|OFF ON A TANGENT
|A Fortnightly Electronic Newsletter from the Hope
College Department of Mathematics
Tomorrow's colloquium will look
at the mathematics behind and Escher print
When: Thursday, April 14 at 4:00 p.m.
Where: VWF 104
Tea time: 3:30 p.m. in VWF 222
M. C. Escher (1898-1972) is one of the world's most famous graphic
artists. He is famous for his so-called impossible structures,
such as a "Waterfall,"
and his transformation prints, such as "Reptiles."
tomorrow's colloquium, Prof. Michael Bolt from Calvin College will join
us and talk about "The Mathematics of Escher's Print Gallery."
One of Escher's more compelling works is his "Print Gallery"
in which a
young man stands in an art gallery, viewing a print that contains the
very gallery in which he is standing. At the center of Escher's picture
is a curious hole that is blank except for the artist's signature.
In 2000, Hendrik Lenstra from U.C. Berkeley and Universiteit Leiden
discovered the mathematical structure behind "Print Gallery," and he
showed there is a unique mathematical solution for what belongs in the
hole. In this talk, we'll see how a team of scientists led by Lenstra
and Bart de Smit (also from Universiteit Leiden) filled in the hole in
"Print Gallery," and generated a number of images and animations that
illustrate other versions of the picture. Their project is described in
the April 2003 issue of the Notices of the AMS. Along the way, we'll
introduce all the complex analysis that is needed for generating images
like Escher's. The mathematics should be understandable to anyone with
a year of calculus.
The last colloquium
scheduled for next week
When: Thursday, April 21 at 4:00 p.m.
Where: VWF 104
Tea time: 3:30 p.m. in VWF 222
Dr. Matt Delong from Taylor University will join us for next week's
colloquium. The title of his talk is "Elliptic Curves: From
Taxi-Cabs to String Theory and Internet Security."
Elliptic curves are beautiful mathematical objects that have
connections to many areas of mathematics and applications to areas as
diverse as string theory and Internet security. In this talk, we will
learn what an elliptic curve is and why it is 'elliptic.' We will see
how a simple geometric construction yields the interesting and useful
arithmetic structure on elliptic curves. Finally, we will see some of
the applications of elliptic curves, including their role in the proof
of Fermat's Last Theorem."
Most of the talk should be accessible to general undergraduate
students; students with background in modern algebra will see some
especially neat connections.
Math 207 students
work as problem solving mentors
The students in Math 207: K-8 Mathematics Software Applications are
mentors for the Math Forum pre-algebra problem of the week web
site. You can see their smiling faces and a link to the current
problem of the week at http://mathforum.org/prealgpow/mentors/bio.ehtml?mentor_group=32.
Problem Solvers of the Fortnight
Congratulations to Brandon Alleman, Amanda Allen, Becca Baker, Brian
Boom, Kevin Butterfield, Michael Cortez, James Daly, Paula Graham,
Andrew Grumbine, Maya Holtrop, Brett Jager, Nik McPherson, Ashley
O'Shaughnessey, Ryan Nelis, Mike Nelsen, Kate Stacey, and Emily Wordell
-- all of whom submitted a correct solution to at least one of the
problems posed in the last edition of America's premiere fortnightly
electronic mathematics department newsletter.
If two diagonally opposite tiles are removed from a checkerboard, they
are of the same color; and since any 2 x 1 domino must cover one red
square and one black, it is impossible to tile this diminished
checkerboard with 2 x 1 dominoes. It is, however, always possible
to use L-shaped corner tiles to tile the checkerboard with one square
removed, regardless of which square is removed. In fact,
regardless of which square is removed, it is always possible to tile a 2n x 2n
board with one square removed, a fact that can be proved by
induction. Problem solvers are invited to drop by Dr. Pearson's
office (VWF 212) to claim their prize.
Problem of the Fortnight
Drum roll, please. . . . Our final problem of the fortnight:
With all the receptions, ceremonies and other events accompanying the
end of the year and graduation, it's a sure bet that a lot of
handshaking will occur in the upcoming weeks. And with that in
mind, we introduce you to our final problem of the problem solving
Marge and her husband Homer went to a party where there were four other
married couples, making a total of 10 people. As people arrived,
a certain amount of handshaking took place in an unpredictable way,
subject only to two obvious conditions: no one shook his or her own
hand, and no one shook the hand of the person to whom he or she was
married. When it was all over, Marge asked everyone how many
hands he or she shook and was surprised by the replies: each of the
nine people she asked gave her a different answer!
How many hands did Homer shake, and how did you figure it out?
In honor of the colloquium on Escher this week, write your solution on
the back of a reproduction of your favorite Escher piece and drop it in
the Problem of the Fortnight slot outside Dr. Pearson's office (VWF
212) by 3:00 Friday, April 22.
Tale of an Epic Journey: Ye Olde
by Mark Yapp, contributing columnist to "Off on a Tangent"
the morn of April 2, 2005, seven young stallions [Brandon Alleman,
Benjamin Crumpler, James Daly, Brian Lajiness, David Visser, Andrew
Wells, and Mark Yapp], two beautiful maidens [Daniela Banu and Jennica
Skoug], and one aged mathematical sorcerer [the editors presume the
author is referring to Dr. Pearson, who disputes being either aged or a
sorcerer] set out on an epic journey that changed the face of history.
They began their journey under dark skies that threatened spilled
blood, and traveled many miles to the flat, desolate lands of Flint,
Michigan. The road was scattered with fallen beasts of the wild,
numbering four score and six. They arrived tired and
travel-weary, but prepared to conquer nonetheless.
For three grueling hours the battle was fought without food or
sleep. It was not apparent who was gaining the upper hand, but
one small group stood out. Three chivalrous warriors that bore
the hand of Hope College stand out in my mind. One was learned in
the practice of alchemy and was deadly with the disk, one was from
lands not yet explored by the men of this region, and one had blonde
hair and enjoyed eating pizza.
Hard they fought, and with weariness they struggled, but they stood
their ground. The end seemed near, and defeat appeared
inevitable, yet a fire remained in the eyes of the warriors. They
rallied with the strength of six legions, and with one last breath,
they conquered the enemy. It was a valiant effort which will be
remembered for ages, one that mothers will tell to their children as
they gaze in awe with shiny eyes, and a little drool drizzling out of
the corners of their small mouths.
[Editor's note: Three teams of students from
Hope participated in the Lower Michigan Mathematics Competition on
April 2. We look forward to hearing the results soon and, if
victorious, the return of the Klein Bottle Trophy to VanderWerf Hall.]
Francesco Severi (1879 - 1961)
Francesco Severi was one of the leaders, along with Castelnuovo and
Enriques of what is often called the "Italian School of Algebraic
Geometry," which greatly influenced modern algebraic geometry (see http://www.answers.com/topic/italian-school-of-algebraic-geometry
for more details).
Severi studied at the University of Turin, where he paid his way
through school by tutoring. After graduating he served in the
Italian army in World War I and later went on to become a professor at
the University of Rome. Toward middle age, he spent less time on
mathematics and more on his other interests: he was president of a
bank, head of the engineering faculty at Padua, and "an expert
agriculturist who managed his own estate." Despite his broad
interests, he published over 400 mathematics papers in his
Leonard Roth, a contemporary, described Severi: "Personal relationships
with Severi, however complicated in appearance, were always reducible
to two basically simple situations: either he had just taken offense or
else he was in the process of giving it---and quite often genuinely
unaware that he was doing so." To read more about Francesco
Severi, who was born on April 13, 1879, please visit http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Severi.html.
||Got a Math Question?
Ask Elvis ...
... email him at email@example.com
I don't know about you, but I have enjoyed the last couple weeks of
nice weather. It is great to be able to go outside to do whatever
it is a dog needs to do and not freeze your tail off. Hopefully,
you have been able to get outside in the sunshine, but at the same time
not neglect your school work. There are only a couple of weeks
left to go now before finals so don't let all your hard work from the
beginning of the semester go to waste by not preparing properly for
I had a chance to show off for a couple of Calc I classes
yesterday. They were working on the famous "Do Dogs Know
Calculus" problem and I thought I would stop by and help them
out. I think they all understand it now. I have also been
working on some mental telepathy activities lately. I got to
impress the classes with these new-found powers.
I have one question to answer this week and it appears below.
I attended the colloquium on
Penrose Tiles. The golden ratio was mentioned in that
lecture. Just what is that?
Dear Colloquially Confused,
The golden ratio, also known as the divine proportion or the golden
mean, is one of those mysterious numbers that occur frequently in
nature. (The golden retriever is one of the mysterious dogs that
frequently occur in nature.) It is denoted by the Greek letter
phi, , in commemoration
of the Greek sculptor Phidias
(ca. 490-430 BC) who used the ratio extensively in his works. The
golden ratio is
The golden ratio has many interesting properties. It can be
represented as a continued fraction
or a nested root
It is also one of the
algebraic real numbers of degree 2 and it is connected with the
Euclidean algorithm for finding the greatest common divisor of two
The legs of a "golden triangle" (an isosceles triangle with
a vertex angle of 36 degrees) are in a golden ration to its base, and
if you recall the solution to the snowplow problem a few weeks ago, the
answer turned out to be 0.618, which is the reciprocal of the golden
To read more about this fascinating and important number,
please see any of the following web sites:
You can check out the book The
Golden Ratio: the story of phi, the world's
most astonishing number by Mario Livio, available in Van Wylen
Finally, if you want to see a representation of the golden ration
tattooed on someone's arm check out http://www.snerk.net/gallery/albums/tattoos/wrist_640x480.jpg
There is currently a
mathematics book sale going on in the Reading Room (VWF 222).
The books are located on the
bookshelves by the windows. These books are priced to sell!
1 book: $0.25
5 books: $1.00
10 books: $1.50
||April is Mathematics Awareness
Month. This year the theme is Mathematics and the Cosmos.
Click on the image to the right to learn more about this event.
The hardest thing
in the world to understand is the income tax.