How to write popular mathematics well? Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead of giving up in the face of a Very Hard Problem, one of us brought up a column popularizing physics that had a brief run in the school paper, and suggested that we author something similar for mathematics. The column will have some particular constraints that seem challenging to satisfy (self-contained week to week, 500-700 words, try to cover at least some of the current research at UW) but this question is a more general one.
In looking for resources and guidance to help with the writing we have come across several good discussions of topic. We have also found examples of good popular writing and a general discussion of presenting mathematics to a non-mathematical audience.
What we have not found, on MathOverflow or elsewhere, is a popular analogue of the well-answered question "How to write mathematics well?". A lot of the tactical advice of Knuth, Halmos, and others goes out the window when you answer their first question, "Who is your audience?" with "a general university educated public".

What is your advice for writing good mathematics for a popular audience? What holds for all styles of writing and what is article or book specific?

 A: I get asked this question a lot, so I'll give my pat (but serious) answer:
Find yourself an editor.
That is, find someone who'll take what you think is crystal clear, engaging prose, and turn it into something that actually is crystal clear and engaging.  I couldn't do my job without my editors.  (For those who don't know me, I work as a freelance mathematics writer.  I report on developments in mathematics for publications such as Science magazine and SIAM News.)  It's very easy, especially when writing about a subject you know inside out, to fool yourself into thinking you've explained things so that any fool can understand it; it's equally easy, when reading someone else's writing, to notice they haven't explained things very well.  A good editor will, at the least, point out your failings.  A great editor will fix them.
Now if I can only find someone to edit this posting....
A: All of us, I think, have occasion now and then to try to explain what we do to a bright
and genuinely interested non-mathematician.  My advice for writing is to keep track of 
what seems to work in conversation and then write that down.
A: I feel I should have a good answer to this question, since I've been writing about mathematics for the general public for many years -- but after some thought, I'm not coming up with any specific advice.
But I can say this. To write about popular mathematics well, you must write well.  Your explication of the main idea can be as clear and correct as you like, but if the sentences lie dead on the page, no one is going to read your column.  And this applies doubly, given that you're asking people to read about a topic that, in most cases, they don't think they care about.
So I think the best advice you're going to get won't come from us, but from people and books that have something to say about English prose generally:  Strunk & White and its successors. 
The short version of all such books:  Vary sentence structure.  Read what you write aloud.  Imitate things you like.  Avoid cliche.  And, above all, cut all words that are not doing work. 
A: How not to write popular math:
Cheating: spreading false math to communicate easily, spreading the false impression that some things are easy when they are not and, on the contrary, clouding behind mysteries things that could be explained. Taking short cuts which are plainly false. Appealing to magic.
Concentrating on personal stories. While it is undoubtable that a good story is always the easiest way to keep the attention high, and that information on people that did math helps in conveying some good math, I found that recently, in way too many popular math books, biographical notes are overwhelming the math content. 
Let me end with an example of both. In many popular books about infinity and math stories about mental illness of Cantor or Godel abounds and are more or less explicitly linked with "devoting thoughts to infinity". This is used to shroud infinity into a cloak of mystery, some magic world in which one can lose his mind, and to raise attention. This is, in my opinion, doing a very bad job both in popularizing math and in writing a story.
A: One piece of negative advice, to avoid a common fault in popular maths books: work out carefully what your audience is and write for that audience. I call that negative advice because it's really the contrapositive that concerns me: don't, for example, carefully explain how to add complex numbers and then a few pages later refer without explanation to a manifold as having trivial homology. (That sounds too obvious to be worth saying. Unfortunately, it isn't.)
A: If you are trying to get good at something, whether it be an art or a sport or a game, certain basic principles always apply.


*

*Aim high.  Don't flatter yourself that you're already pretty good.  If you have not already worked hard at honing your skill, chances are you suck.  Study the masters.  Why are they so much better than you are?  Identify what makes their performances shine, and train yourself at the same techniques.

*Ferret out your weaknesses ruthlessly and work hard at eliminating them.  If you can't see your own weaknesses, find a coach who can.  (As Barry Cipra mentioned, a good editor can fulfill this role.)  Don't get defensive if someone criticizes you; welcome all criticism and try to extract something useful from any feedback you get.

*Practice regularly.  This means taking the time to work at improving your skill even when there is no immediate reward.
For popular mathematical writing in particular, I would add another tip: Don't assume that your audience will find what you have to say interesting.  I'm even tempted to recommend that you assume that your audience will find you boring, but that would be going a little too far.  The point is, you must take it upon yourself to make your writing interesting.  Aim to write something that the readers won't be able to put down, that they will want to tell all their friends about, that will change their whole outlook on the subject.  You won't always succeed, but you will certainly fail to do your best if you don't aim high.
A: Perhaps you should be asking how NOT to write popular mathematics well. The moment you create a template based on these excellent answers, you would lose your authentic tone or signature if you will.
So it depends on what popular mathematics you would like to read. For a student like me I always enjoy author who can maintain certain lucidity with his natural tone and who does not insult the reader's intelligence. By latter I mean who is adept at simplifying without dumbing down for audience. (For instance, I believe when Jeopardy! initially aired they wanted to "dumb down" the show but Merv Griffin did not want so.)
Recently I read Stalking Riemann Hypothesis by Dan Rockmore and because of the nature of the content author intentionally paused to explain concepts such as random matrices, Tracy-Widom distribution, eigenvalue etc. which would be accessible to even high schools students yet capture attention of mature readers. Rockmore used brief biographical snippets of people involved which made his book as enjoyable as Keith Devlin's The Millennium Problems and Mathematical Mountaintops by J.L.Casti.
To recapitulate 1) find a style that works for you; and 2) choose an interesting topic and your audience.
To permit me a transcendentalist quote:
"Do not go where the path may lead; go instead where there is no path and leave a trail."
A: As a graduate students I was forced (not really, I actually enjoyed) to read several long
papers/essays how to write mathematics by people like Halmos who really knew how to it well, but my favorite advice came out of Chinese fortune cookie: "Good writing is clear thinking made visible".
A: I think the answer is simple. 
1) Read, and as widely as possible.
2) Write, as much as possible.
3) Compare what you learn (what works and doesn't) in reading to writing and vice versa.
A: This is a great question, with many possible answers. Here are some of the guidelines that I try (and often fail) to follow in my writing:


*

*With mathematical writing, as with any writing, think clearly about the story you are trying to tell. Consider why this might be interesting and ruthlessly remove even the most fascinating parts that do not push the story forward.

*Watch your language, watch for any words that have a use or meaning in mathematical writing, there are some surprises ("admits" for example) try to remove these, or justify why you need them.

*Do not be frightened to go slow, mathematical ideas we understand well can often seem trivial. Yet also respect your audience, tackle the big ideas. Remember that you are communicating the general ideas and not the specific technical details. 

*Be concrete, an example can often show an idea off well, and do not forget counterexamples, these can often be more informative than the ones that fit it.

*Finally, whenever possible, use a picture, they can allow people far deeper into the mathematical ideas than words alone. 


Good luck with your wonderful experiment.
A: I would say the following: Don't try to give "applications" for math you develop. Do not try to write the n + 1st book on "101 occurences of math in daily life".
I think, many authors fool themselves with the idea, people would get interested in math if just they had enough examples of real world applications. I think that is nonsense. If you see math just as a vehicle to solve problems, you are not really interested in math itself, you are interested in the results that math gives you and you don't care about how to derive them. Nobody cares about a science that just exists to help other fields. It's just something, people that suck in math want to talk you into: If they just knew, how they could USE all that math stuff, they would be interested on the spot. As if!
Also, don't underestimate your readers. Don't go too fast but be somewhat remanding. Math is not for everyone, but rewards people that are willing to put in some effort. A good book should reflect that.
So, what should a good book about math for non-mathematicians be like? Well, in my opinion it should not so much be about solving of practical problems. Rather, if it shall give a feeling of what math is like, it should introduce into mathematical thinking. In the MO thread "What book would you write if you just had the time", someone talked about a book about category theory for non-mathematicians. Though this may be a bit to heavy, I think this goes into the right direction:
Making people understand, how math can make them able to get a better understanding of the world, even without solving actual problems. Because that is what math is about in my opinion.
A: There is a discussion of some of these issues in articles on my web page on `Popularisation and Teaching'
http://www.bangor.ac.uk/r.brown/publar.html
and the 1989 article 'Making a mathematical exhibition' discusses the conclusions we came to after 4 years preparing an exhibition on `Mathematics and knots', which you can see as part of 
http://www.popmath.org.uk
The point was that we were using knots to say some things about mathematics to the general public; so you have to decide what it is you are trying to convey. 
The advantage of knots as a basis for discussion is that everyone can understand the basic ideas and problems. 
As I popular and important book which I think has not been mentioned on MO as an example of good writing, here is The Nothing That is: A Natural History of Zero 
Robert Kaplan (Author), Ellen Kaplan (Introduction) (1999). Grothendieck wrote to me in 1982: "The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps ..." (I had mentioned the resistance to the idea of groupoid.)  The story is interesting partly because of the resistance to the idea of nothing, which was associated in some minds with the devil, and chaos, and also in the way that a conceptual change had such profound results.  One reason for these results was that it led to a notation which better reflected the operations one wanted to do on numbers, particularly multiplication, and the ability to calculate insurance better led, it might be thought, to the prosperity  of Venice. All from counting the number of elements in an empty box! But more likely, from the mark of a thumb in sand as a place maker on an abacus, so associated with calculation. 
So part of the story of maths is that of  conceptual revolutions, which enabled difficult things to become easy, which of course then enables the practice of more difficult things, and that is one of the roles of mathematics. 
Many people want to hear of such conceptual revolutions, rather than of the solution of some problem famous for its difficulty, and perhaps also for the difficulty of understanding its significance for the general world. 
My own take on the importance of mathematics is that it develops rigorous languages for expression, description, deduction, verification, and calculation. There is also the notion of mathematical structure as a method of modelling the real world. 
A: *

*Work with a concrete example instead of in general or the abstract. Ash & Gross do a wonderful job of this in Elliptic Tales.

*Start with a question. (one which normal people might want to know about—not like a pure-number or pure-pattern question.) Tell me why I should care.

*Draw pictures.      

*Tell me what it's like, not just what it is. Plenty of facts that follow logically from definitions are not obvious (or, they're only obvious after X hours / days / years of concentrated thought). As an introduction, you could legitimately describe (for example) a category of smooth compact manifolds as "squishy", or a point-set category as "stippled", without being either dishonest or pedantic. A category of polytopes is "solid" whereas a conformal category "is flexible, but makes things line up".  (Being truthful with these adjectives pays too. Many people repeat the phrase "rubber-sheet geometry" for topology, but rubber is elastic, and AT invariants wouldn't be apparent with rubber whereas  Hooke's-law type behaviour would be.) Choosing good metaphors, like finding good mathematics, requires taste and skill.

*Tell me what it does for you, not what it is. "Compact manifolds help me think about more shapes than I could otherwise, without having to invent too many new tools." "Fourier analysis helps me think about waves."

*You can lie a little. (Especially if you follow it up with a remark like "The above isn't exactly true, but I hope it gives you the impression. See [ABC] for the real story.") The Bohr atom is a lie, and not a terribly bad one—especially situated between the plum-pudding model and the spdf model.

*Think about the material you're competing with for their interest. Look at the other books in the bookstore or library. Or the magazine rack, or television ….

*What is it emotionally that originally grabbed you about [whatever you're writing about]? A mathematical book is necessarily going to be cerebral to some degree, but a person is not a person without feelings. Emotional disingenuity makes prose ugly.

*Assume you don't have the reader's full brainpower. Assume they're distracted by other things in their lives and therefore won't pause at the end of a sentence to reflect on its obvious implications. (Here's a place where concision is bad: do some work for the reader; fill in what are the ramifications of what you just said. Ash & Gross again achieve a "Goldilocks" balance with respect to the problem of too many vs too few sentences.)

*Assume the reader might skip around, rather than reading sequentially.

*Never assign exercises. You're not the boss.

*More generally, the reader is the boss. You have to "earn" their time. You have to figure out what they're interested in and meet them there.

*Keep it short. More insights per minute = more respect for the reader's time. (I 'll draw the analogy to Edward Tufte's dictum in The Visual Display of Quantitative Information. Focussed, short documents are good. Information density is good. When we define that to be information meaningfully imparted (received and successfully processed) by the reader, per word.)

*Variables can be words rather than letters, and equations can be paragraphs. base^exponent is better than m^n (especially three pages later when you refer back to m). And whilst the quadratic equation is simpler as a formula, the Birch/Swinnerton-Dyer conjecture would be easier understood as a paragraph than as .

*Jargon is your enemy. Nobody will understand phrases like "complex simple ____". The need to rephrease this is an invitation to think more deeply through what you would normally take for granted.

*Omit needless words (like "interesting"), but do include transition sentences and overview sentences.

*Read style guides and good non-mathematical authors. Mathematics has a lot, lot, lot of substance to offer the general public. (As opposed to other factoids they might read, which stick to the ribs as much as a Jolly Rancher.) That's a strength you get for free just by the selection of topic, and the work already done by researchers. But the "substance over style" culture has downsides. In particular it's the main reason most people hate mathematics (they never penetrate through the style to suck out the juicy substance).

*Pretend that every time you write a sentence, you are also implicitly saying "This is important". If you wouldn't say "This is important", then the idea can be sent to a footnote or appendix, or excised.

*Il semble que la perfection soit atteinte non quand il n'y a plus rien à ajouter, mais quand il n'y a plus rien à retrancher. ―St-Exupéry, Terre des Hommes, 1939

*Play around.


That's a lot of advice, and while I may not be qualified to give it, I think you'll find I've merely collated the advice of people who really do know what they talk about.

Added: Bill Thurston, Three-dimensional geometry and topology, reader's preface:

The most efficient logical order for a subject is usually different from the best psychological order in which to learn it. Much mathematical writing is based too closely on the logical order of induction in a subject, with too many definitions before, or without, the examples which motivate them, and too many answers before, or without, the questions they address.

A: Well, here are some things to avoid when writing popular math:


*

*Journalistic gullible – there is nothing worse than reading pages and pages empty of meaning;

*Treating the readers as mentally retarded people – it’s insulting. As they say: things should be explained as simply as possible, but not simpler!

*Being afraid to challenge the readers – if they read a math book, they expect it;

*Talking to the reader from above – avoiding the “professor” tone.


I hope this helps.
