| OFF ON A TANGENT |
A Fortnightly Electronic Newsletter from the Hope
College Department of Mathematics
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Tomorrow's colloquium will be tiles of fun
When: Thursday, March 31 at 4:00 p.m.
Where: VWF 104
Tea time: 3:30 p.m. in VWF 222
Tomorrow's colloquium, "Penrose Tilings," will be presented by Prof.
David Austin of Grand Valley State University. It will take place
at 3:30 p.m. in VWF 104.
The history of tilings of the plane is both long and distinguished.
For instance, it has long been known that if we want to tile the plane
using regular tiles then we must use either triangles, squares or
hexagons. In the 1970's, Roger Penrose found a set of tiles that
tile the
plane in remarkable ways. Even more stunning was the realization a few
years later that Penrose tilings can be used to explain new phenomena
in crystallography.
This talk will review some of what is known about Penrose tilings
and demonstrate a novel computer program that can be used to construct
tilings. Most of the talk will only rely on high school geometry making
it accessible to most students. There will also be lots of cool
pictures! Come join us for this talk and "take a walk on the
tiled side."
More geometry in
next week's
colloquium
When: Thursday, April 7 at 4:00 p.m.
Where: VWF 104
Tea time: 3:30 p.m. in VWF 222
Next week we will again be joined by a Grand Valley professor for
our colloquium. The title of Prof. Reva Kasman's talk is "Why
can't I fold a square into an icosahedron?" If you have ever
tried to wrap an icosahedron-shaped gift and not been able to fold the
paper just right and had trouble, you're not all thumbs---mathematics
is to blame!
Dr. Kasman will look at what polyhedra can be created by gluing
together the boundary of a polygon. She will also see how this
changes when we restrict ourselves to convex polygons. The
seminar will examine the computational geometry involved in
investigating these issues. It will be a very accessible talk for
students.
If possible, bring scissors, tape, pencils and rulers to this
seminar---but remember, no running with scissors! The colloquium
is scheduled for Thursday, April 7 at 4:00 p.m. in VWF 104.
Graduate school information session planned
A panel discussion on graduate school will take place on Thursday,
March 31 between 11:00 and 11: 50 a.m. in the Herrick Room of DeWitt.
Panelists include Jack Mulder (Philosophy), Jennifer Young
(English), Sheila Bluhm Morely (Sociology) and Chris Ritsema
(Accounting). They will speak on topics such as how to pick graduate
programs, how to finance graduate education and the application
process. They will also answer student questions.
Problem Solvers of the Fortnight
Congratulations to James Daly for plowing through the last Problem of
the Fortnight and clearing the way to the solution. James
correctly determined that the snow began to fall at 11:23 a.m.
Since the volume of snow removed at any instant is constant, the
product of momentary depth and distance traveled is constant, and hence
the distance traveled is inversely proportional to depth. But
depth is directly proportional to total time elapsed since the snow
falls at a constant rate. This gives the integral equation:
where A is the duration of
snowfall in hours before noon. Solving this equation for A gives A = 0.618, and so the snow started
to fall just before 11:23 a.m.
James and the two individuals who submitted solutions on snowflakes are
invited to drop by Dr. Pearson's office (VWF 212) to claim their prizes.
Problem of the Fortnight
In honor of this week's colloquia on puzzle games and Penrose tiles,
the Problem of the Fortnight asks you to take a walk on the tiled side
and see if you can piece together a solution to either of the following
problems:
- A checkerboard is an 8 by 8 grid. If two diagonally
opposite corner squares are removed from a checkerboard, can the
remainder be tiled by 1 by 2 dominoes?
- If any one square is removed from a checkerboard, can the
remainder be tiled by L-shaped corner tiles always, never, or does it
depend on which square is deleted?
Write your solution on the back of a checkerboard and drop it (gently)
in the Problem of the Fortnight slot outside Dr. Pearson's office by
3:00 on Friday, April 8.
Mathography:
Stefan Banach (1892 - 1945)
Born on March 30, 1892 in the small village of Ostrowsko, some 50 km
south of Kraków, Poland, Stefan Banach began his school life
with great promise, achieving top grades, especially in math and
science, which were his best subjects. As he grew in years, his
academic excellence slipped, but he managed to pass his final school
examinations, albeit without distinction, an honor bestowed upon
roughly a quarter of the pupils.
After graduating from his school in Krakow, Banach and another student
Witold Wilkosz (who himself went on to become a professor of
mathematics) wanted to study mathematics but "both felt that nothing
new could be discovered in mathematics so each chose to work in a
subject other than mathematics. Banach chose to study
engineering, Wilkosz chose oriental languages." Although Banach
and Wilkosz studied subjects other than math, each remained keenly
interested in it and together they formed a mathematical society.
Banach's mathematical career did not follow the usual course: he taught
mathematics at Lvov Technical University for a time before submitting a
doctoral dissertation.
Banach made significant contributions to mathematics, particularly
analysis. To recognize his achievements, his name is today
attached to a certain kind of vector space known as a Banach space and
also to a very interesting paradox, known as the Banach-Tarski
paradox. A Banach space is a real or complex normed vector space
that is complete as a metric space under the metric
d(x,
y) = ||x -
y||
induced by the norm. The completeness is important as this means that
Cauchy sequences in Banach spaces converge. The Banach-Tarski
paradox "shows that a ball can be divided up into subsets which can be
fitted together to make two balls each identical to the first" and was
a major contribution to the axiomatic set theory being developed at the
time. To read more about Banach visit http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Banach.html.
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Got a Math Question?
Ask Elvis ...
... email him at elvis@hope.edu
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Dear Friends,
I hope you all had a great break and got to do some of your favorite
activities. I didn't travel far, but did enjoy getting
reacquainted with some of my favorite trees in Saugatuck State Park and
the Allegan State Forest. On one trip down there, I heard an
interesting program on the radio. There were a couple of
mathematicians on NPR's program "All Things Considered" talking about
geometry. It seems that Daina Taimina crochets some quite complex
shapes that are physical models of the hyperbolic plane.
While I found the program interesting, Tim thought it was so cool that
he rolled down the windows and had the radio blaring so loud that it
drowned out the guy next to us that was blasting an Eminem song.
If you want more information about these shapes and see some pictures,
visit http://www.npr.org/templates/story/story.php?storyId=4531695.
I have one question in this issue to answer. It was prompted by
last week's Mathography column. If you have questions about
anything in this issue or any other mathematical topic, don't hesitate
to send me an email.
Dear Elvis,
The Mathography article in the
last newsletter contained information about Zorn's Lemma and the axiom
of choice. Just what are lemmas and axioms? Are
these just other words for theorems?
~Ammel
Dear Ammel,
Before we prove anything in mathematics we need some things to start
with. We decide that certain things are true and offer no proof
for them. These things are called axioms. A more modern word for
axiom is postulate. For
example, you have probably heard of the parallel postulate in
geometry. (Exactly one line can be drawn through any point not on
a given line parallel to the given line.) This is postulate is
assumed to be true in the study of Euclidean geometry, however
non-Euclidean geometries have different forms of this postulate.
Now when you prove a mathematical statement, it could be called a theorem (or sometimes proposition). For example, the
Pythagorean Theorem has hundreds of different proofs. A short
theorem that is used to help prove a larger one is called a lemma. A statement that is an
immediate consequence of a theorem is called a corollary. Often, corollaries
state the theorem in simpler language and in a way that can be more
applicable.
Finally, another word for a mathematical statement is conjecture. This is a
statement that seems true, but there is no proof yet written. A
famous example of a conjecture is the Goldbach Conjecture. It
states that every even integer greater than two is the sum of two
primes. Write a correct proof for this and there are a couple of
institutes that will give you a million dollars.
Elvis
A
mathematician is a machine for turning coffee into theorems.
Paul Erdös
