The pyramids of Egypt (Figure 1) are one of the wonders of the world--ancient or modern--but the fact these pyramids date from about 4500 years ago makes them all the more remarkable.

Figure 1 (Egyptian pyramids. Courtesy of Wikipedia)

Mathematicians have idealized what one sees in the Egyptian pyramids; they are not really square-bottomed convex (no holes, tunnels, or notches) polyhedra with equilateral triangles as faces! Figure 1 takes a step toward what mathematicians typically do, modeling or idealizing things things they see in the natural or man-made world and studying these mathematical constructs. Mathematicians perhaps more than anything else study patterns. In Figure 1 you see a "hidden line" drawing on a flat surface of something 3-dimensional which suggests that the base is a square and the triangular faces are isosceles triangles; one might idealize further and think of the four triangular faces as equilateral triangles.

Figure 2 (A hidden line drawing of a pyramid with a square base and 4 isosceles triangle faces. Courtesy of Wikipedia.)

There are other ways to understand polyhedra from a mathematical perspective including using a dots and lines diagram called a *graph* (see Figure 3). This diagram sacrifices the three-dimensional quality of Figure 1 and retains only combinatorial information, but has the advantage of avoiding possible confusion about what one is seeing in hidden line drawings of more complex polyhedra. The drawings in Figure 3 also make it easy to see that not all the faces are alike in the sense that there are four 3-sided faces and one 4-sided face. Also the vertices are not alike because one vertex has 4 segments at it (4-valent vertex) while the other vertices only have three edges at them (3-valent vertices).

Figure 3 (Two isomorphic (same structure) drawings of a graph of a "square pyramid.)

One might think about the fact that the "statistics" here, namely four faces are 3-gons and one face a 4-gon, and four vertices that are 3-valent and one vertex that is 4-valent, are that some kinds of polyhedra have the property that their face and vertex combinatorial structures are alike. This, as abstract mathematics, is the germ of the notation of "self-dual" polyhedron or graph. One can also have pairs of polyhedra, called *dual polyhedra* where the counts of the number of sides of the faces and the valences of the vertices on of the polyhedra interchanges the analogous counts for the dual. In this sense as seen in Figure 4 the regular hexahedron (cube) and the regular octahedron are dual polyhedra.

(a)

(b)

Figure 4 (Diagrams showing (a) polyhedron duality (Courtesy of Wikipedia) and (b) graph theory duality--cube and octahedron.)

However, while there was already in the ancient world interest in cubes and octahedra, the mathematical notion of duality was only properly explored in the 18th century and later by Joseph Diaz Gergonne (1771-1859), Jean-Victor Poincelet (1788-1867), Jakob Steiner (1796-1863), and Johannes Max Brückner (1860-1934).

Another mathematical way of representing a square pyramid (Figure 2) on a flat piece of paper is using the drawing in Figure 5, where some of the edges of the polyhedron have been cut so that the flat faces of the polyhedron can be flattened out for display in the plane. Somewhat surprisingly it is still an unsolved question whether this can be done for any convex (no notches, holes, or tunnels) 3-dimensional polyhedron. This problem was posed by the English geometer Geoffrey Shephard (1927-2016). Again, making the definitions of what is often called a "net" precise has a long complex history beginning with Albrecht Dürer (1471-1528). Many easy-to-state problems concerning folding of polygons and unfolding of polyhedra are still unsolved!

Figure 5 (A plane drawing of a square pyramid known as a net. There are other nets of the square pyramid. Can you draw them? Courtesy of Wikipedia.)

Making the Giza pyramids was a great feat of engineering but Egyptian mathematics, though not as well known as that of Greece, was very sophisticated even if they (and the Greeks) did not have as slick a way to represent numbers as we have today. What did the Egyptians know about polyhedra? We are not really sure! However, some think that the Moscow Mathematical Papyrus gives a worked-out example of the way to find the volume of a special kind of truncated pyramid. There is nothing to lead us to believe that the Egyptians knew anything about what today we would call regularity matters for polyhedra, in particular for convex polyhedra. Lots can be said about knowledge in the ancient world with hindsight. Thus, the distinction we make today between convex and non-convex polygons and polyhedra seems not to have been on the radar of the ancient world.

The Egyptian pyramids don't truly have flat faces but from a distance that is the way the pyramids appear. And the pyramids have five faces, four triangle faces and one 4-sided base, which is almost square. However, what caught the interest of the Greek geometers was that some polyhedra had "homogeneous" properties in the sense that either all the faces were alike or all the vertices were alike. But the Greeks approached these questions from a metrical (involving angles and distances) rather than a combinatorial or graph theory of view.

The Greeks took a great interest in the "regularity" of polygons and polyhedra (all sides of the same length and all angles between consecutive sides having equal angles). However, whereas in modern times we pay attention to convex vs. non-convex polygons which don't intersect themselves (simple polygons), and non-convex polygons which intersect themselves, there seems to be no specific mention of convexity or what today would be called *simplicity* of polygons in ancient mathematics.

In light of the ancient origin of regularity ideas, remarkably it took until 1969 to resolve the mathematical question of getting a complete list of those 3-dimensional convex polyhedra all of whose faces are regular polygons. The resolution of the question by Victor Abramovich Zalgaller built on the list of 92 solids (see below for where this number comes from) was proposed by the American mathematician Norman Woodason Johnson. Johnson renewed interest in "regularity" issues involving convex polyhedra by asking the simple question:

What are all of the convex polyhedra whose faces are all regular convex polygons perhaps with many different numbers of sides?

Figure 6 (A fuzzy photo of Norman Johnson--other photos below. Courtesy of Wikipedia.)

We will see that Johnson called attention to this problem, made the right conjecture about what was going on, but did not show that his conjecture was correct. That honor fell to Russian mathematician Viktor Zalgaller!

Figure 7 (Photograph of V.A. Zalgaller)

Sadly, Norman Johnson died on July 13, 2017, at the age of 86. In addition to studying regular-faced convex polyhedra he also made other important contributions to geometry, was a dedicated teacher and a lovely human being. Before looking in more detail at what Norm Johnson did, let us first look at the issue of polyhedra from a historical and modern perspective.

We have already seen that in ancient Egypt there was an interest in polyhedra. Even more work was done in ancient Greece about polyhedra, and there was attention paid to them in Euclid's *Elements*. Yet, it is still remarkable that no mention of the notion of convexity appears in Euclid, presumably because convex polygons and convex polyhedra are the objects that interested Greek scholars and no framework existed for contrasting them with more "exotic" polygons and polyhedra.

In many regards the mathematical study of regular faced polyhedra began in Euclid's Elements. There it is observed that there are precisely 5 (convex) regular polyhedra in the sense that their faces are regular p-gons with q polygons at a vertex. It was shown that these polyhedra, often called the Platonic solids, are five in number. Furthermore it is shown they can be constructed with straight edge (ruler) and compass. Today, the fact that there exist five Platonic solids is most easily established by using that landmark of geometry which Leonhard Euler (1707-1783) noted (without a correct proof) about 1750. Using the fact that Vertices + Faces - Edges = 2 for all convex polyhedra, and as shown by Cauchy more generally for plane-connected graphs, one can easily show there are exactly 5 combinatorially regular polyhedra.

Before continuing with our historical development, a few words may be helpful about the concepts of polyhedra and polygons that define the "faces" of a polyhedron. Today we think of convex polyhedra from a variety of points of view, in particular, from a metric point of view (congruence of faces; size of angles; lengths of sides) and a combinatorial point of view (number of sides of each face and the valences of the vertices). Even today, providing definitions for polygons and polyhedra that are useful within mathematics and are "efficient" in conveying the core idea of the concept is not totally easy. The core idea is that one has a surface with vertices, edges, and faces where the edges are straight lines and the faces are parts of flat planes, and are convex polygons. A very familiar example of a polyhedron is the hexahedron (cube) which has 6 faces where each of the faces is a square.

Figure 8 (Representation of a cube in a plane.)

As noted previously, the drawing in Figure 3 is an attempt to help represent something that is 3-dimensional by drawing it in a suggestive way on a flat surface. Sometimes some of the lines which are "hidden" from view by the front of the polyhedron are shown dotted or thinner in such diagrams. (In Figure 8 different people will sometimes see different faces as the "top" face, a phenomenon psychologists have called attention to with the study of optical illusions.) Can you decide which edges would be (for you) the hidden line edges? (Hint: There are 3 such edges.) Now imagine the top face of the cube is subdivided as shown in Figure 9 by the addition of 4 other vertices and 8 additional edges which lie in the plane formed by the face ABCD of the original cube. Is this a "legal" convex polyhedron? While the original cube had 8 vertices, 6 faces, and 12 edges, how should the counts of vertices, edges and faces be carried out here? While 3-dimensional objects like that depicted in Figure 9 are interesting, it is convenient to use a definition of convex polyhedron which would not count this example as a convex polyhedron. One could get a convex polyhedron back by interpreting the diagram in Figure 9 as having a truncated pyramid (with the 8 vertices, a, b, c, d, A, B, C, and D pasted on the top face of the cube (ABCD)! Now you should be able to count the number of vertices, edges and faces of this strictly convex polyhedron. (Hint: It has 10 faces and four 4-valent vertices.)

Figure 9 (When additional edges and faces are added within a face of the diagram in Figure 8 the result is no longer thought of as a convex polyhedron.)

Suffice it to say that in modern times we approach the issue of the concepts of polygon and polyhedron from the point of view that allows for a variety of mathematical developments into more and more domains using the concept of a convex polygon and convex polyhedron as jumping off points and using both metric and graph-theoretic (combinatorial) points of view as tools to get insights.

Historically, the next developments concerned an accomplishment of Archimedes (c. 287-212 BC), but one we found out about due to Pappus (c. 290 - c. 350 AD). Archimedes came after Euclid but not all of his works have come down to us. In particular, Archimedes seems to have investigated what today are called the *Archimedean polyhedra* or solids. When described "informally," these are convex polyhedra whose faces are all regular polygons and with the same pattern of faces surrounding each vertex. We often use ideas about symmetry in talking about geometrical objects including polyhedra. In an interesting story about how mathematics grows and evolves, Pappus describes--using modern terminology--13 convex 3-dimensional polyhedra whose faces are all regular convex polygons, and the pattern of polygons surrounding each vertex. However, despite Pappus and Archimedes being distinguished mathematicians, they missed one convex polyhedron that obeys this condition. This is because if one starts with the polyhedron in Figure 10, often called the rhombicuboctahedron,

Figure 10 (A representation of the rhombicuboctahedron. Courtesy of Wikipedia.)

one can rotate one of the "caps" of the polyhedron as suggested in Figure 11

Figure 11 (The two caps of the rhombicuboctahedron separated from the octagonal prism in its "middle." Courtesy of Wikipedia.)

to obtain another convex polyhedron (Figure 12) which is not the same as the start polyhedron and is often called the *pseudo-rhombicuboctahedron*.

Figure 12 (When one cap is rotated with respect to the two other parts one gets a different polyhedron from the rhombicuboctahedron often called the pseudo-rhombicuboctahedron. Courtesy of Wikipedia.)

Note that the two convex polyhedra in Figure 10 and Figure 12 have the same pattern of surrounding polygons around each vertex, yet the two polyhedra are not equivalent to each other. It was only many years after Archimedes, and a brief comment in the work of Johannes Kepler (1571-1630) about a 14th "Archimedean" polyhedron, that it was noticed in the 20th century that the pseudo-rhombicuboctahedron exists! Modern discussions of the Archimedean polyhedra typically state there are 13 but this is "true" only if one uses a group-theoretic (isometric) approach to these solids rather than using the viewpoint of Archimedes and Pappus.

By the way, for some so-called Archimedean polyhedra there are two versions of the solids involved, left-handed ones and right-handed ones. We will not be concerned with these "chirality" issues here. It is also worth noting that there are two infinite families of convex polyhedra which are also satisfy the condition of local surroundability. These are the prisms (two regular *n*-gons and *n*-squares (4-gons) for *n* at least 5, the prism with 2 regular 4-gons and 4 additional squares is already counted among the 5 Platonic (regular) polyhedra) and the antiprisms (two regular *n*-gons and 2*n* equilateral triangles for *n* at least 4). These two families usually are not "folded" into the Archimedean classification and Pappus does not mention them explicitly. Again from a modern perspective, when looked at through the lens of graph theory, the graphs of the Platonic solids, the Archimedean solids, and the prisms have the same degree (valence) at every vertex. Sometimes this condition, that all vertices have the same valence (degree), is said to make them "regular graphs," a word already overused from the metric point of view! I prefer to say that these solids are *k*-valent, that is, all the vertices have the same valence *k*, and that *k* can only take on the values 3, 4 or 5. This fact (theorem) is a consequence of Euler's formula (V+F-E = 2) for graphs that are connected (one piece) and planar (can be drawn in the plane so that edges meet only at vertices).

In their search to understand patterns in a wide variety of domains, mathematicians (and others) create "words" designed to give meaning to the concepts and patterns that emerge from the practice of mathematics. Notation and nomenclature have long "bedeviled" mathematics, even experts. When some family of objects is interesting, there are usually different ways of naming or notating these objects from different points of view. Here is one logician's thought on this:

"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race." (Alfred North Whitehead, *An Introduction to Mathematics*)

The distinguished historian of mathematics Florian Cajori wrote a book about the history of mathematical notation, *A History of Mathematical Notations*, in 1928. The book says it is Volume 1 but no second volume seems to have appeared.

Giving names to polyhedra raises challenges. What name should be used for the individual convex regular 3-dimensional solids, otherwise known as the Platonic solids? Many accounts use these names:

tetrahedron, cube, octahedron, dodecahedron and icosahedron

Four of these five names are based on words with Greek roots that count the number of faces of the polyhedron involved. Thus tetra (4), octahedron (8), dodecahedron (12) and icosahedron (20). The exception is the cube, which has 6 faces, and to be consistent one might call it the hexahedron (6). In this case the list would read:

tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron

where the list is not in alphabetical order but in order of the number of faces of the polyhedra in this collection. However, this approach often creates confusion for beginners or in cases where the names of the solids are not well rooted in the Platonic solids context. After all there are many convex hexahedra which have 6 faces. Putting the word regular in front of each item uses more words/letters but makes things clearer but one still has to be cautious because the word convex is not explicit. Thus, the solid in Figure 13 is a "regular" polyhedron, one of the so-called Kepler-Poinsot regular polyhedra:

Figure 13 (One of the Kepler-Poinsot solids, the small stellated dodecahedron. Courtesy of Wikipedia)

This non-convex regular polyhedron has 12 vertices, 12 non-convex faces each of which has six sides, and 30 edges. Note that when viewed as a graph (dots and lines drawing) the counts are different! The faces of this non-convex polyhedron are self-intersecting regular pentagons. This self-intersecting polygon was known to Greeks as a "symbol" called the pentagram. However, when it came to regular polyhedra the Greeks did not seem to consider the possibility that pentagrams could be put together to form a regular polyhedron.

Why do the count by faces rather than by numbers of vertices? Here, for example, is a list of convex 3-dimensional polyhedra organized by the number of vertices in the polyhedron.

When Johannes Kepler did his study of polyhedra he included some thoughts on extending the collection of "regular" polyhedra. Today there is a collection of polyhedra often called regular that are usually referred to as the Kepler-Poinsot polyhedra, since the union of the efforts of Kepler and Louis Poinsot (1777- 1859) provided a full list of these polyhedra, as determined by Augustin-Louis Cauchy. However, additional insights were provided by Joseph Bertrand (1822-1900).

Figure 14 (Schematics for the Kepler-Poinsot indicating their Schläfli symbols. Courtesy of Wikipedia.)

The modern names of the polyhedron in Figure 13, as well as the other Kepler-Poinsot polyhedra in Figure 14 were provided by the 19th century English mathematician Arthur Cayley (1821-1895). Additional insights into these polyhedra and how to name "regular" polyhedra in 3-dimensions and higher dimensions has also been provided by John Horton Conway. As you can see with hindsight and using an historical perspective, the issues of "attribution" become complex. Mathematics and in particular the theory of polyhedra does indeed stand on the shoulders of giants.

Attempts to provide nomenclature that were "clearer" have been carried out by many people. Loosely speaking these efforts were based on attempts to use systems based on numbers rather than words and to move to systems that were more quantitative than qualitative. Examples of such systems were developed or inspired by the Swiss mathematician Ludwig Schläfli (1814-1895) and the Dutch mathematician Willem Abraham Wythoff (1865-1939). The initial systems were typically expanded so as to be useful in describing more situations. Thus, H.S.M. Coxeter (Norman Johnson's doctoral thesis advisor) used numerical "codes" as well as diagrams to classify polyhedra. He also developed a graph theory way to code polyhedra which are known as Coxeter diagrams.

Figure 15 (Ludwig Schläfli. Courtesy of Wikipedia.)

Here, by way of example, are the symbolic notations often used to help name the small stellated dodecahedron.

Schläfli symbol: 5/2, 5

Wythoff symbol: 5 | 2 5/2

While much work on polyhedra was done before Euler, it was Euler who discovered (in modern terminology) that convex polyhedra obey this remarkable relation between their "parts:"

Vertices (V) + Faces (F) - Edges (E) = 2

Over a period of time this powerful tool made it possible to discover many new facts about polyhedra and their properties. Where to go after studying the Platonic solids and Archimedean solids? Perhaps one place to go, and this work was done by Eugène Charles Catalan, was to look at the "dual" concept to the Archimedean solids. The Archimedean solids allow faces of different kinds of regular polygons but require that the pattern around each vertex be the same. While one can study duality from a purely geometrical point of view, perhaps one easier route to duality lies via plane graphs. Figure 16 shows how to take a plane graph and interchange the role of faces and vertices as noted earlier in this essay (Figure 3 shows a self-dual graph).

Figure 16 (Given a graph G (solid lines) its dual graph is shown (dotted).Courtesy of Wikipedia.)

Faces with *k* sides become vertices with valence *k*, and vertices of valence *s* become faces with *s* sides. Catalan decided to try to dualize the Archimedean solids geometrically but an easier approach is to start with an Archimedean solid and dualize its graph.The Archimedean solids are convex. From a symmetry perspective they can be thought of as having distance-preserving transformations that take any vertex to any other vertex. This property is known as *vertex transitivity*. So the "dual" approach would look for convex face-transitive polyhedra. An example of one of the Catalan solids is shown in Figure 17.

Figure 17 (The Catalan solid associated with the Archimedean solid in Figure 10. Courtesy of Wikipedia.)

Figure 18 (Hidden line version of the Catalan solid in Figure 17. Courtesy of Wikipedia.)

Norm Johnson got his undergraduate education at Carleton College in Minnesota finishing his degree in 1953. His graduate work in mathematics was carried out at the University of Toronto. Much of his career was spent teaching at Wheaton College in Massachusetts, which does not have a doctoral program in mathematics. For many years Norm contributed insightful comments on symmetry aspects of polyhedra to various discussion groups on the internet. He had lots of patience in providing self-contained answers to questions about polyhedra in an environment where questions were not always posed with the precision that avoids confusion being sown. In particular Norm was an authority on "uniform polyhedra," where he looked at polyhedra not only in three dimensions but higher dimensions as well. Uniform polyhedra require "regular" faces (this allows both convex and self-intersecting polygons as faces). Additionally, uniform polyhedra are ones where the isometries (symmetries which preserve distance) of the polyhedron can move any vertex of the polyhedron to any other, a property called *vertex-transitivity*. For many years Johnson was working on a comprehensive treatment of uniform polyhedra, where one of the yardsticks for the polyhedra was not only being in 3-dimensional space but in higher-dimensional space as well. It appears that some of this work will appear in a posthumous book authored by Norm, entitled *Geometries and Transformations*. I am not exactly sure whether plans exist to publish more of Johnson's unpublished work on uniform polyhedra.

Now let us briefly look at the contribution of Norman W. Johnson, who reinvigorated the study of convex polyhedra with regular polygons as faces. "Fall out" from Johnson's work also had important implications for renewed interest in other aspects of polyhedra and polytopes. Much of this stems from Johnson's remarkable paper of 1966 in the *Canadian Journal of Mathematics*. This paper looked at the question of enumerating the convex 3-dimensional polyhedra all of whose faces are regular polygons, but leaving out the well-known Platonic solids, the Archimedean solids, and the "generic" prisms and antiprisms.

Initially, it was not even clear that there were only a finite number of these. In a paper jointly written with Branko Grünbaum, among other results it was shown that there could only be a finite number of these solids. As noted above Johnson listed 92 solids, and gave them systematic names which tried to hint at the way solids of this kind were related to other solids. And Zalgaller showed that Johnson's list was complete. Today, these 92 solids are known as the Johnson solids in his honor. If one makes models of polyhedra one can try one's hand at construction of examples of regular-faced convex solids. Sometimes one finds a nice model and looks it up in the list of Johnson solids but doesn't find it. The reason for this is that there are a lot of polyhedra which are "near-misses" for the Johnson solids. Physical models have sufficient play in the plastic or paper used so that perhaps the polygons in the model are "warped" and not really convex. There may be other distortions in which the way the faces of the model come together which means they are not REALLY convex regular-faced polyhedra and thus not in the Johnson solid list. After all, if one has faith in the concept of mathematical proof, we know there can only be 92 Johnson solids!

Norman Johnson was the doctoral student of the prestigious British-born Canadian mathematician Harold Scott MacDonald Coxeter, Donald to his friends. Coxeter had 17 doctoral students, including my doctoral thesis adviser Donald Crowe. The photo in Figure 19 shows Donald Crowe together with Norman Johnson, and another Coxeter student F. Arthur Sherk.

Figure 19 (Photo of Donald Crowe, Norman Johnson, and F. Arthur Sherk)

Figure 20 (Photo of Donald Crowe with my "academic uncle" Norman Johnson.)

This group picture (and the two other photos above) of some of Coxeter's students together with Coxeter's daughter Susan Coxeter Thomas, was taken (if memory serves) at the Coxeter Legacy Conference in Toronto in May 2004 after Coxeter died. Coxeter's life spanned nearly all of the 20th century, he having been born in 1907 and died in 2003.

Figure 21 (Front Row: Susan Coxeter Thomas, William (Willy) Moser, Asia Weiss, Second Row: F. Arthur Sherk, Seymour Schuster, Norman Johnson, Barry Monson, Donald Crowe and Cyril Garner.

Norman Johnson's work inspired a renewed interest in enumerating and understanding the properties of regular polyhedra of many kinds. He legacy will long continue. Norm, thanks for all the nifty mathematics!

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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 these materials. Some of the items above can be found via the ACM Portal, which also provides bibliographic services.

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