If one has a pattern on a band, frieze, or strip (disregarding color if as often happens more than one color appears), regardless of the artistic content of what one sees, there are exactly 7 different patterns that are possible…

Remembering Donald W. Crowe

Maker of Connections between Symmetry, Art and Mathematics

Joe Malkevitch
York College (CUNY)

Introduction

Andrew Wiles came to the attention of the public and the mathematics community when he solved a problem that had eluded solution for hundreds of years, Fermat’s Last Theorem. He showed that there were no positive integer solutions to the equation

$$x^n + y^n = z^n$$

for $n$ a positive integer that is 3 or greater. This is a sharp contrast with what happens when $n =2$, where the corresponding equation has infinitely many solutions, $x=3$, $y=4$ and $z=5$ being a well known solution. While it is relatively easy to state the question, the proof Wiles used to resolve it is not easy to understand. The complex mathematics that Wiles needed (with assistance from his onetime PhD student Richard Taylor to complete the proof) did provide the answer to this long standing unsolved problem.

Maryna Viazovska (from Ukraine, 2022) came to attention of the mathematics community, and much less to that of the general public when she became the second woman to win the Fields Medal, one of the highest awards that a mathematician can win. She did very innovative work that gives insight into dense packing of identical spheres in higher dimensional spaces, a problem whose nature is still not fully understood or explored in arbitrary dimension.

Here I will pay tribute to someone who contributed to mathematics in a less meteoric way, Donald Warren Crowe (1927-2022). However, I hope you will find this window into mathematics a compelling one. Below are a sample of images of Don as he and his beard grayed:

Modern mathematics extends greatly beyond the common wisdom that it is the subject concerned with numbers (arithmetic and algebra) and shape (geometry and topology). Another way to capture its domain is that it is the science of studying patterns. Relatively few mathematicians know the name Donald Warren Crowe (1927-2022), but he deserves to be better known for having helped put together a community of people interested in patterns, symmetry, art, and the presence of symmetry in the mathematics and artwork of cultures all over the globe and throughout the history of humankind.

Donald Crowe’s most cited work on MathSciNet (an abstracting source for research in mathematics run by the American Mathematical Society) is the book Symmetries of Culture, which he jointly developed with the anthropologist Dorothy Washburn.

The book was originally published in 1988 by Washington University Press but a version was relatively recently republished in 2020 by Dover Press. The book is notable in being both a fascinating blend of coffee-table book, with striking images of examples of patterns and designs from the artistic works of indigenous peoples, especially those who live in the southwestern part of the United States, and a tutorial about classifying symmetric patterns on fabrics or pottery.

The genesis of this book takes us along the path of Don Crowe’s life story, which I will now sketch briefly.

Brief Biography of Donald Crowe

Don was born in Lincoln, Nebraska in 1927, his father having been an academic but not in mathematics. Don attended the University of Nebraska and the University of Minnesota, studying mathematics, physics and philosophy. He returned to Lincoln to get a Masters Degree. He eventually made his way to the University of Michigan, where his interest in geometry led him to do some work with the graph theorist (dots and lines geometric objects) Frank Harary (1921-2005). When the eminent geometer Harold Scott MacDonald Coxeter (1907-2003) came to the University of Michigan from the University of Toronto, Don found Coxeter’s mathematical interests appealing. So he followed Donald (as Coxeter was known to his friends) back to Toronto and eventually completed his doctoral dissertation for Michigan under Coxeter’s direction.

His thesis generalized ideas about polygons using the quaternion number system, where the multiplication operation does not obey the commutative law that $ab = ba$. It is noteworthy that during a period when few mathematicians called themselves geometers (in contrast to analysts, algebraists, topologists) Coxeter breathed new life into geometry, partly with his book Introduction to Geometry (1961, 2nd ed. 1968). Coxeter was the subject of the book King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry (2006) by Siobhan Roberts, who drew for her biography on thoughts about Coxeter and his work from John Horton Conway (1937-2020). Conway himself made important contributions to revitalizing geometry in the 20th century. Ironically, Coxeter will probably be most remembered not for his contributions to geometry but to algebra. Thus, many who know his name know it because of the notion of a Coxeter Group, which bears his name. In addition to Crowe, Coxeter had 16 other doctoral students. He had relatively few doctoral “grandchildren” because most of his doctoral students, while creative contributors to mathematics, found work at liberal arts colleges rather than schools granting doctorate degrees. Eventually Don became a professor of mathematics at the University of Wisconsin in Madison and taught geometry courses, though he also taught abstract algebra and group theory courses related to geometry.

In 1969, Don, together with his two University of Wisconsin colleagues Anatol Beck and Michael Bleicher, published a book for a general education course in mathematics called Excursions into Mathematics. One chapter that Don contributed to this book dealt with the remarkable formula, seemingly not discovered by any ancient civilization, due to Leonard Euler. This formula, often called Euler’s Polyhedral Formula, states that for a convex bounded 3-dimensional polyhedron
$$V\text{(number of vertices)} + F \text{(number of faces)} – E \text{(edges)} =2.$$
This formula also has a version for connected graphs drawn in the plane. While Excursions into Mathematics is not directly related to symmetry patterns, clearly Don had been thinking about the relationship between symmetry, geometry transformations, patterns and “regular” polyhedra. In 1973, Don had assisted Claudia Zaslavsky, a precollege mathematics teacher and mother of the distinguished mathematician Thomas Zaslavsky, by writing a chapter for her important book, Africa Counts. This chapter dealt with patterns in African art.

Motivating the geometry of patterns via art of different cultures

Don loved to travel and after completing his doctorate in 1959 he traveled to Nigeria to have a visiting professorship at University College in Ibadan, Nigeria. It was his experience in teaching in Nigeria that helped further his interests in art, mathematics and symmetry. While in Africa he participated in archeological digs in Ghana. He returned from Africa with samples of African fabrics that he collected while he was there. His great intellectual curiosity built on what he learned while in Nigeria. These experiences involving the decorative arts in the form of fabrics and raffia clothes would later cause him to become involved in writing scholarly articles in mathematics journals about, and promoting the use of, symmetry and pattern at the intersection of artworks and mathematics. For many people, understanding the geometry of patterns is an appealing alternative entry point to mathematics, rather than learning about how to carry out arithmetic and algebraic computations. The photos below show some samples of art that Crowe photographed or collected on trips that he took inside or outside the United States.

These examples of symmetric artistic creations include subpieces that can be interpreted as a design in a band, strip, or frieze, in contrast to a designs that fill the whole plane, which are often said to be wallpaper patterns. (Of course, real world artists and artisans cannot fill an infinite strip, quadrant of a plane or the whole plane but often the substance of their work suggests that they are showing a finite portion of an idealized infinite canvas.) The cases of an infinite strip or the whole plane are of particular interest because, as seen above, they approximate symmetric patterns seen in textiles, on pottery or on the frieze of a building.

One of the most remarkable theorems of elementary geometry, though it is rarely mentioned in the mathematics education experiences Americans have in schools, is that if one has a pattern on a band, frieze, or strip (disregarding color if as often happens more than one color appears), regardless of the artistic content of what one sees, there are exactly 7 different patterns that are possible. That is, if one has a floral design, collection of letters, or animal design on a strip (frieze) it can be assigned a unique class to which it belongs. One way to construct frieze patterns and to help understand geometric transformations, symmetry, and shape is to use simple geometric figures and/or representations of letters of the alphabet to generate patterns. In the figure below, we have a part of a frieze of squares and another using the letter H (capital H rather than a lowercase h), but rendered in the spirit of graph theory using dots and edges.

In the first row we have a “run” along a strip of an implied infinite collection of squares, while in the second row we have an infinite collection of stylized representations of the letter H. Often the repeated shape, in the case that the frieze does not consist of one connected subject, is called a motif. While these two frieze patterns have different artistic content, they both have the same symmetry pattern. Each of these rows is a frieze which has as symmetries translations, vertical mirrors, horizontal mirrors, rotations of the whole frieze by 180 degrees, and glide reflections (a translation followed by a horizontal reflection).

T: translation
R: rotation by 180 degrees (or half-turn)
H: horizontal reflection
V: vertical reflection
G: glide reflection

Many systems of names have emerged for designating the 7 types of frieze patterns, but one can use the symbols above to note the presence of particular types of geometric symmetry. Again, it can be shown that in some cases the presence of certain symmetries must imply the presence of others.

Using the everyday artistic content of lower or upper case letters of the alphabet, one sees below representatives of the 7 types of frieze patterns that are possible.

….. L L L L ……translation

…..D D D D … translation, horizontal mirror

….V V V V ……translation, vertical mirror

……H H H H ……translation, vertical mirror, horizontal mirror

…..N N N N …. translation, half-turn (180 degree rotation)

….p b p b p b …translation, glide reflection

…..b p q d b p q d…… translation, glide reflection, vertical mirror

To help you check if you understand the system being used in classifying friezes, here is another collection of 7 patterns which includes each of the 7 patterns. Can you see which type each pattern below belongs to guided by the examples above?

…… S S S S S S S….

……d q d q d q d q……

…… M M M M M M ….

…….q d b p q d b p …

…….b b b b b b …….

…….C C C C C ….

………X X X X X…..

One approach to symmetry, which Donald Crowe emphasized for the benefit of those whose goal was to put a name on the type of pattern they were looking at, was to use a series of questions. In this way, after a small set of questions one could say that the pattern under consideration was pinned down. Unfortunately, as sometimes happens in scholarly areas that overlap in their interests, the names that crystallographers, anthropologists, and mathematicians use are not always the same. While one can convert between the different systems, there are pros and cons to the different naming systems. Suffice it to say that one does not need these naming systems to admire the beauty of real world examples illustrating symmetric patterns. Here are two examples of patterns teachers can use to entice students to see the nifty connections between patterns, art and mathematics.

Relatively recently, Branko Grünbaum (1929-2018) and Geoffrey Shephard (1927-2016) showed that the seven types of frieze patterns can be refined to 15 types of frieze patterns in the case where the motif of the pattern was discrete (rather than one connected design that could be broken up into a “fundamental region” that was repeated by translation). For two examples of connected frieze patterns, consult the figure below.

Crowe also attempted to popularize and make geometers aware of frieze patterns generated from arrays of numbers—a somewhat different riff on the notation of a frieze pattern due to the innovative work of John Conway and Coxeter. It turns out there is a surprising connection between this kind of frieze pattern and subdividing a convex polygon into triangles in different ways.

Crowe’s Legacy

One reason Donald Crowe achieved fame was that he was one of the 17 doctoral students of the remarkable mathematician Harold Scott MacDonald Coxeter. During a period where geometry had been eclipsed by other branches of mathematics, Coxeter was praised for work and writing that suggested that geometry was not dead and that there were many important geometric issues still worthy of attention by researchers, teachers and students. Often important mathematical ideas and notions are born in a geometric setting but reach their full maturity when the geometric problems can be treated using algebraic techniques. Thus, it is not an accident that Euclid (synthetic geometry) came before Descartes (analytical geometry).

The term ethnomathematics describes the branch of mathematics that deals with mathematical ideas inspired by, or born of, the mathematics and art that has been developed in many cultures over long periods of time. Examples of Don’s contributions to this part of mathematics are papers listed in the references about Bakuba art and the fact that his interests in this area inspired other people to teach and write about these topics. For example, Darrah Chavey was one of Don’s doctoral students, and Chavey collaborated with Philip Straffin at Beloit College in Wisconsin on research in ethnomathematics and teaching courses related to this topic.

As noted at the start, Don’s most cited publication is joint with the anthropologist Dorothy Washburn. Washburn, who obtained her doctorate at Harvard in anthropology, developed an approach to frieze patterns and wallpaper patterns that was independent of the work on these matters in the mathematical community. Her work illustrated the idea that what seems like theoretical mathematical ideas can with ingenuity be applied in novel settings. Washburn invented a method of using patterns as a way to understand cultural diffusion. Thus, in two neighboring villages, say in West Africa, one might see different patterns used in the textiles of these villages during a certain time period. However, at a later time period one might see patterns from the first village appear in the work of the artisans of the other village. Careful analysis of patterns might help provide insight into trade patterns that would allow one to try to infer information about how different tribal groups interacted over time.

Don’s collaborations with Dorothy Washburn and Claudia Zaslavsky were typical of his attempts to further the cause of mathematics and its applications as a scholar, teacher and educator. His legacy will live on!

Dedication

Donald Warren Crowe was a mensch. He attempted to make the world a better place and help others be the best people they could be. It was my great good fortune to have had him as a teacher, mentor (he was my doctoral thesis advisor (1968)), and friend. I miss him every day.

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