## In Praise of Collaboration

# In Praise of Collaboration

*Take a look at an extraordinary collaboration in discrete geometry and related geometrical mathematics, the collaboration of Branko Grünbaum and Geoffrey Colin Shephard.*

**Joe Malkevitch**

**York College (CUNY)**

### Introduction

*Point* and *line* do many activities together—their collaborations create a rich texture for many mathematicians and geometers, as well as artists/designers/weavers, to enjoy, study and admire. But *point* and *line* are generally undefined terms in the axiom systems that are at the foundations of different types of geometry (Euclidean, projective, hyperbolic, taxicab, etc.) as a mathematical investigation. Despite this, the study of the interactions of points and lines are definitely worthy of exploration.

Figure 1 (A Pappus configuration in the Euclidean plane. There are 9 points and 9 lines with three points per line and three lines passing through each point. In the Euclidean plane the line displayed in the space between the two labeled lines MUST pass through the three points indicated.)

Figure 2 (The finite Fano Plane, seven points and seven lines in a finite plane where every line intersects each other line in one point and cannot be drawn with straight lines with exactly 3 points per line in the Euclidean plane. This fact is related to the Sylvester-Gallai Theorem.)

Scholarly work in experimental physics is typically published with the names of a large team of scientists, often not at the same institution, who collaboratively worked together on the experimental design and the carrying out of the project involved. By comparison, mathematics is sometimes viewed as a rather solitary activity—a subject which, if one is armed with a pencil and paper, one can make major or minor original contributions to. Even before the advent of the Web and the internet, mathematics had examples, usually of pairs of people, who enriched each other’s work by a collaboration. My goal here is to make the results of one such collaboration, by Branko Grünbaum (1929-2018) and Geoffrey Shephard (1927-2016) in the area of discrete geometry more widely known. (Grünbaum and Shephard are sometimes abbreviated G&S below—shades of the great operetta collaboration of Gilbert and Sullivan!) At the same time I want to point out collaboration as a model for doing effective and important research in mathematics.

### Mathematical collaborations

The list of superstar contributions to mathematics that are attached to the names of individual people date back in time within many cultures—names include:

Euclid, Archimedes, Newton, Euler, Gauss, Kovalevskaya, Noether, Lagrange, Ramanujan, …

These distinguished individuals, whose names belong to a list that stretches over long periods of time and in many countries/cultures, did not work in a vacuum. They were inspired by others and in turn inspired others to do important mathematics. Famously, Newton pointed out:

“If I have seen further it is by standing on the shoulders of giants.”

However, *mathematical* collaborations prior to the 20th century seem to be relatively rare. There are examples of mathematicians consulting with one another via correspondence, but this did not typically result in joint publications with the letter exchangers working together. In past eras scholars often could only communicate easily via written letters because travel between distant places (e.g. England and Germany) was costly and time consuming.

One often-cited example of a productive collaboration was between Geoffrey Hardy (1887-1947) and John Edensor Littlewood (1885-1977). Hardy is perhaps most famous for his book A Mathematician’s Apology but also made significant contributions to number theory, many of these with Littlewood.

One of the conjectures made by Hardy and Littlewood is still unsolved to this day. The conjecture states that the number of primes in the interval $(m,m+n]$ (this is the interval not containing its left endpoint but containing its right endpoint) will be less than or equal to the number of primes in the interval $[2,n]$ (this interval includes both its left and right endpoints).

Hardy and Littlewood certainly had a successful collaboration and it seemed to survive with time. According to their contemporary Harald Bohr, they operated their collaboration using the following rules:

- It didn’t matter whether what they wrote to each other was right or wrong.

- There was no obligation to reply, or even to read, any letter one sent to the other.

- They should not try to think about the same things.

- To avoid any quarrels, all papers would be under joint name, regardless of whether one of them had contributed nothing to the work.

There is considerable discussion about whether these rules are *ethical*, but presumably they did guide Hardy and Littlewood.

Below, I would like to take a look at an extraordinary collaboration in discrete geometry and related geometrical mathematics, the collaboration of Branko Grünbaum and Geoffrey Colin Shephard. I will begin by giving some biographical information about these two distinguished mathematicians, then discuss their collaboration and some of its accomplishments.

### Brief biography of Geoffrey Colin Shephard

Shephard was born in 1927. He studied mathematics at Cambridge and received his undergraduate degree in 1948. His doctorate degree was also from Cambridge University (Queens College), completed in 1954. His doctoral thesis adviser was John Todd. The title of his thesis was *Regular Complex Polytopes*. After working at the University of Birmingham he later moved to the University of East Anglia in 1967 and retired from there in 1984. He had at least two doctoral students—the geometer Peter McMullen (who works on problems about regular, primarily convex polyhedra in 3-dimensions and higher) and the specialist in convexity geometry, Roger Webster. You can get some idea of the mathematical topics that Shephard made scholarly contributions to from this list generated by MathSciNet:

- Combinatorics
- Convex and discrete geometry
- Geometry
- Group theory and generalizations
- History and biography
- Linear and multilinear algebra; matrix theory
- Manifolds and cell complexes
- Number theory

Shephard, who was an active member of the London Mathematical Society (LMS), provided money for LMS to fund what is known as the Shephard Prize. Here is its description:

"The Shephard Prize is awarded by the London Mathematical Society to a mathematician or mathematicians for making a contribution to mathematics with a strong intuitive component which can be explained to those with little or no knowledge of university mathematics, though the work itself may involve more advanced ideas."

The prize has been awarded to Keith Ball and Kenneth Falconer (both of whom are fellows of the Royal Society).

### Brief biography of Branko Grünbaum

Grünbaum was born in Osijek, in what today is known as Croatia, which at various times was part of the conglomerate country known as Yugoslavia. Though he was of Jewish background, he lived with his Catholic maternal grandmother during World War II, a period during which the Nazis occupied Yugoslavia. In high school he met his future wife, Zdenka Bienenstock, from a Jewish family, who survived the war hidden in a convent. Grünbaum began studies at the University of Zagreb but unsettled times in Yugoslavia resulted in Grünbaum and his bride-to-be emigrating to Haifa in Israel in 1949. Grünbaum found work and eventually resumed his mathematical studies at Hebrew University, earning a Master’s degree in 1954. This was also the year he married Zdenka, who trained to be a chemist. He did work for the Israeli air force in operations research while continuing his studies culminating with his doctorate in 1957. While many think of Grünbaum’s doctoral thesis adviser Areheh Dvoretsky (1916-2008) as a functional analyst, it was his budding talent as a geometer that Dvoretsky fostered. His doctoral dissertation was in Hebrew and entitled On Some Properties of Minkowski Spaces. At the completion of his military service Grünbaum came to America to do postdoctoral work at the Institute for Advanced Study with his family. By 1960 he had gone on to the University of Washington in Seattle, where the distinguished geometer Victor Klee (1925-2007) was already teaching. Eventually, after returning for a period to Israel, Grünbaum made his way back to a long career at the University of Washington where he taught until his retirement in 2001, and was part of the group of scholars there working in discrete geometry. He had 19 doctoral students and many more mathematical descendants. It was problems that he posed that I solved in my own doctoral dissertation. Grünbaum’s academic descendants include some of the most prominent discrete geometers of our time, in particular, Noga Alon and Gil Kalai. Over the years he won many prizes including the Lester R. Ford Award (1976) and the Carl B. Allendoerfer Award (2005).

The list of research publication areas of Grünbaum from MathSciNet can be compared with the one earlier for Shephard:

- Algebraic topology
- Combinatorics
- Computer science
- Convex and discrete geometry
- Differential geometry
- Functional analysis
- General
- General topology
- Geometry
- History and biography
- Manifolds and cell complexes
- Mathematical logic and foundations
- Ordinary differential equations
- Topology

### Some of Geoffrey Shephard’s "solo" work:

Before his collaboration with Grünbaum, Shephard was particularly interested in the properties of convex sets, not merely convex polyhedra. For example he did important work related to convex polytopes and geometric inequalities. He also wrote several papers about the Steiner Point. Shephard also studied the issue of when various convex sets could be written as a Minkowski sum. The idea was to see when convex sets could not be "factored" into "simpler" sets, an approached modeled on the idea of the importance of prime numbers in number theory. A convexity problem due to Shephard that is still actively being looked at concerns centrally symmetric convex closed and bounded sets $U$ and $V$ in $n$-dimensional Euclidean space. Suppose the volumes of the projections of $U$ on a hyperplane always are smaller or equal than that of $V$. Shephard conjectured that in this case the volume of $U$ is less than or equal to that of $V$.

One particularly intriguing question has recently been associated with Geoffrey Shephard’s name. The problem involves an idea which is often linked to the name of Albrecht Dürer.

Dürer (1471-1528), in his work Underweysung der Messung, explored ways to represent 3-dimensional objects in paintings in a "realistic" manner. This lead to his investigating questions about perspective and in particular perspective involving 3-dimensional polyhedra. He had noticed that one way to represent a polyhedron on a flat piece of paper was to show the polygons that make up the polyhedron as a drawing in the plane with each polygon shown by a flat region which shared one or more edges with other polygons that made up the polyhedron. Dürer produced examples of such drawings for Platonic and other solids which consisted of regular polygons. Figure 8 shows an example of the kind of diagram that interested Dürer, in this case for the regular dodecahedron, a Platonic Solid with 12 congruent regular polygons as faces.

Such drawings have come to be called nets but there is considerable variance in exactly what is meant by this term. It would be best to restrict this term to a collection of polygons that arises from a (strictly) convex bounded 3-dimensional polyhedron $P$ (sometimes called a 3-polytope) which arises from the cutting of the edges of the vertex-edge graph of the polyhedron along edges that are connected, contain no circuit and include all of the vertices of the polyhedron (i.e., a spanning tree of the polyhedron). If the initial polyhedron had $V$ vertices then the number of edges of the spanning tree whose cuts enable the unfolding of the polyhedron will be equal to $V – 1$. Since each of the edges of the *cut-tree* is incident to exactly 2 faces, it follows that the boundary polygon of a net will be a simple polygon with $(2V – 2)$ sides. Note that while the original polyhedron may be strictly convex, the polygonal boundary of the net while a simple polygon (when the unfolding forms a net), is typically not convex and will often have pairs of edges that lie along a straight line. (See Figure 9 for all of the nets of a regular $1\times 1\times 1$ 3-dimensional cube.)

Figure 9 (The 11 nets of the 3-dimensional cube. Courtesy of Wikipedia.)

Even for some tetrahedra (4 vertices, 4 faces and 6 edges) it is known that one can cut along 3 edges of a spanning tree and "unfold" the resulting polygons so that the result overlaps in the plane. However, it turns out that no polyhedron has ever been found where cutting edges of SOME spanning tree will not result in an assemblage of polygons without overlap. The complexity of the issues involved in such folding and unfolding problems has been explored by Eric Demaine (MIT) and Joseph O’Rourke (Smith) and others. Shephard specifically looked at the question of when such an unfolding would result in a convex polygon in the plane but in light of his work on this collection of ideas I like to refer to the following still open problem as:

#### Shephard’s Conjecture

Every strictly convex bounded 3-dimensional polyhedron can be unfolded to a net (non-overlapping polygons one for each face of the polyhedron) by cutting edges of the polyhedron that form a spanning tree. (Often one gets different results depending on what spanning tree one cuts.)

Intriguingly, experts lack consensus as to whether the result should be true or false. There are many polyhedra where nearly all spanning trees lead to overlapping unfoldings while there are some infinite classes of convex polyhedra where a net has been shown to exist. One source of confusion about the concept of a net is that if one starts with a polyhedron $P$ and cuts along a spanning tree to result in the net $N$, it is possible that the net can be folded by using the boundary edges of the net to a polyhedron different from the one one started with—Alexandrov’s Theorem. Thus, to recover the polyhedron one started with one has to provide gluing instructions about which edges to glue to which other edges. There are "nets" that fold (edges glued to edges) to several non-isomorphic convex polyhedra, and there are "nets" that, when glued along boundary edges one way, give rise to a convex polyhedron but a different gluing of boundary edges yields a non-convex polyhedron.

### Some of Branko Grünbaum’s "solo work"

Grünbaum’s webpage, still in place after his death in 2018, includes a list of his published works from 1955 to 2003, though he published many more additional works before his death in 2018. It is intriguing to compare the first 10 and last 10 titles in this bibliography:

- On a theorem of Santaló.

Pacific J. Math. 5(1955), 351 – 359. - A characterization of compact metric spaces. [In Hebrew]

Riveon Lematematika 9(1955), 70 – 71.

- A generalization of a problem of Sylvester. [In Hebrew]

Riveon Lematematika 10(1956), 46 – 48.

- A proof of Vázsonyi’s conjecture.

Bull. Research Council of Israel 6A(1956), 77 – 78.

- A simple proof of Borsuk’s conjecture in three dimensions.

Proc. Cambridge Philos. Soc. 53(1957), 776 – 778.

- Two examples in the theory of polynomial functionals. [In Hebrew]

Riveon Lematematika 11(1957), 56 – 60.

- Borsuk’s partition conjecture in Minkowski planes.

Bull. Research Council of Israel 7F(1957), 25 – 30.

- On common transversals.

Archiv Math. 9(1958), 465 – 469.

- On a theorem of Kirszbraun.

Bull. Research Council of Israel 7F(1958), 129 – 132.

- On a problem of S. Mazur.

Bull. Research Council of Israel 7F(1958), 133 – 135.

- A starshaped polyhedron with no net.

Geombinatorics 11(2001), 43 – 48. - Isohedra with dart-shaped faces (With G. C. Shephard)

Discrete Math. 241(2001), 313 – 332.

- Convexification of polygons by flips and by flipturns.

(With J. Zaks)

Discrete Math. 241(2001), 333 – 342.

- The Grunert point of pentagons.

Geombinatorics 11(2002), 78 – 84.

- Levels of orderliness: global and local symmetry.

Symmetry 2000, Proc. of a symposium at the Wenner–Gren Centre, Stockholm. Hargitai and T. C. Laurent, eds. Portland Press, London 2002. Vol. I, pp. 51 – 61. - No-net polyhedra.

Geombinatorics 11(2002), 111 – 114.

- Connected (n4) configurations exist for almost all n – an update.

Geombinatorics 12(2002), 15 – 23.

- "New" uniform polyhedra.

Discrete Geometry: In Honor of W. Kuperberg’s 60th Birthday

Monographs and Textbooks in Pure and Applied Mathematics, vol. 253.

Marcel Dekker, New York, 2003. Pp. 331 – 350. - Convex Polytopes. 2nd ed., V. Kaibel, V. Klee and G. M. Ziegler, eds. Graduate Texts in Mathematics, vol. 221. Springer, New York 2003.

- Families of point-touching squares. Geombinatorics 12(2003), 167 – 174.

Grünbaum had a particularly remarkable talent of seeing new geometry ideas "hiding in plain sight" where other geometers had not noticed these issues. A good example of this is extending the definition of what should be meant by a regular polyhedron which led to many interesting new *regular* polyhedra. Today these new regular polyhedra are known as the Grünbaum/Dress polyhedra because Andreas Dress noticed one example missing in Grünbaum’s initial enumeration.

You can find some samples of the remarkable work of Grünbaum in this earlier Feature Column article.

### An amazing collaboration

In trying to determine how Shephard and Grünbaum started their collaboration, one might look to MathSciNet for their earliest joint paper, but in the case at hand this would not return information about a joint venture they were part of. Grünbaum’s highly influential book *Convex Polytopes* was published in 1967 and one has to look at the table of contents and preface (or the list of Related names on MathSciNet) to realize that some of the chapters were prepared by other people. One of these people was Geoffrey Shephard. In 2005 this book won the American Mathematical Society Steele Prize for Mathematical Exposition.

While their earlier research had some aspects of convexity and the properties of polyhedra in common, Grünbaum’s work in many cases had an enumerative flavor. This approach to geometry was less present in Shephard’s work. It was aspects of their common roots but non-identical styles of work that perhaps made their collaboration so fruitful. Shephard points to the fact that G&S met for the first time face-to-face in Copenhagen, Denmark in 1965 at a convexity conference. At a meeting organized by the London Mathematical Society in July of 1975 it seems G&S set in motion the idea that they work together on a 12-chapter book about geometry, with discussion of many ideas about what it might contain. They conceived the idea of starting by writing a single chapter of such a book on patterns and tilings. It turned out that 12 chapters were necessary to do justice to the these ideas. This took about 11 years to develop and swelled to about 700 pages of writing, and it was this work that became their book *Tilings and Patterns*. The originally planned geometry book never got to be written!

There are many aspects of tiling and pattern issues that predate the work of G&S on this topic. However, many of these results are not part of the "standard" topics included in geometry curriculum in America. While many people have noticed with pleasure the multitudinous frieze patterns that they see on buildings, fabrics, and in art, they are unaware of a surprising mathematical result related to these frieze patterns. While one realizes that there are infinitely many different artistic designs that can constitute the "content" of a frieze (ranging from letters of the alphabet, stylized flowers, etc.) there is a mathematical sense in which there are only seven types of such friezes. The 7 types of patterns are shown below (Figure 10).

Figure 10 (Diagrams illustrating the 7 different kinds of friezes. Image courtesy of Wikipedia.)

Rather amazingly, G&S, by developing the definition of a discrete pattern, were able to breathe fresh life into this rather old topic. By using their "new" notion of a discrete pattern they were able to "refine" the sense in which there are 7 types of friezes into a finer classification under which there are 15 such "discrete frieze patterns."

In Figure 11 below, using a small asymmetric triangle as a "building block" one can see how the 7 frieze patterns above can be refined into 15 patterns as pioneered by the work of G&S. The rows of the diagram show how each of the 7 types of friezes sometimes can be refined from one "frieze type" to several discrete pattern types—the letter labels are one method of giving names to these patterns). In thinking about the way the new system refines the old one, notice that in some of these patterns there is a "motif" in a single row and in some of them they can be thought of as being in two rows. The new approach of G&S involves the interaction between the symmetries of the motif and the symmetries related to the "global" pattern arising from translating along the frieze. A few more details can be found in the discussion further along.

Figure 11 (15 types of discrete frieze patterns; diagram courtesy of Luke Rawlings.)

The diagram shown in Figure 12 deals with some of the challenges that researchers such as Grünbaum and Shephard who were interested in the classification and enumeration of "symmetric" tilings and patterns face. Interest in symmetry has complicated roots related to work in art and fabrics and also from the science side—the study of minerals and crystals and flowers.

What do we see in Figure 12? We see a portion of what is meant to be a part of an infinite object, extending what is there both in the vertical and horizontal directions. We see color and perhaps we are not sure what the foreground and background are. In Figure 13, do we have a white design on a black background or the other way around?

In Figure 12 it appears that the background might be the "rouge red" but one could perhaps give other interpretations. We see things that seem to be regions where the alternating rows are "separate" bands or friezes. The design that one sees in rows 2 and 4 perhaps remind one of something one might see on a Greek temple. The flower-like pattern of rows 1, 3 and 5 seems to have rotational symmetry as well as mirror or reflection symmetry. Geometers have developed a system of classifying symmetry in the Euclidean plane using distance preserving functions (mappings) which involve translations, rotations, reflections, and glide-reflections. These distance-preserving geometry transformations are known as the isometries. But in addition to such distance-preserving transformations diagrams like the ones in Figure 13, one might have color interchange transformations. There is another interesting difference between the rows that make up this symmetric pattern. The first and second rows both have translational symmetry in the horizontal direction and thus can be viewed as patterns onto themselves as friezes (band, strip) patterns. But the first row can be viewed as having a "discrete" motif which "generates" the pattern, something not true for the 2nd row. While there was a long tradition of looking at friezes and wallpapers which had a periodic structure and classifying them, rather remarkably the idea of looking at "patterns" with a motif like the one in row 1 had never been systematically discussed before Grünbaum and Shephard. Historically rather than looking at motifs, scholars had subdivided the "infinite" design in row 2 by breaking it up into a fundamental region (often there were different choices about how to do this) which was "propagated" via transformations into the periodic frieze or wallpaper.

G&S discovered that whereas there was a long, if not always "accurate" tradition of classifying and thinking about polyhedra, both convex and non-convex polyhedra, the situation for tilings (and the even less studied idea of patterns) was much less developed. There was some work on tilings that were the analogues for tiles of the Platonic Solids (work by Pappus) and Archimedean solids (work by Kepler) but little else. There were questions about the rules that one should allow for a tile. Thus, most examples of tilings looked at regions that were touching edge to edge as the "legal" tiles, but did it make sense to allow a region like Figure 14 as a tile or as a motif for a pattern?

Figure 14 (Is this "shape" allowed as a tile in a tiling of the plane?)

When one uses tiles to fill the plane, such as the tiling by squares which meet edge to edge shown in Figure 14, one has to think about whether or not the square tile is an "open" set (does not include its boundary points) or a "closed" set (contains its boundary points). For enumerations, note that one can get infinitely many "different tilings" with squares that have tiles that don’t all meet edge to edge by sliding some finite or even an infinite number of columns in Figure 15 half the length of one of the sides of the square up or down. What G&S did was to start from scratch in their joint work to give a coherent and accurate account of the "foundations" of a theory of tiling. They also extended classic enumerations by enumerating how many different "patterns" there could be when the motif used was dots, segments or ellipses. Rather surprisingly this had not been done previously. Along the way they had to invent new words for a variety of concepts that helped organize their new theory.

Figure 15 (One of the three "regular" Euclidean tilings of the plane known from ancient times. Image courtesy of Wikipedia.

Figure 16 (A tiling of the plane with regular hexagons, squares and equilateral triangles. Courtesy of Wikipedia.)

While the study of polyhedra has deep roots and multiple "reinventions" of various aspects of the theory, the seemingly neighboring idea of a tiling, while widely present in the art, fabrics and designs of various countries going back thousands of years, had a less obvious footprint in being investigated mathematically. Perhaps this is because mathematical tools to investigate infinite graphs, one way of conceptualizing about tilings, and to discuss symmetry involving something that was "infinite" (the idea of an isometry group) were developed only much later than the tools the Greeks found to study polyhedra. While in modern times symmetry is studied using the tools of graph theory, group theory and the ideas of geometric transformations, in the early study of polyhedra (and tilings) examining the pattern of the polygons surrounding a vertex was the fundamental approach. Thus, even the identity that today is often called Euler’s polyhedral formula, that for a convex bounded polyhedron the number of vertices, faces and edges of the polyhedron are related by

$$V + F – E = 2,$$

which might well have been discovered in ancient times was "discovered" in about 1750 by Leonard Euler. This formula, which involves the 0, 1, and 2- dimensional faces of a 3-dimensional polyhedron can be generalized to $n$ dimensions. G&S were able to find "rules" for fruitful definitions of tiling and pattern and this was helped by their being able to find a tiling version of the $V + F – E = 2$ result. Being able to draw on graph theory and combinatorics was a big factor in their successful leap forward beyond prior work. They also were able to draw insights from the study of things which were "asymmetric," such as fractals.

Another thread tied together by the work of G&S was where the notion of an aperiodic tiling fit in. While tilings that had translational symmetry in two different directions was the major topic that had been looked at by mathematicians and crystallographers, Roger Penrose had made the spectacular observation that there were tiles which enable one to tile the plane but these tiles did not allow a tiling that was periodic—had translational symmetry in two directions. The work of G&S clarified and laid the foundation for more discoveries related to non-periodic and aperiodic tilings. Some confusion exists about the concepts of non-periodic and aperiodic tilings. While the definitions one sees are still in flux, here is the intuitive idea. There may be a non-periodic tiling of the plane with one or more tiles, meaning that the tiling involved does not have translational symmetry but when one has a set of tiles that tile the plane aperiodically, there is no way to use the tiles to get a periodic tiling. Historically, in the attention that the mathematics community gave to infinite tilings (patterns) in the plane there was translational (shift) symmetry in two directions involved. There are tiles which are convex quadrilaterals that can tile the plane periodically (all triangles and quadrilaterals tile the plane) and also with rotational symmetry but do not tile the plane aperiodically.

#### Tilings and Patterns

For Geoffrey Shephard, MathSciNet informs us that his "Earliest Indexed Publication" was in 1952 and his "Total Publications" were 149 in number. For Grünbaum MathSciNet informs us that his "Earliest Indexed Publication" was in 1955 and his "Total Publications" were 271 in number. Many of these were joint publications, and in fact, individual and joint published works of G&S exceed these numbers. While MathSciNet lists research papers written by both Grünbaum and Shephard, perhaps their most famous collaboration was the book *Tilings and Patterns*. This book originally appeared in 1987 and was relatively recently reprinted and updated by Dover Publications. Remarkably the book contained many results not previously published in research papers which complement their joint work in scholarly journals. *Tilings and Patterns* literally rewrote the landscape of the theory of tilings. Grünbaum and Shephard’s innovation was to try to combine the local and global approaches to tilings in the past so that a more comprehensive look at the phenomenon of tilings and patterns was possible. While the word pattern is widely used in mathematics, its use in conjunction with the word tilings in mathematics is rather novel. Yes, patterns appear in many tilings, but by placing the emphasis on a discrete "motif" G&S opened many new doors.

The collaborative work of Branko Grünbaum and Geoffrey Shepherd has catapulted the areas of discrete geometry involving tilings and patterns to a much broader audience and resulted in many new discoveries based on their ideas. More importantly, their collaboration should be an inspiration to the mathematics community that collaboration is exciting and personally rewarding and can result in giant leaps forward for progress in mathematics and its applications.

### Dedication

I would like to dedicate this column to the memory of the remarkable geometers Branko Grünbaum and Geoffrey Shephard. It was my very good fortune to have had interactions with them both!

#### References

Note: No attempt has been made to give a complete set of the joint papers of Grünbaum and Shephard but several particularly important and interesting examples of their collaborative efforts are listed.

- Brinkmann, G. and P. Goetschalckx, S. Schein, Comparing the constructions of Goldberg, Fuller, Caspar, Klug and Coxeter, and a general approach to local symmetry-preserving operations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473 (2017), 20170267.
- Conway, J. and H. Burgiel, C. Goodman-Strauss, The Symmetry of Things, A.K. Peters, Wesley, MA., 2008.
- Delgado, O. and D. Huson, E. Zamorzaeva, The classification of 2-isohedral tilings of the plane, Geometriae Dedicata 42 (1992) 43-117.
- Delgado, O. and D. Huson, 4-regular vertex-transitive tilings of E3. Discrete & Computational Geometry, 24 (2000) 279-292.
- Demaine, E. and J. O’Rourke, Geometric Folding Algorithms, Cambridge University Press, NY, 2007.

Dress, A., A combinatorial theory of Grünbaum’s new regular polyhedra, Part I: Grünbaum’s new regular polyhedra and their automorphism group, Aequationes Mathematicae. 23 (1981) 252-65. - Dress, A., A combinatorial theory of Grünbaum’s new regular polyhedra, Part II: Complete enumeration, Aequationes Mathematicae 29(1985) 222-243.
- Escher, Maurits C., and D. Schattschneider, Visions of symmetry. Thames & Hudson, 2004.
- Friedrichs, O. Delgado and D. Huson, Tiling space by platonic solids, I. Discrete & Computational Geometry 21, (1999) 299-315.
- Grünbaum, B., Convex Polytopes, John Wiley, New York, 1967. (Parts of this book were developed by V. Klee, M. Perles, and G. C. Shephard.)
- Grünbaum, B., Convex Polytopes, Second Edition, Springer, New York, 2003. (This includes work from the first edition by V. Klee, M. Perles, and G. Shephard and addition input for V. Kaibel, V. Klee and G. Ziegler.)
- Grünbaum, B. Arrangements and Spreads, American Mathematical Society, for Conference Board of the Mathematical Sciences, Providence, 1972.
- Grünbaum, B., The angle-side reciprocity of quadrangles. Geombinatorics, 4 (1995) 115-118.
- Grünbaum, B., Configurations of Points and Lines, American Mathematical Society, Providence, 2009.
- Grünbaum, B.,Side-angle reciprocity – a survey. Geombinatorics, 2 (2011) 55-62.
- Grünbaum, B. and G.C. Shephard, The eight-one types of isohedral tilings in the plane, Math. Proc. Cambridge Phil. Soc. 82(1977) 177-196.
- Grünbaum, B. and G.C. Shephard, Tilings and patterns, Freeman, 1987. (A second updated edition appeared in 2016, published by Dover Press, NY. Errors from the first edition were corrected and notes were added to indicate progress in understanding tilings and patterns.) There also exists a shorter version of the original published by Freeman in 1989, with only the first seven chapters of the original version.)
- Grünbaum, B. and G.C. Shephard, Interlace patterns in Islamic and Moorish art, Leonardo 25 (1992) 331-339.
- Huson, D., The generation and classification of tile-k-transitive tilings of the euclidean plane, the sphere and the hyperbolic plane, Geometriae Dedicata 47 (1993) 269-296.
- McMullen and G.C. Shephard, Convex Polytopes and the Upper Bound Conjecture, London Math. Society, Cambridge U. Press, Cambridge, 1971.
- O’Rourke, J. How to Fold It, Cambridge University Press, NY 2011.
- Rawlings, L., Grünbaum and Shephard’s classification of Escher-like patterns with applications to abstract algebra, Doctoral thesis, Mathematics Education, Teachers College, 2016.
- Roelofs, H. The Discovery of a New Series of Uniform Polyhedra, Doctoral dissertation, 2020.
- Schattschneider, D., The plane symmetry groups: their recognition and notation. American Mathematical Monthly 85 (1978) 439-450.
- Schattschneider, D., The mathematical side of MC Escher, Notices of the AMS 57 (2010) 706-718.
- Shephard, G.C., Convex polytopes with convex, nets, Math. Proc. Camb. Phil. Soc., 78 (1975) 389-403.
- Shephard, G.C., My work with Branko Grünbaum , Geombinatorics, 9(1999) 42-48. (Note the whole Volume 9, Number 2, October 1999 issue of Geombinatorics is a special issue in honor of the 70th birthday of Branko Grünbaum.)
- Stevens, P., Handbook of regular patterns: An introduction to symmetry in two dimensions, MIT Press, Cambridge, 1981.
- Washburn D. and D.W.Crowe, Symmetries of culture: Theory and practice of plane pattern analysis. University of Washington Press, 1988. (Reprinted 2021 by Dover Publications.)
- Washburn D, and D.W. Crowe, editors, Symmetry comes of age: the role of pattern in culture, University of Washington Press, 2004.

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society’s MathSciNet can be used to get additional bibliographic information and reviews of some of these materials. Some of the items above can be found via the ACM Digital Library, which also provides bibliographic services.