IN MEMORIAM
R. H. BING
It was a dark and stormy night
when R.H. Bing volunteered to drive some stranded mathematicians
from the foggedin Madison airport to Chicago. Freezing rain pelted
the windscreen and iced the roadway as Bing drove on  concentrating
deeply on the mathematical theorem he was explaining. Soon the windshield
was fogged from the energetic explanation. The passengers too had
beaded brows, but their sweat arose from fear. As the mathematical
description got brighter, the visibility got dimmer. Finally, the
conferees felt a trace of hope for their survival when Bing reached
forward  apparently to wipe off the moisture from the windshield.
Their hope turned to horror when, instead, Bing drew a figure with
his finger on the foggy pane and continued his proof  embellishing
the illustration with arrows and helpful labels as needed for the
demonstration.
Two
of Bing’s mathematical colleagues, Armentrout and Burgess, independently
told us versions of this memorable evening. Those of us who knew Bing
well avoided raising mathematical questions when he was driving.
Bing’s love for working on mathematical
problems was matched by his success at solving them. He was a mathematician
of international renown and was one of the most distinguished UT professors
of mathematics in The University's history. His professional vita is
impressive  including authorship of seminal research papers in several
branches of topology and honors such as membership in the National Academy
of Sciences and presidencies of both the Mathematical Association of
America and the American Mathematical Society. But we will remember
him most for his zest for life which infected everyone around him with
a contagious enthusiasm and good humor.
R.H. Bing started and ended in Texas.
He was born on October 10, 1914, in Oakwood, Texas, and there he learned
the best of the distinctively Texas outlook and values. What he learned
in Oakwood guided him clearly throughout his life. He had a strong Texas
drawl which became more pronounced proportionate to his distance from
Texas;
and he spoke a little louder than was
absolutely necessary for hearing alone. He might be called boisterous
with the youthful vigor and playful curiosity that he exuded throughout
his life. He was outgoing and friendly and continually found ways to
make what he did fun. You could hear him from down the hall laughing
with his T.A.’s while grading calculus exams or doing other work
that deadens most people. He did not sleep well and when he woke at
4 a.m., he would get up and work. He especially enjoyed working on things
requiring loud hammering at that hour on the grounds that if you are
going to be up at 4, the family should know about it. He practiced the
traditional Texas value of exercising independent judgment, both in
general matters and in matters mathematical, and treated people kindly
and gently  unless he knew them, in which case it was more apt to
be kindly and boisterously.
Both of Bing’s parents were involved
in education. His mother was a primary teacher and his father was the
superintendent of the Oakwood School District. Bing’s father died
when R.H. was five, so Bing most remembered his mother's impact on his
character and interests. Bing attributed his love for mathematics to
his mother’s influence. He recalled that she taught him to do mental
arithmetic quickly and accurately and to enjoy competition both physical
and mental.
After high school, Bing enrolled in
Southwest Texas State Teachers College in San Marcos (now Southwest
Texas State University) and received his B.A. degree in 1935 after two
and a half years there. Later in life, Bing was named as the second
Distinguished Alumnus of Southwest Texas State University. The first
person so honored was Lyndon Baines Johnson. Bing’s college education
had prepared him as a high school mathematics teacher. He also was a
high jumper on the track team and could jump his own height  which
was over six feet.
Bing’s final academic position
was as the Mildred Caldwell and Blaine Perkins Kerr Centennial Professor
in Mathematics at The University of Texas at Austin, but his first academic
appointment was as teacher at Palestine High School in Palestine, Texas.
There his duties included coaching the football and track teams, teaching
mathematics classes, and teaching a variety of other classes, one of
which was typing. His method of touch typing involved anchoring his
position over the keys by keeping some constant pressure on his little
fingers. This habit was hard to break, apparently, because later he
said that when he used an electric typewriter or computer keyboard (neither
of which he did often) he tended to produce large numbers of extraneous
"a’s".
Nowadays one frequently hears complaints
about a school system which gives the football coach the added assignment
of teaching a mathematics class. One wonders if those football boosters
of a bygone day in Palestine complained to the local school board about
a real mathematics teacher coaching the football team.
In an effort to improve public school
education in the ‘30’s, the Texas Legislature had approved
a policy whereby a teacher with a Master's degree would receive more
pay than a teacher with a Bachelor’s degree. So, many teachers
saved (and scrimped) during the ninemonth session and went to summer
school during the three summer months in an effort to upgrade their
talents and their salaries. Bing was among them.
R.H. had begun public school teaching
in 1935, and by taking summer school courses at The University of Texas
at Austin, he had earned a Master of Education degree in
1938. During one summer, Bing took
a course under the late Professor R.L. Moore. Moore was inclined to
deprecate the efforts of an older student such as Bing was, so Bing
had to prove himself. But he was equal to the task.
Bing’s summers in Austin provided
him with more than mathematics, however, for during one class Bing met
Mary Blanche Hobbs. They took evening drives up to Mt. Bonnell and must
have enjoyed them because they married in 1938, and in 1974, they built
a home on Mt. Bonnell.
Bing continued to take some summer courses
while teaching in the high schools. In 1942 Moore was able to get Bing
a teaching position at The University which allowed him to continue
graduate study to work towards a doctorate, and to try his hand at research.
[This practice of allowing a person of instructor rank or higher to
work towards an advanced degree was allowed in those days.]
An unofficial rating scheme sometimes
used by R.L. Moore and his colleagues went something like this: You
could expect a student with Brown’s talents and abilities every
year; you could expect a student with Lewis’s talents and abilities
once every four years; but a student with Smith’s talents and abilities
came along only once in twelve years. Bing’s talents and abilities
threw him in the twelveyear class, or in an even higher class, since
he is one of the most distinguished mathematicians ever to have received
his degree from The University of Texas at Austin. Several of Moore’s
later graduate students have written that in the days after Bing, Moore
used to judge his students by comparing them with Bing  probably
not to their advantage.
Bing received his Ph.D. in 1945 
writing his dissertation on "Planar Webs." Planar webs are
topological objects now relegated to the arcana of historical topological
obscurity. The results from his dissertation appeared in one of his
earliest papers in the Transactions of the American Mathematical Society.
He told us recently that the Transactions had sent him fifty reprints
at the time and if we were interested we could have some because he
still
had fortynine or so left.
But Bing did not have long to wait for
recognition of his mathematical talent. He received his Ph.D. degree
in May 1945, and in June 1945, he proved a famous, longstanding unsolved
problem of the day known as the Kline Sphere Characterization Problem.
When word spread that an unknown young mathematician had settled this
old conjecture, some people were skeptical. Moore had not checked Bing’s
proof since it was his policy to cease to review the work of his students
after they finished their degrees. Moore believed that such review might
tend to show a lack of confidence in their ability to check the work
themselves. So when a famous professor wired Moore asking whether any
firstclass mathematician had checked the proof, Moore replied, "Yes,
Bing had."
Primarily because of the renown among
mathematicians generated by his having solved a famous conjecture, Bing
was offered positions at Princeton University and at the University
of Wisconsin, Madison. Moore naturally wrote letters of recommendation.
One comment he made was that, although the Kline Sphere Characterization
Problem was a much better known topic than that of planar webs, Moore
felt that it was Bing’s work on planar webs that demonstrated that
Bing had the mathematical strength to be an outstanding mathematician.
One of the leading topologists of the
time was at Princeton, but Bing did not wish to follow in anyone’s
footsteps, so in 1947 he accepted a position at Wisconsin. He remained
at Wisconsin for 26 years except for leaves: one at the University of
Virginia (194950), three at the Institute for Advanced Study in Princeton
(195758, 196263, 1967), one at The University of Texas at Austin (197172),
and brief teaching appointments elsewhere. He returned to The University
of Texas at Austin in 1973; but it was during his tenure at the University
of Wisconsin, Madison that his most important mathematical work was
done and his prominent position in the mathematical community established.
Bing’s early mathematical work
primarily concerned topics in general topology and continua theory,
a branch of topology that describes properties of connected compact
sets. He proved theorems about continua that are surprising and still
central to the field. Among these results is Bing’s characterization
of the pseudo arc as a homogeneous indecomposable, chainable continuum.
This result contradicts most people's intuition about the pseudo arc
and directly contradicted a published, but erroneous, "proof"
to the contrary. He continued to do some work in continua theory throughout
his career; including directing a Ph.D. dissertation in the subject
at UT in 1977.
General topology is not really general.
Instead it refers to a special branch of topology that considers questions
about topological spaces that may lack many of the geometrical aspects
of subsets of the Euclidean spaces. Around 1950, one of the great unsolved
problems in this field was the problem of giving a topological characterization
of the metrizability of spaces. In 1951, Bing gave such a characterization.
A Japanese and a Russian mathematician proved similar, independent results
at about the same time, so now the result is referred to as the BingNagataSmirnov
Metrization Theorem. That 1951 paper of Bing has probably been referred
to in more papers than any other of his papers, even though he later
was identified with an altogether different area of topology.
Nowadays if you refer to "Bingtype
topology," you are referring to a certain style of geometric analysis
of Euclidean 3space that came to be associated with Bing because of
the fundamental work he did in the area and the distinctive style with
which he approached it. The first paper Bing wrote in this area appeared
in 1952 and contains one of his bestknown results. The result in this
paper describes a method of shrinking geometric objects in unexpected
ways. When Bing first worked on the question considered in this 1952
paper, he naturally did not know whether it was true or false. He claimed
that he worked two hours trying to prove it was true, then two hours
trying to prove it was false. When he originally worked on this problem,
he used collections of rubber bands tangled together in a certain fashion
to help him visualize the problem. The mathematics that Bing did is
very abstract, but he claimed to get ideas about these abstruse problems
from everyday objects. A final note about this problem involves a paper
which Bing wrote in 1984 containing one of his last results. If one
shrinks the rubber bands in the manner described in Bing’s 1952
paper, each rubber band becomes small in diameter, but very long. It
became interesting to know whether one could do a similar shrinking
without lengthening the bands  in other words, could you do the same
thing with string as Bing had proved could be done with rubber. Bing’s
original procedure had been studied by numerous graduate students and
research mathematicians for more than 30 years, and yet no one had been
able to significantly improve Bing’s shrinking method. It was left
for
Bing himself to prove that "Shrinking
without lengthening" (the title of this final paper) is possible.
Bing’s results in topology grew
in number and quality. He proved several landmark theorems and then
raised lots of related questions. Because of his habit of raising questions,
many other mathematicians and students were able to prove good theorems
in the area of mathematics which he pioneered. He emphasized the importance
of raising questions in one’s papers and encouraged his students
and colleagues to do so. He felt that mathematicians who read a paper
are often more interested in what remains unknown than they are interested
in what has been proved.
The period from 1950 until the mid60’s
was Bing’s most productive period of research. He published about
115 papers in his lifetime  most during this period at the University
of Wisconsin, Madison.
R.H. and his wife, Mary Blanche, had
a son, Robert H., by the end of World War II. In Madison the son was
joined by their three daughters, Susan Elizabeth Hannah, now of Milwaukee,
Virginia Gay Hundley of Princeton, N.J. and Mary Patricia Bing of Union
Grove, Wisconsin. There are six grandchildren. The entire family has
always been very close and supportive and full of fun. It was traditional
in the Bing family to give R.H. toys for Christmas and his birthdays.
He felt that part of his job was to give his wife a purpose in life.
Under this guiding principle he provided her with many challenges. His
granddaughter Beth remembers an occasion when Bing decided that it would
be fun to surprise Mary by lining up some hundred toys in a big curving
line all over the house. No doubt she was thrilled. R.H. and Mary were
also dedicated to their activities with the Presbyterian Church, where
R.H. served as an elder. One of the Bing daughters remembers sitting
with R.H. in church one Sunday and noticing how absorbed he appeared
to be in the sermon. She was not quite so confident in where his thoughts
were directed, however, when he reached forward to erase an errant symbol
on an imaginary blackboard in the air.
His research success brought him honors,
awards, and responsibilities. He was quickly promoted through the ranks
at the University of Wisconsin, becoming a Rudoph E. Langer Research
Professor there in 1964. He was a Visiting Lecturer of the Mathematical
Association of America (195253, 196162) and the Hedrick Lecturer for
the Mathematical Association of America (1961). He was chairman of the
Wisconsin Mathematics Department from 1958 to 1960, but administrative
work was not his favorite. He was President of the Mathematical Association
of America (196364). In 1965, he was elected to membership in the National
Academy of Sciences. He was Chairman of the Conference Board of Mathematical
Sciences (196667) and a U.S. Delegate to the International Mathematical
Union (1966, 1978). He was on the President's Committee on the National
Medal of Science (196667, 197476), Chairman of the Division of Mathematics
of the National Research Council (196769), Member of the National Science
Board (196875), Chairman of the Mathematics Section of the National
Academy of Sciences (197073), on the Council of the National Academy
of Sciences (197780), and on the Governing Board of the National Research
Council (197780). He was a Colloquium Lecturer of the American Mathematical
Society in 1970. In 1974 he received the Distinguished Service to Mathematics
Award from the Mathematical Association of America. He was President
of
the American Mathematical Society in
197778. He retired from The University of Texas at Austin in 1985 as
the Mildred Caldwell and Blaine Perkins Kerr Centennial Professor in
Mathematics. He received many other honors and served in many other
responsible positions throughout his career. He lectured in more than
200 colleges and universities in 49 states and in 17 foreign countries.
Bing believed that mathematics should
be fun. He was opposed to the idea of forcing students to endure mathematical
lectures which they did not understand or enjoy. He liked to work mathematics
out for himself and thought that students should be given the opportunity
to work problems and prove theorems for themselves. During his years
in Wisconsin, Bing directed a very effective training program for future
topologists. The first year graduate topology class which he often taught
there would sometimes number 40 or more students. He directed the dissertations
of 35 students and influenced many others during participation in seminars
and research discussions.
Bing enjoyed teaching and felt that
experiments in teaching were usually successful  not because the
new method was necessarily better, but because doing an experiment showed
an interest in the students which they appreciated and responded to.
Here are a couple of the experiments he tried while teaching at UT.
Bing thought that a person who could solve a problem quickly deserved
more credit than a person who solved it slowly. He would say that an
employer would rather have an employee who could solve two problems
in as much time as it took for someone else to solve one. So in some
of his undergraduate classes he introduced "speed points."
For a fifty minute test, he gave an extra point for each minute before
the fifty minutes elapsed that the test was submitted. He noticed that
often the people who did the work the quickest also were the most accurate.
Speed points were somewhat popular and sometimes he would let the class
vote on whether speed points would be used on a test.
Another experiment in test giving was
not popular. One day Bing had prepared a calculus test that he realized
was too long. Instead of deleting some questions, however, he decided
to go ahead and give the test, but as he phrased it, "Let everyone
dance to the tune of their own drummer." That is, each person could
do as many, or as few, of the problems as he wished and would be graded
on the accuracy of the problems submitted. The class was quite angry
when the highest score was obtained by a person who had attempted only
one problem.
In the 197172 school year, Bing accepted
an offer to visit the mathematics department at UT. In 1973, the mathematics
department, under Leonard Gillman’s chairmanship, persuaded Bing
to accept a permanent position here. When he arrived in 1973, Bing was
the highest paid professor in the state of Texas. He soon showed that
he was worth the money.
Bing believed that part of the fun of
life was to take on a variety of challenges. When he accepted the position
at UT, he came with the idea of building UT’s mathematics department
into one of the top 10 state university mathematics departments in the
country. While he was at Texas from 1973 until his death in 1986, he
helped to improve the research standing of the department by recruiting
new faculty and by helping to change the attitudes and orientation of
the existing faculty. Raising research standards was the watchword of
that period and is the guiding principle for the mathematics department
now. Bing was chairman of the department
from 1975 to 1977 but used his international prominence in recruiting
efforts throughout his stay at UT. Although Bing’s goal of putting
the UT department in the top ten has not yet been realized, the mathematics
department is considered one of the most improved departments over the
period of Bing’s tenure at UT. The 1983 report of the Conference
Board of Associated Research Councils listed Texas as the second most
improved mathematics department in research standing during the period
19771982, ranking it number 14 among state university mathematics departments
at that time.
Bing accomplished much during his life
and left us with many ideas, personal and mathematical, to consider
and enjoy. He left topologists a treasuretrove of theorems and techniques
and left the UT mathematics department with a goal and thirteen years
of good progress toward it. He was a man of strong character and integrity
who liked to understand things for himself. For example, he never claimed
to understand a theorem unless he personally knew a proof of it. He
made decisions based on his own experience  relying on his independent
judgment of a person or a cause whenever possible, rather than averaging
the opinions of others. He was a kind man and respected people for their
own merits rather than measuring them on a single scale.
R.H. Bing died on April 28, 1986. He
suffered from cancer and heart troubles during his last years; but he
never complained about his health problems nor did he allow discomfort
to dampen his enthusiasm and good spirits. He was an exemplary person.
His friends, his family, and his students have been enriched beyond
bound by his character, his wisdom, and his unfailing good cheer and
continue to be enriched by his memory.
<signed>
William H. Cunningham , President
The University of Texas at Austin
<signed>
H. Paul Kelley , Secretary
The General Faculty
This Memorial Resolution was prepared
by a special committee consisting of Professors Michael Starbird (Chairman),
William T. Eaton, Cameron Gordon, and Robert Greenwood with the assistance
of Professor S. Singh, Southwest Texas State University.
