Off on a Tangent
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
April 19, 2017 Vol. 15, No. 12

This week's colloquia will take graph theory to the extreme!

Title: What is Extremal Graph Theory?
Speaker: Dr. Lauren Keough, GVSU
Time: 4:00 p.m. on Thursday, April 20, 2017
Place:  VanderWerf 102

Abstract: Graph theory is the study of relationships that come in pairs.  There are many such relationships occurring naturally, think of matching medical students to residencies, friendship on social networks, or even pairing animals with the regions in which they live.  From these relationships we can draw graphs. For example, for each person on a social network draw a dot, and draw a line segment between two dots if the people are “friends”.  Graph theory is, broadly, the study of these pictures with these dot lines.  So, what could extremal graph theory be? The word “extremal” may sound familiar because of Calculus 1 –perhaps you remember finding  “local and absolute extrema." What does extremal mean in graph theory? Come to find out! We’ll survey results from extremal graph theory dating back to people who were thinking about problems like this all the way back in 1907.

Hope students compete in the Lower Michigan Mathematics Competition

Seven students (clockwise around the blue sign when viewed from above, starting at right:  Richard Edwards, Russell Houpt, Chris McAuley, Elizabeth Lindquist, Amanda Rensmo, Jimmy Cerone, and Yong Chul Yoon) traveled to University of Michigan Flint on Saturday, April 8 to compete in the Lower Michigan Mathematics Competition.  There were 26 registered teams from 11 Michigan colleges and universities.  It was a beautiful day and the competition included some very clever questions (including one about muffin preferences).  They even got to see the lovely Flint River which runs through campus.

The results should be announced later this month. To view the problems the students worked on click here.

Math in the News: The most beautiful mathematical equation

When I teach calculus or precalculus and we first come to exponential equations, I usually tell the students that this is my favorite equation. It is so simple, yet can be very surprising and beautiful. I usually get a chuckle or two from students that might not think one can have a favorite type of equation or there is beauty in what they have been studying. Well there is beauty in equations and research has proven this.

This past Sunday, New York Times Op-Ed writer Richard Friedman wrote about the world's most beautiful mathematical equation.  He discussed research done that shows that mathematicians' brains respond to the beauty of an equation much the same way that they respond to the beauty of art or music. While not all equations, like not all art or music, is considered beautiful to everyone. The research showed that the most beautiful equation was Euler's Identity.  You can read Friedman's article here and see an explanation of Euler's Identity here. Finally (and I don't advocate that you go this far), take a look at what one mathematics major did to show his love of Euler here.

Problem Solvers of the Fortnight

Figure 1
In the last problem of the fortnight we saw the following exchange between Math Man and Vectoria:

"Hey there, Mathlete!"
"Oh, hi, Vectoria.  Did you have a nice spring break?"
"Yeah, it was great!  My family and I made a spur-of-the-moment decision to go someplace, and we pseudo-randomly chose to visit Monte Carlo!"
"Wow!  Sounds like kind of a gamble just to up and go halfway around the world, but I bet your vacation was a winner!"
"It was.  One we decided where to go, we were 'all in,' as the kids say these days.  A couple days were cool and rainy, but you play the hand you're dealt, right?"
"Sure enough," Math Man says rather absent mindedly, as his eyes return to the paper he was looking at before Vectoria stopped by.
"What'cha workin' on?"
"Oh, just something," Double M says, "or, rather, s-u-m thing," he says, spelling it out and chuckling to himself at his clever turn of phrase.  "That was just a little joke.  I'm working on a math problem that involves a sum."
"I should have known it would be s-u-m thing like that" says Vectoria, spelling it out to volley his joke back to him.  "What's the problem?  You know I love those things."
"Well . . . A red circle is inscribed in an equilateral triangle of side length w, like this," says Math Man, pointing to figure 1.  "In each of the three corners of this triangle, a smaller equilateral triangle is drawn tangent to the red circle, and a cyan circle is inscribed in this smaller triangle, like so," he says, pointing to figure 2.  "The next step in this process is shown here," says Math Man, showing her figure 3.  "The question is: If this process of drawing smaller equilateral triangles with inscribed circles is continued indefinitely, what is the total area of all of the circles in terms of w?"

Congratulations to Richard Edwards, Ford Fishman, Ce Gao, Tyler Gast, Calvin Gentry, Ethan Heyboer, Karthik Karyamapudi, Philip LaPorte, Dane Linsky, Colin McMurrary, Alex Osterbaan, John Peterson, Mark Powers, Kip Schomber, Caleb Stuckey, Sam VanderArk, Allison VanZanten, Evan Vogel, Jim Williamson, and Yizhe Zhang  -- all of whom correctly solved the problem in the last issue of Off on a Tangent.

Figure 2

Figure 3

Problem of the Fortnight

"Hi, V.  What'cha doin'?"  Math Man says to Vectoria, who seems preoccupied by the clock on the enormous clock tower at DeVos Fieldhouse.
"Oh, hi, Math Man.  I was just seeing what time it was. . . .  "
"Um, well, I'm kinda surprised you don't know this, but if the big hand is on the 12 and the little hand is on the one, we say it's one o'clock."
"Thanks, Double M!  That clears up a whole lifetime of confusion about telling time on analog clocks," said Vectoria sarcastically.
"Okay, okay.  I deserve that.  But why were you just staring at the clock?
"Well, I started thinking."
"Really?  What about?"
"Well, how long do you think that minute hand is?" asked Vectoria.
"Oh, probably about 6 feet."
"And how long do you think the hour hand it?
"Maybe 4 feet."
"If that's so, then how fast is the distance between points at the end of the the minute hand and the hour hand changing precisely at one o'clock?"
"Wow.  I don't think I've ever wondered that just looking at a clock, but it's kind of a cool question, now that you mention it."
"I thought so, too," said Vectoria, "but I haven't been able to solve it.  Wanna work on it with me?
"Sure!" said Math Man.  "I thought you'd never ask!"

Help Vectoria and Math Man by submitting your solution (not just the answer) on the back of a picture of one of the world's famous clock towers by 3:00 p.m. on Friday, April 28.  As always, be sure to include your name as well as the name(s) of your math professor(s) -- e.g. Stew D. Uss, Professor Hal Palott -- on your solution.  Good luck and have fun!   

Italy, and the spring and first love all together should suffice to make the gloomiest person happy.
Bertrand Russell

Off on a Tangent