We all do it but we think we are bad at it

 kw: book reviews, nonfiction, mathematics, geometry

The book is Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else, by Jordan Ellenberg, a professor of mathematics who happens to write very well, who has also advised in the production of several films. Thinking about what kind of image to use to illustrate this review, I tried a few prompts with DALL-E3, and settled on this one:

The old geometer in his workroom, surrounded by the tools of his craft, clip art

After several trials, this image in dark sepia appealed to me the most.

The term Geometry means "measuring the Earth", but in practical terms it means to measure earthly things. That's something we all do frequently. For example, my wife and I are in the process of getting a bathroom remodeled, because ancient pipes have given out and the walls and floor need to be destroyed to get at them. We need to buy tile and supplies for the new floor, and a different kind of tile (or other covering) for the walls. What do we do? We take measurements and calculate areas. That's a geometrical task.

When we were in seventh and eighth grade, most of us had Geometry one year and Algebra the next. However, I suspect few of us had an Algebra teacher who showed us that geometry and algebra tackle the same problems from different directions; they are equivalent. What made geometry class so hard for me was the proofs. The Proofs! One needs a certain kind of mind to formulate a Proof. I don't have it. With a lot of painful study, I can figure out all the steps in a proof I have been shown, but I could never have produced it myself, and I soon forget it.

This book is not about proofs (but a few simple ones are shown, perhaps as a kind of inoculation: "See, that isn't so bad, is it?" To which I answer, "Yup, it's bad."). No, it is about geometrical thinking. For example, about trees. Biologically, a tree is a branching structure with a root, trunk, branches, twigs, and leaves. Conceptually, a biological tree is a metaphor for a way of arranging information, such as a family tree, a categorization of machine tools, or the districts and blocks and buildings in a neighborhood. A tree has one specific, critical characteristic: branches don't join back to branches. Otherwise, you have a network.

In a chapter on genealogy (#8, You are your own negative-first cousin, and other maps) the author states that your parents "don't share a known ancestor (unless you are from a truly aristocratic clan)." Not so fast, Dr. Ellenberg. Generally, we consider that a "family tree" is a genuine tree. With you as the root, your parents as the first two branches, your grandparents as the next, and so forth, the tree branches and branches but branches don't re-combine. However, sooner or later, they have to, as author notes in passing. If we look deeper into the situation we can go one better than that. 

In most jurisdictions, it is legal for first cousins to marry. In some, only certain first cousins can marry: they can have a common grandparent as long as the siblings in their parents' generation are of opposite sex. Thus, my mother had a sister and a brother. It would not be legal to marry the daughter of my mother's sister, but is quite legal to marry the daughter of my mother's brother. Such "available" cousins are called "kissing cousins".

Among my ancestors is a certain Joseph Macy, born on Nantucket Island in 1765, son of Joseph Macy and Mary Starbuck. Among his great-great grandparents—there are 16 of them—the surname Coffin appears three times and the name Starbuck appears twice. So he was his own third cousin three different ways. There were ten families that settled Nantucket in the 1600's. Within just a couple of generations, marriages between second or third cousins became necessary, and marriages between first cousins were getting common. So the genealogy of the Nantucket settlers is a bit of a treelike network. There are other interesting cross-links elsewhere in my family "tree", but I'll leave them for a later discussion. I'll just leave you with this thought. Joseph Macy was descended from Charlemagne; and so, of course, am I. The relationship is distant: 39 generations. That many generations ago, I had, formally speaking, 549,755,813,764 ancestors. It's about 550 billion. That implies a lot of cross-linkages in everyone's family tree, once you get back one or two dozen generations. As it happens, according to the data I have at present, I have five ancestors descended from Charlemagne.

Dr. Ellenberg likes long chapter titles. One of the shorter titles is "His style was invincibility" (Ch 5), in which we are introduced to an unbeatable Checkers player named Marion Franklin Tinsley. His story is the backdrop to investigating combinatorial math, using games. Tic Tac Toe is a simple game with a simple strategy, and is a "known game" because of its simplicity. There are 765 possible positions (the "state space"), and about 27,000 possible games, or ways to pass through a series of states in the state space. If two players play perfectly, every game is a draw. What about Checkers? The state space of Checkers is about 500 billion billion, or a 5 followed by 20 other digits. With a state space this large, nobody can learn the whole of it, and neither can any computer so far produced. Just for the record, the state space of the positions in Chess is a 2 followed by 46 other digits. This means that it is possible for a human player to still win either Chess or Checkers against a computer program, but it is very hard, because there are various heuristics (rules of thumb based on experience) that measure the strength of one position relative to another, and a machine can check millions of possibilities while a human player is thinking over five or ten. Then there is Go, for which the number of legal board positions is a number with more than 170 digits; this is an estimate! Considering that the number of atoms in the Universe is a 100-digit number, you would need billions of billions of billions (string out four more "billions") of Universes to contain a computer memory big enough to encompass the state space of Go.

But the process of analyzing any of these games is to build a tree! The starting position of the game is the root of the tree. In a game of Checkers, the number of possible starting moves by the first player is 14. Similarly for the initial moves of the second player, so the two-move state space is 14x14 = 196. These moves vary in how "strong" they are, how likely to lead to a Win. If you have a way to evaluate the strength of each move, you can label the nodes in the tree accordingly.

With each move the state space for that move is another factor that's usually between 12 and 16, until pieces start getting captured (which reduces the branching), and it can exceed 16 when some pieces get longer multi-jump options. It doesn't take long for the number of twigs on this tree to become millions, billions, etc.

The last chapter has the title "How math broke democracy (and might still save it)". It is about Gerrymandering, the practice of setting up voting districts in a state or county so as to favor one political party. This graphic from Statista shows examples from 2020. Redistricting, and thus Gerrymandering, follows each census; it's in the Constitution (the redistricting part!). 

There are many definitions of "fair", and none of them is "fair enough" to satisfy everyone involved. There are many proposals for rules or laws to produce a more "fair" outcome, but again, every such proposal generates lots of heat but hardly any light. 

In 2019 the Supreme Court punted on a possible method to ameliorate the problem. The author was involved, as a co-creator of an Amicus brief. Gerrymandering cannot be eliminated, not so long as people have emotions and a lust for power. The chapter does not end with a conclusion, but with a few tantalizing possibilities. None of them is likely to receive the support of a large enough majority to become embalmed in law. It is clear from the passionate tone of the chapter that this matter is of great importance to the author. Here, Geometry is Power.

Geometry is all around us. We all do it. We all think we are bad at it. It takes a peculiar kind of mind to revel in the proofing process, but just as very few people design the cars that millions of us drive, or the phones we consult so compulsively, we know what we need to know, and as this book shows, we know more than we think we do.

We all say we hate it but we all do it

 kw: book reviews, nonfiction, science, mathematics, geometry

…No, my subject is not Sin (but that would be equally true), but Geometry. But before we go on, let me tell a story.

My brother and two close friends took several courses in college together, including a math class that emphasized proofs. I'll conceal identities here, and just represent my brother as Art, and his friends as Bob and Cal. They all did pretty well in their "math proofs" class. Art studied diligently and did well, while Bob struggled mightily to keep a "B" grade, but Cal did the best with the least work. Another fellow student told them one day, "Bob walked into a room and saw a big machine with a large gear on one end. He was told he had to make it run. He looked it over, then put his shoulder against the gear and heaved a great heave, making the gear turn. Art came in next. He nosed around and found a crank that fitted into the gear's shaft. He put it in, and turned the gear. Then Cal came in. He found a cord with a plug, plugged it in, pushed a button, and the machine began to run." When it comes to mathematics courses that require proofs (algebra) or demonstrations (geometry), I am definitely in Bob's league, at best.

Now, when I see a clear geometric demonstration, I can often comprehend it almost instantly. But I could never have produced that demonstration.

These two fellows (one an Arab, one a European), shown in a 15th Century drawing, are having a go at some demonstrations. The Westerner is trying his hand at squaring the circle (which is known to be impossible), while the Arab is, more practically, extending a demonstration of the Pythagorean Theorem.

In a memoir, we read that Abraham Lincoln said of himself that he "nearly mastered the six books of Euclid." In Shape, The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else, by Jordan Ellenberg, we find that Honest Abe struggled for months to square a circle. Of course, he failed, and apparently he never came across a proof or demonstration that doing so is impossible.

All the demonstrations and constructions in Euclid's work must be done with compass and straightedge. The ancient compass was so constructed that once you set the two points you could scribe a circle about one of them, but the two legs collapsed when lifted from the paper; you cannot "set" that kind of compass. Further, the straightedge must have no markings on it; its only use is to draw straight lines through points already marked on the page. It so happens that if you are allowed to make a single mark on the straight edge, effectively turning it into a ruler, you can construct an extended radius, such as what we see in "b" in the illustration above, that has the required length of one side of the square. Otherwise, "No markee, no squaree." This point is not mentioned in Shape, but I suspect that Dr. Ellenberg knows it.

If Euclid were to drop into the office of any modern professor of geometry, he would recognize nearly nothing, except the (probably translated) books on a shelf, nearly out of sight, that he wrote about 2,300 years ago. Much of the "geometry" carried on these days is topology, which isn't about circles, squares or triangles, but about holes. Yes, holes. To a topologist, anything with no holes but with a defined edge is a "circle", including things we'd call squares or triangles. But if you punch a hole in it it's now something else. Most topological work is done in three (or more) dimensions. You may be familiar with the statement, "A topologist has trouble telling the difference between his cup of coffee and the donut he wants to dip into it." Both a cup with a handle, and a donut, are solid shapes with a single hole.

A fun discussion in Chapter 2 involves the title, "How Many Holes Does a Straw Have?" I asked my wife. She said, "One." That is a proper topological answer. If you shorten the straw, and make its wall thicker, it becomes a donut. Distort it some more, also pressing out a cavity (but not a hole!) in one section, and you get a coffee cup. This is easier to do with clay than with paper! But just for fun, the author shows how people defend the answer "Two", because many people would say, "It has a hole in each end"; and even the answer "None", because some would say you started with a flat sheet of paper and rolled it up.

I have to say, I was puzzled that the word "cavity" never appeared in that chapter. Topologically, a cave (source of the word "cavity") has no holes if it has no "other end". In topology, you only have a hole if you can go into one side (or end) and come out the other. So the animal known as Hydra has a cavity, but no holes, while most animals have a single hole called the Alimentary Canal, with a mouth at one end and an anus at the other. So we are donuts. Very lengthy donuts.

Well, that's not where this book is ultimately going. The earlier chapters help us get used to some geometrical ideas, and we soon get to maps. Here are two maps, and they are related:

These maps are the same (but for drawing idiosyncrasies) as on p. 394 of the book. The author calls the second (green) one a "chart", but it is also a map. It is the "first inversion" of the blue map; the green lines represent the relationships between the blue-outlined areas.

Have you ever played Nim? You begin by stacking an arbitrary number of coins in two or more piles. In one version, each player can take either one, two, or three coins, all from one pile. Other versions exist with different "taking" rules. Players alternate taking coins until one coin is left. The player who must then take that last coin loses. The author shows a simple proof that the first player will always lose if the other player makes no mistakes.

These two maps (or "map" and "chart") illustrate something about electoral districts in a state. One method for detecting Gerrymandering (where it isn't obvious, and I'll explain more anon) involves playing Nim with the line segments in the green map until a player can't remove any segment without breaking the map into two pieces. One such game ends after four moves, to look like this:

The green map from before is now a Tree, a connected map with branching but no loops and no holes.

I won't take this further here, except to say this Nim game brings about several possible ways to break up a district made of smaller units into two districts. It is one step in the process of making an electoral map having districts that are "more fair", but then we get into a discussion of what "fair" means.

For example, "proportional representation" is often talked about. To get away from R vs D or L vs R (or X vs Y, which could be sexist), I'll refer to the two "interested" parties as H and O (Lionel fans, take note). Consider a state that has ten districts, and having 60% H voters and 40% O voters. Assume they are all pretty evenly spread throughout the state. How would you draw district boundaries to "ensure" that 6 H and 4 O district representatives will be elected? If you could, would that be "fair"? That would simply guarantee that the H's in the legislature would always win every vote, unless the O's could sometimes convince a couple of H's to vote their way. So, effectively, the O citizens in the state would be without representation.

Gerrymandering, so-called because a proposed district map drawn by one Eldridge Gerry included a district shaped like a lizard, is thought of as unfair mapmaking designed to ensure that a certain party will always win. When you know where the voters live, and voter registration is how you know, you can wiggle the boundaries around to get most opponents into the smallest number of districts, and create many more districts that will just barely elect your own members. The illustration below was cropped from an article titled "The most Gerrymandered districts in America":

Even though the author's passion clearly lies in "dealing with Gerrymandering", he acknowledges that while it is often visible, it is dramatically hard to quantify. The more so because we cannot yet clearly define what "fair" means.

I have an idea: a Federal law that requires one election in four to be automatically reversed the day after the poll results are revealed. To reduce the amount of "gaming the system" that would be indulged in, a pair of fair coins would be flipped on the day in question. This ceremony would be conducted in public, with much publicity. If both coins come up Heads, all elections are reversed. Otherwise, their results stand. Of course, that means that sometimes two Reversal Years might occur in a row, and perhaps even three. It would also happen that five, ten, or more years may pass with no Reversal Years. That's OK. The added uncertainty might not make for better legislating, but it would definitely make it more interesting! I have about as much confidence that such a procedure could become law as I have of the Sun setting in the East tomorrow evening.

Now I want to back up to the idea of the Tree. The author has an interesting statement about trees and related graphs in a footnote on p. 106: "…there's a more general notion than a tree, called a directed acyclic graph…a DAG is like a tree where some branches are allowed to fuse together… Think of a particularly aristocratic family where your parents may share a great-grandparent or two." This isn't as rare as he thinks. Inbreeding happens whenever the "breeding pool" gets too small. 

For example, I have an ancestor, a Quaker, whose parents left Nantucket in the fifth generation after its settlement by ten families. Four generations back, her father could have been descended from eight of the ten families if no marriages between cousins or second cousins occurred, as could her mother. However, her father is descended from only seven, and her mother is descended from six. Between the two, going back to the settlers' generation, the Starbuck couple appears three times, the Coffin couple appears three times, and the Garner couple appears twice. And the couple themselves were second cousins (or a little closer than that, considering). First-cousin marriage is legal in seven U.S. states, and second-cousin marriage is legal in all. First-cousin-once-removed marriage is legal in 42 states. Thus, many family "trees" are really "family directed acyclic graphs". Fortunately the software at sites such as Ancestry.com is written to accommodate relationships of all kinds, perhaps even including the one described in the song "I'm My Own Grandpa."

Also, realistically speaking, when you go back more than a dozen generations or so, you'll find all kinds of links between relatives. Anyone living today who is descended from Charlemagne (crowned in the year 800), is a 38th or 39th or 40th generation descendant. Take a "tree" back 38 generations, and there are theoretically almost 275 billion ancestors. But the population of Europe in 800 AD was around 25-30 million. Think that over…

When I started the book, I had no idea it would go in these directions. It is too much fun to think about all these things. This is an author I'll keep in a tickler file.

When the math you used could mean life or death

kw: book reviews, nonfiction, mathematics, geometry, analysis, renaissance

Who would have thought that for a period of decades a student's adherence to certain mathematical methods could get him in trouble with the Inquisition, imprisoned, or even burnt at the stake. Galileo was placed under house arrest for the last two decades of his life, not only for advocating the motion of the Earth, but also for the kind of mathematical analyses he published!

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander, chronicles the development of a "new" kind of mathematics, one that had actually existed alongside Euclidean geometry for centuries, but had been little used and was denigrated by Aristotle and others. It flowered along with the Italian Renaissance, but ran afoul of the reactionary politics of the Jesuits.

To most mathematicians of the early Renaissance, mathematics was geometry, and all proofs and analyses that proceeded by any method other than straightedge-and-compass derivation from first principles were suspect. It is rather amazing to read how the Society of Jesus, originally rather blind to mathematics because of the proclivities of its founder, Ignatius of Loyola, took up Euclidean geometry as a point of pride within a generation after his death.

In their to-the-death struggle to throw back the influence of the Protestant Reformation, the Jesuits, brought into being as the Reformation was blossoming throughout Europe, realized that geometrical proofs provided a perfect model for their rigid theology and social structure. The Reformers declared that all persons had a right to know and understand Scripture, and offshoots such as some Anabaptists, and free-land proponents such as the Diggers, began to question the "divine right of the King" and the "natural order" of aristocracy. Dogma was being replaced by opinion. Long-held traditions were in danger of being overthrown. Chaos was imminent. The execution of the English king Charles I emphasized the danger.

If one accepts the validity of the methods of Euclid, there is no room for opinion. A geometrical constructive proof, proceeding by pure deduction, leads step by step to a conclusion that cannot be denied. But it had become evident to the disciples of Pythagoras, nearly a full twenty centuries earlier, that some propositions one could state, could not be proved. They had begun by proclaiming that all problems were subject to "rational" proof; by "rational" they meant using only ratios of whole numbers. An early demonstration that the hypotenuse of a square could not be exactly expressed as a ratio, that it was "incommensurable", led to the breakdown of the Pythagorean system and eventually to the disbanding of the Pythagoreans.

By Aristotle's time, about 200 years later, inductive methods based on "indivisible" quantities had shown some promise, and had been used to demonstrate certain propositions that geometric methods could not solve. But Aristotle, at first intrigued, later decried such methods. Euclid he could understand; the new methods seemed to allow a certain leeway for error. In his way he was as rigid as any Jesuit of the Sixteenth Century.

I have often been astounded that the Medieval Roman Catholic Church based so much of its philosophy on Aristotle, whose only brush with Theism is some vague statements about an "unmoved mover." I was further amazed to read of the process that led to this, via Thomas Aquinas. The Jesuits believed that Aristotle had it right. Mathematical induction by "indivisibles" (also called "infinitesimals" after about 1730) was unreliable. The Church needed … NEEDED! … a rigidly reliable theology and rule of society that disallowed dissent as thoroughly as a Euclidean proof disallows "alternate opinion". Galileo was only the most prominent of a large number of Italian mathematicians to learn of inductive methods, and use them to great effect, so much so that these methods swept through Europe. But over about a century's time the Jesuits drove "indivisibles" out of Italy. Indivisibles and inductive methods flourished elsewhere, in all the countries of Europe.

Reasoning similar to that of the Jesuits led Thomas Hobbes to found his political philosophy on Euclidean geometry. He strongly felt that the chaos following the Reformation simply cried out for a more totalitarian form of government. His exceedingly famous book Leviathan proposes the most profoundly totalitarian political system ever devised. When he learned that three very significant propositions were incommensurable via Euclidean methods, he realized that this left a great loophole in his philosophy.

Three problems: Squaring the Circle (making a square with the same area as a given circle), Trisecting an Angle, and Doubling a Cube (constructing a length that can be used to construct a cube with twice the volume of a given cube). None of these can be done using Euclidean geometric methods. This has been proven, using mathematical methods developed centuries after the time of Hobbes. He spent the rest of his life trying to square the circle, and eventually lost his reputation as a mathematician. He ran afoul of Gödel's Incompleteness Theorem: that every mathematical system can be used to formulate problems that cannot be solved withing the confines of that system. This includes geometry. But Kurt Gödel was two centuries in Hobbes's future.

In the opening chapters of the book, it seemed to me that "indivisibles" and "infinitesimals" were described as being in opposition. It took careful reading to understand that they were synonyms separated by a century or two of usage. They form the foundation of The Calculus, as developed by both Newton and Liebnitz. The modern world would not exist without the analytical methods of calculus. From a modest number of "demonstrations" using induction—based on lines being composed of an infinite number of "indivisible" points, planes being composed of indivisible lines, and volumes being composed of indivisible planes—calculus and modern analysis in general have become supercharged, and now include both inductive and deductive methods.

I spent much of my adult life as a working mathematician, and I find it fascinating that such a life-and-death struggle had to be won, and won decisively, for the modern, technological world to appear. I have just touched on a few of the trends and a handful of the players in the saga of Infinitesimals. I have to mention John Wallis, whose 25-year battle with Hobbes "saved" inductive mathematics in England. How much longer would the modern era have been delayed otherwise? He originated the symbol for infinity: ∞. Infinitesimals is quite an amazing story, very well told.

Pathological parallax

kw: politics, dissension, geometry

You may not know the word parallax, but you know the principle: your left and right eyes work together to produce a 3D view. One eye alone cannot reliably determine depth. Each eye sees the world a little differently, but the combination of their views is more accurate. As a political principle, it is a good metaphor of the way diverse political views can work together to guide national policy. This is better than if only one view prevails; that is like trying to drive with one eye closed. It is risky.

There is a second danger. In metaphor it is like this: my brother likes doing stereo photography, and he does it by taking pictures of a static scene from two places, usually a few inches apart. But if the scene is larger or farther away, he will use wider spacing. There is a knack to this, and sometimes the results can be rather odd. A recent stereo pair he sent me, of a cave in Mexico, is an example of taking the two pictures from points that were too far apart. It is nearly impossible to "fuse" them to one image with 3D depth visible. It usually just looks like a confused mess. It is like looking at the tip of your nose with crossed eyes; it just isn't a clear view.

This kind of cross-eyed view has become the norm in modern politics. One side tries to close the other side's eye, because they have become so polarized that no combined view is possible. Many conservatives almost deify Ronald Reagan, who is famous for saying, "When the car of State has gone off the road into the ditch on the left side, it takes a truck on the right side to pull it back onto the road." I, too, prefer Reagan politics to the kind we have had under Obama. Most forget that, as much as Reagan and Tip O'Neill scrapped in public about policy, they worked together to accomplish almost all the facets of "Reaganomics" that led to the prosperity of the 1990s.

Can common ground be found now? I think it will take a sweeping "vote-em-all-out" election next November, to bring in a Congress that can take proper account of the fused vision of all the "eyes" along the political spectrum. Political eyes-crossed posturing simply damages the country.

Mobile Polyhedra

kw: mobiles, polyhedra, geometry

Some months ago a young friend showed me how to make the origami modules for a novel kind of dodecahedron. It has a slightly stellated look, but is not a stellated solid. After some thought, I figured out how to make a number of other polyhedra. The original module has the appropriate angles for the dodecahedron and other shapes that use about thirty modules, but I had to modify it for other shapes with different numbers of modules.

After several months of fooling around I had nine polyhedra of various kinds, some a solid color and some multicolor. I decided to construct a mobile. The backdrop for this image was several black plastic bags. I'll have to get some dark cloth if I want to do any more pics like this. The mobile consists of two smaller mobiles connected by a three-foot rod. Since this image was taken I have replaced the long rod with one of larger diameter that doesn't sag as much. That keeps the whole thing within a foot of the ceiling, making it suitable for hanging in my office (this is my basement workshop).

The larger sub-mobile consists of the original green dodecahedron (30 modules), a lavendar icosahedron (also 30), a pink rhombic dodecahedron (24), a light blue icosadodecahedron (60) and a dark blue rhombic triacontahedron (60).

The smaller sub-mobile consists of a yellow icosahedron (30, smaller modules), a cuboctahedron (24), a shape I think is a snub cube (48) and a darker-hued icosadodecahedron (60). The latter took the most planning, to get each planar section a single color. Purists put these together with friction only. These are glued for durability.

Once I make the appropriate drawings I'll submit a post with directions for the modules and information about making some of the shapes.

He could make anyone love geometry

kw: book reviews, nonfiction, biographies, geometry, polyhedra

One of these times, I'll post a picture gallery of some of my mobiles. Although Calder is a favorite artist of mine, I don't make mobiles in the abstract way he made famous. As a lover of geometry, I make them very regular. I even have a spreadsheet with formulas for working out the length ratios for various kinds. For example, sometimes I use Fibonacci ratios for the lengthening support wires, sometimes a more pure geometric sequence. They also represent fractals, with a dimension in the range 1.4-1.7.

My favorite, which hangs in my office, is made with skeletal polyhedra. All the 5 "regular solids", of course, plus four other mixed-face shapes. Now, I find myself wondering why I never learned before of the king of polyhedra, Donald Coxeter. The recent biography, "King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry" by Siobhan Roberts, is shelved as a math book, but clearly belongs in the BIOG section of the library.

HSM Coxeter ("Donald" is from his third name, MacDonald) lived nearly the entire 20th Century, and three years into the 21st. By the time he was seventy, people that met him were astonished that this legend still lived. By the time he was 95, their children or grandchildren shared the experience. He lived 96 years, and just two days before he died, he put the finishing touches on his last monograph, titled "An Absolute Property of Four Mutually Tangent Circles," a breakthrough paper published in 2005.

Not for him, the usual "burn out young" mathematical career. He was productive for some eighty years...phenomenally productive. A classical geometer in an age of "new math" and "down with triangles", he is solely responsible for the resurgence of geometry that has revivified math, cosmology and physics since the 1970s. It is to "Coxeter groups" that we owe the efficiency of modems, and his masterwork "Regular Polytopes", published in 1948, underlies recent theories of the "shape" of the Universe.

What is a polytope? It is a generalization of a polyhedron into more than three dimensions. This image shows Coxeter with George W. Hart at a conference at which many polyhedral models, most inspired by his work, were displayed. A plane figure is a polygon (gon = side); a 3-D shape made of several or many polygons is a polyhedron (hedron = face). The suffix tope means place.

The five "regular solids" are the Tetrahedron, composed of four equilateral triangles, the Cube or Hexahedron, composed of six squares, the Octahedron, composed of eight equilateral triangles, the Dodecahedron, composed of twelve pentagons, and the Icosahedron, composed of twenty equilateral triangles.

This figure shows one way to represent a 4-D polytope called the 11-cell. It is a regular polytope whose eleven topes or cells are icosahedra. The colored square next to each of the eleven shapes shows which face it attaches to on the other ten, and the numbering is used to indicate orientation. You gotta think in four dimensions to figure it out, which I can't do. Click for a large view.

Somehow, Coxeter and others can think in almost any number of dimensions, and get useful results. His "Coxeter graphs" are a compact notation of the symmetry of a kaleidoscope in any number of dimensions, that would show some regular polytope (I'm really getting to the limit of my understanding here...).

The one called "The Coxeter Graph", however, as shown here in four different arrangements, has a use I really don't comprehend, but was a discovery that tickled him so much that he wrote of it under the title "My Graph".

Though I have mentioned a few uses of his work, he didn't care much for its usefulness. He was the epitome of a pure mathematician, and sometimes retorted rather sharply when a practical application of his work was described.

Although he was typically cordial, even affable, he bore fools not at all, and reading between the lines of the text, one finds that he could be a bit of a jerk. Or perhaps that term is too harsh. Perhaps he is more like a colleague of mine, who tends to roam the halls when thinking hard. He'll run right over you if you don't step quickly; he simply doesn't see you. Coxeter's geometric vision was vastly wide, but in the rest of life, his tunnel vision was legendary.

These scattered reflections sadly belie the quite comprehensive narrative that Ms Roberts has produced. A biography of Coxeter has been long overdue, and I am grateful to make his posthumous acquaintance.

Invertebrate geometer - Mystery Solved

kw: puzzles, geometry, flowers, solutions

In a post on August 2, 2006, I showed a picture of a Phlox flower that looked like a propellor, each petal was chewed on one side only. This image is the clue to solving why a critter (probably a caterpillar) would favor only the left side of each petal.

The opened flower is chewed in the same propellor pattern. The bud next to it, about a quarter opened, lets us see what happened. When a bud gets partway open, and it is still mostly dark, the caterpillar eats the exposed part of each petal quickly, then flees as the sun rises.

The geometric "knowledge" thus resides in the flower bud's DNA, not in the eater.

Invertebrate geometer?

kw: puzzles, curiosity, geometry, flowers

I took this picture the morning of August 1, 2006. Our little Fall Phlox plant produces a flower or two daily; a bud is visible behind this chewed flower. Each afternoon, I see the new flower(s) in pristine beauty. The next morning, they are variously chewed.

This particular morning some fastidious creature—probably some kind of caterpillar—has eaten only the left half of each petal. The other flower, on the other side of the plant, was chewed in a similar pattern.

In the past, I've seen the ends of the petals removed, or two of the five completely eaten. A curious circumstance, this one!