Tidbits of Geometry

Since I grew up up reading Martin Gardner, I’ve often imagined it would be fun to write about math and physics in a way that nonexperts might enjoy. Right now I’m trying my hand at this on Google+. You can read that stuff here.

Google+ encourages brevity—not as much as Twitter, but more than a blog. So I’m posting things that feature a single catchy image, a brief explanation, and URL’s to visit for more details.

Lately I’ve been talking about geometry. I realized that these posts could be cobbled together into a kind of loose ‘story’, so here it is. I couldn’t resist expanding the posts a bit, but the only really new stuff is more about Leonardo Da Vinci and the golden ratio, and five puzzles—only one of which I know the answer to!

The golden ratio

Sure, the golden ratio, Φ = (√5+1)/2, is cool… but if you think ancient Greeks ran around in togas talking about the “golden ratio” and writing it as “Φ”, you’re wrong. This number was named Φ after the Greek sculptor Phidias only in 1914, in a book called The Curves of Life by the artist Theodore Cook. And it was Cook who first started calling 1.618… the golden ratio. Before him, 1/Φ = 0.618… was called the golden ratio! Cook dubbed this number “φ”, the lower-case baby brother of Φ.

In fact, the whole “golden” terminology can only be traced back to 1826, when it showed up in a footnote to a book by one Martin Ohm, brother of Georg Ohm—the guy with the law about resistors. Before then, a lot of people called 1/Φ the “Divine Proportion”. And the guy who started that was Luca Pacioli, a pal of Leonardo da Vinci who translated Euclid’s Elements. In 1509, Pacioli published a 3-volume text entitled De Divina Proportione, advertising the virtues of this number.

Greek texts seem remarkably quiet about this number. The first recorded hint of it is Proposition 11 in Book II of Euclid’s Elements. It also shows up elsewhere in Euclid, especially Proposition 30 of Book VI, where the task is “to cut a given finite straight line in extreme and mean ratio”, meaning a ratio A:B such that A is to B as B is to A+B. This is later used in Proposition 17 of Book XIII to construct the pentagonal face of a regular dodecahedron.

The regular pentagon, and the pentagram inside it, is deeply connected to the golden ratio. If you look carefully, you’ll see no fewer than twenty long skinny isosceles triangles, in three different sizes but all the same shape!

They’re all ‘golden triangles’: the short side is φ times the length of the long sides.

And the picture here lets us see that φ is to 1 as 1 is to 1+φ. A little algebra then gives

which you can solve to get

and thus

For more, see:

• John Baez, Tales of the Dodecahedron.

Da Vinci and the golden ratio

Did Leonardo da Vinci use the golden ratio in his art? It would cool if he did. Unfortunately, attempts to prove it by drawing rectangles on his sketches and paintings are unconvincing. Here are three attempts you can see on the web; click for details if you want:

The first two make me less inclined to believe Da Vinci was using the golden ratio, not more. The last one, the so-called
Vitruvian Man, looks the most convincing, but only if you take on faith that the ratio being depicted is really the golden ratio!

Puzzle 1. Carefully measure the ratio here and tell us what you get, with error bars on your result.

It would be infinitely more convincing if Da Vinci had written about the golden ratio in his famous notebooks. But I don’t think he did. If he didn’t, that actually weighs against the whole notion.

Indeed, I thought the whole notion was completely hopeless until I discovered that Da Vinci did the woodcuttings for Pacioli’s book De Divina Proportione. And even lived with Pacioli while this project was going on! So, we can safely assume Da Vinci knew what was in this book.

It consists of 3 separate volumes. First a volume about the golden ratio, polygons, and perspective. Then one about the ideas of Vitruvius on math in architecture. (Apparently Vitruvius did not discuss the golden ratio.) Then one that’s mainly an Italian translation of Piero della Francesca’s Latin writings on polyhedra.

De Divina Proportione was popular in its day, but only two copies of the original edition survive. Luckily, it’s been scanned in!

• Luca Pacioli, De Divina Proportione.

The only picture I see that might be about using the golden ratio to draw the human figure is this:

The rectangles don’t look very ‘golden’! But the really important thing is to read the text around this picture, or for that matter the whole book. Unfortunately my Renaissance Italian is… ahem… a bit rusty. The text has been translated into German but apparently not English.

Puzzle 2. What does Luca Pacioli say on this page?

The picture above is on page 70 of the scanned-in file. Of course some scholar should have written a paper about this already… I just haven’t gotten around to searching the literature.

By the way, here’s something annoying. This picture on the Wikipedia article about De Divina Proportione purports to come from that book:

Again most of the rectangles don’t look very golden, even though it says “Divina Proportio” right on top. But here’s the big problem: I can’t find it in the online version of the book! Luca Luve, who spotted the online version for me in the first place, concurs.

Puzzle 3. Where is it really from?

Luca Pacioli

Luca Pacioli had many talents: besides books on art, geometry and mathematics, he also wrote the first textbook on double-entry bookkeeping! This portrait of him multitasking gives some clue as to how he accomplished so much. He seems to be somberly staring at a hollow glass cuboctahedron half-filled with water while simultaneously drawing something completely different and reading a book:

Note the compass and the regular dodecahedron. The identity of the other figure in the painting is uncertain, and so is that of the painter, though people tend to say it’s Jacopo de’ Barbari.

Piero della Francesca

This creepy painting shows three people calmly discussing something while Jesus is getting whipped in the background. It’s one of the first paintings to use mathematically defined rules of perspective, and it’s by Piero della Francesca, the guy whose pictures of polyhedra fill the third part of Pacioli’s De Divina Proportione.

Piero della Francesca seems like an interesting guy: a major artist who actually quit painting in the 1470’s to focus on the mathematics of perspective and polyhedra. If you want to know how to draw a perfect regular pentagon in perpective using straightedge and compass, he’s your guy.

Constructing the pentagon

I won’t tell you how to do it in perspective, but here’s how to construct a regular pentagon with straightedge and compass:

Just pay attention to how it starts. Say the radius of the circle is 1. We bisect it and get a segment of length 1/2, then consider a segment at right angles of length 1. But

so these are the sides of a right triangle whose hypotenuse has length √5/2, by the Pythagorean theorem!

Yes, I know I didn’t explain the whole construction… just the start. But the golden ratio is √5/2 + 1/2, so we’ve clearly on the right track. If you’re ever stuck on a desert island with nothing to do but lots of sand and some branches, you can figure out the rest yourself.

Or if you’ve got the internet on your desert island, read this:

• Pentagon, Wikipedia.

But here’s the easy way to make a regular pentagon: just tie a simple overhand knot in a strip of paper!

The pentagon-decagon-hexagon identity

The most bizarre fact in Euclid’s Elements is Proposition XIII.10. Take a circle and inscribe a regular pentagon, a regular hexagon, and a regular decagon. Take the edges of these shapes, and use them as the sides of a triangle. Then this is a right triangle!

How did anyone notice this??? It’s long been suspected that this fact first came up in studying the icosahedron. But nobody gave a proof using the icosahedron until I posed this as a challenge and Greg Egan took it up. The hard part is showing that the two right triangles here are congruent:

Then AB is the side of the pentagon, BC is the side of the decagon and AC’ is the radius of the circle itself, which is the side of the hexagon!

For details, see:

• John Baez, This Week’s Finds in Mathematical Physics (Week 283).

and

• Pentagon-decagon-hexagon identity, nLab.

The octahedron and icosahedron

Platonic solids are cool. A regular octahedron has 12 edges. A regular icosahedron has 12 vertices. Irrelevant coincidence? No! If you cleverly put a dot on each edge of the regular octahedron, you get the vertices of a regular icosahedron! But it doesn’t work if you put the dot right in the middle of the edge—you have to subdivide the edge in the exactly correct ratio. Which ratio? The golden ratio!

This picture comes from R. W. Gray.

According to Coxeter’s marvelous book Regular Polytopes, this fact goes back at least to an 1873 paper by a fellow named Schönemann.

Puzzle 4. What do you get if you put each dot precisely in the center of the edge?

The heptagon

The golden ratio Φ is great, but maybe it’s time to move on? The regular pentagon’s diagonal is Φ times its edge, and a little geometry shows the ratio of 1 to Φ equals the ratio of Φ to Φ+1. What about the regular heptagon? Here we get two numbers, ρ and σ, which satisfy four equations, written as ratios below! So, for example, the ratio of 1 to ρ equals the ratio of ρ to 1+σ, and so on.

For more see:

• Peter Steinbach, Golden fields: a case for the heptagon, Mathematics Magazine 70 (Feb., 1997), 22-31.

He works out the theory for every regular polygon. So, it’s not that the fun stops after the pentagon: it just gets more sophisticated!

Constructing the heptagon

You can’t use a straightedge and compass to construct a regular heptagon. But here’s a construction that seems to do just that!

If you watch carefully, the seeming paradox is explained. For more, see:

• Heptagon, Wikipedia.

Trisecting the angle

When I was a kid, my uncle wowed me by trisecting an angle. He wasn’t a crackpot: he was bending the usual rules! He marked two dots on the ruler, A and B below, whose distance equaled the radius of the circle, namely OB. Then the trick below makes φ one third of θ.

Drawing dots on your ruler is called neusis, and the ancient Greeks knew about it. You can also use it to double the cube and construct a regular heptagon—impossible with a compass and straightedge if you’re don’t draw dots on it. Oddly, it fell out of fashion. Maybe purity of method mattered more than solving lots of problems?

Nowadays we realize that if you only have a straightedge, you can only solve linear equations. Adding a compass to your toolkit lets you also take square roots, so you can solve quadratic equations. Adding neusis on top of that lets you take cube roots, which—together with the rest—lets you solve cubic equations. A fourth root is a square root of a square root, so you get those for free, and in fact you can even solve all quartic equations. But you can’t take fifth roots.

Puzzle 5. Did anyone ever build a mechanical gadget that lets you take fifth roots, or maybe even solve general quintics?

Fluid Flows and Infinite-Dimensional Manifolds I

Or: waves that take the shortest path through infinity

guest post by Tim van Beek

Water waves can do a lot of things that light waves cannot, like “breaking”:

In mathematical models this difference shows up through the kind of partial differential equation (PDE) that models the waves:

• light waves are modelled by linear equations while

• water waves are modelled by nonlinear equations.

Physicists like to point out that linear equations model things that do not interact, while nonlinear equations model things that interact with each other. In quantum field theory, people speak of “free fields” versus “interacting fields”.

Some nonlinear PDE that describe fluid flows turn out to also describe geodesics on infinite-dimensional Riemannian manifolds. This fascinating observation is due to the Russian mathematician Vladimir Arnold. In this blog post I would like to talk a little bit about the concepts involved and show you a little toy example.

Fluid Flow modelled by Diffeomorphisms

The Euler viewpoint on fluids is that a fluid is made of tiny “packages” or “particles”. The fluid flow is described by specifying where each package or particle is at a given time . When we start at some time on a given manifold , the flow of every fluid package is described by a path on parametrized by time, and for every time there is a diffeomorphism defined by the requirement that it maps the initial position of each fluid package to its position at time :

This picture is taken from the book

• V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics, Springer, Berlin, 1998. (Review at Zentralblatt Mathematik.)

We will take as a model of the domain of the fluid flow a compact Riemannian manifold . A fluid flow, as pictured above, is then a path in the diffeomorphism group . In order to apply geometric concepts in this situation, we will have to turn or some closed subgroup of it into a manifold, which will be infinite dimensional.

The curvature of such a manifold can provide a great deal about the stability of fluid flows: On a manifold with negative curvature geodesics will diverge from each other. If we can model fluid flows as geodesics in a Riemannian manifold and calculate the curvature, we could try to infer a bound on weather forecasts (in fact, that is what Vladimir Arnold did!): The solution that you calculate is one geodesic. But if you take into account errors with determining your starting point (involving the measurement of the state of the flow at the given start time), what you are actually looking at is a bunch of geodesics starting in a neighborhood of your starting point. If they diverge fast, that means that measurement errors make your result unreliable fast.

If you never thought about manifolds in infinite dimensions, you may feel a little bit insecure as to how the concepts that you know from differential geometry can be generalized from finite dimensions. At least I felt this way when I first read about it. But it turns out that the part of the theory one needs to know in order to understand Arnold’s insight is not that scary, so I will talk a little bit about it next.

What you should know about infinite-dimensional manifolds

The basic strategy when handling finite-dimensional, smooth, real manifolds is that you have a complicated manifold , but also locally for every point a neighborhood and an isomorphism (a “chart”) of to an open subset of the vastly simpler space , the “model space”. These isomorphisms can be used to transport concepts from to . In infinite dimensions it is however not that clear what kind of model space should be taken in place of . What structure should have?

Since we would like to differentiate, we should for example be able to define the derivative of a curve in :

If we write down the usual formula for a derivative

we see that to make sense of this we need to be able to add elements, have a scalar multiplication, and a topology such that the algebraic operations are continuous. Sets with this structure are called topological vector spaces.

A curve that has a first derivative, second derivative, third derivative… and so on at every point is called a smooth curve, just as in the finite dimensional case.

So should at least be a topological vector space. We can, of course, put more structure on to make it “more similar” to , and choose as model space in ascending order of generality:

1) A Hilbert space, which has an inner product,

2) a Banach space that does not have a inner product, but a norm,

3) a Fréchet space that does not have a norm, but a metric,

4) a general topological vector space that need not be metrizable.

People talk accordingly of Hilbert, Banach and Fréchet manifolds. Since the space consisting of smooth maps from to is not a Banach space but a Fréchet space, we should not expect that we can model diffeomorphism groups on Banach spaces, but on Fréchet spaces. So we will use the concept of Fréchet manifolds.

But if you are interested in a more general theory using locally convex topological vector spaces as model spaces, you can look it up here:

• Andreas Kriegl and Peter W. Michor, The Convenient Setting of Global Analysis, American Mathematical Society, Providence Rhode Island, 1999.

Note that Kriegl and Michor use a different definition of “smooth function of Fréchet spaces” than we will below.

If you learn functional analysis, you will most likely start with operators on Hilbert spaces. One could say that the theory of topological vector spaces is about abstracting away as much structure from a Hilbert space and look what structure you need for important theorems to still hold true, like the open mapping/closed graph theorem. If you would like to learn more about this, my favorite book is this one:

• Francois Treves, Topological vector Spaces, Distributions and Kernels, Dover Publications, 2006.

Since we replace the model space with a Fréchet space , there will be certain things that won’t work out as easily as for the finite dimensional , or not at all.

It is nevertheless possible to define both integrals and differentials that behave much in the expected way. You can find a nice exposition of how this can be done in this paper:

• Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bulletin of the American Mathematical Society 7 (1982), pages 65-222.

The story starts with the definition of the directional derivative that can be done just as in finite dimensions:

Let and be Fréchet spaces, open and a continuous map. The derivative of at the point in the direction is the map

given by:

A simple but nontrivial example is the operator

given by:

with the derivative

It is possible to define higher derivatives and also prove that the chain rule holds, so that we can define that a function between Fréchet spaces is smooth if it has derivatives at every point of all orders. The definition of a smooth Fréchet manifold is then straightforward: you can copy the usual definition of a smooth manifold word for word, replacing by some Fréchet space.

With tangent vector s, you may remember that there are several different ways to define them in the finite dimensional case, which turn out to be equivalent. Since there are situations in infinite dimensions where these definitions turn out to not be equivalent, I will be explicit and define tangent vector s in the “kinematic way”:

The (kinematic) tangent vector space of a Fréchet manifold at a point consists of all pairs where is a smooth curve

With this definition, the set of pairs forms a Fréchet manifold, the tangent bundle , just as in finite dimensions.

The first serious (more or less) problem we encounter is the definition of the cotangent bundle: is isomorphic to its dual vector space. This is still true for every Hilbert space (this is known as the Riesz representation theorem). It fails already for Banach spaces: The dual space will still be a Banach space, but a Banach space does not need to be isomorphic to its dual, or even the dual of its dual (though the latter situation happens quite often, and such Banach spaces are called reflexive).

With Fréchet spaces things are even a little bit worse, because the dual of a Fréchet space (which is not a Banach space) is not even a Fréchet space! Since I did not know that and could not find a reference, I asked about this on Mathoverflow here and promptly got an answer. Mathoverflow is a really amazing platform for this kind of question!

So, if we naively define the cotangent space as in finite dimensions by taking the dual space of every tangent space, then the cotangent bundle won’t be a Fréchet manifold.

We will therefore have to be careful with the definition of differential forms for Fréchet manifolds and define it explicitly:

A differential form (a one form) is a smooth map

where is the tangent bundle, such that restricts to a linear map on every tangent space .

Another pitfall is that theorems from multivariable calculus may fail in Fréchet spaces, like the existence and uniqueness theorem of Picard-Lindelöf for ordinary differential equations. Things are much easier in Banach spaces: If you take a closer look at multivariable calculus, you will notice that a lot of definitions and theorems actually make use of the Banach space structure of only, so that a lot generalizes straight forward to infinite dimensional Banach spaces. But that is less so for Fréchet spaces.

By now you should feel reasonably comfortable with the notion of a Fréchet manifold, so let us talk about the kind of gadget that Arnold used to describe fluid flows: diffeomorphism groups that are both infinite-dimensional Riemannian manifolds and Lie groups.

The geodesic equation for an invariant metric

If is both a Riemannian manifold and a Lie group, it is possible to define the concept of left or right invariant metric. A left or right invariant metric on is one that does not change if we multiply the arguments with a group element:

A metric is left invariant iff for all :

Similarly, is right invariant iff:

How does one get a one-sided invariant metric?

Here is one possibility: If you take a Lie group off the shelf, you get two automorphisms for free, namely the left and right multiplication by a group element :

given by:

Pictorially speaking, you can use the differentials of these to transport vector s from the Lie algebra of – which is the tangent space at the identity of the group, – to any other tangent space . If you can define an inner product on the Lie algebra, you can use this trick to transport the inner product to all the other tangent spaces by left or right multiplication, which will get you a left or right invariant metric.

To be more precise, for every tangent vectors of a tangent space there are unique vectors that are mapped to by the differential of the right multiplication , that is

and we can define the inner product of and to have the value of that of and :

This works for the left multiplication , too, of course.

For a one-sided invariant metric, the geodesic equation looks somewhat simpler than for general metrics. Let us take a look at that:

On a Riemannian manifold with tangent bundle there is a unique connection, the Levi-Civita connection, with the following properties for vector fields :

If we combine both formulas we get

If the inner products are constant along every flow, i.e. the metric is (left or right) invariant, then the first three terms on the right hand side vanish, so that we get

This latter formula can be written in a more succinct way if we introduce the coadjoint operator. Remeber the adjoint operator defined to be

With the help of the inner product we can define the adjoint of the adjoint operator:

Beware! We’re using the word ‘adjoint’ in two completely different ways here, both of which are very common in math. One way is to use ‘adjoint’ for the operation of taking a Lie bracket: . Another is to use ‘adjoint’ for the linear map coming from a linear map between inner product spaces given by Please don’t blame me for this.

Then the formula above for the covariant derivative can be written as

Since the inner product is nondegenerate, we can eliminate and get

A geodesic curve is one whose tangent vector is transported parallel to itself. That is, we have

Using the formula for the covariant derivative for an invariant metric above we get

as a reformulation of the geodesic equation.

For time dependent dynamical systems, we have the time axis as an additional dimension and every vector field has as an additional summand. So, in this case we get as the geodesic equation (again, for an invariant metric):

A simple example: the circle

As a simple example we will look at the circle and its diffeomorphism group The Lie algebra of can be identified with the space of all vector fields on If we sloppily identify with with coordinate , then we can write for vector fields and the commutator

where is short for the derivative:

And of course we have an inner product via

which we can use to define either a left or a right invariant metric on , by transporting it via left or right multiplication to every tangent space.

Let us evaluate the geodesic equation for this example. We have to calculate the effect of the coadjoint operator:

If we write for the vector fields , and , this results in

where the last step employs integration by parts and uses the periodic boundary condition for the involved functions.

So we get for the coadjoint operator

Finally, the geodesic equation

turns out to be

A similar equation,

is known as the Hopf equation or inviscid Burgers’ equation. It looks simple, but its solutions can produce behaviour that looks like turbulence, so it is interesting in its own right.

If we take a somewhat more sophisticated diffeomorphism group, we can get slightly more complicated and therefore more interesting partial differential equations like the Korteweg-de Vries equation. But since this post is quite long already, that topic will have to wait for another post!

Math 2.0

Building on the Elsevier boycott, a lot of people are working on positive steps to make expensive journals obsolete. My email is flooded with discussions, different groups making different plans.

Email is great, but not for everything. So Andrew Stacey (the technical mastermind behind the nLab, Azimuth Wiki and Azimuth Forum):

and Scott Morrison (one of the brains behind MathOverflow, an important math question-and-answer website):

have started a forum to talk about the many issues involved:

• Math 2.0.

That’s good, because these guys actually do stuff, not just talk! Andrew describes the idea here:

The purpose of Math 2.0 is to provide a forum for discussion of the future of mathematical publishing. It’s something that I’ve viewed as an important issue for years, and have had many, many interesting conversations about, but somehow nothing much seems to happen. I’m hoping that the momentum from Tim Gowers’ recent blog posts might lead to something and I’d like to capitalise on that.

However, most of the discussion currently is happening in the comments on blog posts. This is hard to follow, and hard to separate out the new suggestions from the discussions on old ones. I think that forums are much better for discussion, hence this one.

The name, Math2.0, is intended to signify two things: that it’s time for an upgrade of the mathematical environment and that I think we can learn a lot from looking at how software—particularly open source software—works. By “mathematical environment”, I don’t mean how we actually do the mathematics but what happens next, particularly communicating the ideas that we create. This is where the internet can really change things for the better (as it has started to do with the arXiv), but where I think that we have yet to figure out how to make best use of it.

This doesn’t just include journals, but I think that that’s an obvious place to start.

So: welcome to Math2.0. Please join in. It’s important.

Andrew Stacey has also emphasized a principle that’s good for reducing chat about starry-eyed visions and focusing on what we can do now:

In all these discussions, there is one point that I would like to make at the start and which I think is relevant to any proposal to set up something new for mathematicians (or more generally, for academics). That is that whatever system is set up it must be:

Useful at the point of use

This is something that I’ve learnt from administering the nLab over the past few years. It keeps going and there is no sign of it slowing down. The secret of its success, I maintain, is that it is useful at the point of use. When I write something on the nLab, I benefit immediately. I can link to previous things I’ve written, to definitions that others have written, and so link my ideas to many others. It means that if I want to talk to someone about something, the thing we are talking about is easily visible and accessible to both (or all) of us. If I want to remember what it was I was thinking about a year ago, I can easily find it. The fact that when I come back the next day, whatever I’ve added has been improved, polished, and added to, is a bonus—but it would still be useful if that didn’t happen.

For other things, then I need more of an incentive to participate. MathOverflow was a lot of fun in the beginning, but now I find that a question needs to be such that it’s fairly clear that I’m one of the few people in the world who can answer. It’s not that my enthusiasm for the site has gone down, just that everything else keeps pushing it out of the way. So a new system has to be useful to those who use it, and ideally the usefulness should be proportional to the amount of effort that one puts in.

A corollary of this is that it should be useful even if only a small number of people use it. The number of core users of the nLab is not large, but nevertheless the nLab is still extremely useful to us. I can imagine that when a proposal for something new is made, there will be a variety of reactions ranging from “That’ll never work” through to “Sounds interesting, but …” with only a few saying “Count me in!”. To have a chance of succeeding, it has to be the case that those few can get it off the ground and demonstrate that it works, without the input of the wider sceptical community.

So: if you’re a mathematician or programmer interested in revolutionizing the future of math publishing, go to Math 2.0, register, and join the conversation! You’ll see there are a number of concrete proposals on the table, including one by Chris Lee, and Marc Harper and myself.

I’ll say more about those later. But I want to add a principle of my own to Andrew’s ‘useful at the point of use’. The goal is not to get a universal consensus on the future of math publishing! Instead, we need a healthy dissensus in which different groups of people develop different systems—so we can see which ones work.

In biology, evolution happens when some change is useful at the point of use—and it doesn’t happen by consensus, either. When some fish gradually became amphibians, they didn’t wait for all fish to agree this was a good move. And indeed it’s good that we still have fish.

Jan Velterop has some interesting thoughts on the evolution of scholarly publishing, which you can read here:

• Richard Poynder, The open access interviews: Jan Velterop, February 2012.

Velterop writes:

As a geologist I go so far as to say that I see analogies with the Permian-Triassic boundary and the Cretaceous-Tertiary boundary, when life on Earth changed dramatically due to fundamental and sudden changes in the environment.

Those boundary events, as they are known, resulted in mass extinctions, and that’s an unavoidable evolutionary consequence of sudden dramatic environmental changes.

But they also open up ecological niches for new, or hitherto less successful, forms of life. In this regard, it is interesting to see the recent announcement of F1000 Research, which intends to address the major issues afflicting scientific publishing.

[…]

The evolution of scientific communication will go on, without any doubt, and although that may not mean the total demise of the traditional models, these models will necessarily change. After all, some dinosaur lineages survived as well. We call them birds. And there are some very attractive ones. They are smaller than the dinosaurs they evolved from, though. Much smaller.

The Beauty of Roots (Part 3)

 

Dan Christensen, Sam Derbyshire and I are writing a paper on a fun topic I’ve discussed here before: the beauty of roots. We’re getting help from Greg Egan, but he’s too busy writing his next novel to commit to being a coauthor! Anyway, I’m giving a talk about this stuff today, and I think you’ll at least enjoy the pretty pictures:

• The Beauty of Roots: easy fun version.

• The Beauty of Roots: version for mathematicians.

The version for mathematicians has some proofs; the easy fun version states a few theorems, but it’s mainly pictures.

For mathematicians, I think the coolest part is the close relation between our main object of interest:

namely the set of all roots of all polynomials whose coefficients are ±1, and the Cantor set, which you get by taking a closed interval and repeatedly chopping out the middle third of each piece, forever:

They’re related because a point in the Cantor set can be seen as an infinite string of 0’s and 1’s, while a power series with coefficients ±1 can be seen as an infinite string of 1’s and -1’s. The sets called ‘dragons’, like the one at the top of this post, are also images of the Cantor set under continuous maps to the complex plane. But to understand how these facts understand our set of roots of polynomials with coefficients ±1, read the mathematician’s version of the talk.

The Cost of Knowledge

As of this moment, 4760 scholars have joined a boycott of the publishing company Elsevier. Of these, only 20% are mathematicians. But since the boycott was started by a mathematician, 34 of us wrote and signed an official statement explaining the boycott:

• The Cost of Knowledge.

It’s also below. Please check it out and join the boycott! I’m sure more than 34 mathematicians would be happy to sign, but we wanted to get the statement out soon.

This is an attempt to describe some of the background to the current boycott of Elsevier by many mathematicians (and other academics) at http://thecostofknowledge.com, and to present some of the issues that confront the boycott movement. Although the movement is anything but monolithic, we believe that the points we make here will resonate with many of the signatories to the boycott.

The role of journals (1): dissemination of research.

The role of journals in professional mathematics has been under discussion for some time now.

Traditionally, while journals served several purposes, their primary purpose was the dissemination of research papers. The journal publishers were charging for the cost of typesetting (not a trivial matter in general before the advent of electronic typesetting, and particularly non-trivial for mathematics), the cost of physically publishing copies of the journals, and the cost of distributing the journals to subscribers (primarily academic libraries).

The editorial board of a journal is a group of professional
mathematicians. Their editorial work is undertaken as part of their scholarly duties, and so is paid for by their employer, typically a university. Thus, from the publisher’s viewpoint the editors are volunteers. (The editor in chief of a journal sometimes receives modest compensation from the publisher.) When a paper is submitted to the journal, by an author who is again typically a university-employed mathematician, the editors select the referee or referees for the paper, evaluate the referees’ reports, decide whether or not to accept the submission, and organize the submitted papers into volumes. These are passed on to the publisher, who then undertakes the job of actually publishing them. The publisher supplies some administrative assistance in handling the papers, as well as some copy-editing assistance, which is often quite minor but sometimes more substantial. The referees are again volunteers from the point of view of the publisher: as with editing, refereeing is regarded as part of the service component of a mathematician’s academic work. Authors are not paid by the publishers for their published papers, although they are usually asked to sign over the copyright to the publisher.

This system made sense when the publishing and dissemination of papers was a difficult and expensive undertaking. Publishers supplied a valuable service in this regard, for which they were paid by subscribers to the journals, which were mainly academic libraries. The academic institutions whose libraries subscribe to mathematics journals are broadly speaking the same institutions that employ the mathematicians who are writing for, refereeing for, and editing the journals. Therefore, the cost of the whole process of producing research papers is borne by these institutions (and the outside entities that partially fund them, such as the National Science Foundation in the United States): they pay for their academic mathematician employees to do research and to organize the publications of the results of their research in journals; and then (through their libraries) they pay the publishers to disseminate these results among all the world’s mathematicians. Since these institutions employ research faculty in order to foster research, it certainly used to make sense for them to pay for the dissemination of this research as well. After all, the sharing of scientific ideas and research results is unquestionably a key component for making progress in science.

Now, however, the world has changed in significant ways.
Authors typeset their own papers, using electronic typesetting. Publishing and distribution costs are not
as great as they once were. And most importantly,
dissemination of scientific ideas no longer takes place via the physical distribution of journal volumes. Rather, it takes place mainly electronically. While this means of dissemination is not free, it is much less expensive, and much of it happens quite independently of mathematical journals.

In conclusion, the cost of journal publishing has gone down
because the cost of typesetting has been shifted from
publishers to authors and the cost of publishing and distribution is significantly lower than it used to be.
By contrast, the amount of money being spent by university libraries on journals seems to be growing with no end in sight. Why do mathematicians contribute all this volunteer labor, and their employers pay all this money, for a service whose value no longer justifies its cost?

The role of journals (2): peer review and professional
evaluation

There are some important reasons that mathematicians haven’t just abandoned journal publishing. In particular, peer review plays an essential role in ensuring the correctness and readability of mathematical papers, and publishing papers in research journals is the main way of achieving professional recognition. Furthermore, not all journals count equally from this point of view: journals are (loosely) ranked, so that publications in top journals will often count more than publications in lower ranked ones. Professional mathematicians typically have a good sense of the relative prestige of the journals that publish papers in their area, and they will usually submit a paper to the highest ranked journal that they judge is likely to accept and publish it.

Because of this evaluative aspect of traditional journal publishing, the problem of switching to a different model
is much more difficult than it might appear at first. For
example, it is not easy just to begin a new journal (even an electronic one, which avoids the difficulties of printing and distribution), since mathematicians may not want to publish in it, preferring to submit to journals with known reputations. Secondly, although the reputation of various journals has been created through the efforts of the authors, referees, and editors who have worked (at no cost to the publishers) on it over the years, in many cases the name of the journal is owned by the publisher, making it difficult for the mathematical community to separate this valuable object that they have constructed from its present publisher.

The role of Elsevier

Elsevier, Springer, and a number of other commercial publishers (many of them large companies but less significant for their mathematics publishing, e.g., Wiley) all exploit our volunteer labor to extract very large profits from the academic community. They supply some value in the process, but nothing like enough to justify their prices.

Among these publishers, Elsevier may not be the most expensive, but in the light of other factors, such as scandals, lawsuits, lobbying, etc. (discussed further below), we consider them a good initial focus for our discontent. A boycott should be substantial enough to be meaningful, but not so broad that the choice of targets becomes controversial or the boycott becomes an unmanageable burden. Refusing to submit papers to all overpriced publishers is a reasonable further step, which some of us have taken, but the focus of this boycott is on Elsevier because of the widespread feeling among mathematicians that they are the worst offender.

Let us begin with the issue of journal costs. Unfortunately, it is difficult to make cost comparisons: journals differ greatly in quality, in number of pages per volume, and even in amount of text per page. As measured by list prices, Elsevier mathematics journals are amongst the most expensive. For instance, in the AMS mathematics journal price survey, seven of the ten most expensive journals (by 2007 volume list price) were published by Elsevier. (All prices are as of 2007 because both prices and page counts are easily available online.) However, that is primarily because Elsevier publishes the largest volumes. Price per page is a more meaningful measure that can be easily computed. By this standard, Elsevier is certainly not the worst publisher, but its prices do on the face of it look very high. The Annals of Mathematics, published by Princeton University Press, is one of the absolute top mathematics journals and quite affordably priced: $0.13/page as of 2007. By contrast, ten Elsevier journals (not including one that has since ceased publication) cost $1.30/page or more; they and three others cost more per page than any journal published by a university press or learned society. For comparison, three other top journals competing with the Annals are Acta Mathematica, published by the Institut Mittag Leffler for $0.65/page, Journal of the American Mathematical Society, published by the American Mathematical Society for $0.24/page, and Inventiones Mathematicae, published by Springer for $1.21/page. Note that none of Elsevier’s mathematics journals is generally considered comparable in quality to these journals.

However, there is an additional aspect which makes it hard to compute the true cost of mathematics journals. This is the widespread practice among large commercial publishers of “bundling” journals, which allows libraries to subscribe to large numbers of journals in order to avoid paying the exorbitant list prices for the ones they need. Although this means that the average price libraries pay per journal is less than the list prices might suggest, what really matters is the average price that they pay per journal (or page of journal) that they actually want, which is hard to assess, but clearly higher. We would very much like to be able to offer more concrete data regarding the actual costs to libraries of Elsevier journals compared with those of Springer or other publishers. Unfortunately, this is difficult, because publishers often make it a contractual requirement that their institutional customers should not disclose the financial details of their contracts. For example, Elsevier sued Washington State University to try to prevent release of this information. One common consequence of these arrangements, though, is that in many cases a library cannot actually save any money by cancelling a few Elsevier journals: at best the money can sometimes be diverted to pay for other Elsevier subscriptions.

One reason for focusing on Elsevier rather than, say, Springer is that Springer has had a rich and productive history with the mathematical community. As well as journals, it has published important series of textbooks, monographs, and lecture notes; one could perhaps regard the prices of its journals as a means of subsidizing these other, less profitable, types of publications. Although all these types of publications have become less important with the advent of the internet and the resulting electronic distribution of texts, the long and continuing presence of Springer in the mathematical world has resulted in a store of goodwill being built up in the mathematical community towards them. This store is being rapidly depleted, but has not yet reached zero. See for instance the recent petition to Springer by a number of French mathematicians and departments.

Elsevier does not have a comparable tradition of involvement in mathematics publishing. Many of the mathematics journals that it publishes have been acquired comparatively recently as it has bought up other, smaller publishers. Furthermore, in recent years it has been involved in various scandals regarding the scientific content, or lack thereof, of its journals. One in particular involved the journal Chaos, Solitons & Fractals, which, at the time the scandal broke in 2008–2009, was one of the highest impact factor mathematics journals that Elsevier published. (Elsevier currently reports the five-year impact factor of this journal at 1.729. For sake of comparison, Advances in Mathematics, also published by Elsevier, is reported as having a five-year impact factor of 1.575.) It turned out that the high impact factor was at least partly the result of the journal publishing many papers full of mutual citations. (See Arnold for more information on this and other troubling examples that show the limitations of bibliometric measures of scholarly quality.) Furthermore, Chaos, Solitons & Fractals published many papers that, in our professional judgement, have little or no scientific merit and should not have been published in any reputable journal.

In another notorious episode, this time in medicine, for at least five years Elsevier “published a series of sponsored article compilation publications, on behalf of pharmaceutical clients, that were made to look like journals and lacked the proper disclosures”, as noted by the CEO of Elsevier’s Health Sciences Division.

Recently, Elsevier has lobbied for the Research Works Act, a proposed U.S. law that would undo the National Institutes of Health’s public access policy, which guarantees public access to published research papers based on NIH funding within twelve months of publication (to give publishers time to make a profit). Although most lobbying occurs behind closed doors, Elsevier’s vocal support of this act shows their opposition to a popular and effective open access policy.

These scandals, taken together with the bundling practices, exorbitant prices, and lobbying activities, suggest a publisher motivated purely by profit, with no genuine interest in or commitment to mathematical knowledge and the community of academic mathematicians that generates it. Of course, many Elsevier employees are reasonable people doing their best to contribute to scholarly publishing, and we bear them no ill will. However, the organization as a whole does not seem to have the interests of the mathematical community at heart.

The boycott

Not surprisingly, many mathematicians have in recent years lost patience with being involved in a system in which commercial publishers make profits based on the free labor of mathematicians and subscription fees from their institutions’ libraries, for a service that has become largely unnecessary. (See Scott Aaronson’s scathing but all-too-true satirical description of the publishers’ business model.) Among all the commercial publishers, the behavior of Elsevier seemed to many to be the most egregious, and a number of mathematicians had made personal commitments to avoid any involvement with Elsevier journals. (Some journals were also successfully moved from Elsevier to other publishers; e.g., Annales Scientifiques de l’école Normale Supérieure which until recent years was published by Elsevier, is now published by the Société Mathématique de France.)

One of us (Timothy Gowers) decided that it might be useful to
publicize his own personal boycott of Elsevier, thus encouraging others to do the same. This led to the current boycott movement at http://thecostofknowledge.com, the success of which has far exceeded his initial expectations.

Each participant in the boycott can choose which activities they intend to avoid: submitting to Elsevier journals, refereeing for them, and serving on editorial boards. Of course, submitting papers and editing journals are purely voluntary activities, but refereeing is a more subtle issue. The entire peer review system depends on the availability of suitable referees, and its success is one of the great traditions of science: refereeing is felt to be both a burden and an honor, and practically every member of the community willingly takes part in it. However, while we respect and value this tradition, many of us do not wish to see our labor used to support Elsevier’s business model.

What next?

As suggested at the very beginning, different participants in the boycott have different goals, both in the short and long term. Some people would like to see the journal system eliminated completely and replaced by something else more adapted to the internet and the possibilities of electronic distribution. Others see journals as continuing to play a role, but with commercial publishing being replaced by open access models. Still others imagine a more modest change, in which commercial publishers are replaced by non-profit entities such as professional societies (e.g., the American Mathematical Society, the London Mathematical Society, and the Société Mathématique de France, all of which already publish a number of journals) or university presses; in this way the value generated by the work of authors, referees, and editors would be returned to the academic and scientific community. These goals need not be mutually exclusive: the world of mathematics journals, like the world of mathematics itself, is large, and open access journals can coexist with traditional journals, as well as with other, more novel means of dissemination and evaluation.

What all the signatories do agree on is that Elsevier is an exemplar of everything that is wrong with the current system of commercial publication of mathematics journals, and we will no longer acquiesce to Elsevier’s harvesting of the value of our and our colleagues’ work.

What future do we envisage for all the papers that would
otherwise be published in Elsevier journals? There are many
other journals being published; perhaps they can pick up at
least some of the slack. Many successful new journals have been founded in recent years, too, including several that are electronic (thus completely eliminating printing and physical distribution costs), and no doubt more will follow. Finally, we hope that the mathematical community will be able to reclaim for itself some of the value that it has given to Elsevier’s journals by moving some of these journals (in name, if possible, and otherwise in spirit) from Elsevier to other publishers. One notable example is the August 10, 2006 resignation of the entire editorial board of the Elsevier journal Topology and their founding of the Journal of Topology, owned by the London Mathematical Society.

None of these changes will be easy; editing a journal is hard work, and founding a new journal, or moving and relaunching an existing journal, is even harder. But the alternative is to continue with the status quo, in which Elsevier harvests ever larger profits from the work of us and our colleagues, and this is both unsustainable and unacceptable.

Signed by:

Scott Aaronson
Massachusetts Institute of Technology

Douglas N. Arnold
University of Minnesota

Artur Avila
IMPA and Institut de Mathématiques de Jussieu

John Baez
University of California, Riverside

Folkmar Bornemann
Technische Universität München

Danny Calegari
Caltech/Cambridge University

Henry Cohn
Microsoft Research New England

Jordan Ellenberg
University of Wisconsin, Madison

Matthew Emerton
University of Chicago

Marie Farge
École Normale Supérieure Paris

David Gabai
Princeton University

Timothy Gowers
Cambridge University

Ben Green
Cambridge University

Martin Grötschel
Technische Universität Berlin

Michael Harris
Université Paris-Diderot Paris 7

Frédéric Hélein
Institut de Mathéatiques de Jussieu

Rob Kirby
University of California, Berkeley

Vincent Lafforgue
CNRS and Université d’Orléans

Gregory F. Lawler
University of Chicago

Randall J. LeVeque
University of Washington

László Lovász
Eötvös Lor´nd University

Peter J. Olver
University of Minnesota

Olof Sisask
Queen Mary, University of London

Terence Tao
University of California, Los Angeles

Richard Taylor
Institute for Advanced Study

Bernard Teissier
Institut de Mathématiques de Jussieu

Burt Totaro
Cambridge University

Lloyd N. Trefethen
Oxford University

Takashi Tsuboi
University of Tokyo

Marie-France Vigneras
Institut de Mathématiques de Jussieu

Wendelin Werner
Université Paris-Sud

Amie Wilkinson
University of Chicago

Günter M. Ziegler
Freie Universität Berlin

Appendix: recommendations for mathematicians.

All mathematicians must decide for themselves whether, or to what extent, they wish to participate in the boycott. Senior
mathematicians who have signed the boycott bear some
responsibility towards junior colleagues who are forgoing the
option of publishing in Elsevier journals, and should do their
best to help minimize any negative career consequences.

Whether or not you decide to join the boycott, there are some
simple actions that everyone can take, which seem to us to be
uncontroversial:

1) Make sure that the final versions of all your papers, particularly new ones, are freely available online— ideally both on the arXiv. (Elsevier’ electronic preprint policy is unacceptable, because it explicitly does not allow authors to update their papers on the arXiv to incorporate changes made during peer review). When signing copyright transfer forms, we recommend amending them (if necessary) to reserve the right to make the author’s final version of the text available free online from servers such as the arXiv, and on your home page.

2) If you are submitting a paper and there is a choice between an expensive journal and a cheap (or free) journal of the same standard, then always submit to the cheap one.

Note

The PDF version of this statement has many useful references not included here.

Archimedean Tilings and Egyptian Fractions

Ever since I was a kid, I’ve loved Archimedean tilings of the plane: that is, tilings by regular polygons where all the edge lengths are the same and every vertex looks alike. Here’s my favorite:

There are also 11 others, two of which are mirror images of each other. But how do we know this? How do we list them all and be sure we haven’t left any out?

The interior angle of a regular -sided polygon is obviously

since it’s a bit less than 180 degrees, or and how much?— well, times a full turn, or But these ‘s are getting annoying: it’s easier to say ‘a full turn’ than write Then we can say the interior angle is

times a full turn.

Now suppose we have an Archimedean tiling where polygons meet: one with sides, one with sides, and so on up to one with sides. Their interior angles must add up to a full turn. So, we have

or

or

So: to get an Archimedean tiling you need n whole numbers whose reciprocals add up to one less than n/2.

Looking for numbers like this is a weird little math puzzle. The Egyptians liked writing numbers as sums of reciprocals, so they might have enjoyed this game if they’d known it. The tiling I showed you comes from this solution:

since it has 3 polygons meeting at each vertex: a 4-sided one, a 6-sided one and a 12-sided one.

Here’s another solution:

It gives us this tiling:

Hmm, now I think this one is my favorite, because my eye sees it as a bunch of linked 12-sided polygons, sort of like chain mail. Different tilings make my eyes move over them in different ways, and this one has a very pleasant effect.

Here’s another solution:

This gives two Archimedean tilings that are mirror images of each other!

Of course, whether you count these as two different Archimedean tilings or just one depends on what rules you choose. And by the way, people usually don’t say a tiling is Archimedean if all the polygons are the same, like this:

They instead say it’s regular. If modern mathematicians were inventing this subject, we’d say regular tilings are a special case of Archimedean tilings—but this math is all very old, and back then mathematicians treated special cases as not included in the general case. For example, the Greeks didn’t even consider the number 1 to be a number!

So here’s a fun puzzle: classify the Archimedean tilings! For starters, you need to find all ways to get whole numbers whose reciprocals add up to one less than . That sounds hard, but luckily it’s obvious that

since an equilateral triangle has the smallest interior angle, of any regular polygon, and you can only fit 6 of them around a vertex. If you think a bit, you’ll see this cuts the puzzle down to a finite search.

But you have to be careful, since there are some solutions that don’t give Archimedean tilings. As usual, the number 5 causes problems. We have

but there’s no way to tile the plane so that 2 regular pentagons and 1 regular decagon meet at each vertex! Kepler seems to have tried; here’s a picture from his book Harmonices Mundi:

It works beautifully at one vertex, but not for a tiling of the whole plane. To save the day he had to add some stars, and some of the decagons overlap! The Islamic tiling artists, and later Penrose, went further in this direction.

If you get stuck on this puzzle, you can find the answer here:

• Michal Krížek, Jakub Šolc, and Alena Šolcová, Is there a crystal lattice possessing five-fold symmetry?, AMS Notices 59 (January 2012), 22-30.

• Combinations of regular polygons that can meet at a vertex, Wikipedia.

Not enough?

In short, all Archimedean tilings of the plane arise from finding whole numbers whose reciprocals sum to But what if the total is not enough? Don’t feel bad: you might still get a tiling of the hyperbolic plane. For example,

so you can’t tile the plane with 3 heptagons meeting at each corner… but you still get this tiling of the hyperbolic plane:

which happens to be related to a wonderful thing called Klein’s quartic curve.

You don’t always win… but sometimes you do, so the game is worth playing. For example,

so you have a chance at a tiling of the hyperbolic plane where five equilateral triangles and a square meet at each vertex. And in this case, you luck out:

For more beautiful pictures like these, see:

• Uniform tilings in hyperbolic plane, Wikipedia.

• Don Hatch, Hyperbolic tesselations.

Too much?

Similarly, if you’ve got reciprocals that add up to more than , you’ve got a chance at tiling the sphere. For example,

and in this case we luck out and get the snub dodecahedron. I thought it was rude to snub a dodecahedron, but apparently not:

These tilings of the sphere are technically called Archimedean solids and (if all the polygons are the same) Platonic solids. Of these, only the snub dodecahedron and the ‘snub cube’ are different from their mirror images.

Fancier stuff

In short, adding up reciprocals of whole numbers is related to Archimedean tilings of the plane, the sphere and the hyperbolic plane. But this is also how Egyptians would write fractions! In fact they even demanded that all the reciprocals be distinct, so instead of writing 2/3 as , they’d write

It’s a lousy system—doubtless this is why King Tut died so young. But forget about the restriction that the reciprocals be distinct: that’s silly. If you can show that for every the number can be written as for whole numbers you’ll be famous! So far people have ‘only’ shown it’s true for up to a hundred trillion:

• Erdös–Straus conjecture, Wikipedia.

So, see if you can do better! But if you’re into fancy math, a less stressful activity might be to read about Egyptian fractions, tilings and ADE classifications:

• John Baez, This Week’s Finds in Mathematical Physics (Week 182).

This only gets into ‘Platonic’ or ‘regular’ tilings, not the more general ‘Archimedean’ or ‘semiregular’ ones I’m talking about today—so the arithmetic works a bit differently.

For the special magic arising from

see:

• John Baez, 42.

In another direction, my colleague Julie Bergner has talked about how they Egyptian fractions show up in the study of ‘groupoid cardinality’:

• Julie Bergner, Groupoids and Egyptian fractions.

So, while nobody uses Egyptian fractions much anymore, they have a kind of eerie afterlife. For more on what the Egyptians actually did, try these:

• Ron Knott, Egyptian fractions.

• Egyptian fractions, Wikipedia.

The Beauty of Roots (Part 2)

Here’s a bit more on the beauty of roots—some things that may have escaped those of you who weren’t following this blog carefully!

Greg Egan has a great new applet for exploring the roots of Littlewood polynomials of a given degree—meaning polynomials whose coefficients are all ±1:

• Greg Egan, Littlewood applet.

Move the mouse around to create a little rectangle, and the applet will zoom in to show the roots in that region. For example, the above region is close to the number -0.0572 + 0.72229i.

Then, by holding the shift key and clicking the mouse, compare the corresponding ‘dragon’. We get the dragon for some complex number by evaluating all power series whose coefficients are all ±1 at this number.

You’ll see that often the dragon for some number resembles the set of roots of Littlewood polynomials near that number! To get a sense of why, read Greg’s explanation. However, he uses a different, though equivalent, definition of the dragon (which he calls the ‘Julia set’).

He also made a great video showing how the dragons change shape as you move around the complex plane:

The dragon is well-defined for any number inside the unit circle, since all power series with coefficients ±1 converge inside this circle. If you watch the video carefully—it helps to make it big—you’ll see a little white cross moving around inside the unit circle, indicating which dragon you’re seeing.

I’m writing a paper about this stuff with Dan Christensen and Sam Derbyshire… that’s why I’m not giving a very careful explanation now. We invited Greg Egan to join us, but he’s too busy writing the third volume of his trilogy Orthogonal.