Cocktail Party Physics

love among the equations

[NOTE: This post originally appeared at our new home at Scientific America.]

Four years ago on this date, the Time Lord and I officially tied the knot. I wrote the piece below last fall, as The Calculus Diaries was coming out, but it didn't really seem to fit anywhere --too "math-y" for the mainstream, too intensely personal for your average science publication, and honestly, still kind of a work in progress. But in the spirit of the blog as "writing lab," it seems appropriate to post it here, on our fourth anniversary, as a way of saying thanks to the man who irrevocably changed my life ... for the better. Here's to many more years to come.

Shortly after becoming engaged, my now-husband and I drove from a conference in San Francisco to our new home in Los Angeles via the scenic route along the Pacific Coast Highway. At sunset, we stopped briefly to refuel just north of Malibu and found ourselves admiring the brilliant orange, red, and purple hues stretching across the darkening horizon, savoring the peaceful sound of ocean waves lapping against the shore. Against this idyllic Hallmark moment, Sean put his arms around me, pressed his cheek to mine, and gently whispered, “Wouldn’t it be fascinating to take a Fourier transform of those waves?” A

Fourier transform is a mathematical equation that takes a complex wave of any kind – water, sound, light, even the gravitational waves that permeate the fabric of space time – and breaks it down into its component parts to reveal the full spectrum of “colors” that are otherwise hidden from human perception.

Another woman might have been taken aback by Sean injecting a bit of cold hard math into the warm hues of a romantic ocean sunset – talk about over-analyzing the scene and spoiling the mood! Me? I found it charming, yet another intriguing color in the spectrum that makes up this multifaceted man with whom I have chosen to share my life. My husband is a theoretical physicist. He spends his days pondering big questions about space, time, and the origins of the universe.

It’s not just Fourier transforms that lurk in the nooks and crannies of our marriage. Our pillow talk includes animated discussions about Boltzmann brains, the rules of time travel, poker, phase transitions, and the possibility of a multiverse: the notion that there are an infinite number of universes out there, beyond our ken, perhaps containing carbon copies of ourselves – the same, and yet somehow different. I have issues with this concept, especially when I’m sleepy: all those universes filled with doppelgangers cluttering up the landscape just strikes me as crowded and untidy. But Sean wrestles with these questions all the time, and is adamant in his defense. “It’s infinity,” he reassures me. “It’s not like we’ll run out of room!” I guess the multiverse has unlimited storage space.

I wasn’t looking to fall in love, and never imagined I would be a wife. Years of failed relationships had convinced me that I had no gift for making love work. My romantic calculations seemed doomed to failure, always slightly off, never quite yielding the right combination, no matter how intricately I manipulated the numbers. By the time Sean entered my orbit, my heart had been broken into little pieces and reassembled so many times, I was convinced the telltale cracks would never fully heal. I gave up on dating, buried myself in work and told myself it was better this way. I built a thick wall around my heart and guarded the perimeter zealously.

Love stole back into my life, ninja-like, while I was looking the other way. Sean is a scientist, and I am a science writer, but our day-to-day lives were like parallel lines that never met. Our paths didn’t cross until we discovered each other’s blogs online. We quickly formed an online friendship, both recognizing a kindred spirit across the vast expanse of Cyberspace. Two months and many emails later, we arranged to meet over dinner at a physics conference in Dallas.

Physicists are often unfairly characterized as absent-minded geniuses, socially inept, with zero fashion sense, a la Sheldon on The Big Bang Theory. It's an exaggeration, but there is a tiny element of truth to that. So I was pleasantly surprised when a tall, lanky man with boyish good looks and an engaging smile appeared in the hotel bar, sporting jeans and a casual-yet-chic jacket. This was not your stereotypical physicist.

He ordered a martini. “I’d like to taste the vermouth,” he instructed the bartender. (He is a man who takes his cocktails seriously.) We chatted about science, art, music, and books, with the odd foray into personal details and more philosophical musings. A first date is usually fraught with self-conscious anxiety, as each person strives to present only the most flattering colors in their personal spectrum -- preferably through a soft-focus lens. But we had an instant rapport, an easy familiarity from our electronic exchanges that translated effortlessly into “meat space.” By the end of the evening, I was smitten, and happily, the feeling was mutual.

We defied the geographical distance, racking up countless frequent flyer miles. Six months after that first encounter, he proposed, and a year later, I found myself married and living in sunny southern California. I felt as if I’d stepped into an alternate universe where the calculations of love had finally worked out in my favor. I had become my own doppelganger.

With my new life came a new appreciation for the secret language of scientists: mathematics. Like many people, I had steadfastly avoided all things math since high school. My eyes glazed over at the merest glimpse of an equation. I was convinced it was irrelevant to my life – or at the very least, unnecessary. But now that life featured a man who left technical papers scattered about the house, filled with mysterious symbols that might encode the secrets of the universe. Our living room boasted a white board with a constantly changing parade of scrawled equations, and our groaning bookshelves now included massive tomes on quantum mechanics and general relativity.

The deep, technical aspects of his work was the one part of Sean’s life that was truly closed to me, although as someone who writes about physics for a living, I certainly grasped the basic concepts -- far more than the average non-physicist. But if I wanted to appreciate the full spectrum of the man I’d married, I would have to learn a little bit more of his language. So I resolved to overcome my longstanding kneejerk rejection of all things numerical and teach myself the basics of calculus.

Sean was patience personified during my quest, explaining basic concepts, leaving practice problems on our white board every morning for me to solve, and artfully dodging the occasional bit of metaphorical heaved crockery whenever I hit a frustrating obstacle (“Integrate that!”). The frustration was real: Our communication gap when it came to math was a yawning chasm at the outset. Often I didn’t even know how to phrase my questions in a way he could comprehend.

Slowly, surely, that gap began to close as he helped me see that equations were all around me. We found calculus in the rides at Disneyland, and the exquisite architecture of Antoni Gaudi. We went to Vegas, learned to shoot craps, and Sean tutored me in the calculus of probability (and a spot of game theory for good measure). Even our quest to buy a house became fodder for exploration.

It turns out that the world is filled with hidden connections, recurring patterns, and intricate details that can only be seen through math-colored glasses. Those abstract symbols hold meaning. How could I ever have thought it was irrelevant? This is what I have learned from loving a physicist. Real math isn’t some cold, dead set of rules to be memorized and blindly followed. The act of devising a calculus problem from your observations of the world around you – and then solving it – is as much a creative endeavor as writing a novel or composing a symphony. It isn’t easy, but there is genuine pleasure to be found in making the effort.

As with mathematics, so with love. There are no hard and fast rules to be blindly followed, no matter what the self-help gurus may tell you. Sometimes you just need to take a Fourier transform of yourself, shatter the walls and break everything down into the component parts. Once you’ve analyzed the full spectrum, you can rebuild, this time with just the right mix of ingredients that will enable you finally to combine your waveform with that of another person.

Does mathematically analyzing a sunset, or the ocean waves, make either any less romantic? Not to me. It only enhances my sense of wonder. When we listen to the rhythmic cycle of waves crashing on the shore, we can hear those waves because our brains break apart that signal to identify the basic “ingredients.” And every time we gaze at a sunset —a spectacular orange-red, or a soft pinkish glow—our brain has taken a Fourier transform so we can fully appreciate those hues.

I will never listen to ocean waves or view the setting sun in quite the same way again. I looked out over the water that evening and saw a picture-perfect ocean sunset, but there was so much more that I missed. Sean looked out onto the same scene and saw the rich complexity of nature expressed in mathematical symbols, the fundamental abstract order lying just beneath the surface. And through his eyes, I can now catch a glimpse of that hidden world -- proof that love can transform you just as surely as the Fourier equation transforms a seemingly simple ray of white light into shimmering technicolor. Happy anniversary, Time Lord!  

UPDATE: Was running around doing anniversary stuff all day yesterday, but as a commenter points out, I failed to identify xkcd, Randal Munroe's brilliant Webcomic, as the source for the two cartoons. Usually I link to image sources somewhere in the text, but failed this time. Although if you didn't recognize the source, you really should be reading xkcd on a regular basis. He updates three times a week. Go! Read him! Wedding photo by Jen Kerker Photography. And the video -- for those who didn't click through to YouTube -- was an award-winning entry to a UK jobs site ad campaign, believe it or not: reed.co.uk's "Love Mondays" series.

i think that i shall never see / a carbon offset as lovely as a tree

I overhead an exasperated parent the other day:  “The kid just won’t stop asking ‘why’ all the time.  Everything I say, she challenges.  She’s driving me absolutely nuts.”

We need more kids like that.

I know, easy for me to say since I’m childless by choice, but I really do believe that the biggest danger we face as a society is a populace that doesn’t want to think for themselves.  I am more than happy to have students challenge me about the material I teach.  In reality, though, they spend a lot more time challenging me on my attendance policy, my lack of understanding that they missed class due to a hangover, my grading policy, and my refusal to allow ‘do-overs’ for tests that didn’t produce the desired scores.  Most disppointingly, so many test answers are phrases regurgitated directly back from my notes with no evidence that the writer spent even a few seconds considering what those words meant and whether they were right.

While professors hate this trend, marketers but be thrilled.  Someone tried to argue the other day that the Tesla Model-S electric car was actually cheaper than a luxury gasoline sedan.  He told me that the New York Times said that the $57,400 list-price Tesla Model-S was actually only $35,000 "when you accounted for tax credits and gas savings". 

Sure enough, there was Elon Musk saying that the Tesla sedan would cost $57,400 list price, $49,900 after tax credits, and $35,000 “after factoring in gas savings”.   Anyone have a bell go off there?  A 'one of these things is not like the others' moment?

You can’t directly compare the first two items with the last.  There is a difference between one-time costs (purchase price and tax credits) and the more-nebulous issue of "gasoline saved" because the latter depends on how much you drive.  It will vary from person to person and it's money you get back, not money you don't have to put out in the first place. 

Probing further, I found an actual quote on a GM site about the Chevy Volt, which quotes Musk as saying.

“The ownership cost of Model S, if you were to lease and then account for the much lower cost of electricity vs. gasoline at a likely future cost of $4 per gallon, is similar to a gasoline car with a sticker price of about $35,000. That’s why we’re positive this car will be the preferred choice of savvy consumers.” 

Arg.  Now we’ve got leasing thrown in and I have no idea how they calculated the cost of the electricity  used to charge the car.  Inquiring minds want to know.  Or, at least, they should.

Goodyear does a pretty nice job in advertisements for low rolling resistance tires.  They tells us that we’d save 2600 miles worth of gas over the life of the tires.   An asterisked statement tells us that this estimate includes a lifetime of 65,000 miles and a 4% increase in fuel efficiency.  They even provide a calculator for you to input your car's mileage so you can see how much money you’d save.

During interviews for my Physics of NASCAR book, I interviewed a Todd Meredith, a manager at Joe Gibbs Racing.  I always ask whether a race shop has any employees with a physics background, since most are mechanical or aerodynamics engineers.  Todd thought for a moment and said,

“Yeah, I think we do… but I bet he doesn’t use anything you taught him.”

“I bet he does,” I shot back, “because we taught him how to think.”

That started a really productive and interesting conversation about the importance of science and math and how they are taught. In the end, racing is real-time advanced problem solving, and all the skills you learn as a scientist, engineer or mathematician are exactly the skills you need to be part of a race team.

Fundamentally, science and math are about learning how to evaluate data to ensure it’s valid, synthesizing multiple data sources, critically analyzing that data and coming to conclusions. We need these skills more now than at any other time in history.  Climate change, energy, manned vs. unmanned space programs, oil well leaks…  even if you don’t understand the details of the science, you HAVE to be able to understand if you’re getting the truth or being sold a load of goods.

It’s hard enough for scientists to find ‘The Truth”.  There are significant disagreements on directions for action in many areas (climate change, transportation, etc.) because we simply don’t have enough data to make unequivocal conclusions.  Although there have been some issues with advocacy getting in the way of science, that’s nothing compared to the number of people who are purposely twisting science to their own goals.  If Shakespeare were alive today, I suspect he would have chosen "public relations persons" rather than lawyers as his occupational target of choice.

The honorable people in marketing and public relations bring to light the most positive accomplishments of their company.  There are those who are less honorable.  How much "news" is actually a minimal re-write of a press release?  Some science "news" sites are simply places for universities and companies to post their news releases, and some science journalism is little more than a thin re-write of said news release.  Science is not the only place this happens.

I heard a NASCAR broadcaster make the following claim: "NASCAR offsets 100% of the carbon emissions from this race via their tree-planting program."  Anytime someone says "100%" or "always" or "without exception", my ears perk up.

OK, NASCAR has been (and is) doing a lot of good things to be environmentally responsible.  They were recycling used oil before it was cool, they are initiating recycling programs at tracks, even getting teams to minimize how much they run generators during race weekends.  All good things, but not very exciting.  Newspapers don't want articles about recycling oil.

But tree planting… beautiful PR idea.  Get your biggest stars out to dig the holes, put the trees at a local school or park.  Bingo.  So how many trees do we have to plant?

Some brilliant NASCAR PR person did major damage to the cause of logic and educating the public by deciding that they would plant ten trees for every “green” flag during a race.  Green flags wave at the start of the race, and after every caution, so the number of green flags depends on the number of accidents and amount of debris on the track. 

This is just wrong on so many, many levels, and it's not like this is difficult science.

Combusting one gallon of gasoline produces 19.4 pounds of carbon dioxide.  One gallon of gasoline weighs about 7 lbs – the additional mass making up the carbon dioxide comes from the oxygen that combines with the gasoline during combustion.  Two octane molecules combine with 25 oxygen molecules, which explains how the weight of carbon dioxide is greater than the weight of the fuel. 

But the important point here is that the amount of carbon dioxide emitted is directly proportional to the number of gallons of gasoline combusted.  Not only does this have absolutely NOTHING to do with how many green flags wave during a race, it’s a number we can calculate with pretty reasonable precision.

Forty-three cars run 500 miles, each getting about 4 miles per gallon.  I am overestimating the amount of fuel used by just a little:  the cars get about double the gas mileage during cautions and some cars don’t run the full race.  I’m calculating the upper bound. It wouldn't be hard to use the data on exactly how many laps were run by each car, but I'm not the one claiming I'm planting enough trees to offset the emissions, so I'm making an estimate.  When you do the calculation, it's about 5,375 gallons of gas in a 500-mile race.

I know – you’re horrified at this tremendous waste of gas.  Get over it: In 2009, the Department of Energy says the US consumed 137.93 billion gallons of gasoline. That means this country uses 4,373 gallons of gasoline every second. I guarantee you I could easily find enough U.S. NASCAR fans who would volunteer to use less gasoline in their everyday lives to compensate. Compare a weekend of NASCAR (including the media coverage and fans getting to the track) with a weekend of the NFL. You can make a better argument for saving fuel and emissions by eliminating a weekend of NFL football than you can a weekend of NASCAR.

Nonetheless, my hypothetical NASCAR race still uses 5,375 gallons of gasoline, which means that 104,275 pounds of carbon dioxide (about 52 tons) are released into the atmosphere.

One of the first things I remember learning about science is that people breathe in oxygen and breathe out carbon dioxide, while plants “breathe in” carbon dioxide and emit oxygen.  Photosynthesis combines water, carbon dioxide and energy from the sun to produce sugar and oxygen gas:

Six carbon dioxide molecules and six water molecules combine to produce one sugar molecule (the C₆H₁₂O₆) and six molecules of oxygen.  Plants covert the sugar into organic matter like stems, leaves and stalks.  

The problem with plants is that they lose their leaves (and, if they live in my house, they die).  The real masters at storing carbon are trees, because trees convert sugars into cellulose -- long chains of the repeating unit C₆H₁₀O₅.  As long as the tree is growing, it’s sucking up carbon dioxide.   Since each of us generates about 2.3 tons of CO₂ each year, we should all be doing out part to encourage trees to grow.

Now for the tricky question:  How much carbon dixoide does a tree absorb?  Turns out that's not as easy a question to answer as I thought.  You can find numbers ranging anywhere from 10-70 lbs of carbon dixoide per tree per year.

Let’s analyze this.  Since carbon dixoide is mostly stored as wood, the amount and type of wood are going to be important.  In other words, when it comes to trees and carbon sequestration, size matters.

Assume a cylindrical tree.  The volume of the tree goes like the diameter of the tree squared times the height.  Compare two trees of the same species the same height, but diameters that differ by a factor of two.  The one with the larger diameter stores four times as much carbon dioxide (pi r squared!).  The difference in storage is even larger, of course, because trees (unlike people) continue to grow upward, as well as getting larger around the trunk as they age.

Trees with denser wood store more carbon dioxide than trees with less dense wood.  Next time you’re at the home improvement store, compare trim made from pine with that made from oak or cherry.  I’ve summarized the mean densities of a couple of different species of trees on the graph below, which shows the height of the tree on the vertical and the type of tree on the horizontal axis.  Sorry about the labels being so small.  From left to right, they are: balsa, red pin, red oak, cherry, ebony and lignum vitae.  The densities are given in kg/cubic meter and range from 160 to almost 1400.  Red pine has a density of 370 – 660 kg/m³, whereas ebony (a hardwood) has a density of 960 – 1120 kg/m³.  The champ in terms of tree density is a tree called Lignum Vitae, which has a density of 1280 – 1370 kg/m³.

 

 

I noted that the larger the tree is, the more carbon it has stored.  So you might think big trees are the best to plant.  But what's really important is how fast the tree is growing - how fast it can convert carbon dioxide into wood.  Big means that the tree has stored a lot of carbon.  Fast growing means that the tree is storing a lot of carbon 

A tree grows slowly in the early years (while it's putting down roots), then has a period of more rapid growth, and finally tapers off as it approaches its maximum height, as I’ve diagrammed below.  The region of maximum growth is at the steepest parth of the curve, so you want to plant trees that are near the maximum in the derivative of their height vs. time curve.  Unfortunately, most nurseries do not put this data on the little tags that hang on the trees when you go to buy them.  Forestry types tell me that Southern Pine may reach its maximum growth period in 20-28 years, while Douglas Firs on the Pacific Coast require 60 years.  Things are a little more laid back on the Left Coast.

Speaking of laid back, consider also that tropical trees account for 95% of all tree-based carbon dioxide sequestration on Earth.  A tree without leaves isn’t doing much photosynthesis, which means it’s not removing carbon dixoide from the atmosphere.  Tropical trees work 12 months of the year, while boreal trees only work 3-6 months of the year.  Tropical trees are mostly hardwoods and grow more quickly than their cold-weather relations, so there are a lot of people who advocate that, if you're going to support tree planting, you should send money to oragnizations that are trying to re-plant tropical rain forests because those trees are more likely to have a larger impact than the pine trees you plant at a local park in New York.

There are some ways of estimating how much carbon dioxide a tree will take in each year, like determining the mass of the tree based on its dimensions.  One is a really great exercise for middle or high school students.  They find that:

• A 10 year old Grevillea robusta (the southern silky oak, an evergreen Australian tree) that is 45 feet tall and 6 inches in diameter would sequester about 64 lbs of carbon dioxide per year.
• A newly planted Acacia angustissima (native to Central America and the US) at 2.5 yrs, 15 ft tall and 3” in diameter would take up about 21.5 lbs of carbon dioxide per year.
• Calliandra calothyrsus (powder puff tree, Mexico, Centra America) that is 10 years old, 15 feet tall and 8” in diameter would remove about 65 lbs of CO₂ per year.

In general, trees planted in the US have a difficult time reaching the numbers tropical trees can achieve.  Most people, noting that environment, rainfall and other conditions matter, will use an estimate that a tropical tree can absorb an average of 50 lbs of carbon dioxide per year.   "Tropical" usually means within 23 degrees North or South of the Equator, but let's count Florida as "tropical".

Let’s be generous and use the 50 lbs of carbon dioxide per tree per year number, which is probably 1.5 to 6 times larger than reality.  To offset the 104,275 lbs of carbon dixoide from the NASCAR race, you would need 2,085.5 year-trees.  "Year-trees" means that the product of the number of trees times the years the trees are sequestering carbon needs to equal two thousand and eighty five.  You could plant 2,085 trees and they will compensate for the race in one year.  Or you could plant a thousand trees and compensate in just over two years.

Last year’s Daytona 500 had nine cautions, so including the green flag that started the race, there would have been 10 green flags and NASCAR planted 100 trees.  The calculations above show that those 100 trees will offset the carbon emissions from the race in just under 21 years.  And I'm being really generous with giving each tree credit for 50 lbs of carbon dixoide per year.  If that number is 25 lbs per tree, we're talking more than 40 years before all the carbon from those five or six hours last February have really been accounted for.  (And yes, I do know that the race went 520 miles due to multiple green-white-checkers and I didn’t add in those extra 20 miles, which is another 215 pounds of carbon dioxide, and ignoring that probably accounts for the laps under caution and the cars that dropped out before the race was over.)

That estimate doesn't include the carbon emissions needed to take care of the trees for that long:  fertilizers, vehicles, replanting trees that die, etc.  

 In 2009, Petaluma Junior High School received 30 trees from NASCAR and Infineon Raceway to offset the carbon emissions from the race in Sonoma.  Sonoma is a road race, so it is shorter (224 miles) and we’re “only” talking about 2,408 lbs of carbon dixoide.  The kids from that junior high school are going to have their own kids in junior high school before the carbon from that race is offset.

Offsetting carbon emissions from planting trees is not the panacea it has been advertised to be.  One website calculates that, in order for the Earth to become carbon neutral, we would have to reforest a land area approximately equal to Spain every year and maintain that land in perpetuity.

The small impact NASCAR's trees will have on the environment is at least good; however, their decision to pursue - and publicize - a non-scientific approach to the problem is grievously wrong.  I'm offended  NASCAR thinks I'm stupid enough to fall for this. 

I'm also a wee bit irritated at some members of the "NASCAR media" who are willing to parrot the 'facts' they are handed without question.  I don't expect them to be experts, or even have time to call up a scientist to ask whether it's accurate. 

The fact of the matter is that you get credit for sequestering carbon as it is sequestered, just like you get credit for paying down your mortgage as you make the payments.  You don’t own the house by taking out a mortgage and you don’t offset your carbon emissions the minute the trees are in the ground.

As I tell my students over and over again:  If you don't understand something, the last thing you should do is repeat it.

Neat trivia I learned while researching, but couldn't slip into the blog:  Forestry professionals use the abbreviation "dbh" when describing trees.  dbh means "diameter at breast height".  Breast height is defined to be 1.4 meter.  There are apparently very strict physical requirements for being a forrestry professional.

 

myths are bustin' out all over

Exciting news broke earlier this week, at least for fans of Discovery Channel's Mythbusters (and oh yes, Jen-Luc Piquant is a mega-fan!). President Obama will make a special appearance on the December 8 episode of the series -- part of ongoing efforts of the administration to promote science, technology and engineering (STEM) education, starting with the Educate to Innovate campaign launched in 2009. In fact, he made the announcement during the first ever White House Science Fair. We are currently taking bets on how quickly Faux News and its noisy acolytes will start braying about how all this "promoting science to a broader audience" is really just a commie/Socialist plot to forcibly redistribute the wealth knowledge to the undeserving Ignorant. Or Muslims.

No doubt adding fuel to the fire, the episode in question will revisit the "myth" of the Archimedes "death ray." For those who don't recall the story, the brilliant Greek mathematician, Archimedes of Syracuse, was also known for building ingenious weapons of war to defend Syracuse from the invading Roman army. There was, for instance, a giant crane capable of capsizing ships, known as the Claw of Archimedes. Another such invention, legend has it, was a large curved parabola-shaped array of mirrors capable of collecting and focusing the sun's rays onto the Roman ships moored in the harbor, laying siege to the city. The heat caused the ships to catch fire and burn, and Syracuse was saved -- for awhile, at least. (Eventually the Romans overcame the city's defenses, and Archimedes was killed in the ensuing chaos. Set mathematics in Western Europe back a good 700 years, at least.)

Long before the Mythbusters appeared on TV, folks were trying to ascertain the validity of that legend. Skulls in the Stars has a classic post detailing the history of such attempts to test similar devices, usually with mixed results. Back in the 18th century, the noted naturalist, he Comte de Buffon (who also devised the "Buffon's Needle" puzzle) assembled an array of ordinary mirrors (40 in all) and managed to set a log pf tarred beechwood on fire from a distance of 66 feet. The more mirrors he used, the more effective the technique was at setting fires from a distance. With 128, he could set fire to a plank of tarred fir from a distance of 150 feet.

Thre was also an article by one John Scott in the late 19th century assessing the evidence for and against the effecitveness of an Archimedes "death ray." It appeared in the Proceedings of the Royal Society of Edinburgh. Scott was more skeptical than Buffon, noting that historical accounts closer to the time of Archimedes say nothing of burning mirrors, Livy and Plutarch among them. But it's a fun story, and I used it to talk about parabolas and finding the area under a curve in the first chapter of The Calculus Diaries. Some legends are worth repeating, whether or not they turn out to be true.

The Mythbusters have already tackled this challenge twice already: once on their own, and the second time with the help of team of scientists from MIT. Conclusion: it's most likely a myth, although in principle it's feasible. The MIT experiment managed to start a small fire on a wooden ship, although it quickly burned out. Considering the time it would take to set fire to a ship using such a technique -- think of how long it took when you, as a kid, tried to set a piece of paper on fire with a magnifying glass on a hot summer day -- flaming arrows would probably be more efficient. But I guess President Obama asked them to revisit the challenge. Why? Who knows? Maybe he's keen on getting a nifty death ray for the White House. (Cue mass hysteria from the paranoid fringe!)

Personally, I'd like to see the Mythbusters tackle a related challenge: the purported "death ray" that strikes poolside at the newly built Vdara Hotel in Las Vegas. It's part of the City Center complex, and the building has a distinctive parabola-like shape. Therein lies the problem. According to recent news reports, a vacationing lawyer was relaxing poolside at the Vdara, when he started to feel very warm, and then smelled something burning. It was... his HAIR! HIS HAIR WAS SMOKING! He jumped up from his seat and doused his head in the pool, then repaired to the bar for a stiff drink. The bartender nodded knowingly when he descibed his plight: "Yeah, we call that the Death Ray." The Las Vegas Review Journal published this helpful schematic to illustrate the principles at work:

I guess there was a reason no one was sitting in what would otherwise be a prime poolside seat. When the lawyer went back to retrieve his newspaper, he found the plastic in which it was wrapped had melted.

But I'm just a tad bit skeptical. I mean, check out this photo of the alleged newspaper:

It clearly spells out the word "VDARA." I smell a hoax. How did the sun's rays manage to carve out just those letters? Was there a "stencil effect" at work, i.e., over one of the curved windows? Inquiring minds need to know! And the Mythbusters are known for their inquiring minds and ingenious experiments. We eagerly await their findings.

UPDATE: Several commenters -- thanks, guys! Knew I could count on you! -- pointed out that the plastic bag itself was stenciled with black letters,the black absorbs the sun's heat faster, and hence the bag melted in just that pattern. Science! Also? We have been called "fluffy" in the comments section. We consider this a compliment. Jen-Luc Piquant humbly suggests that if you're looking for a detailed explication of this effect, with fancy diagrams and equations and all, a blog that proudly calls itself Cocktail Party Physics probaby isn't your best bet. Do that sort of thing at a cocktail party and you'll soon find yourself alone in a corner, doodling on a napkin and talking to the catering staff (who are paid to be there and already bored), while the other guests avoid you like the plague. Just sayin'. (Except if it's a cocktail party with physicists, in which case it's good form to provide a white board.) However, you can find just that sort of thing over at the most excellent blog, Dot Physics (formerly of SEED Science Blogs, now housed at Wired), and we thank said commenter for the link. Check it out!

 

heads will blog... about calculus

If you're wiped out from reading yesterday's monster post on poker-playing physicists, and just want to watch/listen to something instead as a bit of a break, you can hear me chat with George Johnson over at Bloggingheads.tv about all things calculus. It's all in there: Archimedes and the method of exhaustion (pioneered by a Greek dude named Eudoxus, who studied under Plato), why killing Archimedes was "the Romans' greatest contribution to mathematics" (h/t to fellow science writer Charles Seife), the poolside "death ray" at the new Vdara Hotel in Vegas, zombies, and Newton's bitter fight with Leibniz over the title "inventor of calculus." Poor George could hardly get a word in edgewise. What can I say? I was feeling chatty. About calculus. Me, the former English major. Because the universe has a wicked sense of humor. Besides, ask any writer who's just finished a book to expound on the subject of said book, and you're bound to get an earful -- it just pours out of us, like lava (or Exorcist-style vomit). If you've got some time this weekend, give it a listen.

 

change and motion

It's here! Today is the official publication date for The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse -- two and a half years of hard work finally comes to fruition. There are zombies, of course, and rollicking good tales about probability theory in craps, surfing in Hawaii (more than you ever wanted to know about the Fourier transform), and ferreting out parabolas and vectors at Disneyland. It's peppered with colorful historical math-y characters. There's even a couple of appendices with actual, you know, equations and stuff, for those readers who crave something a bit mathier, while still making sure the book appeals to my fellow math-phobes. I like to think of The Calculus Diaries as a gateway drug to the hard stuff. Sure, you think you're just taking a light hit, reading about catenary curves and architectural arches, but soon the Math Monkey is on your back, and you're staying awake for three days straight, solving differential equations and obsessing over whether P = NP. Not that I would know anything about that. I just sell the stuff.

Calculus is all about change and motion, and this seems as good a time as any to announce another impending change. For the last two years, I've been director of the Science & Entertainment Exchange, a fledgling program of the National Academy of Sciences to foster creative collaborations between scientists and the entertainment industry. We've had a very successful first two years, and all that hard work is beginning to bear some tangible fruit -- I'm excited about the potential of the program as it moves into Phase II. But I won't be actively involved for that phase. Effective September 24th, I'll be stepping down as director to return to writing full-time. The new director will be Marty Perreault, an industrial engineer by training who's been working at the Academy for several years, most recently heading up their President's Circle program, and recently relocated to Los Angeles.

This was not an easy decision. The short version: I'm exhausted. There just aren't enough hours in the day. I managed to write The Calculus Diaries on evenings and weekends while running the daily operations of the Exchange, but it's been tough juggling two very demanding, very different kinds of work. It became abundantly clear that I would have to choose between them. Everyone recites the mantra, "Follow your heart!", but what if your heart is divided between two equally amazing options? Still, as fulfilling as the Exchange has been, when the chips are down, writing wins out in the end. I miss being able to regularly delve deeply into nifty science, and am eager to get back in the writing game with renewed vigor. I'll still be tangentially invested in the future fortunes of the Exchange -- just on an informal, watching-from-the-sidelines basis, cheering the program on.

The next couple of months will be far from restful, as I wind down my tenure as Exchange director and work to promote the new book; you can keep track of various book-related appearances here. For those who care, I also set up a separate page on Facebook for The Calculus Diaries, making sure the feed isn't just all promo, all the time -- because don't you just hate that? There's a generous sampling of fascinating math-bits gleaned from all over the Interwebs as things catch my fancy. A sampling of cool links is below, just to whet your appetite for more things mathematical.

The Mathematics of Viking Jewelry. The beautiful bracelets and necklaces made by Viking artisans from rods of gold and silver are all twisted together into double helices. That's true regardless of whether the jewelry was found in Ireland, Scotland, the Orkney Islands or Scandinavia -- an impressive degree of regularity. Is it just a coincidence? Or could math be to blame? A couple of mathematicians at Denmark's Technical University think they've cracked the case in a paper recently submitted to the arXiv: Kasper Olsen and Jakob Bohr say that two wires become maximally twisted when no more rotations can be added with deforming the double helix. You guessed it: Viking jewelry is maximally twisted.

Mathematicians: Behind the Music. Forget mad scientists, how about mad mathematicians? Adam (Paco) Hanlon brings a refreshing dose of snark to this historical overview of some of the more colorfully insane math-y sorts during Nerd Nite NY. You gotta love any talk with slides captioned, "David Hilbert: Trash-Talking Mathematician Badass." (If Hilbert were alive today, I bet he'd put that on his business cards.)

 

The Mathematics of Balloon Animals. Man, there's a math paper for everything these days. In "Computational Balloon Twisting: The Theory of Balloon Polyhedra," the authors introduce the idea of ‘Bloons’: mathematical idealisations of real-world balloons (eg a doggie balloon, pictured with its associated 'bloon' below). Per Improbable Research: "The research not only provides '…algorithms to find the fewest balloons that can make exactly a desired graph or, using fewer balloons but allowing repeated traversal or shortcuts, the minimum total length needed by a given number of balloons.'”

A Math-y Take on the Perfect Roller Coaster Loop. Physics Buzz reports that Swedish mathematicians have used the same equations that describe how the planets orbit the sun to design a bunch of roller coaster loops that would give riders the visual experience of a loop without any whiplash. Good times!

Tasting the Limit, with Pie! Back in March, Ethan Siegel of Starts With A Bang took a break from blogging about astrophysics to pump out a tasty twist on infinite series and the limit -- illustrated with yummy photos of various kinds of pies. It's a fun explication, but it might make you hungry for something sweet. Forewarned!

Sunday with the St. Petersburg Paradox. Matt Springer of Built on Facts has a regular feature called the Sunday Function that quickly became one of my favorites. In this post, he takes a look at the St. Petersburg Paradox: a hypothetical gambling scenario where you win money based on the outcome of a coin toss. Per Matt: "Play this game enough times and it doesn't matter how much it costs to play, you will certainly come out ahead in the long run." There's a catch, of course: you have to play for, like, infinity. But it's a fascinating discussion, all the same.

Futurama Unveils Prisoner of Benda Theory. Psst! Did you know that Futurama writer Ken Keeler has a PhD in mathematics? Well he does! And he put his training to good use a couple of weeks ago, devising (and proving) a theorem based on group theory to explain a plot twist in the episode “The Prisoner of Benda.” (APS News leaked a spoiler back in May in an interview with executive producer David X. Cohen.) Basically, the Professor and Amy use a new invention to switch bodies. But the same two brains can’t switch twice, so they need an equation to prove that, "with enough people switching, eventually everyone will end up in their rightful form." Also? Most disturbing animated sex scene ever. Just sayin'.

NP Hard Problems Frustrate Hitler. Earlier this month, the Internet was abuzz with a new paper claiming to have solved the P vs NP conundrum. It's an esoteric math question relating to the speed at which a computer can accomplish a task such as factorising a number (eg, the Traveling Salesman problem). If P = NP, every problem that can be checked quickly can also be completed quickly. If it doesn't -- as the new proof claims -- well, that could have huge repercussions for internet security, since we keep our data safe from hackers using factoring of very large numbers. But it has an even bigger impact on Hitler, who needs to find the best route to get a KFC DoubleDown in Reichstag and doesn't have eternity to wait. ("Everyone who failed Theory of Computation, leave the room!")

Cracking Plato's Mathematical Code. Science historian and philosopher Jay Kennedy claims that Plato understood the mathematical structure of the universe quite well, 2000 years before Newton and Galileo. He bases this conclusion on analysis of Platonic texts using stichometry: the measure of ancient texts by standard line lengths. Per the Guardian: "What he found was that within a margin of error of just one or two percent, many of Plato's dialogues had line lengths based on round multiples of twelve hundred. The Apology has 1,200 lines; the Protagoras, Cratylus, Philebus and Symposium each have 2,400 lines; the Gorgias 3,600; the Republic 12,200; and the Laws 14,400." Paging Dan Brown! I smell a sequel to The Da Vinco Code!

Herbie Hancock Says Do the Math! What's the secret to a successful music career? Herbie Hancock insists it's math and science, in this LA Times exclusive interview. A taste: "I've always been interested in science. I used to take watches apart and clocks apart, and there's little screws, and a little this and that, and I found out if I dropped one of them, that thing ain't gonna work. When I was a kid, I put things back together and they never worked anyway! But just, like, going into those details, it's kind of a scientist's thing. And I have that kind of [mind], it's part of my personality.... So when synthesizers came in, they used terminology I knew. I knew what an amplifier was and I knew what it did.... I knew about wave forms. I knew what a 'sawtooth' was. I mean, if you studied physics, you'd learn those things. I was really good in math and I was good in science." Read the whole thing.

Marco Fusinato Fuses Math, Music and Art. "Music is what numbers feel like," io9 declared in this post about artist Marco Fusinato. He "brings together avant garde music and art in his work, creating imagery that looks like the results of a mad scientist's musings on how sound functions. In this series of drawings, called Mass Black Implosion, he's transformed scores for avant garde works into suggestions for what he calls "free noise," by changing the order of the notes and suggesting new relationships between them. Basically he's suggested a way to make something abstract even more abstract. In the process, he's created charts that are gorgeously strange." The whole gallery of images is worth a gander.

The Calculus of Saying I Love You. This ranks as possibly my favorite blog post ever, courtesy of Inkling Magazine, via the irrepressible Anna Gosline. She applies basic calculus to the conundrum to determine the optimal moment when the engineer her roommate is dating shoud declare his affection. The solution? "The Engineer should in fact solve for zero in the second derivative to the love-time function and say 'I love you' when love has stopped accelerating. This solves the concerning problem of having to wait until his love has stopped growing. Because zero growth in the love function is likely to make any woman, chemist, calculus enthusiast or otherwise, pretty goddamn pissed off." Wise words!

Mathematically Modeling Marital Breakups. Apparently math can also shed light on what happens when love sours. A new paper appeared in May purporting to analyze marital breakups (see graph below). Per the article: "The results of the mathematical analysis showed when both members of union are similar emotionally they have an 'optimal effort policy,' which results in a happy, long-lasting relationship. The policy can break down if there is a tendency to reduce the effort because maintaining it causes discomfort, or because a lower degree of effort results in instability. Paradoxically, according to the second law model, a union everyone hopes will last forever is likely break up, a feature Rey calls the failure paradox.'"

Teaching Calculus in Haiti.One of the Spousal Unit's former students, Eugene Lim, took some time off from his postdoc to spend a summer teaching calculus at a university in Haiti. He wrote a guest post for Cosmic Variance about his experiences as a volunteer teacher -- "a powerful and affecting look into conditions there, and the spirit of the students."

Math Skills Declined Back in 1804? Apparently the challenge of inspiring apathetic students in the face of declining math and science literacy isn't an especially new problem. Skulls in the Stars unearthed a fascinating article by one Reverend John Toplis, A.M., “On the decline of mathematical studies, and the sciences dependent upon them,” which appeared in Philosophical Magazine 20 (1805), 25-31. Among other things, Toplis lamented the pending loss of England's competitive edge: "We have long ceased to study those sciences in which we took the lead and excelled, and are content to follow, at a very humble distance, the steps of the philosophers of the continent, in those which they have in a manner discovered and made plain by their glorious exertions. We, after having discovered and conquered regions in science, suddenly quit them to be possessed and cultivated by other nations, that we may pick up a few gleanings in the countries found out and cultivated by their exertions." Plus ca change, anyone?

Math Class Needs a Makeover. I wish I could be a high school student again just to take Dan Meyer's high school math classes. He gave a fantastic TED talk on what's wrong with math education in this country, echoing many of my own thoughts and experiences -- namely, that we focus on the computational aspect (numbers crunching), which most of us forget when we leave high school, but is easy enough to relearn as an adult -- provided you also have a solid grounding in reasoning ability. And that's what high school students aren't getting. Meyers is a funny man, admitting that as a high school math teacher, "I sell a product to a market that doesn't want it, but is forced by law to buy it." But as I learned while writing The Calculus Diaries, math is still relevant outside of high school -- and it's far, far cooler than I ever imagined it could be. Folks like Meyers could change our minds about math for the better.

 

support your local science writers (and bloggers!)

I see that Not Exactly Rocket Science guru Ed Yong is resurrecting his meme from a year or so ago, asking regular readers to introduce themselves in the comments. All the cool kids are doing it! This seems an awesome idea, and we missed the meme the first time around, so we hereby invite our readers -- especially the lurkers! -- to say howdy and tell us about themselves. And I invite my co-bloggers -- yoo-hoo! -- to engage in the comment thread as well. (Exceptions include GPS Tracking Systems, Cellulite Creams, Athens Greece Hotels, and the countless other Affirmation SPAMMERS -- you know who you are! -- peddling Viagra and designer shoes at discount prices, who leave fake comments just to snag a few free links. (Gollum voice) We hates them, precioussss, oh yesss, we do. They only pretend to like usss to flog their products, and waste our time by forcing usss to delete their inane fake observations. (/Gollum voice)

Speaking of flogging products: While you're all figuring out what to say, might as well indulge in a bit more shameless self-promotion. (Don't worry, we do very little of that around here.) Regular readers know posting has been a lot more sporadic at the cocktail party over the last year and a half. My main excuse is a demanding job as director of the Science and Entertainment Exchange. But writers compulsively write. So I juggle that with blogging at Discovery News, penning the odd book review, and have I mentioned lately that I wrote a book in my spare time? (Bora! mentioned it for me last month, along with a fantastic list of other books past, present and future from science writers/bloggers.)

The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse comes out August 31, and I revamped my author Website in honor of the occasion -- it's now vaguely steam-punky and much easier to navigate (there are still a few minor errors because I did the html coding of the main text myself; will fix them soon). So that's what I've been up to, for those who think we're just slackers. As I said at one of my panels at Skepchickon this weekend: I am very tired. I've stayed active on Twitter and Facebook, though -- and for the latest breaking book-related news, you can always become a fan of The Calculus Diaries on Facebook.

Brian Switek of Laelaps knows my pain. He wrote his first book, Written in Stone, in his spare time, too. I think I'm almost as excited for Brian's book as I am about The Calculus Diaries, because I've watched him struggle, persevere and succeed in achieving his dream. He's a promising young science writer and I'm proud to know him, in the blogosphere and IRL. He recently posted advance praise ("blurbs") for Written in Stone, and an impressive array it is, too. That's something every author I know is grateful for: the support of colleagues when it comes time to publish a book. So this seems a good time to thank the amazing folks who lavished advance praise on The Calculus Diaries. It takes a huge amount of effort and discipline to bring an entire book to the publication stage, and it can be pretty darned lonely at times, because everyone's off having a life except you. It warms a writer's heart when such esteemed colleagues find merit in his/her work, and take time out of their own busy schedules to say so. Publicly. So thank you! Publicly! Your support means the world to me. 

"The Calculus Diaries is a great primer for anyone who needs to get over their heebie-jeebies about an upcoming calculus class, or for anyone who’s ever wondered how calculus fits into everyday life and wants to be entertained, too!”
--Danica McKellar, New York Times bestselling author of Math Doesn’t Suck and Hot X: Algebra Exposed

"Zombies? Surfing? Gambling? Nobody told me calculus could be like this. To my twelfth-grade math teacher: I demand a do-over!" --Carl Zimmer, author of Parasite Rex and The Tangled Bank: An Introduction to Evolution

"Back in the day, when I was close to flunking out of calculus class because I couldn't understand why it was worth my valuable time to actually understand it, I needed someone like Jennifer Ouellette to gently explain how I wrong I was.  She's like every English major's dream math teacher: funny, smart, infected with communicable enthusiasm, and she can rock a Buffy reference.  In this book, she hastens the day when more people are familiar with an integral function than with Justin Bieber." -- Peter Sagal, host, NPR's "Wait, Wait Don't Tell Me," and author of The Book of Vice.

"In this wonderful and compulsively readable book, Jennifer Ouellette finds the signature of  mathematics -- and especially calculus, of course -- in the most unexpected places, the gorgeously lunatic architecture of Spain's Antonin Gaudi, the shimmering arc of waves on a beach. Just following her on the journey is the half the fun. But the other half is learning about the natural beauty and elegance of calculations. Ouellette's ever clear and always stimulating voice is a perfect match to the subject - and The Calculus Diaries is a tour-de-force." -- Deborah Blum, author of The Poisoner's Handbook: Murder and the Birth of Forensic Medicine in Jazz Age New York.

"As amusing as it is enlightening, The Calculus Diaries is no dry survey of abstractions. It’s a guide to everyday life—to car trips and roller-coaster rides, diet and exercise, mortgages and the housing bubble, even social networking. As Ouellette modestly recounts her own learning curve, she and her husband become characters alongside eccentrics such as Newton and Gaudi and William the Conqueror. Like a great dance teacher, Ouellette steers us so gently we think we’re gliding along on our own." —Michael Sims, author of Adam’s Navel: A Natural and Cultural History of the Human Form and Apollo’s Fire: A Day on Earth in Nature and Imagination.

"Jennifer Ouellette's calculus confessional is a delight, and an example of the finest kind of science writing.  Her book reveals to its readers the gritty inner workings of the most important idea humans have ever thought.  (Yes, calculus is that big:  it's all about understanding how things change in space and time, and there just isn't much more important than that.)  Ouellette's wit, her elegant wielding of metaphor, and her passion for both math and funky culture produce this crucial insight:  every equation tells a story, she says, and she's right, and the tales she tells here will captivate even the most math-phobic." -- Tom Levenson, author of Newton and the Counterfeiter; Head of the Program on Writing and Humanistic Studies and Director of the Graduate Program in Science Writing at MIT

"If you ever thought that math was useless, read this book. Want to survive a zombie attack? Win at craps? Beat a zombie at craps? Well, listen to Jennifer Ouellette. The math she describes might just be your best hope if you don't want your brains to be gobbled by the undead." -- Charles Seife, author of Zero: Biography of a Dangerous Idea.

"Like the movies Batman Begins, Spider-Man, or Superman, The Calculus Diaries is the story of how an insightful, creative, and hard-working young person acquires superpowers and uses them for the benefit of society. Only this tale is true: Jennifer Ouellette can't fly or spin a web, but she can spin a yarn. The Calculus Diaries documents the author's seduction by mathematics and her conquering of it--Eureka!--to see the world with sharper vision. For too many people math, calculus in particular, is an albatross. But Ouellette reveals math for what it is, a powerful tool for solving problems and the exquisite language we use to describe nature. Reading this book will make you smarter. And more powerful." -- Eric Roston, author of The Carbon Age

"A charming and gentle introduction to important mathematical concepts and their relevance to everyday life." -- Leonard Mlodinow, author of The Drunkard's Walk: How Randomness Rules Our Lives

it's a math, math world

While I've been dutifully plugging away at copy edits for The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse, Jen-Luc Piquant has been hanging out over at Script Frenzy, which (among other things) randomly generates possible screenplay scenarios for the blockbuster movies of the future. How's this for an elevator pitch: "After crash-landing in the desert, a fallen angel hijacks a bus full of tae bo instructors." Or this: "While lost in a wormhole, a god of the underworld discovers a shocking use for spray cheese." I'm sure wackiness ensues on both counts -- and the one with the tae bo instructors should have some awesome fight choreography.

It's fun playing with Script Frenzy's random generator, but for now I'll leave the screenplays to Jen-Luc and stick with my writerly strengths. The good news is that we're moving into high gear for The Calculus Diaries, which will be published August 31. The copy edits are done (with galleys/ARCs imminent), the book has a rudimentary Amazon page, I set up a fan page for it on Facebook, and hey, there's even a nifty cover:

 

I love the roller coaster imagery with the math-y stuff overlaid (courtesy of the Spousal Unit). It's apt: The entire book is about making the real world your mathematical playground, whether you're shooting craps in Vegas, enjoying the rides at Disneyland (Space Mountain meets vector calculus!), turning your househunting adventure into a multivariable optimization problem, ruminating on Fourier transforms while learning to surf in Hawaii, or expounding on the calculus of the living dead. (Why yes, I'd be happy to show you the derivation for the coming zombie apocalypse. Wait! Come back! Just kidding! Although if anyone's interested, it's covered in Appendix 2.)

Since the book won't be out for a few months yet, I thought now would be an excellent opportunity to highlight some other good math-centric books out there that might appeal to a popular audience. So here's a sampling of books I really enjoyed, in authorial alphabetical order. Feel free to add your own in the comments! And also to buy these other books!

Math and the Mona Lisa, by Bulent Atalay. In this erudite-yet-accessible tome, Atalay goes where The Da Vinci Code feared to tread: away from paranoid conspiracy theories and into the realm of actual established fact. He explores phi, the "golden ratio," and its sibling, the Fibonacci series, in the context of the mathematical symmetries that can be found in math and art. From Publishers Weekly:

Physics professor Atalay uses as his touchstone Leonardo da Vinci, of whom he says in his prologue: "Had [da Vinci] been able to publish the scientific ruminations found in his manuscripts in his own time, our present level of sophistication in science and technology might have been reached one or two centuries earlier." This assertion sets the buoyant tone for the rest of the book. The author marvels at the symmetries to be found in art and the natural world, discussing the Fibonacci series (0, 1, 1, 2, 3, 5, 8...) and the golden ratio related to it designated by the Greek letter phi (1.618...) with illustrated examples ranging from da Vinci's three portraits of women to the Great Pyramid and the Parthenon.

The Calculus Wars, by Jason Bardi. This was one of several books I cite in my own bibliography: an eminently readable historical account of the knock-down, drag-out fight that ensued when Isaac Newton and Gottfried von Leibniz each claimed credit for inventing calculus Per Publishers Weekly:

Those interested in a lucid, nontechnical account of the battle between [Newton and Leibniz] over who invented calculus will welcome science writer and debut author Bardi's cautionary tale. As early as 1665, Newton composed a manuscript detailing his method of calculus with examples, but after his unpleasant experience with a 1672 paper on optics that aroused the ire of Robert Hooke, an eminent member of the Royal Society who accused the younger man of plagiarism, Newton became shy of publishing. Between 1672 and 1676, Leibniz independently discovered calculus, using notation that has since become standard. When Leibniz published his results, Newton's allies rushed to discredit Leibniz in what developed, in Bardi's words, into "the greatest intellectual property debate of all time." [...] Bardi provides a timeless lesson about human pride as he describes the series of misunderstandings and miscommunications that led to the clash between these two great minds, "perhaps the greatest of their day."

A Tour of the Calculus, by David Berlinski. This is pretty much the classic popular book on the subject, recommended to me by several regular commenters at the cocktail party. The prose can be a bit pedantic, unfocused, and overly florid at times, and some non-mathy readers won't care for the inclusion of equations in the text. (Once I really got into the calculus, I found his inclusion of the equations quite helpful, but as a casual reader, I probably would have skipped them.) There are some excellent explanations, some lovely turns of phrases, and one charming chapter on continuity using a stroll around the city of Prague as a metaphor. Berlinski later annoyed atheists with The Devil's Delusion, and wrote a follow-up calculus book I've yet to read: Newton's Gift: How Sir Isaac Newton Unlocked the System of the World. Here's a take on A Tour of the Calculus, courtesy of Booklist:

Even those who flailed through calculus class sense the power and perfection of that branch of mathematics, and Berlinski rekindles the interest of lapsed students in this pleasing excursion through graphs and equations. Berlinski's goal is to explain the mystery of motion and the area and volume of irregular shapes, issues that gave rise to Leibnitz and Newton's invention of calculus. He makes his points one concept at a time, but not so dryly as asking and answering, "What is a function?" No, ... Berlinski tangibly grounds the abstract notions, so that attentive readers can ease into and grasp the several full-blown proofs he sets forth, as of the mean-value theorem. Though the math-shy won't necessarily jump to the blackboard to begin differentiating and integrating polynomial equations, Berlinski's animated presentation should tempt them to sit forward and appreciate the elegance of calculus--and perhaps banish recollections of its exam-time terrors.

The Unknowns: A Mystery, by Benedict Carey. I met Carey last year when we were both instructors at the Santa Fe Science Writing Workshop (good times, y'all). He's a reporter with The New York Times, and happens to really like math, which is what inspired him to write this Young Adult novel about a couple of kids in a small not-quite-town called Adjacent (get it?) who must rely on mathematical clues to solve their hometown mystery. Per Booklist: 

Math is the key to solving the mystery in this fast-paced adventure about a group of seventh-grade misfits who discover secrets surrounding the energy plant next to their trailer park. After their teacher disappears, the kids stumble across mysterious clues that she left behind. Di, Tom, and three more classmates band together, and their sleuthing takes them through claustrophobic man-made tunnels, secret underground workstations, and horrifying mountains of trash before they finally expose a team of powerful and corrupt grown-ups. Computers are part of the detective work: essential clues are on the teacher’s flash drive, and the details about how the kids crack the code and get the password will hook young readers. Science and math buffs will love the equations and charts, but even those bored by the technical details will be swept up in the fast talk and exciting action.

The Unfinished Game: Pascal, Fermat and the 17th Century Letter that Made the Modern World, by Keith Devlin. I found this book incredibly useful while writing my chapter on probability. I used the game of craps, courtesy of the New York, New York casino in Vegas, but Devlin's book provided some excellent historical anecdotes about how probability theory developed over the years (blame wealthy and titled patrons with gambling addictions -- even Galileo dabbled in the subject). Per Publishers Weekly:

Prior to the development of statistics in the late seventeenth and eighteenth centuries, even rationalists were convinced that no human could speculate on the future. Devlin ... shows us how that belief was transformed through the 1654 correspondence between mathematicians Blaise Pascal and Pierre de Fermat. Devlin uses the critical letter from Pascal to Fermat in which he discusses "the problem of points"-that is, how to determine the probable outcome of a game of chance-as a framework for a history of probability theory and risk management, fields which now dominate our social, political and financial lives. Devlin interweaves the specific issues discussed in that famous letter with the work of other mathematicians, like the London businessman John Graunt, whose ingenious, groundbreaking work analyzing London parish death records helped predict a breakout of bubonic plague and essentially founded the science of epidemiology. Devlin also introduces the remarkable Bernoulli family, eight of whom were distinguished mathematicians, and the Reverend Thomas Bayes, whose formula has enabled the calculation of risk in a variety of fields.

The Numbers Behind NUMB3RS: Solving Crime with Mathematics, by Keith Devlin and Gary Lorden. It won't surprise regular readers to know I'm a huge fan of the TV series, Numb3rs. In fact, one of my very first blog posts at the cocktail party reported on a AAAS session on Numb3rs, featuring co-creators Cheryl Heuton and Nick Falucci, and star David Krumholtz. Don Eppes (Rob Morrow) is a practical-minded FBI agent intent on catching the bad guys by any means necessary to ensure the safety and security of America and its citizens. Charlie Eppes (Krumholtz) is the quintessential absent-minded professor, a mathematical genius whose precociousness earned him tenure at a prestigious California institute -- loosely based on CalTech -- at the ripe old age of 26. The two brothers are repeatedly thrown together when Charlie's math skills turn out to be critical to helping Don solve the crime du jour. In the process, they develop a growing appreciation for their respective strengths, and realize how well they complement each other.

Devlin teams up here with one of the show's technical consultants, Caltech's Gary Lorden, to explore some of the underlying mathematical concepts in more detail than can be realistically achieved in the weekly episodes. It's an entertaining and engaging read. And hey, if you're not already a fan, why not check out Seasons 1 through 5, now available as a DVD boxed set? Surely you could spare a couple of weeks for marathon viewing sessions.

Logicomix: An Epic Search for Truth, by Apostolos Doxiadis. The science blogosphere started raving about this book last year (at least that's when it crossed my consciousness), and the Spousal Unit has been hoarding our household copy, so I haven't had a chance to read it yet. But the Spousal Unit loved it, and so does pretty much anybody else who enjoys this sort of thing, so I have high hopes for the reading experience. Per Publishers Weekly:

An ambitious full-color exploration of the life and ideas of philosopher and mathematician Bertrand Russell, the book meticulously interconnects Russell's life, the timelessness of his ideas and the process of creating the book. While a comic about the quest for the foundations of mathematics may seem arduous, it is engrossing on many levels; the story moves, despite heavy philosophical and technical information, as the images, dialogue and narration play off each other. ... One of the most prominent themes is the conflict and symbiosis between madness and logic. The fear of madness haunts Russell because of childhood trauma, as he neurotically pushes himself toward what he conceives of as its opposite, a system for certainty. Inventive, with both subtle and overt narrative techniques, the comic form organizes the complex ideas into a simpler system, combining to form a smart and engaging journey through the ambiguity of truth.

The Complete Idiot's Guide to Calculus, by Michael W. Kelley. Okay, this is less of a popular book and more a supplementary textbook to help high school students get through calculus, and it's definitely not for rank beginners. Frankly, when I first dipped a toe into the murky waters of the subject, Kelley's book was a bit beyond me; he might want to re-title it The Half-Wit's Guide to Calculus. But once I got the basic concepts down and needed to seriously start grappling with the actual math, this book proved an invaluable resource. And I just discovered that Kelley has another book called The Humongous Book of Calculus Problems for People Who Don't Speak Math. According to the jacket copy: "The best-selling author of The Complete Idiot’s Guide to Calculus has taken what appears to be a typical calculus workbook, chock full of solved calculus problems, and made legible notes in the margins, adding missing steps and simplifying solutions. Finally, everything is made perfectly clear." Huzzah! I just ordered it myself.

The Manga Guide to Calculus, by Hiroyuki Kojima. This is just one in an entire math and science series from No Starch Press. From the book jacket: "Noriko is just getting started as a junior reporter for the Asagake Times. She wants to cover the hard-hitting issues, like world affairs and politics, but does she have the smarts for it? Thankfully, her overbearing and math-minded boss, Mr. Seki, is here to teach her how to analyze her stories with a mathematical eye." Of course, Noriko soon figures out that calculus isn't for math nerds anymore: she use it to understand patterns in the real-world that are relevant to the hard-hitting journalist she aspires to do (probability, supply and demand curves, even the density of a Japanese liquor known as Shochu). I had a lot of fun perusing this book. If The Calculus Diaries has a manga soul mate out there, this could definitely be a contender. If only it had been written by a fellow mathephobe....

The Golden Ratio, by Mario Livio. I read this in conjunction with Atalay's book above, since taken together they tell you everything the lay reader needs to know about the golden ratio. From Publishers Weekly:

Theoretical astrophysicist Livio gives pi's overlooked cousin phi its due with this lively account, the first on the subject written for the layperson. Phi is the golden ratio of antiquity (1.6180339887), a never-ending number so lauded for its harmonious qualities that in the 16th century it was dubbed the divine proportion. It is related to phenomena as diverse as the petal arrangements of roses, the breeding patterns of rabbits and the shape of our galaxy. Phi is also claimed to have been crucial in the design of the Great Pyramids, the composition of the Mona Lisa and the construction of Stradivarius violins. Livio carefully investigates these and other claims and does not hesitate to debunk myths perpetuated by overzealous enthusiasts he calls "Golden Numberists." This is an engaging history of mathematics as well, addressing such perennial questions as the geometric basis of aesthetic pleasure and the nature of mathematical objects. Useful diagrams and handsome illustrations of works under discussion are amply provided.

Math Doesn't Suck and Kiss My Math, by Danica McKellar. The actress best known as Winnie from The Wonder Years is also an accomplished mathematician, and a few years ago she decided to share her hard-earned wisdom with young girls who might be struggling with (or afraid of) their math classes. The first covers the basics of middle school math (fractions, decimals, percents and so forth), while Kiss My Math helps prepare girls for pre-algebra, and I believe she has a third book on algebra proper due any day now. McKellar has been criticized in some circles for her unapologetic, blatantly "girly" approach to all things math: her examples focus on shopping ("You can never have too many shoes"), boy troubles, sharing with siblings, even a math "horoscope," along with Cosmo-style personality quizzes. But the criticism is a little unfair: I have teenaged nieces and believe, McKellar is speaking their language, plus she offers her own personal stories and those of other girls to serve as role models/inspiration. Buried amidst all that sugar is quite a lot of practical mathematical advice. I originally bought the books for my nieces, but despite the deliberately juvenile tone, Kiss My Math, in particular, actually turned out to be useful when I had to sharpen my own sadly atrophied algebra skills (along with geometry and trig) in preparation for delving into calculus. Everything we learn in math class is supposed to be prepare us for calculus and beyond, after all. I can't wait for McKellar's take on calculus.

The Drunkard's Walk, by Leonard Mlodinow. Probably one of the most readable accounts of randomness, probability, statistics, and so forth you're ever likely to encounter. Mlodinow makes the mathematical concepts crystal clear, even to a reader like me, whose eyes tend to glaze over then things get too technical. That's probably because he tells such terrific stories. This book has one of the most detailed and colorful accounts of Cardano, who invented the concept of the sample space, for example, as well as an excellent summary explication of the problem of points (the focus of Devlin's Unfinished Game). Per Publisher's Weekly:

Mlodinow, a visiting lecturer at Caltech ... leads readers on a walk through the hills and valleys of randomness and how it directs our lives more than we realize. Mlodinow introduces important historical figures such as Bernoulli, Laplace and Pascal, emphasizing their ideas rather than their tumultuous private lives. Mlodinow defines such tricky concepts as regression to the mean and the law of large numbers, which should help readers as they navigate the daily deluge of election polls and new studies on how to live to 100.

Innumeracy and A Mathematician Reads the Newspaper, by John Allen Paulos. Paulos is a professor of mathematics at Temple University, as well as a regular contributor to such esteemed publications like The New York Times and Newsweek. Innumeracy, first published in 1988, is his j'accuse to a public he feels has remained too long ignorant to math, and indifferent about their own ignorance. And you know, it's a valid point. He can be a tad bit curmudgeonly at times, but it's usually with a wryly humorous twist. Per the jacket copy:

Why do even well-educated people understand so little about mathematics? And what are the costs of our innumeracy? John Allen Paulos argues that our inability to deal rationally with very large numbers and the probabilities associated with them results in misinformed governmental policies, confused personal decisions, and an increased susceptibility to pseudoscience of all kinds. Innumeracy lets us know what we're missing, and how we can do something about it. Sprinkling his discussion of numbers and probabilities with quirky stories and anecdotes, Paulos ranges freely over many aspects of modern life, from contested elections to sports stats, from stock scams and newspaper psychics to diet and medical claims, sex discrimination, insurance, lotteries, and drug testing.

The second book is Paulos' "irreverent investigation of the often faulty use of statistics and fact in newspaper articles." If you're one of those people who groans inwardly and experiences repeated "face palm" moments when perusing your daily paper or watching broadcast news, Paulos is a kindred spirit. But you know, being a mathematical smarty-pants didn't save him from losing a bunch of money in the stock market. He learned from the experience, though, and the result is yet another book, published in 2004, that is probably also worth a gander: A Mathematician Plays the Stock Market.

Zero: The Biography of a Dangerous Idea, by Charles Seife. I've known Seife for more years than I care to admit, since we were both regularly covering meetings of the American Physical Society. This book is proof that lightning can strike on a science writer's first foray into book writing. It's well nigh flawless (and garnered a PEN award as evidence of that perfection): clear, succinct yet elegant prose laying out the history of the number 0, why it's so important, and most importantly, why the non-scientific among us should care. One of the first books I ever read that made me want to know more about the fascinating history of math. Per Publishers Weekly:

Seife takes readers on a historical, mathematical and scientific journey from the infinitesimal to the infinite. With clever devices such as humorously titled and subtitled chapters numbered from zero to infinity, Seife keeps the tone as light as his subject matter is deep. By book's end, no reader will dispute Seife's claim that zero is among the most fertile--and therefore most dangerous--ideas that humanity has devised. Equally powerful and dangerous is its inseparable counterpart, infinity, for both it and zero invoke to many the divine power that created an infinite universe from the void. ... In addition to offering fascinating historical perspectives, Seife's prose provides readers who struggled through math and science courses a clear window for seeing both the powerful techniques of calculus and the conundrums of modern physics: general relativity, quantum mechanics and their marriage in string theory. In doing so, Seife, this entertaining and enlightening book reveals one of the roots of humanity's deepest uncertainties and greatest insights.

The Witch of Agnesi, by Robert Spiller. This is part of a series of murder mysteries by Spiller, a math teacher turned novelist; other titles include A Calculated Demise: The Hypatia Murders and Irrational Numbers. His protagonist is Bonnie Pinkwater, a widowed high school math teacher who finds herself getting involved in investigating murder after murder in tiny East Plains, Colorado. There's not a whole lot of mathematical discussion in these, but there is a bit of math-y history, and Pinkwater is a terrific, feisty character. My only quibble is after three books (with more on the way), it kind of strains credibility that such a small town has such a high murder rate. The corpses keep piling up, and sometimes they are the corpses of Pinkwater's young students.

Quicksilver, by Neal Stephenson. Book 1 in Stephenson's justly lauded Baroque Cycle. Snow Crash remains my all-time favorite Stephenson novel, but this one might run a close second. How can you go wrong with a riveting historical thriller/mystery centered on who gets credit for inventing calculus? It's a weighty tome, and a firmer editorial hand would have helped some of the more cloying sections, but on the whole, it's an impressive achievement, and I have every intention of reading the other two books in the cycle, once my insane schedule permits. Per the Amazon review:

The novel, divided into three books, opens in 1713 with the ageless Enoch Root seeking Daniel Waterhouse on the campus of what passes for MIT in eighteenth-century Massachusetts. Daniel, Enoch's message conveys, is key to resolving an explosive scientific battle of preeminence between Isaac Newton and Gottfried Wilhelm Leibniz over the development of calculus. As Daniel returns to London aboard the Minerva, readers are catapulted back half a century to recall his years at Cambridge with young Isaac. Daniel is a perfect historical witness. Privy to Robert Hooke's early drawings of microscope images and with associates among the English nobility, religious radicals, and the Royal Society, he also befriends Samuel Pepys, risks a cup of coffee, and enjoys a lecture on Belgian waffles and cleavage-—all before the year 1700. In the second book, Stephenson introduces Jack Shaftoe and Eliza. "Half-Cocked" Jack recovers the English Eliza from a Turkish harem. Fleeing the siege of Vienna, the two journey across Europe driven by Eliza's lust for fame, fortune, and nobility. Gradually, their circle intertwines with that of Daniel in the third book of the novel.

Geek Logik, by Garth Sundem. This is one of those one-off whimsical humor books, but in the case of Geek Logik, Sundem wants to show people like me how to use the principles of basic algebra to "take the guesswork out of life" -- and foster critical thinking skills. I found it quite helpful, actually, because one of the trickier aspects of writing The Calculus Diaries was figuring out how to translate my real-world questions into the more rigorous form of a mathematical equation. Sundem uses humor to show you how to do this for questions like, "Should I call in sick," "Should I join a gym," "Do I have a snowball's chance in hell with her," and "How many beers should I have at the company picnic?" (Regarding the latter, the Spousal Unit and I seriously discussed including a segment on the calculus of inebriation -- but then we got, um, distracted by the experiment involving lots of cocktail sampling and wine tasting. We suspect it would take the form of an exponential decay curve, though.) Apparently you can even devise an algebraic equation to determine whether or not to get a tattoo. Bonus: there's a little working calculator embedded in the cover. Sundem followed up with The Geek's Guide to World Domination: Be Afraid, Beautiful People.

And there you have it: lots of lovely reading material to keep you busy until The Calculus Diaries comes out in the fall. It's a math, math, world out there, people, with numbers and geometry to be found everywhere in nature, as this gorgeous movie by Cristobal Villa (via Eterea Studios) makes clear:

shameless self-promotion

So, the science blogosphere is all a-flutter about this weekend's Science Online 2010, pretty much the Woodstock of Science Blogging, or as I like to call it, Bora!-Fest 2010. And oh yes, Jen-Luc-Piquant and I will be there -- I missed last year's confab due to a slight case of launching a national outreach program. I'm thrilled at the prospect of seeing all my blogging buddies again, and making new ones. Bora! even interviewed me (and numerous others) after Science Online 2008.

This time around, I'll be co-chairing a panel -- on science, Hollywood, the changing landscape of online multimedia, and what that might mean for blogging -- with the lovely Tamara Krinsky, all-around Renaissance woman of science and entertainment. Ironically, even though Tamara and I live in the same city (Los Angeles) and have several acquaintances in common, we met through Bora!, who lives 3000 miles away. That's why he's King of the Blogosphere. Our panel, called Science and Entertainment: Beyond Blogging, will be on Saturday from 2 to 3 PM and in the words of Michael Bay, it is going to be "AWESOME!"

The official description:

Over the past several years, the Internet has tangibly changed the way that movies and TV shows are produced and marketed. Blogs will call out ridiculous scientific errors found in stories and the critique can go viral very quickly; therefore, science advising is on the rise in an attempt to add some semblance of plausibility to your favorite flicks. As tools on the web continue to evolve, filmmakers and television creators are finding new ways to connect with and market to their viewers. For some shows, this has meant tapping into the science featured in their content, ranging from an exploration of the roots of the science that has been fictionalized to the expansion of a scientific topic explored in a documentary. In this session, we’ll look at how online video and social networking tools are playing a part in connecting science, Hollywood and its fans.

But we're actually hoping not to just lecture; we want you to come and join us for what promises to be a fascinating conversation. Sure, we'll both give brief presentations as an overview of the type of online multimedia currently available -- I'm excited about what's been happening in that sphere, and the stuff Tamara has told me about what's on the horizon -- and then we will open the floor to discuss how blogging fits into this new multimedia space. As that space continues to evolve and change, how should blogging change with it? What are the untapped potentials? What new multimedia formats and partnerships might be formed between science bloggers and the entertainment industry? I don't want to see blogging being left in the dust, like print media, because it gets too entrenched in the status quo. Not that this would happen: bloggers are some of the earliest adopters out there, after all. Basically it's an excuse for me to pick everyone's brain because I tend to be a bit slow up the uptake with the latest Web-trends.

In other shameless self-promotion news, I was interviewed on yesterday's Armed With Science podcast by John Ohab of the Department of Defense. Jessica of Bioephemera introduced us, and I'm so glad, because I love what he's doing with that podcast. Joining me for the interview was none other than Eureka co-creator Jaime Paglia, who gave his own take on how his show, in particular, portrays science, scientists, and the inevitable tension between idealism and pragmatism that scientists often must grapple with (e.g., they want to make discoveries and invent things to help humanity, but those same breakthroughs can be weaponized, particularly since much of their funding comes from defense). The Eureka writers are a smart, savvy bunch -- they all take after Jaime -- and now they have their own blog, too, Eureka Unscripted, for anyone who's interested in some insider baseball.

Finally, The Damn Book is finally -- finally!! -- in production. I've been working on revisions intermittently for so long, my friends started saying, "But you were revising the manuscript a month ago, weren't you? Isn't it done yet?" It's never done. But at some point you have to let your baby go out into the world, warts and all. There's no cover, or even a Web page for the book yet, but here's a sneak peek at the catalog copy and my favorite of the many excellent illustrations Jason Torchinsky created for the book:

The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse

Jennifer Ouellette never took math in college, mostly because she—like most people—assumed that she wouldn’t need it in real life. But then the English-major-turned-award-winning-science-writer had a change of heart and decided to revisit the equations and formulas that had haunted her for years. The Calculus Diaries is the fun and fascinating account of her year spent confronting her math phobia head on. With wit and verve, Ouellette shows how she learned to apply calculus to everything from gas mileage to diet, the rides at Disneyland to shooting craps in Vegas—proving that even the mathematically challenged can learn the fundamentals of the universal language.

Those of you who've been following the cocktail party for awhile now, know the book arose out of a series of calculus-related posts I wrote starting in 2006. Here's a sampling for those who missed them:

I Could Be Mathier

To Infinity and Beyond

Derive This

Picture This

Calculus is Craptastic

The Devil's Playground

Game Changer

The book will be much more polished; blog posts are always the first drafts, part of my "writing lab." It's fun to see how far I've come since those early days, both in my rudimentary understanding of calculus, and in my ability to write about such an abstract subject coherently. And oh yes, there will be zombies -- even an equation involving zombies in Appendix 2: Calculus of the Living Dead. Because sometimes you just have to nut up or shut up, and do some calculus before the zombies wipe out the human race.

miles (per gallon) to go before understanding fuel economy

The rocket scientist just bought a new car.  He claims he was thinking about trading in his old Pontiac Aztek well before I bought my new car, but even if we were thinking about the need for new cars at the same time, I guarantee we had distinctly different thought processes in deciding what kind of cars to buy.  I have a much more aerodynamic dog than he does (hound dog vs. golden retriever), so a convertible made a lot better sense for me than for him. (Sorry, Katie...) 

One obvious consideration in buying a car is fuel economy.  Fuel economy is the figure of merit for everything from advertising to the "Cash for Clunkers" program.  C for C only allowed you to trade in cars with an EPA fuel efficiency rating of 18 mpg or less.  If you had a Toyota Camry and you wanted to get a new Prius, you were on your own.

So which is better:  going from a car that gets 34 miles per gallon (mpg) to one that gets 50 mpg, or changing from a car that gets 18 mpg to one that gets 28 mpg? For the precise among you, let's define 'better' to mean saving you the most money in fuel costs.  So the first one is better, right?  You've got a 16 mpg improvement vs. a 10 mpg improvement.

Not so fast.  Let's say you drive 18,000 miles each year.  The amount of gas you need to drive 18,000 miles if you get 18 miles per gallon of gas is 18,000 miles/18 miles per gallon = 1,000 gallon.

Miles per gallon	Gallons of gas needed for 18,000 miles
18	1,000
28	643
34	529
50	360

If I change from a car that gets 18 mpg to one that gets 28 mpg, I save 1000-643 = 369 gallons of gas.  If I change from 34 mpg to 50 mpg, I save 529-360 = 169 gallons of gas.  Sort of counter intuitive, huh?  That's what Richard P. Larrick and Jack B. Soll of Duke University found in a study in which they asked people to rank order five pair of proposed mpg changes in order of "their benefit to the environment (i.e. which new car would reduce gas consumption the most compared to the original car?)"  Sixty percent of the respondents rank ordered the changes according to the difference in mpg.

If you plot (as the authors did in their paper) the amount of gas used for a fixed driving distance vs. the miles per gallon your car gets, the plot is definitely not linear. I've made a similar plot for three different driving distances - 10,000 miles, 15,000 miles and 20,000 miles and indicated on the graph where the two mileage differences I compared above are located.

This is an example of linear reasoning being applied to a case in which it doesn't apply.  In some countries, the figure of merit is a quantity like how many gallons you need to drive 1000 miles.  (Actually, the quantity is liters per 1000 kilometers since just about everyone in the civilized world except us uses the metric system.)  I calculated the gpm (gallons per mile) in the table below and you can compare the fuel efficiency much more easily.  

Miles per gallon	Gallons per 1000 miles
18	55.5
28	35.7
34	29.4
50	20.0

Larrick and Sole argue that gallons per mile and not miles per gallon should be used to help people understand energy efficiency, but I would argue we need to take it one step further.  I used to like to ask my graduate students the following question:  "What one skill do you want your students to have when then enter the intro course in your discipline in college?" 

For me, the answer is the ability to really read graphs.  That's a little bit of a cheat for an answer because there's a lot of math tied up in understanding graphs.  But a graph gives you so much more information than a number does.  In this case, we're really not interested in just a number:  we're interested in the slopes between the sets of points -- or (dare I say) the derivatives of the curves. On the graph below, I've identified the slopes between 15 mpg and 20 mpg and between 35 mpg and 40 mpg with dark lines.  The lines are a lot steeper for the low-mpg interval than for the high-mpg interval, even though we're looking at a 5-mpg difference in both cases.

Another important thing to notice is that how much you drive in a year makes a pretty significant difference.  This graph tells me that the more miles you drive, the more important it is for you to improve the fuel economy of your car.  The little old lady with the Mercedes who only takes it out on a Sunday for church is not making too much of a dent in the petroleum stores.  The person who drives from Allen to downtown Dallas five days a week, 50 weeks a year really needs to worry about fuel economy.

The final thing I would hope people notice from the graph is the total amount of fuel.  Changing from a 10 mpg to a 12 mpg vehicle definitely saves fuel; however, you're still using a lot more fuel than if you went to a 15 mpg or 30 mpg vehicle.

Of course, there's a caveat - there always is.  Although the study focused on environmental impact in terms of fuel used, don't forget that there can be significant differences in terms of emissions.  In fact, the two best things you can do with your current car to help the environment are to make sure your car is tuned up so that if combusts the fuel efficiently, and maintain the right tire pressure.

My hopes for getting students to learn how to use and digest information from graphs is probably doomed for disappointment.  People want the easy solution:  just give me one number I can base my decision on.  Just as No Child Left Behind has reduced school performance to standardized test scores, and all universities care about anymore is how many students are enrolled and whether the students think the professors are entertaining - never mind what they learn, the average person wants things to be laid out.  I'm with Einstein:  Everything should be made as simple as possible.  But no simpler. 

Maybe we need a test before you're allowed to buy a new car.  On second thought, that would probably kill the Big Three faster than they've been able to do it themselves.

game changer

Oh blogosphere, I miss you so! There are so many cool science-y items and fascinating debates to tempt me away from the task at hand, and I hate being cut off from it all for so long, but I must stay the course and finish The Damn Book, as it has come to be known. (Yes, I have reached the dark night of the writer's soul, but we shall overcome.) I'm nearly done, the manuscript is due September 18, and after that, I will return to blogging (and life in general) in full force, rather than the intermittent appearances I've been making for the last few months. In the meantime, we've been debating two options for The Damn Book's eventual title, and I figure, why not let my readers weigh in?

1. The Calculus Diaries. This was my original title, way back in the book proposal stage. It's direct, high-concept, and captures the nature of the book pretty well -- i.e., it's about me learning calculus by seeking it out in the real world. The question is, does putting "calculus" right there, front and center, turn off the potential general reader (as opposed to those who are already mathematically inclined)?

2. Dangerous Curves: How I Learned to Stop Worrying and Love the Calculus. This is the current working title. It's clever, playing on the "area under the curve"/"face of the function" aspects of calculus; and it's more subtle in its appeal to the general reader. However, it might be a little TOO clever, in that it's not as immediately apparent what the book is about. You kinda have to know a wee bit about calculus already to get the title.

I could go either way at this point, so feel free to cast your vote in the comments, with a brief explanation as to why you'd make that choice.

I have also reached that point of the writing process where I must sum up my conclusions/lessons learned from the long journey into calculus, and condense it into a compelling, readable epilogue. Much soul-searching has taken place during the last couple of years. I have talked to dozens of people (scientists, educators, my fellow "mathogynists") about their attitudes towards math (calculus in particular), how it is taught, how they learned it (assuming they did), etc., and compared it to my own experience. Here are some of my random musings thus far (much of which probably will not end up in the book, although it all feeds into the final product).

First, where does this knee-jerk dislike of math come from? Two years later, I can only say, who the hell knows? There is no one single factor, as far as I can tell. For many people, their struggles with math set in with high school algebra. Co-blogger Allyson performed so well in her other high school subjects that she was placed in advanced math classes. Alas, she was ill-prepared for that level, and the placement set her up for failure. “I distinctly recall the humiliation of my eyes welling in frustration at algebra,” she said. By the time her friends were taking calculus, Allyson had been demoted to what one might charitably call “math for dummies,” learning how to calculate compound interest and how to do her taxes – useful skills, no doubt, but she remains haunted by the memory of her failure.

Another co-blogger, Lee, had a similar experience, with more dire consequences: her inability to grasp algebra – despite top grades in all her other classes -- kept her from becoming a marine biologist, and she has a visceral hatred of mathematics to this day. “It wrecked my self-confidence in a way nothing else ever did, and still knots my stomach,” she told me. “I’m not totally innumerate, but anything that looks like an equation makes me break out into a cold sweat and run screaming in the other direction.”(My bloggy buddy Brian over at Laelaps has written extensively about his struggles with math, which keep getting in the way of his desire to be a scientist.)

On the surface, at least, I have no good reason for my own negative reaction to mathematical symbols. I did very well in my high school geometry and algebra classes, yet somehow I never self-identified as someone with a penchant for math. The truth is that fear and loathing in math class does not necessarily arise from a lack of aptitude, but from a belief in a lack of aptitude. Where did I acquire that belief?

No doubt part of it stems from gender bias. There is a well-documented prejudice against women in math and science dating back thousands of years, although history gives us the rare exceptions, such as the plucky Sophie Germain. The century before, a French noblewoman named Emilie du Chatelet translated Newton's Principia and became the lover of Voltaire before dying in childbirth in her early 40s. Victorian England had Mary Somerville, another self-taught female mathematician who defied the cultural stereotypes of her age. Her passion for math was so strong, she was revising a paper the day before she died at 92.

Such women often have been dismissed as mere statistical anomalies, but evidence is mounting that there is, in fact, no innate difference in the mathematical ability of girls and boys. Any gap in performance is due primarily to sociological factors. This is a controversial statement, as evidenced by the heated comment threads that ensue whenever someone in the blogosphere dares to bring up the touchy subject of women in math (and science). We would prefer to believe that the overt sexism experienced by Sophie Germain et al are a thing of the past, and simply not an issue in this enlightened age, but the reality is that these attitudes persist. Women have come forth with innumerable horror stories ranging from mere discouragement to overt sexual harassment.

A geometry teacher tells the entire class that the girls would probably do the worst in his course because they lacked spatial reasoning ability. A guidance counselor shunts female students into “practical math” classes where they learn how many ham slices each guest would need at a wedding. A physics professor insists on checking his female students’ work before they can leave the lab, yet doesn’t feel the need to check the work of his male students. A computer science professor dismisses any questions from female students as “lazy little girl whining.” And a calculus teacher thinks it’s perfectly appropriate to measure his female students’ bodies and use those measurements as part of his volume calculations in class. One woman on a comment thread over at Tiny Cat Pants last year told of her high school math teacher who made the three female students sit in the front row, “because girls have a harder time with math than boys do.” It was really a flimsy excuse to ogle their cleavage and brush his crotch up against them suggestively during exams. Quoth the commenter: “Guess which three people in that class were not about to be stuck in a basement computer lab with that dude?”

While I have no doubt those things happened (and still do), I never experienced anything so horrific; my math teachers were kind and, if not openly encouraging, they certainly were not discouraging or hostile, nor was I ever sexually harassed. My parents were supportive of my intellectual pursuits, if a bit bemused by my headier inclinations. Nobody ever told me explicitly that girls weren’t as good as boys at math, yet somehow I absorbed that message anyway. Carol Tavris, a cognitive psychologist and author of several popular books (The Mismeasure of Woman should be mandatory reading for young women), explained to me that there are subtle, situational social cues that seep into our consciousness, like osmosis, even if we never encounter overt negative messaging about gender.

The phenomenon is known in psychological circles as stereotype threat, and it has been confirmed in more than 100 scientific articles. For example, a 2007 study in Psychological Science found that female math majors who viewed a video of a conference with more men than women reported feeling less desire to participate in the conference, and less of a sense of belonging, than female math majors who viewed a gender-balanced version of the video. The male math majors were immune to those subtle situational cues. 

That’s stereotype threat in a nutshell. The 2004 film Mean Girls perfectly captures the subtle influence of cultural factors. Home-schooled in Africa for most of her early life by her anthropologist parents, Cady (played by Lindsay Lohan) didn’t absorb the subliminal message that women can’t do math, or that liking the subject is uncool – in fact, she appreciates the universality of math, because “it’s the same in every country.” Then she begins attending a regular high school and the inevitable peer pressure kicks in. She is urged by her peers not to join the high school math team (“It’s social suicide!”), and pretends to be bad at math to win over the cute boy in her calculus class. This being a movie, she gets over it, and ends up copping to her love of math and snagging the cutest boy in school. Alas, peer pressure doesn’t excuse me, either.  I was a painfully shy, socially awkward, brainy sort in high school, and pretending suddenly to be bad at math would not have transformed me magically into the homecoming queen.

Tavris also cited our fascination in the US with the notion of innate ability. We are born with certain built-in talents, this reasoning goes; you either have a gift for math, or you don’t, and no amount of hard work can make up for that lack of innate ability. The reality is much more complicated. My friend Deborah self-identified as being good at math early on in her education. Her fourth-grade teacher held multiplication table competitions in class. Deborah was highly competitive, so she worked very hard on memorizing her multiplication tables and practicing at home. As a result, she excelled in these competitions and became known as being “good at math.” This had a significant impact on her later on: whenever she struggled with an especially tough problem, she pushed through, thinking, “I should be able to do this because I’m good at math.” Yet her belief in her innate ability, and success at math, were actually the product of a lot of hard work.

Another part of the problem has to be the way the subject matter is presented and/or taught. Guest blogger Alex Morgan offered a clue when it wrote about his own daughter's ambivalence towards math, despite having a certain aptitude for the subject. She just doesn't like it! And after perusing her math homework, Alex found he didn't much blame her. The material was dry, uninspiring, and completely divorced from any real-world experience. (Hence my use of real-world environments in which I seek out possible calculus problems in The Damn Book.)

Students need to feel inspired, particularly when it comes to a difficult subject. While I was at the Kavli Institute for Theoretical Physics last year as journalist in residence, I got to know UC-Santa Barbara mathematician Bisi Agboola, who generously shared his own story with me. Bisi was educated in the UK and failed most of his math classes through their equivalent of high school. “I found it dull, confusing and difficult.” As a child, he was determined to find a career where he wouldn’t need any math, finally announcing to his skeptical parents that he would be a woodcutter. He was crushed when they pointed out that he would need to measure the wood.

But one summer he encountered a Time-Life book on mathematics –- Mathematics by David Bergamini -– that offered “an account of the history of some of the main ideas of mathematics, from the Babylonians up until the 1960s, and it captured my imagination and made the subject come alive to me for the very first time.” It changed his mind about this seemingly dry subject. He realized there was beauty in it. He wound up teaching himself calculus, and told me he is convinced most physicists also do this. Today he is a PhD mathematician specializing in number theory, and exotic multidimensional topologies. Ironically, he still doesn’t much like basic arithmetic: “I find it boring.”

Some students respond well to how calculus (and physics) is traditionally taught, others don't. The Spousal Unit sent me a link to a fairly new blog called Gravity and Levity, written by a physicist with a way with words:

For those of us who immediately liked physics class in high school, physics was a game.  It was like a little logic puzzle where the rules of the game were given to you (usually on a formula sheet) and you were asked to use them cleverly to come up with a solution.  A friend of mine once put it succinctly: “Physics is all about finding out which variables you know and which variable you want, and then searching through your formula sheet for an equation that has all of those letters in it.”  That, more or less, was the physics game.  You rearrange some symbols on a paper and you come up with an answer.  Instant gratification.

Those of us who went on to study physics in college almost invariably did so because we liked the game.  I personally loved it, and I was good at it.  But as I went further into physics, it began to be more than a game.  Little by little, all the equations and “rules of the game” started coming together into a coherent perspective on the universe and how it works.  ...  Over years of study my interest in physics gradually but completely shifted from “the game is fun” to “I want to know how to think about what the universe is made of and how it works.”

I hated the game; I couldn't really see the point -- at least until I got interested in physics and realized that calculus was relevant to my world. I'm one of those people who really needs to understand the context, and the "why" of things. (Although I must admit to a fondness for word games. I killed at Boggle and Scrabble in college.)

The point is, different people learn in different ways, and that makes coming up with a standardized educational system particularly challenging. The best model I heard about was from a woman scientist I met at a party, whose daughter went to an elite private school on the East Coast. There were several teachers for each subject, and students could transfer out of a class if the teaching style didn't agree with them, and try another instructor. The daughter's attitude towards her history class did an about-face once she found an instructor who inspired her. Just like science communication, learning (and teaching) science, or any subject, is about making that critical connection. I just don't see how it would be possible to scale up that approach to a nation-wide level; there's a reason this system was implemented at a pricey private school.

So: stereotype threat, gender bias, peer pressure, bad teaching, poor subject presentation -- all of these play a role in discouraging people (especially those of the female persuasion) from taking/liking math. There are countless efforts worldwide to combat these sweeping socio-cultural factors, and we should continue to fight the good fight in that regard. That said, Sophie Germain and Mary Somerville didn't let socio-cultural factors keep them from pursuing their love of math. What made the difference? Personal mentorship helped at some point, but it started with inspiration and falling in love with the subject in the first place.

Ultimately, when I closely examine my dubious history with math, I can't really cast too much blame on those Big Picture factors for my prolonged ignorance/avoidance of math. They discouraged me because I let them discourage me -- because I wanted an excuse to avoid calculus. I can't say I was intellectually lazy, because I avidly pursued knowledge in any subject that caught my interest, and worked very hard in those areas. Like Alex's daughter, I just wasn't all that interested, and hence was more than happy to be pushed away from math and science. What is the cure for willful ignorance and very deliberate avoidance?

I'm not sure there was a single game-changing moment, since the process of thawing towards math was gradual. It started with going to work for The American Physical Society, then becoming a science writer specializing in physics. And one day, out of curiosity, I asked a physicist named Alan Chodos (associate executive officer of the APS) about why objects fall at the same rate regardless of mass -- it seemed really counter-intuitive to me, although I had no doubt it was true. He insisted that I didn't have to take the matter on faith, and walked me (kicking and screaming) through the basic algebraic equation. Suddenly the numbers had relevance to something I'd actually experienced. And my kneejerk defenses started lowering bit by bit. I can't say I love math and calculus, but I understand the basics now, and I no longer break into a cold sweat at the sight of an equation. Believe me, that's tremendous progress.

It just goes to show that real learning is personal and individual. A good teacher can change someone's life and undo years of willful ignorance. I owe Alan a great debt, I realized. The least I can do is say thank you in public.