my ideal bookshelf

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So my wife bought this book: My Ideal Bookshelf, in which famous (and not so famous) people, many of them writers and artists, describe the 10 \( \pm \) books that influenced them and that would be essential on their shelves.  You know, kind of a what would you take to a desert island exercise.

Which got me thinking:  what would my books be?  Here's what I came up with. 

Slaughterhouse-Five, by Kurt Vonnegut, Jr.  Well, duh.  When I was 15 or so someone in my high school class (Chris Hurst, maybe?) was reading this in class one day.  "What's that?" I asked.  I've read it several times since then, even teaching a course about it last semester.  Of all Vonnegut's books, this is the one that holds up best, with Cat's Cradle a close second, and as I've aged I've come to understand and appreciate it on a much deeper level.  As a teenager I saw it as an entertaining science-fiction story, but now I know that it's a rather disturbing, semi-autobiographical tale of post-traumatic stress disorder and the lengths that former soldiers sometimes go to in dealing with the horrors of war.  It really should be required reading now as we welcome home soldiers from Afghanistan and Iraq; instead it's still getting removed from high school library shelves.  Go figure.

The Once and Future King, by T.H. White  OK, I'll admit it.  One of my favorite movies is Excalibur.  It's really pretty bad, but I can't help it.  I suspect this derives from my love of White's classic retelling of the legend of King Arthur.  I'm really not much of a fantasy nerd, but there's something about Camelot that sucks me in.  I was never really a fan of Book I: The Sword in the Stone, which was made into an animated film by Disney and consists of fanciful stories of Merlin turning Arthur and Kay into various animals to teach them life lessons.  Once Arthur pulls Excalibur from the stone, though, it really gets interesting.  I haven't read this in about 25 years, but it's still one of my favorite books.

A Supposedly Fun Thing I'll Never Do Again, by David Foster Wallace  I'm not a fan of Wallace's fiction.  I tried reading Infinite Jest back in the 90s; I got through about 200 pages and gave up (I'm afraid that might happen with my current book, War and Peace, but the jury's still out).  Wallace's true talent was as an essayist, and while there are pieces in Consider the Lobster that are as brilliant as anything in A Supposedly Fun Thing... (especially the lead essay about the adult film awards), I will always think of this book as the quintessential collection of his best work.  Read and stand in awe.  Or get annoyed by all the footnotes.  It's pretty much a bimodal distribution.

Cloud Atlas, by David Mitchell  I haven't seen the movie yet, but I've read this book twice.  The first time was on the Kindle app on my iPad and I had two thoughts:  (1) that it is among the most brilliant pieces of fiction I've ever read, and (2) one shouldn't read it on a Kindle.  The structure of the book doesn't lend itself to the format, so I bought a copy of the old-fashioned codex version and re-read it.  Much better.  I like all the stories, but my favorite is An Orison of Sonmi-451, a disturbingly plausible dystopian story of the not-too-distant future. 

Lolita, by Vladimir Nabokov  Let's be clear:  Humbert Humbert is a disgusting human being.  But Nabokov writes such perfect sentences that you can't stop reading this story of pedophilia and self-delusion.  You know the story, so I won't say more.

Cohomology of Groups, by Kenneth S. Brown  This one will appeal only to me, but that's ok.  I spent a lot of time studying this book, the foundational text in the discipline.  I was introduced to spectral sequences in Rational Homotopy Theory and Differential Forms, by Griffiths and Morgan, but Brown's book really cleared up a lot of things for me.  My 20-year-old copy is banged up but still holds a central place on my professional bookshelf.

Assassination Vacation, by Sarah Vowell  This is that rare book that makes me laugh out loud while teaching me something.  Vowell tells the stories of the assassinations of Presidents Lincoln, Garfield, and McKinley, and does it via her adventures to various sites associated with each of them and their killers.  Her nephew, Owen, makes a lot of appearances, too.  Vowell loves America and isn't ready to give up on its promise just yet, while recognizing that we're pretty f*cked up in a lot of ways.  This is my favorite among her books and she is among my favorite writers.  She'd have to be on the shelf, right?

The Times Atlas of the World  This is the Times of London, and it's the heavy atlas on our bookshelf at home.  I love maps and this atlas is beautiful.  That's it, really.  Nothing deep.

Fahrenheit 451, by Ray Bradbury  This book is a little too real.  The firemen burn books because the people asked for them to be taken away.  Everyone has wall-sized television screens on at least two walls through which they "engage" in interactive shows.  Guy Montag's wife wants all four walls to be interactive.  State power always wins (beware the mechanical hound).  Bradbury was a master of his craft, a towering figure in American letters.  If you ever saw him on the old Politically Incorrect, you also know he could be a jackass.  But I suppose that's true of many great artists (see also: Picasso, Pablo), and it's important to separate the person from the product sometimes. 

I only have nine.  I'm sure I could think of others, but this is a pretty good list so I'll stop.  What are your essential books?

of maps and Platonic solids

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photo(5).JPG

I love maps.  One of my favorite blogs is Strange Maps, and I try to visit it only when I have some time to kill.  I have a USGS quad of the Mt. Tam/Muir Woods area in my office.  At home, I have Axis Cartography's San Francisco map hanging in a prominent location.  Give me an atlas, and I will be occupied for hours, poring over the maps inside. 

I think most mathematicians (geometers and topologists in particular) have an appreciation for maps.  There's a lot of interesting mathematics going on in them that most people never really think about.  The main issue is one of projection--the Earth is round and maps are flat.  Anyone who has tried to wrap a ball at Christmas knows that there's no way to do the job smoothly.  Any projection from the curved surface of the Earth to a plane will necessarily create some distortion; how much and which kind is a function of the projection chosen.  Atlases get around this problem by cutting the surface into chunks and drawing the pieces.  At close range, the Earth looks flat (hence the old popular belief), and it is, approximately.  That allows cartographers to create accurate maps of cities and states with little distortion of the shapes.

But this isn't a post about the mathematics of map projections.  The university's library book store is closing this week and everything is on sale.  I wandered in to browse and discovered flat file drawers full of USGS quads.  Most of them were of pieces of Florida (naturally), but I also found some from Washington and Montana.  I grabbed 40 without even looking at them; at 10 cents apiece I couldn't resist.  What will I do with them?  I don't know.  Some of them are rather beautiful and even deserve to be framed.  In particular, I found a few orthophotographic maps of portions of the Gulf Coast and some farmland that had been plowed for crops.  They look more like pieces of abstract art than maps.

In my haste, I managed to grab multiple copies of some maps.  What to do with them?  Fold origami, of course.  The photo above shows two dodecahedra that I made from cutting one map into 24 3" by 4" rectangles, folding them into pentagonal units, and then assembling them into the polyhedra.  They're sitting on my 13" laptop, so you can get a sense of the finished size (about 3.5 inches in diameter).  I folded both of these in about an hour total; it's a lot of fun.  

I'm teaching my one-credit origami class again this term and I was trying to think of a big project to do as a class instead of the level 3 Menger sponge we did last time.  I had the following crazy idea.  Each person creates a tessellated rectangle 4.5 ft by 6 ft.  I showed the students how to fold a basic unit to do this from a 6" square of paper; creating the large rectangle will take 432 such units.  We could then take these large rectangles, fold them into pentagonal units, and put them together to make very large dodecahedra.  Large here means about 5 feet tall (I think).  Will it even work?  Who knows?  We're trying a scale model first with 18" by 24" rectangles.  I'll keep you posted.

By the way, Platonic solids are very interesting in their own right.  There are only 5--tetrahedron, cube, octahedron, dodecahedron, and icosahedron.  Euclid proved this in his Elements, but my favorite proof uses Euler's formula from topology.  My students will learn all this as I teach them to fold all five of these solids--the dodecahedron as above and the others using business cards. 

So you see, math can be fun.

my favorite Christmas present

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the painted cover

Look, I'm a real nerd.  I'm not even going to pretend that I'm not.  So I'm not really sure why my wife was a bit nervous to give me this really excellent (although perhaps a bit strange) present at Christmas.

You are looking at the cover of a book entitled A New Second Course in Algebra.  My wife found it at a craft show in Gainesville.  The seller had a collection of similar folk art pieces--books on which she had painted images on the covers.  I like folk art anyway, especially portraits and landscapes done in the 18th and 19th centuries.  This painting reminds me of those, and the imperfect checkerboard border is a nice touch.  Plus, the idea that a young man would wear a sweatshirt (long-sleeved T?) with the title of a high school algebra book screenprinted on it (anachronisms FTW!) is a delightful touch. 

But the cover is just the beginning of this gem.  Let's take a look inside.  (Clicking on the images below will pop them up into a lightbox gallery that you can scroll through.)

Who owned this book?  Was it Connie Flynn (a boy)?  Or was it Marie David?  I'll guess the latter, since boys didn't (don't) usually write such things in their books, and because the handwriting in the various margins is a nice script, more likely to be a girl's.  Note also the lighter heart-shaped box with Nancy Canby +  -?-  -?-, probably Marie's friend looking for love.   

The book is from the 1940s.  In those days, this is probably about as far as most students got in mathematics in high school.  Sure, some would study trigonometry and perhaps pre-calculus, but for most students a second course in algebra was sufficient, even for those planning on going to college.  I won't go on my rant about how too many students are being pushed into calculus in high school these days (see the second half of this if you want to read that).  Instead, I'd like to write a few words on the virtues of a more careful study of algebra, including the dying art of logarithms.

Back in the 1600s, multiplication was a real problem, especially for astronomical and surveying calculations.  The computer didn't exist and wouldn't for another 300 years, so mathematicians and their ilk were forced to perform tedious calculations by hand.  It was not unusual for a collection of calculations to take months, just because multiplying two large numbers by hand takes a lot of work.  For example, say you need to multiply two 6-digit numbers.  Using the algorithm you learned in grade school, this requires 36 separate multiplications and then about a dozen additions to get the final result.  Even if you're fast and know your multiplication tables well (which the average 17th century human did not) this will take you a couple of minutes.  Now, imagine you're constructing a table of values for something (e.g., planetary motion or sea navigation) and you have to do this a few thousand times.  First, your hand will cramp up; second, you are bound to make errors; third, it will take months of hard work. 

A crucial insight by John Napier, a Scottish laird, which was refined by the famous English mathematician Henry Briggs, made this problem much more tractable.  The insight:  addition is easier than multiplication.  Well, duh, you say, and so what?  The idea is that if you have a fixed base number \( x\), then for any exponents \( a \) and \( b\) we have \[ x^a x^b = x^{a+b}. \] 

Now define, for a fixed base \( x \), the base \( x \) logarithm of the positive number \( N \) to be the exponent \( a \) that makes the following equation hold: \[ x^a = N. \] 

We write this as \( a = \log_x N \).  Now suppose you want to multiply two numbers \( N \) and \( M\).  Then, writing \( N = x^a \) and \(M = x^b \), we have \[ NM = x^a x^b = x^{a+b}. \] 

Taking the logarithm of both sides of this equation we see that \[ log_x(NM) = a+b = \log_x(N) + \log_x(M). \]

This is magical!  We've turned multiplication into addition.  This suggests the following procedure:  fix a base \( x \), say 10, and calculate the logarithms of a regular selection of the numbers less than \( x\).  Create a table of these values.  Then to multiply two numbers \( N \) and \( M \), write each of them in scientific notation: \( N = u \times 10^r \) and \( M = v \times 10^s \), where \( u, v < 10 \) and \( r, s \) are integers.  Then \[ \log_{10} (NM) = \log_{10} (u) + \log_{10} (v) + r + s, \]

where I've done a little algebra to get this simple form.  Look up the logarithms of \( u \) and \( v \) in the table and add them.  Then find this sum in the table and go backwards to find the product \( NM \). 

The base 10 (a.k.a. common) logarithm table is the last picture in the gallery above.  Also, I love the doodles you can see--the little face on the top near the center, and the girl's face on the right side.  Anyway, let's do an example:  Let's multiply 2345 by 762. Here's how it goes.  We have \( 2345 = 2.345 \times 10^3 \) and \( 762 \times 10^2 \).  Find the logarithms of \( 2.345 \) and \( 7.62 \) in the table:  \( \log(2.345) = .3701 \) and \( \log(7.62) = .8820 \).  Add these: \( 1.2521 \) and then add the exponents 3 and 2 to get \( \log(2345\cdot 762) = 6.2521 \).  Look up \( .2521 \) in the table:  \( 1.7868 \).  Then we get (approximately) \[ 2345\cdot 762 = 1.7868 \times 10^6 = 1,786,800.\] 

The real answer is \( 1,786,890 \), so we missed a little bit.  Note also that I had to use something called interpolation to deal with the extra decimal places; this invariably leads to errors.  The remedy for that is to have more accurate tables.

This was still in the Algebra II curriculum when I was in high school in the mid-80s.  No longer.  When I teach calculus, I try to explain to my students the historical motivations for logarithms, but I tend to get a bunch of blank stares.  All they know about the (natural) logarithm function is that it is the inverse of the natural exponential function and that \[ \ln x = \int_1^x \frac{1}{t}\, dt.\] 

They usually remember (after I remind them) that the log of a product is the sum of the logs, and that this can be somewhat useful for differentiating products of functions, but that's it.   

There's been a lot of talk lately about whether or not algebra is necessary.  I mean, it's hard, and we don't want to make it a stumbling block, right?  I won't make a counterargument here (but here's a good one).  But I do know from years of experience teaching college students that their difficulties with calculus arise from their mediocre algebra skills, not the calculus itself.  Perhaps it's time to return to a more solid algebra I - geometry - algebra II - trig/precalculus high school sequence, including all kinds of fun stuff like logarithms, to better prepare students for success later. 

I'll conclude by pointing out that logarithms are not just an abstraction used to simplify calculations, but they are built into the way our brains work.  Our ears perceive sound on a roughly logarithmic scale, and the Weber-Fechner law quantifies this.  Or, check out this, in which scientists from MIT determined that children, when asked what number is halfway between 1 and 9 usually answer 3, which is halfway logarithmically.  This suggests that logarithmic scales are hardwired in us, so maybe we should understand them better.