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Invited Paper Session Abstracts - Surprising Discoveries by Amateur Mathematicians

Part A: Thursday, July 30, 8:00 a.m. - 10:50 a.m., Philadelphia Marriott Downtown, Grand Ballroom D
Part B: Thursday, July 30, 1:30 p.m. - 4:20 p.m., Philadelphia Marriott Downtown, Grand Ballroom D

This session will focus on sometimes overlooked non-professionals who have solved interesting mathematical problems or made significant contributions to mathematical knowledge. These persons had no formal education in higher mathematics and pursued mathematical investigations in their own way. Martin Gardner inspired such amateurs throughout his career. Indeed, he himself never completed a math course past high school, yet contributed new mathematical results, many of them published in award-winning MAA papers. From the 19th century and earlier, we will learn of the mathematical contributions of Benjamin Franklin, Mary Somerville, Florence Nightingale, Thomas Kirkman, Henry Dudeney, and Alicia Boole Stott. From the 20th century to the present, in addition to Gardner, we will learn of patent officer Harry Lindgren, artist George Odom, postal worker Robert Ammann, surgeon Jan Gullberg, artist Anthony Hill and others. On Saturday, the Martin Gardner Lecture will feature three other amateur mathematicians who made surprising discoveries: M.C. Escher, Marjorie Rice, and Rinus Roelofs.

Doris Schattschneider, Professor Emerita of Mathematics, Moravian College
Colm Mulcahy, Spelman College

Part A

Thursday, July 30, 8:00 a.m. - 10:50 a.m., Marriott Philadelphia Downtown, Grand Ballroom D

Is Mathematics too Serious a Matter to Be Left to Mathematicians?

8:00 a.m. - 8:20 a.m.
Peter Renz, Retired Editor (W. H. Freeman and Co., Birkhaüser Boston, Academic Press)


Clemenceau observed, “War is too serious a matter to be left to soldiers.” Look at the literature of mathematics and you will see the immense contributions made by amateurs to the understanding and appreciation of our subject – so my answer to this question is “Yes.” “Those who know the truth are not equal to those who love it, and they who love it are not equal to those who delight in it,” as Confucius wrote in his Analects, Chapter VI, praising amateurs above those who simply master a subject.

Many professionals we know love and delight in our subject, but my talk is about the amateurs who influenced my understanding of the subject or those whose work I have some connection with. These include: Martin Gardner, James R. Newman, Harold Jacobs, Jan Gullberg, and Lynn Gamwell. I mention other high-level amateurs such as Alfred Loomis, a lawyer, who went on to invent the LORAN navigational system and Ludwig Wittgenstein, whose dabbling in logic and the philosophy of mathematics had immense effect, and whose comments proved a source of revelation to me. These amateurs help us see things from different points of view, and that is how we learn new things.


Benjamin Franklin, 230 Years Later

8:30 a.m. - 8:50 a.m.
Paul C. Pasles, Villanova University


Centuries after the death of Benjamin Franklin, his mathematical ideas continue to be cited in academic journals and books. How did a self-educated amateur birth original ideas in such wide-ranging areas as demography, decision-making, and most famously, the art of magic squares? This talk will examine Franklin’s self-education and early influences, his contributions to quantitative thinking, and his influence on others.


‘The Philosopher in His Study, the Literary Lady in Her Boudoir’: How Mary Somerville Transcended the Amateur Status of 19th-Century Scientific Women

9:00 a.m. - 9:20 a.m.
Brigitte Stenhouse, The Open University


The childhood mathematical education of Mary Somerville (1780-1872) was, according to her autobiography, restricted to learning ‘the common rules of arithmetic’ at a local village school and an accidental discovery of algebra in a ladies magazine. Ten years later, as a young widow in Scotland, Somerville began submitting solutions to mathematical puzzles printed in The New Mathematical Repository, which in 1812 led to a correspondence relationship with Professor of Mathematics at the Royal Military College in Marlow, William Wallace. Barred from universities and scientific societies owing to her gender, Somerville had to rely on her social mobility to make acquaintances with her mathematical contemporaries, thus gaining personal access to their research and publications, and eventually becoming recognised and celebrated for her 1831 adaption of Laplace’s Mecanique Celeste. Nevertheless, society continued to view scientific women in general as amateurs, as demonstrated by the title of this talk (quoted from a review of Somerville’s second book). In this presentation we consider how Somerville educated herself in mathematics and reflect on how she navigated polite society in a way that maintained her womanhood whilst transcending the label of amateur.


The Reverend Thomas P. Kirkman: What Did He Do Besides Inventing the Fifteen Schoolgirls Problem?

9:30 a.m. - 9:50 a.m.
Ezra (Bud) Brown, Virginia Tech


The Reverend Thomas Pennyngton Kirkman (1806-1895) was a Church of England cleric by profession, spending 52 years as rector of the Parish of Southwark with Croft. He had little or no formal training in mathematics, and published his first mathematical paper when he was 40.

Kirkman was the first to describe many structures in discrete mathematics, and he also wrote extensively on other subjects such as hypercomplex numbers, algebraic geometry, finite groups, and knot theory. His published work spans nearly half a century, and includes over sixty substantial mathematical papers as well as dozens of minor publications. He was highly respected by such major figures as Hamilton and Cayley, and he was elected to the Royal Society of London.

This talk is about his life, the catalyst that attracted him to mathematics, his work, and the not-so-pretty story about why he fell into obscurity -- except for the Fifteen Schoolgirls Problem.


Florence Nightingale’s Notes on Victorian Officials’ Misunderstanding of Basic Mathematical Calculations and Management of Data

10:00 a.m. - 10:20 a.m.
Noel-Ann Bradshaw, London Metropolitan University


Florence Nightingale is well known in the mathematics community for her general contribution to statistics and her visualisation of data using polar area diagrams. However the details of her work are often not fully appreciated. This talk, drawing on Nightingale’s own writings, will show how she not only challenged but also corrected British officials’ use and management of data during and after the Crimean War. Her work influenced policy makers in the British Army and Government, resulting in improved conditions for both soldiers and the working classes. Her notes on the use and management of data are as relevant in today’s data-focussed society as they were in Victorian Britain.


Henry Dudeney: Amateur Mathematician?

10:30 a.m. - 10:50 a.m.
Charles Ashbacher, Charles Ashbacher Technologies


Henry Dudeney, an Englishman who spent his entire working life in the English Civil Service, also created and published many mathematical puzzles. Like his grandfather, most of the mathematics that he used in the creation of his puzzles was self-taught. He learned chess at a very early age, which did a great deal to inspire him in the creation of mathematical puzzles. He published the first known crossnumber puzzle, which is similar to a crossword puzzle, but the entries are numbers. Dudeney was also a pioneer in the area of verbal arithmetic or alphametics.

Like the Americans Sam Loyd and Martin Gardner, Dudeney’s puzzles and problems appealed to an audience wider than the mathematical community, inspiring many people to think about and study mathematics. Therefore, Dudeney not only made money publishing his math puzzles, he also served as an educator and inspiration in mathematics. Which leads to the serious questioning as to whether he was truly an amateur mathematician.


Part B

Thursday, July 30, 1:30 p.m. - 4:20 p.m., Marriott Philadelphia Downtown, Grand Ballroom D

Alicia Boole Stott in the Fourth Dimension

1:30 p.m. - 1:50 p.m.
Moira Chas, Stony Brook University


The untimely death of George Boole in 1864, left his widow, Mary Everest and their five young daughters in a precarious economic situation. Despite this, all of them, mother and daughters, pursued with remarkable persistence artistic creation or scientific understanding of some sort.

Alicia (1860-1940), the middle daughter, without any formal mathematical training, rediscovered the convex regular four dimensional polytopes (these are the four dimensional analogues of the Platonic solids in dimension three and the regular polygons in dimension two) and studied their sections, that is, intersections with three dimensional planes. In dimension four, there are six convex regular polytopes: the simplex (whose faces are tetrahedra), the 4D-cube, the 16-cell (dual of the 4D cube, whose faces are tetrahedra), the 24-cell (self-dual, faces are octahedra), the 120-cell (whose faces are dodecahedra) and the 600-cell (dual of the 120-cell, faces are tetrahedra). She also devised a method to construct semi-regular polytopes in all dimensions, starting with regular ones.

In this talk we will discuss Alicia Boole Stott’s geometrical ideas, as well as recently discovered facts about her mathematical path.


The Exquisite Geometric Dissections of Harry Lindgren

2:00 p.m. - 2:20 p.m.
Greg N. Frederickson, Purdue University


Harry Lindgren (1912-1992) was an Australian patent examiner who published in 1964 the first book devoted exclusively to geometric dissections, reaching well beyond what was known so far. To find dissections with few pieces, he created ingenious techniques: He cut a geometric figure, such as a regular polygon or star, into pieces that he rearranged into a pattern that could tessellate the plane, and overlaid it with a tessellation for a second figure of the same area and an identical repeating pattern to give a dissection. He similarly showed how to fill two infinite strips of constant width with appropriate repeating patterns, so that laying one such strip across another identified a dissection. Lindgren also used the inherent structure of a regular polygon in terms of rhombuses and equilateral triangles. He derived rational trigonometric relationships between the dimensions of pairs of regular polygons and stars of equal area. He explored classes of identical geometric figures which could easily be dissected into a larger similar figure. This talk illustrates many lovely examples of Lindgren's work.


Martin Gardner - "Are You a Mathematician?"

2:30 p.m. - 2:50 p.m.
Dana Richards, George Mason University


Martin Gardner is best known as a writer of recreational mathematics. It is often said he was not a mathematician; he said so himself. However he often contributed original results. These are found in math journals, his column and in the magic literature.


LOOK! George Phillips Odom Jr. and a Search for an Understanding Order

3:00 p.m. - 3:20 p.m.
Dick Esterle,


Upon receiving George Odom’s method for constructing a pentagon, Canadian geometer H.S.M.Coxeter published it in Odom’s name in 1983 where it appeared as a picture accompanied by one word: “Behold!” This presentation will examine some of this American artist and amateur geometer’s insights into the golden mean in drawings and models as well as searches for an “overriding system of polyhedral permutations” in 3 dimensional space. A solitary individual, Odom corresponded with H.S.M. Coxeter, Magnus Wenninger and R. Buckminster Fuller.


Robert Amman (1946 - 1994): Postman and More

3:30 p.m. - 3:50 p.m.
Marjorie Senechal, Smith College


Googling "Robert Ammann, mathematician" today brings up 694,000 hits in 0.6 seconds. Not bad for one whose only published math paper was a co-authored work he disavowed! Yet this off-and-on early-computer programmer-turned-postal-worker was respected by such luminaries as Martin Gardner, Roger Penrose, N. G. de Bruijn, and Branko Grünbaum. And surely he is the only such to be celebrated in an off-Broadway play! Who was Robert Ammann, and what is his legacy?


Anthony Hill and The Crossing Number

4:00 p.m. - 4:20 p.m.
Marcus Schaefer, DePaul University


Suppose we are tasked with visualizing a network of n people with connections drawn between any two of them. To improve the readability of the visualization, we want to minimize the number of crossings in the drawing. How many crossings are necessary to do so, and how does this number depend on n? This question was first asked not by a mathematician, but by a famous British constructivist artist, Anthony Hill. His conjectured answer has driven research in the crossing number of graphs (networks) for many decades. We will talk about the early history of the crossing number problem, and how it was shaped by Hill.