welsh corgi, showed me that he could naturally
solve an optimization problem by finding the quickest route from the
shore to a ball thrown in Lake Michigan. The resulting article
"Do Dogs Know Calculus?" (College Mathematics Journal, Vol. 34, No. 3,
2003) explained how Elvis, rather than plunging into the water
immediately and swimming straight to the ball, instead runs down the
shore for a certain distance before entering the water. By determining
Elvis's running and swimming speeds, I showed that Elvis was generally
entering the water quite close to the optimal point of this
minimization problem.
However, two French mathematicians were
not convinced. Perruchet and Gallego wrote a follow-up article, "Do
Dogs Know Related Rates Rather than Optimization?" (College Mathematics
Journal, Vol. 37, No. 1, 2006) in which they showed that by thinking of
the situation as a local related rate problem (Elvis at every point
moves so as to close his distance with the ball as quickly as possible)
rather than a global optimization problem, one arrives at the same
solution. They also argued that their related rate model was a more
plausible model for a dog's strategy, since it was gained without
assuming canine knowledge of the entire route.
The answer to the question of which
strategy Elvis uses (local related rates, or global optimization) was
unwittingly determined soon after, when on a hot day, I threw the ball
while standing in the water - with Elvis beside me. When I threw the
ball a short distance away, Elvis swam directly to it. But when I threw
it a far distance, Elvis swam in to the shore, ran along it, and then
swam back out to the ball. WHILE SWIMMING TO THE SHORE ELVIS WAS NOT
MOVING TOWARDS THE BALL AS QUICKLY AS POSSIBLE, so he was indeed
solving a global optimization problem.
But does Elvis know when it is best to
swim all the way, and when it is best to run along the shore? Since a
choice must be made of which strategy to use, Roland Minton from
Roanoke College and I have written a follow up paper (to be published
soon) entitled, "Do Dogs Know Bifurcations?" It details the (rather
interesting) mathematical analysis needed to solve the problem and then
shows how well Elvis did.
I think I finished my latest writing project just in time. Spring
is here, and soon my ACL will be good as new, and then I'll be able to
play dogball and fetch sticks and go to the beach!
It's been great to hear from all of you -- keep those emails
coming! Best wishes to all of you as you finish the school year!
Your pal,
Elvis
Join us for Tea Time on Thursdays
before colloquia
