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.
J. Ernest Wilkins: His Life and Mathematics
The 1920s was a time in the history of the United States of America often looked upon as a decade of luxury and success, but this was not the case for all it’s citizens. In fact, in 1923, 29 African-Americans were lynched in this country . Throughout the United States, African-Americans were forced to experience grave injustices because of the color of their skin. Very few African-Americans were able to rise above this discrimination and succeed in the academic and especially the mathematical world. One such man was J. Ernest Wilkins.
J. Ernest Wilkins, Jr. was born in Chicago, Illinois in 1923 to Lucille Robinson and J. Ernest Wilkins, Sr. Both Lucille and Wilkins, Sr., received Bachelor’s degrees from the University of Chicago, Lucille in Education and Wilkins, Sr. in Mathematics. Wilkins, Sr. went on to become an attorney and was later appointed by President Truman as the Assistant Secretary of Labor and in 1958 to the Civil Rights Commission .
It can be inferred from the small amount of information available about the parents of Wilkins, Jr., that most likely the importance of learning and education was stressed in the home.
During his teenage years, Wilkins, Jr. was credited by national newspapers as “the negro genius” . At 13, Wilkins entered the University of Chicago, and four years later he received his BS in Mathematics .. During his undergraduate work, Wilkins was ranked in the top ten in the prestigious Putnam Competition . At nineteen, in 1942, Wilkins earned his Ph.D. at the University of Chicago [3,8]. His dissertation was entitled Multiple Integral Problems in Paramagnetic Form in the Calculus of Variations . Wilkins became only the seventh African American to obtain a doctorate in Mathematics. Later, in 1957 and 1960 respectively, Wilkins earned a Bachelor’s and Master’s degree in Mechanical Engineering .
After leaving the University of Chicago, Wilkins became a visiting member of the Institute for Advanced Study at Harvard University. Following this appointment, it was difficult for Wilkins to find a job at a research university . It is possible that this rejection was a result of racial discrimination in the United States at that time.
Another experience that allows us to understand more about the obstacles and discriminations Wilkins faced as an African-American mathematician, was described by Lee Lorch, a Caucasian-American human rights activist, in 1947
Wilkins was a few years past the Ph.D. ... He received a letter from the AMS Associate Secretary for that region urging him to come and saying that very satisfactory arrangements had been made with which they were sure he'd be pleased: they had found a ``nice colored family" with whom he could stay and where he would take his meals! The hospitality of the University of Georgia (and of the American Mathematics Society) was not for him - he refused. This is why the meeting there was totally white .
Because of this encounter, Wilkins has never since attended a meeting of the American Mathematical Society in the Southeast.
Wilkins’ first teaching position was instructing mathematics at the Tuskegee Institute in Alabama from 1943-1944. Following this, he was an Associate Physicist to Physicist on the Manhattan Project and worked at the Metallurgical Laboratory at the University of Chicago. During the time Wilkins spent at the Metallurgical Laboratory, research was being conducted on the atomic bomb .
Wilkins went on to hold jobs from many different companies and universities. From 1946 to 1950, Wilkins was a Mathematician for the American Optical Company, and for the next decade held several positions at the Nuclear Development Corporation of America. He was the Assistant Chairman of the Theoretical Physics Department, General Atomic Division of General Dynamics Corporation from 1960 to 1970, and a Distinguished Professor of Applied Mathematics at Howard University from 1970 to 1977. Wilkins worked for the next nine years for EG&G Idaho, Inc. and proceeded to enter retirement from1985 to 1990. Since this time, J. Ernest Wilkins has been a Distinguished Professor of Applied Mathematics and Mathematical Physics at Clark Atlanta University [1,4,8].
Some of the most prestigious honors awarded to Wilkins include being elected a Fellow of the American Association for the Advancement of Science, election to the National Academy of Engineering, Fellow and later President of the American Nuclear Society, chairman of the Army Science Board, Outstanding Civilian Service Medal from the Army. J. Ernest Wilkins has published over eighty papers in mathematics and mathematical physics. He has written over twenty unpublished but unclassified Atomic Energy Commission Reports [8, 10]. One of Wilkins’ greatest contributions to the education of minorities in mathematics in the United States was his help in the establishment of a doctoral program in Mathematics at Howard University. This program was the first to be set up at a predominately African-American school in the United States .
The most important work done by J. Ernest Wilkins is considered to be “the development of radiation shielding against gamma radiation, emitted during electron decay of the Sun and other nuclear sources” [1,4]. Wilkins developed ways to mathematically calculate the amount of gamma radiation absorbed by a given material. The technique he developed is extremely useful among researchers in space and nuclear science projects .
Although, undoubtedly, this research has been the most important of Wilkins’ lifetime, this paper will focus on another area of his extensive research. In the late 1940’s, Wilkins published two papers concerning mathematical or geometric surfaces, The Contact of a Cubic Surface with a Ruled Surface and Some Remarks on Ruled Surfaces [12,13].
Geometric solids, such as spheres or cylinders, occur in three-dimensional space. The geometric surface encloses the space or is the boundary of the solid. The two surfaces focused on by Wilkins in the above research are ruled and cubic surfaces. Ruled surfaces were first explored by Jesuits in the seventeenth century . Some basic examples of a ruled surface include cylinders and cones. These are classified as such because they are created by sweeping a straight line around a curve. A cylinder, for example, is formed when a normal line to a circle sweeps around the circumference of that circle. Cones are created by sweeping a line in a circular motion from a single point. It is simple to visualize the line that sweeps around the curve in both the cone and cylinder, but there are some other ruled surfaces that are not as easy to identify. There is an equation that can be used to discover whether or not a surface is ruled.
The equation used to define a ruled surface is x(s,v)=α(t)+v*w(t), tєI and vєR where α(t) is the curve, and w(t) is the vector which sweeps around the curve. It is easy to explore this equation by using an hyperboloid . The curve in the hyperboloid on which the line sweeps is a circle. In this case, we will use the unit circle, x2+y2=1. This can be written as α(s)=(cos(s), sin(s), 0). To find w(s), we use the equation w(s)= α‘(s) + e3 where e3 is a unit vector of the z-axis. α’(s)=(-sin(s), cos(s), 0), so w(s)=(-sin(s), cos(s), 0) + (0,0,1) = (-sin(s), cos(s), 1). Going to the original equation, we get x(s,t)=(cos(s), sin(s), 0)+v(-sin(s), cos(s), 1). By simplifying, x(s,t)=(cos(s)-v*(sin(s), sin(s)+v*cos(s),v). If we substitute these formulas for x,y, and z, in x2+y2-z2 = 1, after, simplification, we get, 1+v2-v2. Because x2+y2-z2=1 and 1+v2-v2=1, we know x2+y2-z2 = 1 must be a ruled surface .
In this picture of the hyperboloid, it is easy to see the line that is sweeping around the circle.
Here is a picture of a building that is an hyperboloid from Japan in the 1940’s.
Another ruled surface is a saddle surface, or hyperbolic paraboloid. Saddles surfaces are called this because their shape resembles that of a saddle used in riding horses or bicycles. The saddle equation, shown here, is defined as kz=x2+y^2 . A saddle is created by sweeping a line about a hyperbola. Another equation for a hyperbolic parabloloid is z=kxy. The parametric equation of the saddle surface is created in much the same way as the hyperboloid. After taking the intersection of the family of curves in the z =0 plane, one gets α(t)=(t,0,0) and w(t)=(0,1/k,t). By using the formula for a ruled surface, we get, x(t,v)=(t,v/(sqrt(1+k2t2)),vkt/( sqrt(1+k2t2))). If we use the equation z=kxy to check the parametric equation, we get vkt/( sqrt(1+k2t2) = k* t*v/(sqrt(1+k2t2)). Since this is obviously a true statement, we know x(t,v) is an accurate parametric equation of this saddle surface . The following is a picture of a saddle surface.
The equation for the helicoid is y=x*tan(z/k). Visually, it is similar to the double helix form of DNA. The helicoid is created by sweeping a vector about a circle similar to the sweeping line of the hyperboloid. The difference is that at the same time as the line is rotating about the curve, the z values of the line are also increasing. In other words, the vector goes up as it sweeps around.
One important fact about ruled surfaces is that they consist only of straight lines. A practical application of ruled surfaces is that they are used in civil engineering. Since building materials such as wood are straight, they can be thought of as straight lines. The result is that if engineers are planning to construct something with curvature, they can use a ruled surface since all the lines are straight.
The other form of surfaces discussed in the papers by Wilkins is cubic surfaces. The cubic surface is a surface that can be defined by a third degree polynomial in three-dimensional space. One example of a cubic surface is x^3+y^3+z^3=1. Algebraic properties are used to study these figures. One important fact about cubic surfaces is the idea that each one has exactly twenty-seven straight lines on it. It is true, however that in regular, three-dimensional space, all of these lines cannot be easily seen. To “see” all of the lines on a surface, it is necessary to use a complex, three-dimensional space .
Cubic surfaces were first researched around the early to mid 1800s. One of the first people to study these surfaces was Arthur Cayley. Cayley was the first to observe the number of lines on the cubic surface, and one unique cubic surface is named after him. Here is a picture of the Cayley cubic.
In his research, Wilkins used the definitions and theorems about both ruled and cubic surfaces to prove theorems about their relatedness. He used a general form of any ruled surface and at times in his second paper, dealt with Cayley’s cubic. Wilkins explored the ways in which the two types of surfaces contacted each other [12,13]. This included the number of places where they were in contact. Wilkins used power series expansion to prove the theorems in his papers.
J. Ernest Wilkins is still impacting the formation of mathematical research in the United States of America. He is influencing young minds at Clark Atlanta University in Atlanta, Georgia as a Professor of Mathematics. Throughout his life, J. Ernest Wilkins has risen above the discrimination of his race and has succeeded in making an impact in the history of mathematical research. Some have described Wilkins as having one of the “most exemplary careers of scholarship and application of an American mathematician/physicist/engineer in the 20th century” .
1. Brown, Mitchell. “J. Ernest Wilkins, Jr.: Physicist Mathematician, Engineer.”
[Online]. Available from: http://www.princeton.edu/~mcbrown/display/wilkins.html
comments: small overview of life, some references, but not available
2. do Carmo, Manfredo, Differential Geometry of Curves and Surfaces, Prentice Hall,
Inc., 1976, 188-195.
comments: good for understanding mathematics behind equations of ruled surfaces.
3. Great African American Inventors and Engineers [Online]. Available from:
comments: brief paragraph about life and contributions.
4. Haynie, Edward. “Ernest Wilkins, Jr.” [Online]. Available from:
comments: small overview of life
5. Historic Contributions of Black Scientists and Engineers [Online]. Available from:
comments: gives information about accomplishments of Wilkins.
6. Hyperboloid, The [Online]. Available from:
comments: gives information about hyperboloids as well as good graphs.
7. J. Ernest Wilkins, Jr. [Online]. Available from:
comments: good overview of life, contains information about education, employment, honors.
8. J. Ernest Wilkins, Jr. – Mathematicians of the African Diaspora [Online]. Available
comments: gives background of family and racial discrimination as well as accomplishments and honors.
9. Ksir, Dr. Amy E., SUNY Stoneybrook.
comments: sent information to me about cubic surfaces.
10. Kenschaft, Patricia. “Black Men and Women in Mathematical Research”, Journal of
Black Studies, December, 1987, 19:2, 170-190.
comments: interesting information about employment and information about time period.
11. Six Types of Ruled Surfaces [Online]. Available from:
comments: good overview of ruled surfaces
12. Wilkins, Jr., J. Ernest. “The Contact of a Cubic Surface with a Ruled Surface,”
American Journal of Mathematics, January, 1945, 67:1, 71-82.
comments: hard to understand
13. Wilkins, Jr., J. Ernest. “Some Remarks on Ruled Surfaces,” Bulletin of the
American Mathematical Society, 1949, 55, 1169-1176.
comments: hard to understand.