J. Ernest
Wilkins: His Life and Mathematics

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1923- present

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 [10]. 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
[8].

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” [8]. At 13, Wilkins
entered the University of Chicago, and four years later he received his BS in
Mathematics [7].. During his
undergraduate work, Wilkins was ranked in the top ten in the prestigious Putnam
Competition [8]. 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__ [1]. 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 [1].

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 [8]. 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 [8].

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 [5].

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
[7].

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 [5].

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 [11]. 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), tє**I** and vє**R **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
[6]. The curve in the hyperboloid
on which the line sweeps is a circle.
In this case, we will use the unit circle, x^{2}+y^{2}=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) + e^{3} where e^{3 }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 x^{2}+y^{2}-z^{2}
= 1, after, simplification, we get, 1+v^{2}-v^{2}. Because x^{2}+y^{2}-z^{2}=1
and 1+v^{2}-v^{2}=1, we know x^{2}+y^{2}-z^{2}
= 1 must be a ruled surface [2].

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=x^{2}+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+k^{2}t^{2})),vkt/(
sqrt(1+k^{2}t^{2}))).
If we use the equation z=kxy to check the parametric equation, we get
vkt/( sqrt(1+k^{2}t^{2}) = k* t*v/(sqrt(1+k^{2}t^{2})). Since this is obviously a true
statement, we know x(t,v) is an accurate parametric equation of this saddle
surface [2]. 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 [9].

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 20^{th} century” [7].

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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:

http://www.uwm.edu/StudentOrg/NSBE/bie.html

comments: brief paragraph about life and
contributions.

4. Haynie, Edward.
“Ernest Wilkins, Jr.” [Online].
Available from:

http://www.onmy.com/haynie/new_page_18.htm

comments: small overview of life

5. Historic Contributions of Black Scientists and
Engineers [Online]. Available
from:

http://bgess.berkeley.edu/bios/text/shtml

comments:
gives information about accomplishments of Wilkins.

6. Hyperboloid, The
[Online]. Available
from:

http://www.math.hmc.edu/faculty/gu/curves_and_surfaces/surfaces/hyperboloid.html

comments:
gives information about hyperboloids as well as good graphs.

7. J. Ernest Wilkins, Jr. [Online].
Available from:

http://www.maa.org/summa/archive/WilkinsJ.htm

comments: good overview of life, contains
information about education, employment, honors.

8. J. Ernest Wilkins, Jr. – Mathematicians of the African Diaspora [Online]. Available

from: http://www.math.buffalo.edu/mad/PEEPS/wilkins_jearnest.html

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:

http://faculty.fairfield.edu/jmac/rs/sixmodels.htm

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.

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