TITLE: How to write popular mathematics well?
QUESTION [74 upvotes]: 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?

REPLY [10 votes]: 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.