Introduction
On April 12, 2014, I went to the Southern California-Nevada Section of the MAA Spring Meeting, held at Concordia University in Irvine, CA. It has been more than ten years since I last went to a meeting. It feels good to go back.
Link on to their website: http://sections.maa.org/socalnv/
Highlights
Hal Stern, University of Redlands talked about how statistics play a significant role in sports, and how it can be useful in measuring and predicting performance.
A student poster session, which lasted an hour. Given how excellent and engaging the student's posters were, I was only able to look at four in the hour given for the poster sessions, and I only wished that I looked at more.
After the poster session, Rachel Levy of Harvey Mudd College, spoke about her journey as a mathematician deals with the media. She states that good incidental communication is important because it form an impression on anyone who is listening.
Levy starts by emphasizing the need for mathematicians to communicate and be aware of how they communicate. Harvey Mudd requires all math majors take a public speaking class. The class has shown to have positive affects on her students. Levy also emphasize the use affirmations, such as saying to students "You are thinking like a math major," leading students to believe that they can join the mathematics community.
Levy stresses the need for positive communication. She challenges the often used saying "So easy even your grandmother can do it," implying the referred groups are seen as novices. This inspired her to start her blog, Grandma got STEM, which highlights grandmothers and their mathematical and scientific accomplishments. After communication with a librarian, her blog gained a significant increase in readers, leading to radio interviews world-wide.
Link to Grandma got STEM: ggstem.wordpress.com
Levy talked about how Twitter can be used to advertise positive messages regarding mathematics, advertising math blogs and events, and send thank you notes.
In the final part of the presentation, Levy describes her dealings with the general press, stating it is a risky proposition, as the press can easily distort the intended message (either intentionally or unintentionally). She gives tips include having your talking points prepared, thinking about the audience, having photographs and videos ready, and making sure the one takeaway point is said during the interview.
This is my favorite part of the spring meeting.
The next talk was given by Perla Myers, University of San Diego. Myers describe her mission to change the prevailing feelings of fear and distraught when people think of mathematics. She specializes in training future teachers to enhance mathematical understanding and introduce activities designed to make learning math enjoyable, such as the use of origami.
Jamie Pommersheim, Reed College, gave he final talk of the day. The topic: dissecting squares into triangles of equal areas.
It is possible to accomplish this task by using an even number of triangles, but what about odd number of triangles? This question was first addressed by Fred Richman, who at first posed this questions to his students. After finding out the difficulty of this task, he turned the question to American Mathematical Monthly publication.
It was later proved by Paul Monsky that splitting the square into an odd number of triangles of equal area was impossible. Pommersheim devoted the rest of his talk to describe why, using two approaches.
The first approach describes Monsky's proof. Pommersheim starts by describing the 2-adic norm which is described by:
|| n || = || 2^t * r/s || = 2^(-t)
Where n is a rational number, and r and s are odd integers. The 2-adic norm of 0 is defined to be 0.
Examples of calculating the 2-adic norms:
|| 6 || = || 2^1 * 3 || = 1/2 (t = 1)
|| 16 || = || 2^4 || = 1/16 (t = 4)
|| 5/8 || = || 1/8 * 5/1 || = || 2^(-3) * 5 || = 8 (t = -3)
Consider a square with corner points (0,0), (1,0), (1,1), and (0,1). Each corner point and any point that helps form triangles within that square is assigned a "color". For each point (x,y), the color is assigned as follows:
The color A is assigned if:
* x has the largest 2-adic norm or
* x at least as big of 2-adic norm of either y or 1.
The color B is assigned if:
* y has the biggest 2-adic norm or
* the 2-adic norm of y is 1 and x has a 2-adic norm is less than 1.
The color C is assigned if both x and y have 2-adic norms less than 1.
For the corner points, the following colors are assigned:
(0,0) has the color C
(1,0) has the color A
(1,1) has the color A
(0,1) has the color B
It is next shown that three collinear points cannot have all three colors A, B, and C. Consider the three points (0,0), (x1, y1), and (x2, y2). Point (0,0) is assigned the color C.
The area of a general triangle can be calculated by:
Area = 1/2 * det([[x1, y1, 1],[x2, y2, 1],[x3, y3, 1]])
Using this formula above, the "area" is x1*y2 - x2*y1. We know the area of any line is 0. Hence, x1*y2 - x2*y1 = 0. And:
Show that a straight line can contain points of only two colours.
x1*y2 = x2*y1
Taking the two 2-adic norms of both sides to get:
|| x1*y2 || = || x2*y1 ||
|| x1 || * || y2 || = || x2 || * || y1||
This implies that both points must be assigned the same color which contradicts the assumption that a line made of three collinear points can have three different colors.
The proof goes on to use Sperner's Lemma, which states (briefly) given any dissection of square there exists of a tricolored triangle. Also, the 2-adic norm of an area of tricolored triangle is greater than 1. However, if the square is divided into an odd number of triangles, the 2-adic norm of each triangle is 1.
Pommersheim shows a second way to demonstrate that squares cannot be cut into an odd number of equal area triangles. He uses finds a polynomial of areas that is associated with each dissection.
For a square dissected into four triangles, the associated polynomial is
D + B - (A + C), which A, B, C, and D represent the areas for each triangle. In this case each area of the triangle is n/4 where n is the area of the square. Clearly, n/4 + n/4 - (n/4 + n/4) = 0, which is the desired result.
For a square dissected into six triangles the polynomial becomes:
(A + C + E)^2 - 4*A*B - (B + D + F)^2 + 4*D*F.
Substituting n/6, the area of each triangle in this case, and the value of the polynomial is 0.
Pommersheim eliminates triangle B. Now we have five triangles, each with area n/5. The polynomial becomes:
(A + C + E)^2 - 4*A*C - (D + F)^2 + 4*D*F
The trouble comes when we evaluate the polynomial with each area n/5, which leaves the value n^2/5. This shows that it is impossible to divide a square into an odd number of triangles of equal area.
There it is. I hope you find this enjoying, informational, and inspiring. I look forward to going to the next one. Until next time,
Eddie
This blog is property of Edward Shore. 2014
A blog is that is all about mathematics and calculators, two of my passions in life.