These are actual student papers that were not designed to be web pages. They may contain historical, grammatical, mathematical, or formatting errors. These papers were graded using the criterion mentioned in the paper directions, and the writing checklist. The test review sheets and the WebCT tests are good indicators of the mathematics that was discussed in class during and/or after each presentation.
Alice T. Schafer
Schafer (maiden name Turner) was born in 1915 in the state of Virginia. Early in her life she lost her parents and was raised by two of her aunts. One of her aunts she lived with in Scottsburg while the other helped her financially. Her family was very supportive of Alice’s decisions. So, when they found out that Schafer was eager to define herself with math they were happy to help.
Alice may have had a supportive family but it wasn’t always easy to get the necessary support to get what she wanted. When she decided that she wanted to obtain a degree in mathematics from the University of Richmond she was slightly set back by her high school principle. When asked to write a letter of recommendation for Alice he replied “girls shouldn’t do math”. Afterwards he never sent the letter. Awesomely, Alice’s performance in school was stronger then the biased opinion of her principal and she was accepted to the University of Richmond with a full scholarship.
The University of Richmond turned out to be less liberal then when they gave her the scholarship. Women were not allowed in the library. As a result every time Alice wanted to do research she had to order the book she needed and had to read it in a study room designated just for women. Also, only women taught her for the first two years of her math degree. This may seem fair but the reason for this was because women were only allowed to teach up to the level of analytic geometry. The men taught the higher level mathematics.
In analysis she had a professor who had the same reasoning as her past principal. Her professor was known of saying that he wanted to fail every woman that attended his classes. However Alice proved her position when she won the Crump prize which her professor took part in grading. Soon after she graduated with a B.S. in Mathematics.
Alice wasn’t stopping with a B.S. in math. So after teaching high school for a few years she began to attend the University of Chicago where she studied metric and projective differential geometry. In these areas she would obtain her masters and doctorate. Afterwards she pursued careers in faculty at Douglass College, University of Michigan, Swarthmore College and Wellesly College. She also wrote on World War 2 in her spare time and after retiring became a lecturer and then the chair of the math department of Marymount.
Alice wasn’t just satisfied with being an accomplished mathematician. She also felt the need to fight against discrimination against women. The urge started in high school but was being full-filled during college. At the University of Richmond she was mainly responsible for the opening of the library to women. Sadly, she was kicked out on her first day in the library for laughing out loud while reading a book.
Alice didn’t just stop with the library incident. More importantly, she helped the start of the AWM (Association for Women in Mathematics) and then was pronounced the second president of the organization in 1972. She took part as an active member from the start and is even a member now.
In the year of 1989 a women in the AWM came up with a prize that would be awarded to high school girls with a high degree of excellence in mathematics with the desire to continue in math throughout college. Because of Schafer’s love of math and her desire to fight against discrimination the award was dedicated to Schafer. It was named the Alice T. Schafer award.
Alice’s thesis was on Two Singularities of Space Curve. We came across a paper, which explains much of her thesis. We have decided to explore some of her ideas mentioned. More specifically we will discuss tangent and normal vectors and follow them up with an explanation on oscillating planes.
The tangent is defined as F’(x) of a parameterized curve. If we were given F(x)=(a cos(t), a sin(t), bt) where t is an element of R. The tangent vector is F’(x)=(-a sin(t),a cos(t),b), note that we have taken the derivative with respect to t. Using this same example we can find the normal vector. The normal vector is perpendicular to the tangent vector and is defined as F”(x). So with the last example the normal vector would be F”(x)=(-a cos(t),-a sin(t), 0). You can see this on the graph at the end of the paper.
The length of the normal is determined by the sharpness of the curve. The sharper the curve the longer the normal or as the curve becomes more like a line the normal shortens. A real life application of this is driving your car on a curvy mountain road. Think about when you drive around sharp turns and long drawn out turns. The force or the normal which pulls you inward is stronger on the sharp turn then it is on the long drawn out curve.
Recall that a plane is built up of at least two vectors. So the tangent vector and the normal vector build up a plane. So as the graph of a curve grows the plane (built from the tangent vector and the normal vector) moves or oscillates with the graph. Let’s think about simple surfaces to start this idea.
The first example we will use is a circle. Look at the picture and see the planes formed.
As seen in the picture the tangent and the normal changes direction with the surface of the circle. As a result of the change the plane also has to change. This is what we describe as the oscillating plane. Note that the circle is a “nice” surface. Meaning that it has no sharp points or edges. Because there are no edges the plane can oscillate smoothly with the path of the surface.
Not all surfaces are “nice” though. So let’s look at one that has some sort of edge or point.
Look at this picture of a cone and try to see what happens with the plane at its point.
As you can see at the point of the cone there is infinitely many tangents. As a result there are infinitely many planes. To stop confusion we just say that the plane is undefined at such a point. To understand this more think about Calculus. Remember that you can’t take a derivative of a point or sharp edge on a graph. So in this case again you cannot define a tangent which leads to an undefined plane.
In conclusion, these terms that we have just talked about build the introduction to Schafer’s thesis. There are many more sections that are much more in depth. We feel the knowledge of the normal and tangent vectors, and oscillating planes on a curve should be good prerequisite insight to her thesis.