Relative Entropy (Part 1)

20 June, 2013

I’m trying to finish off a paper that Tobias Fritz and I have been working on, which gives a category-theoretic (and Bayesian!) characterization of relative entropy. It’s a kind of sequel to our paper with Tom Leinster, in which we characterized entropy.

That earlier paper was developed in conversations on the n-Category Café. It was a lot of fun; I sort of miss that style of working. Also, to get warmed up, I need to think through some things I’ve thought about before. So, I might as well write them down here.

The idea

There are many categories related to probability theory, and they’re related in many ways. Last summer—on the 24th of August 2012, according to my notes here—Jamie Vicary, Brendan Fong and I worked through a bunch of these relationships. I need to write them down now, even if they’re not all vitally important to my paper with Tobias. They’re sort of buzzing around my brain like flies.

(Tobias knows this stuff too, and this is how we think about probability theory, but we weren’t planning to stick it in our paper. Maybe we should.)

Let’s restrict attention to probability measures on finite sets, and related structures. We could study these questions more generally, and we should, but not today. What we’ll do is give a unified purely algebraic description of:

• finite sets

• measures on finite sets

• probability measures on finite sets

and various kinds of maps between these:

• functions

• bijections

• measure-preserving functions

• stochastic maps

Finitely generated free [0,∞)-modules

People often do linear algebra over a field, which is—roughly speaking—a number system where you can add, subtract, multiply and divide. But algebraists have long realized that a lot of linear algebra still works with a commutative ring, where you can’t necessarily divide. It gets more complicated, but also a lot more interesting.

But in fact, a lot still works with a commutative rig, where we can’t necessarily subtract either! Something I keep telling everyone is that linear algebra over rigs is a good idea for studying things like probability theory, thermodynamics, and the principle of least action.

Today we’ll start with the rig of nonnegative real numbers with their usual addition and multiplication; let’s call this [0,\infty) . The idea is that measure theory, and probability theory, are closely related to linear algebra over this rig.

Let C be the category with of finitely generated free [0,\infty) -modules as objects, and module homomorphisms as morphisms. I’ll call these morphisms maps.

Puzzle. Do we need to say ‘free’ here? Are there finitely generated modules over [0,\infty) that aren’t free?

Every finitely generated free [0,\infty)-module is isomorphic to [0,\infty)^S for some finite set S . In other words, it’s isomorphic to [0,\infty)^n for some n = 0, 1, 2, \dots . So, C is equivalent to the category where objects are natural numbers, a morphism from m to n is an m \times n matrix of numbers in [0,\infty) , and composition is done by matrix multiplication. I’ll also call this equivalent category C.

We can take tensor products of finitely generated free modules, and this makes C into a symmetric monoidal †-category. This means we can draw maps using string diagrams in the usual way. However, I’m feeling lazy so I’ll often write equations when I could be drawing diagrams.

One of the rules of the game is that all these equations will make sense in any symmetric monoidal †-category. So we could, if we wanted, generalize ideas from probability theory this way. If you want to do this, you’ll need to know that [0,\infty) is the unit for the tensor product in C. We’ll be seeing this guy [0,\infty) a lot. So if you want to generalize, replace C by any symmetric monoidal †-category, and replace [0,\infty) by the unit for the tensor product.

Finite sets

There’s a way to see the category of finite sets lurking in C, which we can borrow from this paper:

• Bob Coecke, Dusko Pavlovic and Jamie Vicary, A new description of orthogonal bases.

For any finite set S , we get a free finitely generated [0,\infty) -module, namely [0,\infty)^S . This comes with some structure:

• a multiplication m: [0,\infty)^S \otimes [0,\infty)^S \to [0,\infty)^S , coming from pointwise multiplication of [0,\infty) -valued functions on S

• the unit for this multiplication, an element of [0,\infty)^S, which we can write as a morphism i: [0,\infty) \to [0,\infty)^S

• a comultiplication, obtained by taking the diagonal map \Delta : S \to S \times S and promoting it to a linear map \Delta : [0,\infty)^S \to [0, \infty)^S \otimes [0,\infty)^S

• a counit for this comultiplication, obtained by taking the unique map to the terminal set ! : S \to 1 and promoting it to a linear map e: [0,\infty)^S \to [0,\infty)

These morphisms m, i, \Delta, e make

x = [0,\infty)^S

into a commutative Frobenius algebra in C . That’s a thing where the unit, counit, multiplication and comultiplication obey these laws:

(I drew these back when I was feeling less lazy.) This Frobenius algebra is also ‘special’, meaning it obeys this:

And it’s also a †-Frobenius algebra, meaning that the counit and comultiplication are obtained from the unit and multiplication by ‘flipping’ them using the †category structure. (If we think of a morphism in C as a matrix, its dagger is its transpose.)

Conversely, suppose we have any special commutative †-Frobenius algebra x . Then using the ideas in the paper by Coecke, Pavlovich and Vicary we can recover a basis for x , consisting of the vectors e_i \in x with

\Delta(e_i) = e_i \otimes e_i

This basis forms a set S such that

x \cong [0,\infty)^S

for some specified isomorphism in C. Furthermore, this is an isomorphism of special commutative †-Frobenius algebras!

In case you’re wondering, these vectors e_i correspond to the functions on S that are zero everywhere except at one point i \in S, where they equal 1.

In short, a special commutative †-Frobenius algebra in C is just a fancy way of talking about a finite set. This may seem silly, but it’s a way to start describing probability theory using linear algebra very much as we do with quantum theory. This analogy between quantum theory and probability theory is so interesting that it deserves a book.

Functions and bijections

Now suppose we have two special commutative †-Frobenius algebra in C, say x and y .

Suppose f : x \to y is a Frobenius algebra homomorphism: that is, a map preserving all the structure—the unit, counit, multiplication and comultiplication. Then it comes from an isomorphism of finite sets. This lets us find \mathrm{FinSet}_0 , the groupoid of finite sets and bijections, inside C.

Alternatively, suppose f : x \to y is just a coalgebra homomorphism: that is a map preserving just the counit and comultiplication. Then it comes from an arbitrary function between finite sets. This lets us find FinSet , the category of finite sets and functions, inside C .

But what if f preserves just the counit? This sounds like a dry, formal question. But it’s not: the answer is something useful, a ‘stochastic map’.

Stochastic maps

A stochastic map from a finite set S to a finite set T is a map sending each point of S to a probability measure on T .

We can think of this as a T \times S -shaped matrix of numbers in [0,\infty) , where a given column gives the probability that a given point in S goes to any point in T . The sum of the numbers in each column will be 1. And conversely, any T \times S -shaped matrix of numbers in [0,\infty) , where each column sums to 1, gives a stochastic map from S to T .

But now let’s describe this idea using the category C. We’ve seen a finite set is the same as a special commutative †-Frobenius algebra. So, say we have two of these, x and y . Our matrix of numbers in [0,\infty) is just a map

f: x \to y

So, we just need a way to state the condition that each column in the matrix sums to 1. And this condition simply says that f preserves the counit:

\epsilon_y \circ f = \epsilon_x

where \epsilon_x : x \to [0,\infty) is the counit for x , and similarly for \epsilon_y .

To understand this, note that if we use the canonical isomorphism

x \cong [0,\infty)^S

the counit \epsilon_x can be seen as the map

[0,\infty)^S \to [0,\infty)

that takes any S -tuple of numbers and sums them up. In other words, it’s integration with respect to counting measure. So, the equation

\epsilon_y \circ f = \epsilon_x

says that if we take any S -tuple of numbers, multiply it by the matrix f , and then sum up the entries of the resulting T -tuple, it’s the same as if we summed up the original S -tuple. But this says precisely that each column of the matrix f sums to 1.

So, we can use our formalism to describe \mathrm{FinStoch}, the category with finite sets as objects and stochastic maps as morphisms. We’ve seen this category is equivalent to the category with special commutative †-Frobenius algebras in C as objects and counit-preserving maps as morphisms.

Finite measure spaces

Now let’s use our formalism to describe finite measure spaces—by which, beware, I mean a finite sets equipped with measures! To do this, we’ll use a special commutative †-Frobenius algebra x in C together with any map

\mu: [0,\infty) \to x

Starting from these, we get a specified isomorphism

x \cong [0,\infty)^S

and \mu sends the number 1 to a vector in [0,\infty)^S : that is, a function on S taking values in [0,\infty) . Multiplying this function by counting measure, we get a measure on S .

Puzzle. How can we describe this measure without the annoying use of counting measure?

Conversely, any measure on a finite set gives a special commutative †-Frobenius algebra x in C equipped with a map from [0,\infty) .

So, we can say a finite measure space is a special commutative †-Frobenius algebra in C equipped with a map

\mu: [0,\infty) \to x

And given two of these,

\mu: [0,\infty) \to x , \qquad \nu: [0,\infty) \to y

and a coalgebra morphism

f : x \to y

obeying this equation

f \circ \mu = \nu

then we get a measure-preserving function between finite measure spaces! If you’re a category theorist, you’ll draw this equation as a commutative triangle:

Conversely, any measure-preserving function between finite measure spaces obeys this equation. So, we get an algebraic way of describing the category \mathrm{FinMeas} , with finite measure spaces as objects and measure-preserving maps as morphisms.

Finite probability measure spaces

I’m mainly interested in probability measures. So suppose x is a special commutative †-Frobenius algebra in C equipped with a map

\mu: [0,\infty) \to x

We’ve seen this gives a finite measure space. But this is a probability measure space if and only if

e \circ \mu = 1

where

e : x \to [0,\infty)

is the counit for x . The equation simply says the total integral of our measure \mu is 1.

So, we get a way of describing the category \mathrm{FinProb} , which has finite probability measure spaces as objects and measure-preserving maps as objects. Given finite probability measure spaces described this way:

\mu: [0,\infty) \to x , \qquad \nu: [0,\infty) \to y

a measure-preserving function is a coalgebra morphism

f : x \to y

such that the obvious triangle commutes:

f \circ \mu = \nu

Measure-preserving stochastic maps

Say we have two finite measure spaces. Then we can ask whether a stochastic map from one to the other is measure-preserving. And we can answer this question in the language of C .

Remember, a finite measure space is a special commutative †-Frobenius algebra x in C together with a map

\mu: [0,\infty) \to x

Say we have another one:

\nu: [0,\infty) \to y

A stochastic map is just a map

f: x \to y

that preserves the counit:

\epsilon_y \circ f = \epsilon_x

But it’s a measure-preserving stochastic map if also

f \circ \mu = \nu

Next…

There’s a lot more to say; I haven’t gotten anywhere near what Tobias and I are doing! But it’s pleasant to have this basic stuff written down.

19 Comments | information and entropy, probability | Permalink
Posted by John Baez

The Selected Papers Network (Part 2)

14 June, 2013

Last time Christopher Lee and I described some problems with scholarly publishing. The big problems are expensive journals and ineffective peer review. But we argued that solving these problems require new methods of

• selection—assessing papers

and

• endorsement—making the quality of papers known, thus giving scholars the prestige they need to get jobs and promotions.

The Selected Papers Network is an infrastructure for doing both these jobs in an open, distributed way. It’s not yet the solution to the big visible problems—just a framework upon which we can build those solutions. It’s just getting started, and it can use your help.

But before I talk about where all this is heading, and how you can help, let me say what exists now.

This is a bit dangerous, because if you’re not sure what a framework is for, and it’s not fully built yet, it can be confusing to see what’s been built so far! But if you’ve thought about the problems of scholarly publishing, you’re probably sick of hearing about dreams and hopes. You probably want to know what we’ve done so far. So let me start there.

SelectedPapers.net as it stands today

SelectedPapers.net lets you recommend papers, comment on them, discuss them, or simply add them to your reading list.

But instead of “locking up” your comments within its own website—the “walled garden” strategy followed by many other services—it explicitly shares these data in a way that people not on SelectedPapers.net can easily see. Any other service can see and use them too. It does this by using existing social networks—so that users of those social networks can see your recommendations and discuss them, even if they’ve never heard of SelectedPapers.net!

The idea is simple. You add some hashtags to let SelectedPapers.net know you’re talking to it, and to let it know which paper you’re talking about. It notices these hashtags and copies your comments over to its publicly accessible database.

So far Christopher Lee has got it working on Google+. So right now, if you’re a Google+ user, you can post comments on SelectedPapers.net using your usual Google+ identity and posting process, just by including suitable hashtags. Your post will be seen by your usual audience—but also by people visiting the SelectedPapers.net website, who don’t use Google+.

If you want to strip the idea down to one sentence, it’s this:

Given that social networks already exist, all we need for truly open scientific communication is a convention on a consistent set of tags and IDs for discussing papers.

That makes it possible to integrate discussion from all social networks—big and small—as a single unified forum. It’s a federated approach, rather than a single isolated website. And it won’t rely on any one social network: after Google+, we can get it working for Twitter and other networks and forums.

But more about the theory later. How, exactly, do you use it?

Getting Started

To see how it works, take a look here:

https://selectedpapers.net

Under ‘Recent activity’ you’ll see comments and recommendations of different papers, so far mostly on the arXiv.

Support for other social networks such as Twitter is coming soon. But here’s how you can use it now, if you’re a member of Google+:

• We suggest that you first create (in your Google+ account) a Google+ Circle specifically for discussing research with (e.g. call it “Research”). If you already have such a circle, or circles, you can just use those.

• Click Sign in with Google on https://selectedpapers.net or on a paper discussion page.

• The usual Google sign-in window will appear (unless you are already signed in). Google will ask if you want to use the Selected Papers network, and specifically for what Circle(s) to let it see the membership list(s) (i.e. the names of people you have added to that Circle). SelectedPapers.net uses this as your initial “subscriptions”, i.e. the list of people whose recommendations you want to receive. We suggest you limit this to your “Research” circle, or whatever Circle(s) of yours fit this purpose.

Note the only information you are giving SelectedPapers.net access to is this list of names; in all other respects SelectedPapers.net is limited by Google+ to the same information that anyone on the internet can see, i.e. your public posts. For example, SelectedPapers.net cannot ever see your private posts within any of your Circles.

• Now you can initiate and join discussions of papers directly on any SelectedPapers.net page.

• Alternatively, without even signing in to SelectedPapers.net, you can just write posts on Google+ containing the hashtag #spnetwork, and they will automatically be included within the SelectedPapers.net discussions (i.e. indexed and displayed so that other people can reply to them etc.). Here’s an example of a Google+ post example:

This article by Perelman outlines a proof of the Poincare conjecture!

#spnetwork #mustread #geometry #poincareConjecture arXiv:math/0211159

You need the tag #spnetwork for SelectedPapers.net to notice your post. Tags like #mustread, #recommend, and so on indicate your attitude to a paper. Tags like #geometry, #poincareConjecture and so on indicate a subject area: they let people search for papers by subject. A tag of the form arXiv:math/0211159 is necessary for arXiv papers; note that this does not include a # symbol.

For PubMed papers, include a tag of the form PMID:22291635. Other published papers usually have a DOI (digital object identifier), so for those include a tag of the form doi:10.3389/fncom.2012.00001.

Tags are the backbone of SelectedPapers.net; you can read more about them here.

• You can also post and see comments at https://selectedpapers.net. This page also lets you search for papers in the arXiv and search for published papers via their DOI or Pubmed ID. If you are signed in, the homepage will also show the latest recommendations (from people you’re subscribed to), papers on your reading list, and papers you tagged as interesting for your work.

Papers

Papers are the center of just about everything on the selected papers network. Here’s what you can currently do with a paper:

• click to see the full text of the paper via the arXiv or the publisher’s website.

• read other people’s recommendations and discussion of the paper.

• add it to your Reading List. This is simply a private list of papers—a convenient way of marking a paper for further attention later. When you are logged in, your Reading list is shown on the homepage. No one else can see your reading list.

• share the paper with others (such as your Google+ Circles or Google+ communities that you are part of).

• tag it as interesting for a specific topic. You do this either by clicking the checkbox of a topic (it shows topics that other readers have tagged the paper), by selecting from a list of topics that you have previously tagged as interesting to you, or by simply typing a tag name. These tags are public; that is, everyone can see what topics the paper has been tagged with, and who tagged them.

• post a question or comment about the paper, or reply to what other people have said about it. This traffic is public. Specifically, clicking the Discuss this Paper button gives you a Google+ window (with appropriate tags already filled in) for writing a post. Note that in order for the spnet to see your post, you must include Public in the list of recipients for your post (this is an inherent limitation of Google+, which limits apps to see only the same posts that any internet user would see – even when you are signed-in to the app as yourself on Google+).

• recommend it to others. Once again, you must include Public in the list of recipients for your post, or the spnet cannot see it.

We strongly suggest that you include a topic hashtag for your research interest area. For example, if there is a hashtag that people in your field commonly use for posting on Twitter, use it. If you have to make up a new hashtag, keep it intuitive and follow “camelCase” capitalization e.g. #openPeerReview.

Open design

Note that thanks to our open design, you do not even need to create a SelectedPapers.net login. Instead, SelectedPapers.net authenticates with Google (for example) that you are signed in to Google+; you never give SelectedPapers.net your Google password or access to any confidential information.

Moreover, even when you are signed in to SelectedPapers.net using your Google sign-in, it cannot see any of your private posts, only those you posted publicly—in other words, exactly the same as what anybody on the Internet can see.

What to do next?

We really need some people to start using SelectedPapers.net and start giving us bug reports. The place to do that is here:

https://github.com/cjlee112/spnet/issues

or if that’s too difficult for some reason, you can just leave a comment on this blog entry.

We could also use people who can write software to improve and expand the system. I can think of fifty ways the setup could be improved: but as usual with open-source software, what matters most is not what you suggest, but what you’re willing to do.

Next, let mention three things we could do in the longer term. But I want to emphasize that these are just a few of many things that can be done in the ecosystem created by a selected papers network. We don’t need to all do the same thing, since it’s an open, federated system.

• Overlay journals. A journal doesn’t need to do distribution and archiving of papers anymore: the arXiv or PubMed can do that. A journal can focus on the crucial work of selection and endorsement—it can just point to a paper on the arXiv or PubMed, and say “this paper is published”. Such journals, called overlay journals, are already being contemplated—see for example Tim Gowers’ post. But they should work better in the ecosystem created by a selected papers network.

• Review boards. Publication doesn’t need to be a monogamous relation between a journal and an author. We could also have prestigious ‘review boards’ like the Harvard Genomics Board or the Institute of Network Science who pick, every so often, what they consider to be best papers in their chosen area. In their CVs, scholars could then say things like “this paper was chosen as one of the Top Ten Papers in Topology in 2015 by the International Topology Review Board”. Of course, boards would become prestigious in the usual recursive way: by having prestigious members, being associated with prestigious institutions, and correctly choosing good papers to bestow prestige upon. But all this could be done quite cheaply.

• Open peer review. Last time, we listed lots of problems with how journals referee papers. Open peer review is a way to solve these problems. I’ll say more about it next time. For now, go here:

• Christopher Lee, Open peer review by a selected-papers network, Frontiers of Computational Neuroscience 6 (2012).

A federated system

After reading this, you may be tempted to ask: “Doesn’t website X already do most of this? Why bother starting another?”

Here’s the answer: our approach is different because it is federated. What does that mean? Here’s the test: if somebody else were to write their own implementation of the SelectedPapers.net protocol and run it on their own website, would data entered by users of that site show up automatically on selectedpapers.net, and vice versa? The answer is yes, because the protocol transports its data on open, public networks, so the same mechanism that allows selectedpapers.net to read its users’ messages would work for anyone else. Note that no special communications between the new site and SelectedPapers.net would be required; it is just federated by design!

One more little website is not going to solve the problems with journals. The last thing anybody wants is another password to remember! There are already various sites trying to solve different pieces of the problem, but none of them are really getting traction. One reason is that the different sites can’t or won’t talk to each other—that is, federate. They are walled gardens, closed ecosystems. As a result, progress has been stalled for years.

And frankly, even if some walled garden did eventually eventually win out, that wouldn’t solve the problem of expensive journals. If one party became able to control the flow of scholarly information, they’d eventually exploit this just as the journals do now.

So, we need a federated system, to make scholarly communication openly accessible not just for scholars but for everyone—and to keep it that way.

77 Comments | publishing, the practice of science | Permalink
Posted by John Baez

Quantum Techniques for Reaction Networks

11 June, 2013

Fans of the network theory series might like to look at this paper:

• John Baez, Quantum techniques for reaction networks.

and I would certainly appreciate comments and corrections.

This paper tackles a basic question we never got around to discussing: how the probabilistic description of a system where bunches of things randomly interact and turn into other bunches of things can reduce to a deterministic description in the limit where there are lots of things!

Mathematically, such systems are given by ‘stochastic Petri nets’, or if you prefer, ‘stochastic reaction networks’. These are just two equivalent pictures of the same thing. For example, we could describe some chemical reactions using this Petri net:

but chemists would use this reaction network:

C + O2 → CO2
CO2 + NaOH → NaHCO3
NaHCO3 + HCl → H2O + NaCl + CO2

Making either of them ‘stochastic’ merely means that we specify a ‘rate constant’ for each reaction, saying how probable it is.

For any such system we get a ‘master equation’ describing how the probability of having any number of things of each kind changes with time. In the class I taught on this last quarter, the students and I figured out how to derive from this an equation saying how the expected number of things of each kind changes with time. Later I figured out a much slicker argument… but either way, we get this result:

Theorem. For any stochastic reaction network and any stochastic state \Psi(t) evolving in time according to the master equation, then

\displaystyle{ \frac{d}{dt} \langle N \Psi(t) \rangle } = \displaystyle{\sum_{\tau \in T}} \, r(\tau) \, (s(\tau) - t(\tau)) \; \left\langle N^{\underline{s(\tau)}}\, \Psi(t) \right\rangle

assuming the derivative exists.

Of course this will make no sense yet if you haven’t been following the network theory series! But I explain all the notation in the paper, so don’t be scared. The main point is that \langle N \Psi(t) \rangle is a vector listing the expected number of things of each kind at time t. The equation above says how this changes with time… but it closely resembles the ‘rate equation’, which describes the evolution of chemical systems in a deterministic way.

And indeed, the next big theorem says that the master equation actually implies the rate equation when the probability of having various numbers of things of each kind is given by a product of independent Poisson distributions. In this case \Psi(t) is what people in quantum physics call a ‘coherent state’. So:

Theorem. Given any stochastic reaction network, let
\Psi(t) be a mixed state evolving in time according to the master equation. If \Psi(t) is a coherent state when t = t_0, then \langle N \Psi(t) \rangle obeys the rate equation when t = t_0.

In most cases, this only applies exactly at one moment of time: later \Psi(t) will cease to be a coherent state. Then we must resort to the previous theorem to see how the expected number of things of each kind changes with time.

But sometimes our state \Psi(t) will stay coherent forever! For one case where this happens, see the companion paper, which I blogged about a little while ago:

• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks.

We wrote this first, but logically it comes after the one I just finished now!

All this material will get folded into the book I’m writing with Jacob Biamonte. There are just a few remaining loose ends that need to be tied up.

42 Comments | chemistry, networks, probability | Permalink
Posted by John Baez

The Selected Papers Network (Part 1)

7 June, 2013

Christopher Lee has developed some new software called the Selected Papers Network. I want to explain that and invite you all to try using it! But first, in this article, I want to review the problems it’s trying to address.

There are lots of problems with scholarly publishing, and of course even more with academia as a whole. But I think Chris and I are focused on two: expensive journals, and ineffective peer review.

Expensive Journals

Our current method of publication has some big problems. For one thing, the academic community has allowed middlemen to take over the process of publication. We, the academic community, do most of the really tricky work. In particular, we write the papers and referee them. But they, they publishers, get almost all the money, and charge our libraries for it—more and more, thanks to their monopoly power. It’s an amazing business model:

Get smart people to work for free, then sell what they make back to them at high prices.

People outside academia have trouble understanding how this continues! To understand it, we need to think about what scholarly publishing and libraries actually achieve. In short:

1. Distribution. The results of scholarly work get distributed in publicly accessible form.

2. Archiving. The results, once distributed, are safely preserved.

3. Selection. The quality of the results is assessed, e.g. by refereeing.

4. Endorsement. The quality of the results is made known, giving the scholars the prestige they need to get jobs and promotions.

Thanks to the internet, jobs 1 and 2 have become much easier. Anyone can put anything on a website, and work can be safely preserved at sites like the arXiv and PubMed Central. All this is either cheap or already supported by government funds. We don’t need journals for this.

The journals still do jobs 3 and 4. These are the jobs that academia still needs to find new ways to do, to bring down the price of journals or make them entirely obsolete.

The big commercial publishers like to emphasize how they do job 3: selection. The editors contact the referees, remind them to deliver their referee reports, and communicate these reports to the authors, while maintaining the anonymity of the referees. This takes work.

However, this work can be done much more cheaply than you’d think from the prices of journals run by the big commercial publishers. We know this from the existence of good journals that charge much less. And we know it from the shockingly high profit margins of the big publishers, particularly Elsevier.

It’s clear that the big commercial publishers are using their monopoly power to charge outrageous prices for their products. Why do they continue to get away with this? Why don’t academics rebel and publish in cheaper journals?

One reason is a broken feedback loop. The academics don’t pay for journals out of their own pocket. Instead, their university library pays for the journals. Rising journal costs do hurt the academics: money goes into paying for journals that could be spent in other ways. But most of them don’t notice this.

The other reason is item 4: endorsement. This is the part of academic publishing that outsiders don’t understand. Academics want to get jobs and promotions. To do this, we need to prove that we’re ‘good’. But academia is so specialized that our colleagues are unable to tell how good our papers are. Not by actually reading them, anyway! So, they try to tell by indirect methods—and a very important one is the prestige of the journals we publish in.

The big commercial publishers have bought most of the prestigious journals. We can start new journals, and some of us are already doing that, but it takes time for these journals to become prestigious. In the meantime, most scholars prefer to publish in prestigious journals owned by the big publishers, even if this slowly drives their own libraries bankrupt. This is not because these scholars are dumb. It’s because a successful career in academia requires the constant accumulation of prestige.

The Elsevier boycott shows that more and more academics understand this trap and hate it. But hating a trap is not enough to escape the trap.

Boycotting Elsevier and other monopolistic publishers is a good thing. The arXiv and PubMed Central are good things, because they show that we can solve the distribution and archiving problems without the help of big commercial publishers. But we need to develop methods of scholarly publishing that solve the selection and endorsement problems in ways that can’t be captured by the big commercial publishers.

I emphasize ‘can’t be captured’, because these publishers won’t go down without a fight. Anything that works well, they will try to buy—and then they will try to extract a stream of revenue from it.

Ineffective Peer Review

While I am mostly concerned with how the big commercial publishers are driving libraries bankrupt, my friend Christopher Lee is more concerned with the failures of the current peer review system. He does a lot of innovative work on bioinformatics and genomics. This gives him a different perspective than me. So, let me just quote the list of problems from this paper:

• Christopher Lee, Open peer review by a selected-papers network, Frontiers of Computational Neuroscience 6 (2012).

The rest of this section is a quote:

• Expert peer review (EPR) does not work for interdisciplinary peer review (IDPR). EPR means the assumption that the reviewer is expert in all aspects of the paper, and thus can evaluate both its impact and validity, and can evaluate the paper prior to obtaining answers from the authors or other referees. IDPR means the situation where at least one part of the paper lies outside the reviewer’s expertise. Since journals universally assume EPR, this creates artificially high barriers to innovative papers that combine two fields [Lee, 2006]—-one of the most valuable sources of new discoveries.

• Shoot first and ask questions later means the reviewer is expected to state a REJECT/ACCEPT position before getting answers from the authors or other referees on questions that lie outside the reviewer’s expertise.

• No synthesis: if review of a paper requires synthesis—combining the different expertise of the authors and reviewers in order to determine what assumptions and criteria are valid for evaluating it—both of the previous assumptions can fail badly [Lee, 2006].

• Journals provide no tools for finding the right audience for an innovative paper. A paper that introduces a new combination of fields or ideas has an audience search problem: it must search multiple fields for people who can appreciate that new combination. Whereas a journal is like a TV channel (a large, pre-defined audience for a standard topic), such a paper needs something more like Google—a way of quickly searching multiple audiences to find the subset of people who can understand its value.

• Each paper’s impact is pre-determined rather than post-evaluated: By ‘pre-determination’ I mean that both its impact metric (which for most purposes is simply the title of the journal it was published in) and its actual readership are locked in (by the referees’s decision to publish it in a given journal) before any readers are allowed to see it. By ‘post-evaluation’ I mean that impact should simply be measured by the research community’s long-term response and evaluation of it.

• Non-expert PUSH means that a pre-determination decision is made by someone outside the paper’s actual audience, i.e., the reviewer would not ordinarily choose to read it, because it does not seem to contribute sufficiently to his personal research interests. Such a reviewer is forced to guess whether (and how much) the paper will interest other audiences that lie outside his personal interests and expertise. Unfortunately, people are not good at making such guesses; history is littered with examples of rejected papers and grants that later turned out to be of great interest to many researchers. The highly specialized character of scientific research, and the rapid emergence of new subfields, make this a big problem.

In addition to such false-negatives, non-expert PUSH also causes a huge false-positive problem, i.e., reviewers accept many papers that do not personally interest them and which turn out not to interest anybody; a large fraction of published papers subsequently receive zero or only one citation (even including self-citations [Adler et al., 2008]). Note that non-expert PUSH will occur by default unless reviewers are instructed to refuse to review anything that is not of compelling interest for their own work. Unfortunately journals assert an opposite policy.

• One man, one nuke means the standard in which a single negative review equals REJECT. Whereas post-evaluation measures a paper’s value over the whole research community (‘one man, one vote’), standard peer review enforces conformity: if one referee does not understand or like it, prevent everyone from seeing it.

• PUSH makes refereeing a political minefield: consider the contrast between a conference (where researchers publicly speak up to ask challenging questions or to criticize) vs. journal peer review (where it is reckoned necessary to hide their identities in a ‘referee protection program’). The problem is that each referee is given artificial power over what other people can like—he can either confer a large value on the paper (by giving it the imprimatur and readership of the journal) or consign it zero value (by preventing those readers from seeing it). This artificial power warps many aspects of the review process; even the ‘solution’ to this problem—shrouding the referees in secrecy—causes many pathologies. Fundamentally, current peer review treats the reviewer not as a peer but as one who wields a diktat: prosecutor, jury, and executioner all rolled into one.

• Restart at zero means each journal conducts a completely separate review process of a paper, multiplying the costs (in time and effort) for publishing it in proportion to the number of journals it must be submitted to. Note that this particularly impedes innovative papers, which tend to aim for higher-profile journals, and are more likely to suffer from referees’s IDPR errors. When the time cost for publishing such work exceeds by several fold the time required to do the work, it becomes more cost-effective to simply abandon that effort, and switch to a ‘standard’ research topic where repetition of a pattern in many papers has established a clear template for a publishable unit (i.e., a widely agreed checklist of criteria for a paper to be accepted).

• The reviews are thrown away: after all the work invested in obtaining reviews, no readers are permitted to see them. Important concerns and contributions are thus denied to the research community, and the referees receive no credit for the vital contribution they have made to validating the paper.

In summary, current peer review is designed to work for large, well-established fields, i.e., where you can easily find a journal with a high probability that every one of your reviewers will be in your paper’s target audience and will be expert in all aspects of your paper. Unfortunately, this is just not the case for a large fraction of researchers, due to the high level of specialization in science, the rapid emergence of new subfields, and the high value of boundary-crossing research (e.g., bioinformatics, which intersects biology, computer science, and math).

Toward solutions

Next time I’ll talk about the software Christopher Lee has set up. But if you want to get a rough sense of how it works, read the section of Christopher Lee’s paper called The Proposal in Brief.

30 Comments | publishing, the practice of science | Permalink
Posted by John Baez

Symmetry and the Fourth Dimension (Part 10)

4 June, 2013

Some people say it’s impossible to visualize 4-dimensional things. But lots of people I know can do it.

How do they do it?

Well, how do you visualize 3-dimensional things? A computer screen is just 2d, but we can draw a 3d object on it by picking some diagonal direction—southeast in the picture below—to stand for ‘forwards, towards our eyes’. Similarly, we can draw a 4d object by picking another diagonal direction—northeast in this picture—to stand for ‘forwards in the 4th dimension’.

Here we are using this trick to draw 0d, 1d, 2d, 3d and 4d cubes. The first dimension, often called the x direction, points along the red arrow. The second, called the y direction, points along the green arrow. The third, the z direction, points diagonally along the blue arrow. And the fourth, sometimes called the w direction, points diagonally along the yellow arrow.

There’s nothing sacred about these names or these particular directions; we can implement this idea in lots of different ways. It’s ‘cheating’, but that’s okay. A vague, misty image can be a lot better than no image at all.

Of course, we need to think about math to keep straight which lines in our picture point in the w direction. But that’s okay too. A mixture of equations and visualization lets mathematicians and physicists make faster progress in understanding the 4th dimension than if they used only equations.

Physicists need to understand the 4th dimension because we live in 4d spacetime. Some mathematicians study much higher-dimensional spaces, even infinite-dimensional ones, and here visualization becomes more subtle, and perhaps more limited. But many mathematicians working on 4d topology take visualization very seriously. If you’ve ever seen the elaborate, detailed gestures they make when describing 4-dimensional constructions, you’ll know what I mean.

Visualizing shapes in 4 dimensions takes practice, but it’s lots of fun! As this series of posts continues, I want to give you some practice while talking about nice 4-dimensional shapes: the 4d relatives of the Platonic and Archimedean solids. In the series so far, we’ve spent a lot of time warming up by studying the 3d Platonic and Archimedean solids and developing a technique for classifying them, called Coxeter diagrams. All that will pay off soon!


Here’s a great series of videos that explains higher dimensions:

• Jos Leys, Étienne Ghys and Aurélien Alvarez, Dimensions.

The only problem is that it’s tough to navigate them. Click on your favorite language and you’ll see part 1 of the series. After you start playing it you’ll see an arrow at the lower right of the video that lets you jump to the next one. This is good if, like me, you’re impatient for the 4th dimension! That starts in part 3.

There’s a guide to all nine parts here:

• Jos Leys, Étienne Ghys and Aurélien Alvarez, Tour.

but you can’t get to the videos from there! They need a bit of help from a good website designer.

The picture above is a shot of the glorious 120-cell… one of the six Platonic solids in 4 dimensions. But more on that later! We’ll start with a simpler one: the 4-cube.

7 Comments | mathematics | Permalink
Posted by John Baez

Symmetry and the Fourth Dimension (Part 9)

2 June, 2013

 

Last time in this series we completed the first phase of our quest: we got lots of polyhedra from Coxeter diagrams in a systematic way. But before we sail off into new seas, let’s review what we’ve done.

To spice things up, I’ll explain everything in a different way than before. If you don’t see how this new way relates to the old one, please ask! There’s a lot to say about this stuff that I’m not saying, so there are plenty of gaps left to fill.

And if you’re wondering what that thing is up there, read on.

Coxeter theory

Harold Scott MacDonald Coxeter, the ‘king of geometry’, developed a beautiful theory relating groups to highly symmetrical polyhedra and their higher-dimensional generalizations, called ‘polytopes’.



His book Regular Polytopes is required reading for anyone who wants to learn about these things. There’s a copy sitting by my bed, and there should be one by yours, too. It took him 24 years to write.

In his honor, finite groups generated by reflections in n-dimensional space are called Coxeter groups. They’re described by diagrams with n dots, called Coxeter diagrams. We’ve been looking at the 3d case, where there happen to be three such diagrams, with mysterious-sounding names:

A3      o—3—o—3—o
B3      o—3—o—4—o
H3      o—3—o—5—o

I want to show you to use these diagrams. So let’s do an example, namely A3:

o—3—o—3—o

This diagram is telling us to draw a bunch of triangles on a sphere, like this:

This shape is called a Coxeter complex. As you can see, it contains lots of great circles. For each one we get a symmetry: the reflection across that circle, as if a mirror cut our Coxeter complex in half through that circle. So, the symmetries of the Coxeter complex form a finite group, generated by reflections. This is a Coxeter group!

But how do we get the Coxeter complex from the diagram? The dots in the diagram are secretly called V, E, and F:

V—3—E—3—F

Among other things, these letters are names for the edges of our favorite triangle in the Coxeter complex:

It doesn’t matter which is our favorite triangle, so I just picked one. Each edge of this triangle gives a symmetry: the reflection that reflects the Coxeter complex across that edge! We call these symmetries V, E and F.

The 3 on the line going between V and E in this diagram:

V—3—E—3—F

is a quick way of saying that

(VE)3 = 1

In other words, doing the reflections V, E, V, E, V, E gets us back where we started.

To see this, let’s call our favorite triangle 1. Reflecting this triangle across the great circle containing the edge V, we get a new triangle, which we’ll call V. Reflecting that across the great circle containing the edge E, we get a new triangle, which we’ll call VE. And so on:

By the time we get to VEVEVE = (VE)3, we’re back where we started! That’s a total of 6 reflections, so the V and E edges of each triangle must meet at a 60° angle.

Similarly, the 3 on the line going between E and V says that

(EF)3 = 1

so doing the reflections E, F, E, F, E, F also gets us back where we started:

On the other hand, there’s no line connecting the dots V and F in our Coxeter diagram:

V—3—E—3—F

But this is an abbreviation for a line with a 2 on it! You see, lines with 2 on them occur so often that Coxeter decided to save time by not drawing such lines. So, we have

(VF)2 = 1

and doing the reflections V, F, V, F also gets us back where started:

That’s a total of 4 reflections, so the V and F edges of each triangle must meet at a 90° angle.

So, I’ve sketched how the Coxeter group and the Coxeter complex arise from the Coxeter diagram. To be a bit more precise, the Coxeter group A3 has generators V, E, F obeying relations

(VE)3 = (EF)3 = (VF)2 = 1

and also

V2 = E2 = F2 = 1

since V, E, and F are reflections. If we draw a sphere with one great circle serving as the ‘mirror’ for each reflection in the Coxeter group, we get the Coxeter complex.

What makes Coxeter complexes special, compared to other ways of tiling a sphere with triangles? One thing is that they have exactly as many symmetries as triangles. If you pick any triangle and call it your favorite, there’s exactly one symmetry—that is, rotation and/or reflection—sending this triangle to any other.

So, the Coxeter complex is actually a picture of its own symmetry group!

The particular Coxeter complex we’ve been looking at has 24 triangles. So, its symmetry group has 24 elements. This group is called the tetrahedral finite reflection group, or A3 for short. There’s a lot to say about it, but not now! There’s another more urgent question.

Polyhedra

How do we get polyhedra from our Coxeter complex?

The easiest one works like this. We take the Coxeter complex and create a polyhedron with a corner in the middle of each triangle, connecting two corners with an edge whenever their triangles touch:

The polyhedron looks better if we view it from a slightly different angle, and let a skilled artist like Tom Ruen do the drawing:

This polyhedron is called the Poincaré dual of the Coxeter complex.

There’s a notation for this particular polyhedron:

•—3—•—3—•

All the dots are black, because this is the fanciest, most interesting polyhedron that comes from our Coxeter complex. We get other polyhedra by blackening just some of the dots.

Earlier in this series, I described how to build these other polyhedra using the concept of ‘flag’. But there’s another way, using the Coxeter complex, which I’ll sketch here.

Suppose we blacken just some of the dots in our Coxeter diagram. Then our favorite triangle belongs to a bunch of triangles, all related by reflections corresponding to the dots we left white. And indeed, all the triangles can be grouped into bunches that all look the same… and there’s a polyhedron that has a corner in the middle of each bunch of triangles.

For example, suppose we leave the E dot white:

•—3—o—3—•

Then our favorite triangle belongs to a bunch of triangles—in this case, just a pair!—that are related by reflections along the E edge of our favorite triangle. If we group our triangles into pairs like this, we get a polyhedron with a corner in the middle of each pair:

Again, it looks better if we let Tom Ruen draw it:

I should do more examples, but I think I’ll wrap up by describing the procedure in more highbrow language. Skip the next paragraph if you don’t know group theory, and move on to the complete list of examples!

If we call the Coxeter group G, blackening all the dots of the Coxeter diagram gives a polyhedron with one corner for each element of G. But if we don’t blacken all of them, the reflections corresponding to the white dots generate a subgroup B of G. Then we get a polyhedron with one corner for each element of G/B. Last time I said each way of blackening some dots describes some sort of ‘flag’. In these terms, B is the subgroup fixing your favorite flag of this sort, and G/B is the set of all flags of this sort. Each of those flags corresponds to what I’m calling a ‘bunch of triangles’ in my current story.

But now let’s see what we get from all this! We get three families of polyhedra, and these are almost all the ‘Archimedean solids’ in 3 dimensions.

The A3 family:     o—3—o—3—o

This family of polyhedra can all be gotten from the tetrahedron by chopping off vertices, edges or faces. They’re associated to the Coxeter complex we’ve just been looking at:

It’s built from triangles whose interior angles are \pi/3, \pi/3 and \pi/2. These numbers come from taking π and dividing it by the numbers on the edges of the Coxeter diagram: 3, 3, and the invisible edge labelled 2.

As mentioned, this Coxeter complex has 24 triangles, and its symmetry group, with 24 elements, is called the tetrahedral finite reflection group, or A3.

Here are all the polyhedra in this family. The list has some repeats, because this Coxeter diagram is its own mirror image!

tetrahedron •—3—o—3—o
truncated tetrahedron •—3—•—3—o
octahedron o—3—•—3—o
truncated tetrahedron o—3—•—3—•
tetrahedron o—3—o—3—•
cuboctahedron •—3—o—3—•
truncated octahedron •—3—•—3—•

The B3 family:     o—3—o—4—o

This family of polyhedra can all be gotten by taking the cube or the octahedron and chopping off vertices, edges or faces. They come from the Coxeter complex whose triangles have interior angles \pi/3, \pi/4 and \pi/2:

This Coxeter complex has 48 triangles, and its symmetry group is a Coxeter group with 48 elements, called the octahedral finite reflection group, or B3.

Here are all the polyhedra in this family. We’ve seen some of these already in the A3 family, and there’s a reason for that: the group B3 happens to contain A3 as a subgroup! It’s twice as big.

cube •—4—o—3—o
truncated cube •—4—•—3—o
cuboctahedron o—4—•—3—o
truncated octahedron o—4—•—3—•
octahedron o—4—o—3—•
rhombicuboctahedron •—4—o—3—•
truncated cuboctahedron •—4—•—3—•

The H3 family:     o—3—o—5—o

This family of polyhedra can all be gotten by taking the dodecahedron or icosahedron and chopping off vertices, edges and faces. They come from the Coxeter complex built from triangles whose interior angles are \pi/3, \pi/5 and \pi/2:

This Coxeter complex has 120 triangles, and its symmetry group is a Coxeter group with 120 elements, called the icosahedral finite reflection group, or H3.

Here are all the polyhedra coming from this Coxeter complex. These are my favorites, because they look the most fancy, and I have a fondness for the quirky charm of 5-fold symmetry:

dodecahedron •—5—o—3—o
truncated dodecahedron •—5—•—3—o
icosidodecahedron o—5—•—3—o
truncated icosahedron o—5—•—3—•
icosahedron o—5—o—3—•
rhombicosidodecahedron •—5—o—3—•
truncated icosidodecahedron •—5—•—3—•

Afterword

The picture at the start of this post was taken by my friend Allen Knutson:

It’s the H3 Coxeter complex, and for some reason it’s in the music library at Cornell University, where Allen teaches math.

The picture of H. S. M. Coxeter is from the cover of a book about him:

• Siobhan Roberts, King of Infinite Space, Walker & Company, 2006.

As usual, the pretty pictures of solids with brass balls at the vertices were made by Tom Ruen using Robert Webb’s Stella software. Tom Ruen also drew the Coxeter complexes; I’m to blame for the crude extra stuff added on.

You can see the previous episodes of this series here:

• Part 1: Platonic solids and Coxeter complexes.

• Part 2: Coxeter groups.

• Part 3: Coxeter diagrams.

• Part 4: duals of Platonic solids.

• Part 5: Interpolating between a Platonic solid and its dual, and how to describe this using Coxeter diagrams. Example: the cube/octahedron family.

• Part 6: Interpolating between a Platonic solid and its dual. Example: the dodecahedron/icosahedron family.

• Part 7: Interpolating between a Platonic solid and its dual. Example: the tetrahedron family.

• Part 8: The missing solids, coming from Coxeter diagrams with both ends blackened.

10 Comments | mathematics | Permalink
Posted by John Baez

42

25 May, 2013

In The Hitchhiker’s Guide to the Galaxy by Douglas Adams, the number 42 is the “Answer to the Ultimate Question of Life, the Universe, and Everything”. But he didn’t say what the question was!

Since today is Towel Day, let me reveal that now.

If you try to get several regular polygons to meet snugly at a point in the plane, what’s the most sides any of the polygons can have? The answer is 42.

The picture shows an equilateral triangle, a regular heptagon and a regular 42-gon meeting snugly at a point. If you do the math, you’ll see the reason this works is that

\displaystyle{ \frac{1}{3} + \frac{1}{7} + \frac{1}{42} = \frac{1}{2} }

There are actually 10 solutions of

\displaystyle{ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{2} }

with p \le q \le r, and each of them gives a way for three regular polygons to snugly meet at a point. But this particular solution features the biggest number possible!

But why is this so important? Well, it turns out that if you look for natural numbers a, b, c that make

\displaystyle{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} }

as close to 1 as possible, while still less than 1, the very best you can do is 1/2 + 1/3 + 1/7. It comes within 1/42 of equalling 1, since

\displaystyle{ \frac{1}{3} + \frac{1}{7} + \frac{1}{42} = \frac{1}{2} }

And why is this important? Well, suppose you’re trying to make a doughnut with at least two holes that has the maximum number of symmetries. More precisely, suppose you’re trying to make a Riemann surface with genus g \ge 2 that has the maximum number of symmetries. Then you need to find a highly symmetrical tiling of the hyperbolic plane by triangles whose interior angles are \pi/a, \pi/b and \pi/c , and you need

\displaystyle{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} < 1 }

for these triangles to fit on the hyperbolic plane.

For example, if you take a = 2, b = 3, c = 7 you get this tiling:

A clever trick then lets you curl up the hyperbolic plane and get a Riemann surface with at most

\displaystyle{ \frac{2(g-1)}{1 - \frac{1}{a} - \frac{1}{b} - \frac{1}{c}} }

symmetries.

So, to get as many symmetries as possible, you want to make 1 - \frac{1}{a} - \frac{1}{b} - \frac{1}{c} as small as possible! And thanks to what I said, the best you can do is

\displaystyle{ 1 - \frac{1}{2} - \frac{1}{3} - \frac{1}{7} = \frac{1}{42} }

So, your surface can have at most

84(g-1)

symmetries. This is called Hurwitz’s automorphism theorem. The number 84 looks mysterious when you first see it — but it’s there because it’s twice 42.

In particular, the famous mathematician Felix Klein studied the most symmetrical doughnut with 3 holes. It’s a really amazing thing, called Klein’s quartic curve:

It has

84 \times 2 = 168

symmetries. That number also looks mysterious when you first see it. Of course it’s the number of hours in a week, but the real reason it’s there is because it’s four times 42.

If you carefully count the triangles in the picture above, you’ll get 56. However, these triangles are equilateral, or at least they would be if we could embed Klein’s quartic curve in 3d space without distorting it. If we drew all the smaller triangles whose interior angles are \pi/2, \pi/3 and \pi/7, each equilateral triangle would get subdivided into 6 smaller triangles, and there would be a total of 6 \times 56 = 336 triangles. But of course

336 = 8 \times 42

Half of these smaller triangles would be ‘left-handed’ and half would be ‘right-handed’, and there’d be a symmetry sending a chosen triangle to any other triangle of the same handedness, for a total of

168 = 4 \times 42

symmetries (that is, conformal transformations, not counting reflections).

But why is this stuff the answer to the ultimate question of life, the universe, and everything? I’m not sure, but I have a crazy theory. Maybe all matter and forces are made of tiny little strings! As they move around, they trace out Riemann surfaces in spacetime. And when these surfaces are as symmetrical as possible, reaching the limit set by Hurwitz’s automorphism theorem, the size of their symmetry group is a multiple of 42, thanks to the math I just described.

Puzzles

Puzzle 1. Consider solutions of

\displaystyle{ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{2} }

with positive integers p \le q \le r, and show that the largest possible value of r is 42.

Puzzle 2. Consider solutions of

\displaystyle{ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} < 1}

with positive integers a, b, c, and show that the largest possible value of \frac{1}{a} + \frac{1}{b} + \frac{1}{c} is 1 - \frac{1}{42}.

Acknowledgments and references

For more details, see my page on Klein’s quartic curve, and especially the section on incredibly symmetrical surfaces. Sums of reciprocals of natural numbers are called ‘Egyptian fractions’, and they have deep connections to geometry; for more on this see my article on Archimedean tilings and Egyptian fractions.

The picture of a triangle, heptagon and 42-gon (also known as a tetracontakaidigon) was made by Tyler, and you can see all 17 ways to get 3 regular polygons to meet snugly at a vertex on Wikipedia. Of these, only 11 can occur in a uniform tiling of the plane. The triangle, heptagon and 42-gon do not tile the plane, but you can see some charming attempts to do something with them on Kevin Jardine’s website Imperfect Congruence:


The picture of Klein’s quartic curve was made by Greg Egan, and you should also check out his page on Klein’s quartic curve.


“Good Morning,” said Deep Thought at last.
“Er…good morning, O Deep Thought” said Loonquawl nervously, “do you have…er, that is…”
“An Answer for you?” interrupted Deep Thought majestically. “Yes, I have.”
The two men shivered with expectancy. Their waiting had not been in vain.
“There really is one?” breathed Phouchg.
“There really is one,” confirmed Deep Thought.
“To Everything? To the great Question of Life, the Universe and everything?”
“Yes.”
Both of the men had been trained for this moment, their lives had been a preparation for it, they had been selected at birth as those who would witness the answer, but even so they found themselves gasping and squirming like excited children.
“And you’re ready to give it to us?” urged Loonsuawl.
“I am.”
“Now?”
“Now,” said Deep Thought.
They both licked their dry lips.
“Though I don’t think,” added Deep Thought. “that you’re going to like it.”
“Doesn’t matter!” said Phouchg. “We must know it! Now!”
“Now?” inquired Deep Thought.
“Yes! Now…”
“All right,” said the computer, and settled into silence again. The two men fidgeted. The tension was unbearable.
“You’re really not going to like it,” observed Deep Thought.
“Tell us!”
“All right,” said Deep Thought. “The Answer to the Great Question…”
“Yes…!”
“Of Life, the Universe and Everything…” said Deep Thought.
“Yes…!”
“Is…” said Deep Thought, and paused.
“Yes…!”
“Is…”
“Yes…!!!…?”
“Forty-two,” said Deep Thought, with infinite majesty and calm. – Douglas Adams

49 Comments | mathematics | Permalink
Posted by John Baez

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