# Branko Grünbaum Remembered--A Great Geometer!

My goal here is to look at changes in the landscape of work in geometry seen through the lens of the work of one geometer, who, as David Hilbert did in all of mathematics, moved from one part of geometry to another and made important contributions as he changed his research attention. ...

Joseph Malkevitch

York College (CUNY)

Email Joseph Malkevitch

### Introduction

When Andrew Wiles found a route to a proof, noticeably more voluminous than the famous margin in which Fermat suggested that he had a proof of what has come to be known as Fermat's Last Theorem,

*a*^{n} + *b*^{n} = *c*^{n} has no solutions which are positive integers, when *n* is an integer greater than 2

he ensured a place for himself in the history of mathematics.

In a recent column I looked at some of the mechanisms that have been used to recognize mathematical accomplishment, but the issues raised there are only part of the story. Mathematics grows and evolves in complicated ways. Individuals with varied interests and backgrounds have contributed to this evolution, while at the same time making names for themselves. At different times in the past it has appeared that certain parts of mathematics were more in the mainstream than others. Furthermore there have been times that there has been a sudden spurt of interest in one area of mathematics while other areas have garnered less interest. Sometimes, using the tools that are available, all the progress that can be be made seems to have been explored and a spurt awaits some person or the development of new tools that can be put to use to explore new ideas or further the study of established ideas.

At the turn of the century from the 19th to the 20th century, the German mathematician David Hilbert (1862-1943) provided a collection of problems that he felt, when solved, would clarify important issues in many parts of mathematics and require the development of new mathematical tools which would be of value over broad areas of mathematics. Hilbert's career was marked by his active work in a specific part of mathematics, and then he would move on to another part of mathematics. The breadth of his important mathematical contributions is noteworthy because some feel that individuals can no longer make, as Hilbert did, seminal contributions to so many parts of mathematics. While the 20th century has seen some individuals make important contributions to many parts of mathematics (for example John Horton Conway), many other important contributors to mathematics were noted for work in one major area. Thus, the 20th century had great logicians, algebraists, and topologists, but perhaps fewer people like Hilbert. So perhaps a new version of polymaths for mathematics in the late 20th and early part of the 21st century are individuals who, even within one large branch of mathematics, made contributions to many parts of that branch. An individual who had these characteristics was Branko Grünbaum (1929-2018).

Figure 1 (Photo of Branko Grünbaum, courtesy of Wikipedia.)

While roots of geometry, algebra, and "analysis" (the study of functions of a real and complex variable) go back thousands of years in many different cultures, my goal here is to look at changes in the landscape of work in geometry seen through the lens of the work of one geometer, who, as David Hilbert did in all of mathematics, moved from one part of geometry to another and made important contributions as he changed his research attention.

### Geometry

Before saying more about Grünbaum's life and contributions to mathematics, let me comment more about the way geometry has evolved in the last 100 years. While geometric ideas were of use to all cultures (Chinese, Mayan, Egyptian, Indian, etc.) going back for thousands of years, in terms of thinking of geometry as an organized area for scholars to study, a great milestone was Euclid's *Elements*. To this day we are not sure whether at the time it was written, Euclid's work was a contribution to physics (phenomena in the world in which we live) or a work about the "axiomatics" of mathematics. By modern standards, the "axioms" that Euclid uses, as remarkable as they are, are not enough to capture the body of knowledge that we would today call Euclidean geometry, in contrast to what today would be the classical Non-Euclidean geometries. Early work in geometry was concerned primarily with metrical aspects of geometry--lengths, measurements of angles and properties of common figures such as triangles and quadrilaterals in 2-dimensions and some of the polyhedra that occur in nature (and have aesthetic appeal because they have symmetry and regularity properties).

While the *Elements* culminate in a discussion of the five regular solids, often called the Platonic solids, that are convex, there is in fact no use of the phrases "convex polygon" or "convex polyhedron" in Euclid. A set of points X is *convex* if for any two of its elements, *x* and *y*, the line segment joining *x* and *y* is also in X.

Figure 2 (A set of dice, with each of the five Platonic solids appearing. All are convex. Image courtesy of Wikipedia.)

Figure 3 (A non-convex shape, diagram courtesy of Wikipedia.)

From a modern perspective, lots of "evolution" has happened to our understanding of regular solids. The Platonic solids are of interest because they are mathematical objects that are convex, with the same number of polygons at each vertex, and whose "faces" are congruent convex regular polygons. As time went on, it became clear that one could broaden the rules in studying polygons and polyhedra so that one got additional "regular" shapes by using more general rules. Thus, the plane polygon below (Figure 4) is not convex and intersects itself, but it has a good claim to being a "regular" pentagon because the angles at the five "vertices" and the side lengths are equal.

Figure 4 (A self-intersecting non-convex regular pentagon. Image courtesy of Wikipedia.)

What is remarkable in terms of intellectual history is that the name of this shape is the pentagram, a term of Greek origin. The Greeks knew about this shape but did not incorporate it into their theory of polyhedra, by noticing the "regular polyhedron," (see Figure 5) often called the small stellated dodecahedron which meets the criterion of "regularity." All its faces are congruent and all its "pointed" vertices are alike! It was Cauchy (among others) who called attention to the way this polyhedron, non-convex though it was, could be added to the Platonic solids, with three others and constitute a collection of 9 regular solids. However, the story does not stop there and includes Branko Grünbaum's amazing extension of the notion of regularity for polyhedra, published in 1979. And the remarkable aspect of mathematics is that one does not have to appeal to authority to know what are the "facts" but can verify them in principle for oneself. Andreas Dress, in looking at Grünbaum's paper, noted that he had missed one polyhedron in his enumeration, and today people often refer to the Grünbaum-Dress polyhedra. In developing his new broader view of regularity Grünbaum built on work of other geometers (notably H. S. M. Coxeter) in extending the rules for what was allowed as a "face" of the regular polyhedron, what "polygons" were allowed to make up the faces, and whether one could have "more" than a finite number of vertices.

Figure 5 (A small stellated dodecahedron--a solid that is classified regular when one allows faces that are self-intersecting and non-convex. Image courtesy of Wikipedia.)

### Complex career

Branko Grünbaum had a remarkable life story, showing how talent interacts with history and other circumstances. Grünbaum was born in 1929 in what today is called Croatia but at one time was called Yugoslavia. His father's family was of Jewish background but his mother was Catholic. This situation made it possible for him to benefit from his mixed background during World War II. His high school "sweetheart" Zdenka Bienenstock, was also born in Croatia, of a Jewish family. She survived the Nazi period by being hidden in a convent. Grünbaum started his mathematical studies at the University of Zagreb. Grünbaum and Zdenka (not yet married) moved to the new country of Israel in 1949, a step made possible when the leaders of Yugoslavia gave people of Jewish background the opportunity to immigrate to Israel. Zdenka and Branko were married in 1954. Grünbaum continued his education in Israel, including getting a doctorate in 1957 from Hebrew University.

Perhaps Grünbaum's interest in geometry was fostered by Stanko Bilinski (1909-1998) whom he studied with at the University of Zagreb while sill in Yugoslavia. (Grünbaum wrote a paper about the so-called Bilinski dodecahedron.) Grünbaum's doctoral thesis advisor was Aryeh Dvoretsky (1916-2008), who is often thought of as an analyst rather than a geometer. It was not common during that period for mathematical researchers to refer to themselves as geometers, with analysis, algebra, and topology being the big fields of investigation by mathematics researchers.

Figure 6 (Photo of Aryeh Dvoretsky.)

The title of his thesis was "On Some Properties of Minkowski Spaces" (in Hebrew). His first publications were in either English or Hebrew and throughout his career displayed a wide range of interests, which perhaps narrowed with time but showed within geometry an unusual range of areas that he devoted time to. Here is a sample of some of the titles of his early papers:

On a theorem of L. A. Santaló (1955 in English)

A characterization of compact metric spaces (1955 in Hebrew)

A proof of Vazonyi's conjecture (1956 in English)

A generalization of a problem of Sylvester (1956 in Hebrew)

A simple proof of Borsuk's conjecture in three dimensions (1957 in English)

Two examples in the theory of polynomial functionals (1957 in Hebrew)

Note: A compilation of his papers up to 2003 was 248 items long. One can also look at his older and newer papers on MathSciNet. Some of the earlier papers were reviewed by such distinguished mathematicians as David Gale and Paul Erdős. A complete list would be much longer, and there are probably manuscripts he did not publish that have new results.

Not long after getting his degree in Israel, he was awarded the opportunity to visit the Institute for Advanced Study in Princeton, New Jersey. He got a visiting position at the University of Washington in 1960, though his plan to return for a position at Hebrew University was complicated by the annulment of his "Jewish" marriage to Zdenka--under Orthodox Jewish views, one is not a Jew unless one's mother is Jewish. Though he and Zdenka remarried before returning to Jerusalem, he was nervous about his status and that of his family in Israel. So eventually he accepted a position at the University of Washington where he spent nearly all of his remaining academic career. Already at the University of Washington was Victor Klee (1925-2007) who like Grünbaum helped change the direction of research in geometry and collaborated with Grünbaum on many projects.

Figure 7 (Photo of Victor Klee, courtesy of Wikipedia.)

### Polytopes

While, as indicated above there were strands of interest in polyhedra and convex polyhedra early in the history of mathematics, it was Grünbaum's 1967 book *Convex Polytopes* that put an organized modern theory of convex polyhedra on the mathematical map. That this was indeed true was recognized by Grünbaum receiving the Steele Prize in 2005 for his book. Grunbaum's book is notable for its use of graph theory and combinatorial ideas in helping to understand the theory of convex polytopes. Convex polytopes, bounded convex polyhedra--where "bounded" refers to the fact that the polyhedron can be enclosed in a sphere of sufficiently large radius--are now vigorously studied from many perspectives. In the plane, the convex region formed between two rays emanating from a point is 2-polyhedral but not 2-polytopal because the region involved is not bounded.

There are two approaches to convex polyhedra in *d*-dimensional space that have emerged to understand them from a mathematical point of view. One approach starts with a point set and takes its convex hull. The *convex hull* of a set of points Y is the intersection (points in common) to all convex sets that contain Y. In Figure 8 the initial point set (red) has as its convex hull the blue together with the red points.

Figure 8 (The convex hull of the red set of points is shown. Diagram courtesy of Wikipedia)

If one starts with a finite set of points and takes their convex hull one gets, by definition, a convex polytope. While the points one starts with might be in 3-dimensional space, if the points are chosen in a special way they may lie in a single plane in this 3-dimensional space. To get a 3-polytope, the finite set of points one chooses cannot lie on a single line or a single plane. If one starts with a set of 8 points (Figure 9) in the plane, their convex hull might lie on a single line, form a triangle or quadrilateral but one could not get a polygon with more than 8 corners/sides. Given a set of points in the plane some of those initial points may wind up as interior points inside the set's convex hull. While the convex hull of the diagram in Figure 9 started with 8 points, the convex hull gives rise to a region with 3 "extreme points," or "corners," and 5 vertices, though 4 of the initial points lie on a single line. Three of the original points lie in the interior of the convex hull. As is often the case in mathematics there are many details to select definitions that "recover" the objects one wants to consider. For example, this region, while "triangular" in shape, can also be regarded as a pentagon since it has a boundary line with 4 points on it.

Figure 9 (The convex hull of 8 points giving rise to a convex polytope of 2 dimensions.)

The other historical approach to "generating" polytopes is to talk about half-spaces (determined by lines in 2 dimensions and planes in 3 dimensions) and look at the points that are common to the half-spaces. This approach is more "algebraic" but in many ways more mathematically powerful than the convex hull approach.

Here is the way half-spaces are represented--the generalization of the linear inequalities one sometimes sees in high school or college mathematics classes.

Figure 10 (Equations for open and closed half-spaces. Courtesy of Wikipedia.)

When *n* is 2 (half-spaces in dimension 2), the two equations involve what are called open and closed half-planes, respectively. The open half-plane does not include the points of the line that define the half-plane, while the closed half-space does include the points on the line that "bounds" the half-plane. One can also replace the "greater than" and "greater than or equal to" symbols in Figure 10 with "less than" and "less than or equal to" signs.

The region formed by the intersection of a collection of half-spaces might not be bounded, so the definition of convex polytope comes with the requirement that the concept being defined is bounded. Much work in the early history of convex sets and polyhedral sets involved understanding the relationships between the convex hull and intersection of half-spaces (creating a bounded set) approaches to *n*-dimensional polytopes. Some of this work was carried along by the development of the optimization technique called *linear programming*. The bounded feasible sets for linear programming problems consist of convex polytopes. The most famous method of solving linear programming problems, the simplex method, involves following paths on a convex polytope. A typical example of a linear programming problem might involve the most profitable way to create a gasoline blend.

Polytopes are a very important class of geometrical objects but they are also important for those interested in topology and the geometry of curves and surfaces. This is in part a consequence of the fact that one can use polygons to approximate closed plane curves and 3-polytopes to approximate spheres and other closed surfaces. For example, one of the most important constants that mathematics has called attention to is the ratio of the circumference (perimeter) of a Euclidean circle to its diameter. The well-known relationship is that circumference = π(diameter) = πD. Since the diameter of such a circle is twice its radius *r* many people know this "fact" better in the form circumference = 2π*r*. One can prove that this ratio is a constant without determining the value of the constant involved, almost universally represented by π. So how can one compute the value of π? One approach would be to look at regular polygons with *n* sides "centered" at the center of the circle that are inscribed inside a circle or circumscribed about a circle, and increasing the number of sides in the hope of getting more and more accurate values for π. What one has to do here is get a "formula" for the perimeter of the regular polygon of *n* sides inscribed and circumscribed about a circle of a particular diameter. In Figure 11 we see polygons of 3, 4, 6 and 8 sides circumscribed about a circle, but in this diagram not all of the circles are the same "size" as measured by their radii.

Figure 11 (Polygons circumscribed about a circle. Image courtesy of Wikipedia.)

In Figure 12 we see regular polygons of 5, 6, and 8 sides inscribed and circumscribed about a circle of the same diameter. As the number of sides of the polygon used to circumscribe and to be inscribed in the circle increases, the perimeter (circumference of the circle) gets "squeezed" between the perimeters of the inscribed and circumscribed regular *n*-gons. This enables one to get increasingly precise estimates as *n* (the number of sides of the inscribing and circumscribing polygons) gets larger and larger.

Figure 12 (The value of π can be approximated using inscribed and circumscribed polygons to a circle. Image courtesy of Wikipedia.)

The "details" of these calculations are surprisingly subtle but that this is true makes it even more remarkable that in the distant past and across many cultures good approximations for π were obtained (though not always using the exact idea above). Similar arguments can be used to get approximations for the volume of *d*-dimensional spheres by using polyhedra.

One aspect of Grünbaum's book especially stands out. This was his treatment of the "combinatorial" aspects of 3-dimensional convex polytopes. Whereas lots of attention had been given to 3-dimensional polyhedra--after all we live in a space of 3-dimensions so making models of such objects is possible--nearly all the attention to such polyhedra involved metric considerations--angles, lengths of sides, volume and surface area, regularity of the polygonal faces, etc.) rather than combinatorial properties. As an example of a combinatorial question about polyhedra, is there a convex 3-dimensional polyhedron with exactly 7 edges? (Answer, no!) However, convex 3-dimensional polyhedra exist with *e* edges for all positive integers greater than or equal to 6, other than 7. Progress had been made in getting insight into combinatorial aspects of 3-dimensional polyhedra by a person who many people think of more as an algebraist than a geometer, the German mathematician Ernst Steinitz (1878-1928).

Figure 13 (Photo of Ernest Steinitz and his tomb. Courtesy of Wikipedia.)

Grünbaum looked at a lot of the work of Steinitz did and reinvigorated it and redirected it by using the language of graph theory, which was not part of the framework in which Steinitz developed his ideas, though diagrams which we would call graphs appear in his work. The most important of these rediscoveries of Steinitz's work was done in conjunction with Theodore Motzkin (1908-1970). In recent years I tried to find out exactly the circumstances of how Grünbaum and Motzkin came to "reinvent" the theorem about 3-polytopes that is now known as Steinitz's Theorem. However, he never recounted to me the details of how their collaboration played out. It is possible that Motzkin, who was from Germany before moving to Israel and later to the United States, brought the geometric work of Steinitz to his attention.

Figure 14 (A photo of Theodore Motzkin.)

This theorem as stated by Motzkin and Grünbaum reduces the study of combinatorial aspects of convex 3-dimensional polytopes to the study of graphs drawn in the plane so that edges meet only at vertices. Here the term graph is used for a diagram consisting of a finite collection of points (vertices) joined by a finite collection of straight line segments or curves, where no point is joined to itself nor does any pair of edges join the same pair of vertices.

Restated in modern graph theory language:

Theorem: (approximately 1920) (Ernst Steinitz as reformulated by Grünbaum and Motzkin).

The graph G is isomorphic to the vertex-edge graph of some 3-dimensional (convex) polytope P if and only if G is planar and 3-connected.

A graph (dots-and-lines diagram) is called *planar* if it can be drawn in the plane so all of its edges meet only at vertices. A specific drawing of a planar graph in the plane is called a *plane graph*. Isomorphic graphs, ones with the same structure, can be drawn in the plane with rather different visual appearances. While both graphs in Figure 15 have 5 vertices and 4 edges, the one on the left is planar but is not shown in a plane drawing because it has two edges that cross, while the graph on the right is isomorphic to the graph on the left and is a plane drawing of this graph, which is a path of length 4.

Figure 15 (A planar but not plane and plane drawing of two graphs isomorphic to a path of length 4.)

A graph is 3-connected if for *any* two vertices of the graph* v *and *w*, there are at *least* 3 paths from *v* to *w *whose only elements (edges or vertices) in common are *v* and *w*. Figure 16 shows a schematic diagram to help you understand what is meant by 3-connected; the diagram shows three paths from *v *to *w *with only* v* and *w *in common. Note that it is possible for the same graph to be simultaneously 3-connected and 4-connected, or 3-connected, 4-connected and 5-connected. The graph of the regular icosahedron is 5-connected but it is also 3-connected and 4-connected. Using Euler's polyhedral formula for plane connected graphs, namely that the number of the vertices (V), faces (F), and edges (E) of a 3-polytopal graph obeys V + F- E = 2, one can show that a plane connected graph can be at most 5-connected.

Figure 16 (A part of a graph showing vertices

*v* and

*w* having at least three paths with only

*v* and

*w* in common.)

To help illustrate the potential power of Steinitz's Theorem let me consider the following example from the attempts to solve the famous 4-color problem. It appeared that one could color the faces (regions in a plane graph) so that faces sharing an edge got different colors with at most 4 different colors. It was shown that if this result could be shown for convex 3-dimensional polyhedra it would be true for all plane graphs. Furthermore the problem was reduced to showing that the result would follow if all convex 3-dimensional polyhedra with exactly 3 edges at a vertex (polyhedra of degree 3; 3-valent polyhedra) could be 4-colored, then the theorem would hold for general polyhedra. A further simplification reduced the 4-color problem to the question of whether every convex 3-dimensional polyhedron, all of whose vertices were of degree 3 (3-valent), had a circuit (called a Hamiltonian circuit) that was a simple tour which visited each vertex once and only once. In fact, any plane graph which has a Hamiltonian circuit can be colored with at most 4 colors. To see this, consider the Hamiltonian (named to honor the Irish mathematician William Rowan Hamilton) circuit of the 3-polytopal graph (it is planar and 4-connected, in fact) shown in Figure 17.

Figure 17 (A 4-valent vertex 3-polytopal graph with a Hamiltonian circuit shown in bold.)

Since the bold edges in Figure 17 constitute a simple closed curve in the plane, the Jordan Curve Theorem says that points not on the Hamiltonian circuit lie in either the interior or exterior of the Hamiltonian circuit. We can color the interior faces by coloring one face, say with color *a*, and then every time we cross any edge it shares with a neighboring region, color the neighbor with color *b*. It is easy to see we can color all interior regions by alternating adjacent regions with colors *a* and *b*. Similarly, the exterior regions can be colored with the colors *c* and *d*, for a total of 4 colors. But does every plane 3-connected 3-valent (hence, 3-valent 3-polytopal graph) always have a Hamiltonian circuit? Despite some claims that this was always true, William Tutte (1917-2002) showed in 1946 that the graph in Figure 18 has no Hamiltonian circuit. This graph with 46 vertices is not the one with the smallest number of vertices with this property. There are such graphs with 38 vertices (3-valent, plane, 3-connected). Thus, the "easy" approach to solving the 4-color problem (now a theorem due to work of Wolfgang Hakan and Kenneth Appel) would not work.

Figure 18 (A graph first constructed by William Tutte illustrating that a plane 3-valent 3-connected graph can lack a Hamiltonian circuit. Diagram courtesy of Wikipedia.)

Suffice it to say that there have been hundreds of papers developing new and sometimes unexpected properties of 3-polytopes that emerged from applying Steintiz's Theorem, not to mention all the "doodles" in the plane that geometers can use to study 3-polytopes!

### Arrangements and spreads

Having devoted much attention, to polytopes, Grünbaum turned his attention in addition to what at the time was an obscure area of geometry--arrangements of lines, curves and "spreads." While polytopes have both metrical and combinatorial geometry issues at their foundations, arrangements of lines are much more centered in combinatorial or discrete geometry. Here is a small example to help you visualize what the subject matter of arrangements of lines is about:

Figure 19 (Some arrangements of lines. Image courtesy of Wikipedia.)

For the arrangement on the left (Figure 19) you can think of the diagram as being generated by the lines determined by pairs of points from an initially presented point set. Note that the lines formed by pairs of points result in new intersection points but in the diagram only the initial points are highlighted. In the drawing on the right (Figure 19) we have 4 lines and three points on each of the lines. Note that we have not added the additional lines that join the points shown in red. We can look at the number of sides of the bounded regions involved but also note that some unbounded regions appear and might be "studied." Arrangements where all of the bounded regions, usually called the cells of the arrangement, formed are triangles, are called *simplicial arrangements*. One of the first tasks Grünbaum looked into when he investigated this area of geometry was enumerating the different non-isomorphic arrangements with *n* lines, in particular, counting the number of "inequivalent" (non-isomorphic) simplicial arrangements. Embedded in his studies was getting insights into ideas related to the Sylvester-Gallai Theorem. When one looks at a collection S of points in the plane which do not all lie on a straight line and considers the lines determined by points in S, there must be lines that have exactly 2 points of S on them. These are usually called ordinary lines. The right-hand diagram in Figure 19 has 6 points which determine 3 ordinary lines. (One can interchange the roles of points and lines and talk about ordinary points; these are arrangements where there are only two lines passing through some of the points.) When looked at just the right way this is not so difficult to prove but without "hints" it is not so obvious how to see that this result is true. (Another perspective on this result is that one cannot have a collection of points S in the plane not already on a line, so that if one takes any two of the points of S creating line *m*, there must be at least one more point of S on *m*.)

The notion that Grünbaum developed of a *spread* will be easiest to develop with a specific example. Suppose one has a plane equilateral triangle T (more generally any triangle) and picks any point *p* on the line segments that bound the triangle T, then there is exactly one point *p** which has the property that the line joining *p* to *p** divides the perimeter of the triangle exactly in half. For an equilateral triangle, if one picks one of the vertices of the triangle, this line will be the point joining the vertex to the midpoint of the opposite side--the median of triangle from that vertex. For an arbitrary triangle this line will be unique but not go through the midpoint of the opposite side, but one "Cevian" of the triangle. The collection of perimeter bisectors of a triangle generates a family of lines, called a spread, that lie within the interior of the triangle. One can study which points lie on exactly one, two, or three perimeter bisectors. Similarly, one could look at the family of area bisectors of a triangle generated by picking a point P on the boundary of the triangle and locating the point P' that is at the other end of a line segment that bisects the area of the triangle. In both of these examples one gets a family of lines associated with the initial set. Grünbaum studied such families of lines or curves associated with, typically, convex polygons and curves. Many new geometric insights have derived from his work.

### Tilings and patterns

The study of tilings dates into antiquity as does the study of polyhedra. It seems to have been the geometer Pappus (circa 290-350) who called attention to the fact that there are three tilings of the Euclidean plane by congruent regular convex polygons--a tiling with congruent equilateral triangles (6 at a vertex), with congruent squares (4 at a vertex) and congruent regular hexagons (3 at a vertex). Over the centuries, just as the Archimedean Solids generalized the Platonic solids, artists, chemists, and mathematicians studied tilings of the Euclidean plane with mixtures of regular polygons.

There have been many famous fruitful collaborations between talented mathematical practitioners (the work of G.H. Hardy and J.E. Littlewood comes to mind). Geometry was very lucky to have the wonderful collaboration of Geoffrey Shephard (1927-2016) and Branko Grünbaum.

Figure 20 (Photo of Geoffrey Shephard. Oberwolfach photo collection.)

While both men did distinguished work on their own both before and after their extremely prolific collaboration, many important results emerged from their joint work. This joint work is largely summarized in their seminal book *Tilings and Patterns*, which is not only richly illustrated with many beautiful tilings, some of their own discovery, but also has new theorems and enumerations that did not also appear in their long series of joint publications.

Here is a small look at an innovative addition to the theory of results from the collaboration of Grünbaum and Shephard. This particular example should and could be something that becomes a part of the geometry taught in K-12 mathematics classrooms. While there are infinitely many artistic versions of band or frieze patterns (see Figure 21) there is a mathematical sense in which every such pattern belongs to one of 7 types.

Figure 21 (Two inequivalent frieze patterns. Courtesy of Wikipedia.)

Note that the two different types of patterns in Figure 21 display some of the distance-preserving transformations, called* isometries*, that mathematicians have studied in detail. These transformations include translation, reflection in a mirror, rotations (about a point), and glide reflections. One of the friezes above has mirror symmetry, the other does not. It turns out that there are exactly 7 different frieze pattern types from a well-defined mathematical point of view. However, notice that the patterns in Figure 21 are one "connected" design, and one might have designs which had "motifs" which were congruent but separated from each other. This notion of a frieze generated by congruent copies of a "discrete" motif is illustrated below with letters of the alphabet. Each "row" is one frieze pattern that goes off to the left and right forever. Towards the bottom are some examples where a frieze might have several rows of symbols (using a motif).

...L L L L L...

....H H H H H.....

....p b p b p b...

....p q p q p q....

...b q b q b q....

...W W W W W...

....C C C C C...

...X X X X X...

...E E E E E...

...A A A A A...

.....p d p d p d ...

.....b p b p b p ....

....d b d b d b d b ...

----------------------------

B B B B B B B

B B B B B B B

-----------------------------

p p p p p

d d d d

----------------------------

Can you see which patterns are of the same type and which patterns are of a different type? Can you see a way of distinguishing for discrete motif patterns such as those above that we could refine the system of using 7 types of patterns to a system of 15 patterns? Figure 22 may help you see what is meant. While both of these frieze patterns have a horizontal reflection (mirror) isometry (in addition to translational symmetry), one can be thought of as generated by a motif of a T on its side, while the other can be thought of as arising from an L shape on its side.

Figure 22 (Two friezes which can be distinguished though they have the same "type." Diagrams courtesy of Luke Rawlings.)

Figure 23 (Four frieze patterns that can be thought of as being different from each other. Each row is part of a different frieze and the motif for the "whole" pattern is shown highlighted in red. All four rows would be assigned the same frieze pattern name when one classifies friezes using symmetry groups. Diagram courtesy of Luke Rawlings.)

Grünbaum and Shephard showed that the 7 frieze patterns could be refined to a system of 15 patterns when the motif was discrete (in contrast to what one sees in Figure 21) and the symmetry of the motif is used as part of the classification system. Grünbaum and Shephard's work here can be extended to the "wallpaper" patterns of the plane and in many other directions. Their research and new questions they suggested will keep geometers busy for a long time to come!

### Next steps

Many of Grünbaum's results tend to involve enumerations. While in some cases his work involves existence theorems and the application of other mathematical tools, many of his results involve counting geometrical objects (tilings, patterns, polygons, arrangements, etc.) that were not considered in the past. Being able to be sure that all of the counts are complete is what needs proving in these situations, often a difficult task. While Grünbaum's "older" work was on convexity, polytopes, arrangements, spreads, patterns and tiling, his most recent book deals with "configurations."

This work, while inspired by geometrical considerations, has a much more combinatorial flavor. Grünbaum trained many graduate students in the emerging new types of geometrical problems that he helped pioneer, including many doctoral degree students. Not only have his students done important work but his "academic grandchildren" also are further widening the domain of geometric ideas based on his legacy. For a snapshot of this work you can look at the abstracts and videos from a conference honoring Grünbaum (and Victor Klee) held at the University of Washington in 2010.

Branko Grünbaum's great vision for a diversified and broadened view of geometry will long continue and his spirit will live on in the many new results that geometers will discover inspired by his example.

**DEDICATION**

This column is dedicated to the memory of the extraordinary man and geometer Branko Grünbaum. Details of the way he personally affected me as a mathematician are described here.

**References**

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.

Grünbaum, B., and V. Klee, M. Perles, G. C. Shephard, Convex Polytopes, Wiley, NY, 1967. (Second edition, Dover, NY, had input from other authors.)

Grünbaum B., Polytopes, graphs, and complexes, Bulletin of the American Mathematical Society, 76 (1970) 1131-201.

Grünbaum, B., Arrangements and Spreads. American Mathematical Society, 1972.

Grünbaum, B., Regular polyhedra--old and new, Aequationes Mathematicae, 16 (1977) 1-20.

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.)

Grünbaum, B. Configurations of points and lines. American Mathematical Soc., Providence, 2009.

Klee, V. (ed.), Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society (Vol. 7) American Mathematical Soc., Providence, 1963. (This volume contains several survey papers with rich geometric content co-authored by Grünbaum on topics like symmetry measures for convex sets, polyhedral graphs, and Helly's Theorem.)

Malkevitch, J. and L. Rawlings, Patterns Everywhere, Consortium, COMAP Newletter, 111, 2016.

Steinitz, E., Vorlesungen über die Theorie der Polyeder: unter Einschluß der Elemente der Topologie (Vol. 41), Springer-Verlag, 1934. (This book had a complex history. Hans Rademacher produced a final version from unfinished work of Steinitz.)

Steinitz, E., Polyeder und Raumeinteilungen, Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries) (IIIAB12), 1922, pp. 1–139.

Rawlings, L.B., 2016. Grünbaum and Shephard's classification of Escher-like patterns with applications to abstract algebra (Doctoral dissertation, Teachers College, Columbia University.)

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 Portal, which also provides bibliographic services.

Joseph Malkevitch

York College (CUNY)

Email Joseph Malkevitch