Alice T. Schafer

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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.