Off on a Tangent
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
 November 3, 2017 Vol. 16, No. 5

Cryptograph will be the topic of next week's colloquium

Title: The Mathematics of Secrecy: A Brief Introduction to Cryptography
Speaker:  Dr.  Charles Cusack
Time:  Thursday, November 9 at 4:00 pm
Place:  VanderWerf 102
Abstract: The main purpose of cryptography is to enable two people to communicate privately over a public channel. In this talk I will give a brief introduction to private-key and public-key cryptography. I will discuss the shift cipher, substitution cipher, Vigenere cipher, one-time-pad, and RSA. Along the way I will discuss modular arithmetic, Euler's phi function, the Euclidean algorithm, and binary exponentiation. I will conclude with a very brief survey of other uses of cryptography.

You can’t tell a gerrymandered district by its shape

Gerrymandering, the process to select political districts so that to make it advantageous to a certain political party, is a hot topic these days.  In the past few months, there have been petitions widely distributed in Michigan to get a ballot proposal to end the practice.

But how do you tell if the boarders of a Congressional district have been determined through gerrymandering? Can you tell by its shape?

According to a couple of mathematicians, the answer is no. The pair recently published a paper on this topic. You can read more about this in the Ohio State News and Dustin Mixon's blog. You can also read there paper here.

Problem Solvers of the Fortnight

In our last problem of the fortnight we had a red circle inscribed in an equilateral triangle of side length w (see 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 (see Figure 2).  The next step in this process is shown in Figure 3.  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 Kevin Catalfano, Philip LaPorte, Nick Lillrose, Dane Linsky, Andrew Nguyen, Alex Osterbaan, Eleda Plouch, Sam VanderArk, and Eildert Zwart
for submitting correct solutions.

Figure 1

Figure 2

Figure 3

Problem of the Fortnight

Starting with the vertices of a square, P1 = (0,1), P2 = (1,1), P3 = (1,0), P4 = (0,0), we construct the following points as shown in the figure on the right (click on the figure to enlarge it).  In the figure, P5 is the midpoint of the line segment P1P2, P6 is the midpoint of the line segment P2P3, etc.  The infinite sequence of points P1, P2, P3, P4, P5, P6, P7, ... approaches a point P inside the square.
  1. If the coordinates of Pn are (xn, yn), find xn and yn for 1 < n < 13.
  2. Show that (1/2) xn + xn+1 + xn+2 + xn+3 = 2 for 1 < n < 9.  Find a similar formula for the y-coordinates and show that it is true for 1 < n < 9.
  3. Find the coordinates of P.
Write your solution -- showing all your work, please! -- on a galaxy, hurricane, or your favorite type of spiral and drop it by the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, November 10.  As always, be sure to include your name and the name(s) of your professor(s) -- e.g. Point Les, Dr. Mini Mize -- on your solution.

Pure mathematics is the world's best game.  It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly.  It's free.  It can be played anywhere - Archimedes did it in a bathtub. 

~Richard J. Trudeau, Dots and Lines

Off on a Tangent