Monday, December 23, 2019

My Year's Best List 2019

The best things that I read, watched, listened to, and otherwise ingested in 2019!

(Previous lists: 201820172016201520142013.)

Best Book—Fiction

Juan Gabriel Vásquez, Lovers on All Saints’ Day (2015)

I bought a used copy of this book during a daylong feeding frenzy at legendary Powell's Books in Portland. Of all the books I shipped home from Powell's, Lovers on All Saints' Day made the strongest impression. I'll quote some of the advertising copy that accurately describes this collection:

Lovers on All Saints’ Day is an emotional book that haunts, moves, and seduces. … In these stories, there are love affairs, revenge, troubled pasts, and tender moments that reveal a person’s whole history in a few sentences. … Set in Europe (the scene of Vásquez’s own self-imposed exile from Latin America) and never before available in English, this collection evokes a singular mood and a tone, and showcases Vásquez’s hypnotic writing.

Here's a New York Review of Books article that covers Lovers on All Saints' Day as well as Vásquez more generally.

Buy it online: Lovers on All Saints' Day

Best Book—Nonfiction

John Carryou, Bad Blood: Secrets and Lies in a Silicon Valley Startup (2018)

Bad Blood isn't remarkable as a piece of writing—it’s workmanlike—but you'll probably read it from cover to cover anyway, because the true story of the Theranos fraud is astonishing. It's like a car crash you can't look away from.

Buy it online: Bad Blood

Best Book—Airplane Reading

Michael Connolly, The Wrong Side of Goodbye (2016). This is No. 19 in the venerable series of Harry Bosch detective novels. It's also an homage to Chandler, and I’m never not interested in a retelling of the basic Chandler plot. Buy it online.

Best Short Stories

Ted Chiang, "Story of Your Life," "How You See Me," "The Lifecycle of Software Objects," and "Anxiety Is the Dizziness of Freedom." Those were my favorites from Stories of Your Life and Others (2002) and Exhalation (2019). Both collections might be overrated on the whole, but see for yourself; both are well worth buying. Buy online: Stories of Your Life and OthersExhalation

Nick Fuller Googins, "The Doors." Paris Review, Spring 2019. Read the beginning or subscribe to Paris Review for the full text.

Peter Orner, "Ineffectual Tribute to Len." Paris Review, Spring 2019. Read the beginning or subscribe to Paris Review for the full text.

Kimberly King Parsons, "Foxes." Paris Review, Summer 2019. Read the beginning or subscribe to Paris Review for the full text.

Karen Russell, "Bog Girl: A Romance." I read this surreal story in Russell's 2019 collection Orange World and Other Stories, but it was first published in the New Yorker in 2016 as "The Bog Girl," which means you can read it online. Buy online: Orange World and Other Stories.

Best Poetry

Philip Larkin, "The Trees." Read it online.

Patricia Smith, "You’re Gonna Write This." Paris Review, Spring 2019. This is a tough read. Read the beginning online or subscribe to Paris Review for the full text.

Brandon Som, from "Tripas." "Tripas" is a long poem, parts of which have been published in several places. Here is a selection in Paris Review; additional selections can be read online here and here.

Best Essays

Leslie Jameson, "I Met Fear on the Hill." Paris Review, Winter 2018. Surprisingly given its title, this is a charming personal essay about the author’s mother, the mother’s ex-husband, and the relationship between the two during the tumultuous Sixties and after.

David Wong Louie, "Eat, Memory"; Clifford Thompson, "The Moon, the World, the Dream"; Amit Majumdar, "Five Famous Asian War Photographs"; and Thomas Powers, "The Big Thing on His Mind," all in Best American Essays 2018Buy it online

Wesley Yang, "We Out Here." Harper's, March 2016. I first read "We Out Here" in Yang's breakout nonfiction collection of 2018, The Souls of Yellow Folk (I reviewed it here). The essay is also available at the Harper's website, so you can read it online.

Best Theater

The Lifespan of a Fact, Studio 54, 254 West 54th Street. A funny and timely play about truth, fiction, and the spaces between. Cherry Jones, Daniel Radcliffe, and Bobby Cannavale were flawless with their rapid-fire lines and their comic and dramatic timing. The production was excellent, including a surprising and uncliched setting in the home of a recently deceased parent. Thematically, The Lifespan of a Fact wasn't so much profound as it was teasing and suggestive…but it was still thought-provoking. Here's the New York Times review.

Best Music

Alfredo Rodríguez Trio in concert at the Miller Theater at Columbia University

Rodríguez is a skilled pianist and a playful composer. Some of his numbers began surprisingly with elements combining—I don't know—art rock and New Age? But eventually they dissolved cleanly into straight-ahead jazz and Afro-Cuban jazz. Rodríguez's hands moved so fast at times that in the lights I could see them falling in and out of sync with the 60 Hz alternating current frequency—yet he could also transport you in the next moment with soft dynamics. Cameroonian bassist Richard Bona joined in for some joyful and funky times. Adding to the star quality, sitting a few seats away from us with his entourage, was Quincy Jones, who produces some of Rodríguez's music.

Here are some 30-second samples of Rodríguez's music from iTunes:

Best Meal in an Airport

The city of Austin, Texas has pretty good barbecue, even at the airport. My best airport meal of 2019 was at Salt Lick Barbecue in AUS: I had the three-meat combination plate, and I washed it down with, uh, the baked brie plate at nearby Second Bar + Kitchen. (I was hungry!)

Best Whisky

This summer during a visit to the Isle of Skye I drank a wee dram of Kilchoman Loch Gorm 2019, and it was the best whisky I tasted in Scotland. (Even in summer, the highlands have a chill that sends you to the liquor shelves.) Ordinarily I don't prefer whisky when it's aged in sherry casks, or port barrels, or such gimmicks; I just like Scotch. But in Kilchoman Loch Gorm, the sherry influence was subtle. I haven't seen this whisky yet in any American stores, but apparently it's gettable if you're willing to spend enough.

Best Movies

(Links point to reviews; some reviews have spoilers.)

Yesterday. This sentimental musical fantasy works thanks to actor Himesh Patel and director Danny Boyle. The screenwriter wrote Love, Actually and Notting Hill, and he follows the same plan here.

La La Land. Well, it’s no Singin' in the Rain—but what is? And there’s one scene that reaches those heights.

Spider-Man: Into the Spider-Verse (review 1, review 2). A true comic book movie, with a beautiful comic book aesthetic. I've seen this animated film twice now with the whole family, and we'd gladly watch it again.

Surprisingly good: Lion King (2019). I saw it one day this year when I was in charge of my niece and nephew and needed something for us to do. Five minutes into the film, I had the same thought I had in 1993 when I first saw the dinosaurs in Jurassic Park: "This is an advance." And I was intrigued by the artistic implications of a pure-CGI film, in which, in the absence of physical constraints on cameras or sets, every single decision is the free choice of the director and other talents involved. Anyway, the movie aspect of it was OK.

Best Building

Mosaic tiled ceiling of the Woolworth Building lobby
This year I saw the model for the beautiful 1928 Fisher Building in Detroit when my wife and I took a leisurely tour of the 1913 Woolworth Building in New York, at the time of completion the tallest skyscraper in the world. Our guide knew everything about the building, its surrounding neighborhood, and its tycoon, the retail magnate F.W. Woolworth. We toured the breathtaking lobby, went up into the mezzanine to see the ornamentations more closely, and went below street level to see some interesting infrastructure. Taking the tour was a nice way to spend a Sunday afternoon, and I recommend the company, WoolworthTours.

Best Artworks 

A gallery of pictures I took this year (alphabetical by artist)

Bisa Butler, Les Sapeurs, a quilt shown in a prize exhibition at the Museum of Arts and Design in New York. The design appeared to be based on a photograph, so I went online and found it here. (Reminds one of Malick Sidibé.)

Edgar Degas, pastels of female nudes in ungainly poses. I saw these while wandering through the Metropolitan Museum of Art (click then scroll down to see). The nudes aren't idealized—caught rather in momentarily awkward or off-balance poses. Degas displayed these works at the last Impressionists exhibition (1886). From an online essay: "Observed in the artificial light of the dressing room, the bodies fascinated champions of the Naturalist aesthetic … Geffroy praised the boldness of the bathers' 'frog-like postures.'"

After seeing Thornton Dial's genius sculptural works in last year's "History Refused to Die" exhibition at the Met, I was noticing the artist more this year. His 1990s works Tiger and Ladies Will Stand By Their Tigers were in an excellent exhibition at the American Folk Art Museum, and his 1990s striped animal was on view at the must-visit National Museum of African American Art and Culture.

Simone Leigh, Sentinel (2019), part of a prize exhibition at the Guggenheim

Maya Lin, Vietnam War Memorial, Washington, D.C.

Sarah Lucas, Pauline Bunny (Tate Britain)

JMW Turner, Norham Castle, Sunrise (Tate Britain). Tate Britain is heaven for a Turner fan. Sunrise with Sea Monsters was out on loan unfortunately—sixteen years ago I saw it at the Clark Art Institute and had hoped to see it again—but there were dozens of Turner canvases to explore, and Norham Castle, Sunrise was my favorite.

Andy Warhol, various portraits in the retrospective at the Whitney. Warhol's images are so ubiquitous it's hard to think of them as art anymore—but they came alive in person.

Charles White, various images in the retrospective at MoMa.

Best of the Year—Period. 

I saw a glacier, and it was amazing.

This is the Blue Glacier, one of several glaciers on the eight-thousand-foot Mount Olympus. I finally get what a glacier is! Pictures had never explained it to me before…it always just looked like snowy valleys or mountainsides. Being there in person however, I could appreciate the "thingness" of it.

The Blue Glacier was the high point of a five-day, 37-mile backpacking trip in Olympic National Park that began at low elevation and ascended through the mysterious and lovely Hoh Rainforest, with its mossy maples and its towering Sitka Spruce and Western Red Cedar.

Me on the left climbing down a washout on the trail. Below me is the guide.

To be honest I don't enjoy backpacking…it's painful, there's never enough food to eat, sleeping outdoors is uncomfortable, you're always having to set up up camp or strike camp or look for firewood or wonder where your next water is coming from, and then there are the toilet and hygiene issues. But I go anyway, because there are things to see and places to be that you can't experience on a day hike. This backpacking trip was easier than most, thanks to our excellent guide from Wildland Trekking, an outdoor company I recommend and would use again.


That's it for 2019! And since I like to sign off with something to watch or listen to, here is just sixty seconds or so of the Alfredo Rodríguez trio improvising during a sound check before a concert in Saint Lucia. Though brief, this snippet shows Rodríguez's personal ebullience and his trio's chemistry.

Sunday, December 22, 2019

Holiday Challenge 2019

Welcome to the 2019 Holiday Challenge! 

Email answers to 

(Previous Holiday Challenges: 20182017201620152014.)

In each puzzle below, the soundtrack to the video is a traditional Christmas song that has been scrambled. Can you identify the song?

In each case, here is how I scrambled the songs:

  1. Select a recognizable passage from the song
  2. Break the passage into a number of segments
  3. Play the segments in random order.

Song 1:

Song 2:

Song 3:

Song 4:

Song 5:

Song 6:

Song 7:


This year's prize is any book that's mentioned in my soon-to-be-published Year's Best List:

You can get some more detail on these books when my Year's Best List comes out (any day now!).

Here's how the prizes will go:

The first person who identifies any of the seven songs will receive a copy of the book of their choice.

Let x be the greatest number of songs identified by anyone by the time the contest ends; then the first person who identifes x of the songs will receive a copy of the book of their choice.

Everybody who attempts the challenge will be entered into a drawing to win a copy of the book of their choice.

Have fun! Contest ends December 31st at 11:59:59 p.m. Eastern time. Email answers to

Acknowledgment. I'd like to thank the folks at for hosting free quality videos and images. The video I used today is this one, and I paid the creator $10.00 for using it.

Thursday, October 17, 2019

Book Review: Algebraic Inequalities: New Vistas

Algebraic Inequalities: New Vistas
Titu Andreescu and Mark Saul
MSRI Mathematical Circles Library
Volume: 19; 2016; 124 pp;  Softcover

If you like algebra, then doing the work in Algebraic Inequalities: New Vistas will make you better at it. If you're not sure if you like algebra, then a better plan might be to start a book club devoted to Algebraic Inequalities, with perhaps an expert to moderate the group. As it happens, such an arrangement forms one of the main purposes of the book. It's part of a series called "Mathematical Circles Library," co-published by the Mathematical Sciences Research Institute (MSRI) and the American Mathematical Society (AMS). What's a Mathematical Circle? From the MSRI website:
Mathematical circles are gatherings that give mathematicians the opportunity to share their deep understanding and appreciation for the subject with school students [[and teachers—JZ]] who are looking for new challenges in mathematics. Using a variety of approaches such as problem-solving, individual and group exploration of noncurricular topics, reading guidance, competitions, and just plain fun, the circles help students acquire useful mathematical ideas and techniques.
Books in the Mathematical Circles Library provide grist for such a gathering, in the form of "collections of solved problems, pedagogically sound expositions, discussions of experiences in math teaching, and practical books for organizers of mathematical circles." See the website for all twenty-two volumes (and counting).

Algebraic Inequalities is coauthored by Titu Andreescu, a professor of mathematics, and Mark Saul, a longtime colleague who serves as Executive Director of the Julia Robinson Mathematics Festival ( Their book contains a wealth of problems, including many gems. Here's one that I enjoyed, from a chapter about algebraic symmetry:

Solve simultaneously:
x + [y] + {z} = 1.1
{x} + y + [z] = 2.2
[x] + {y} + z = 3.3

where [x] means "the greatest integer not exceeding x" and {x} means "the fractional part of x," that is, {x} = x − [x].

True to the description on the book's back cover, "The exposition is lean. Most of the learning occurs as the student engages in the problems posed in each chapter." I'd phrase that more strongly: do the problems. One cannot improve at math by passively reading. Ditto for physics by the way, as I said in the introduction to my textbook, Force and Motion: An Illustrated Guide to Newton's Laws:

Readers have told me that they were surprised by the brevity of the chapters in Force and Motion. I say what needs to be said, and then it's time for the reader to get to work. The exposition in Algebraic Inequalities is even leaner; it can afford to be, since (by comparison with Newton's Laws) the content itself is leaner.

"Algebraic Inequalities: New Vistas" is the kind of title that could earn you funny looks from your fellow subway riders, but within mathematics this topic is anything but esoteric: algebraic inequalities are prevalent in advanced math and science. For instance, the Heisenberg uncertainty relation ΔxΔp ≥ ℏ/2 is an inequality, one that's often proved using the Cauchy-Schwarz inequality—Chapter 9 in Algebraic Inequalities.

Reading this book made me nostalgic for the inequalities I'd proved in the past, when proving things was part of my job description. There have even been inequalities on this blog—most recently, a + b − Sqrt[a2 + b2ab] < Sqrt[ab] in the Cutting a Triangle problem. Could I have proved this inequality more easily using the tools described in Algebraic Inequalities?

Going back further, Curious Cuboids features an inequality in three variables and also uses the Cauchy-Schwarz inequality and the inequality of arithmetic and geometric means (Chapters 2 and 3 in Algebraic Inequalities).

A major reason why inequalities recur on Zimblog is that optimization does. Students who've taken calculus often come away thinking that optimization is fundamentally about taking a derivative; but fundamentally, optimization is better thought of as proving an inequality of one kind or another. Among the vistas which Algebraic Inequalities: New Vistas opens up for the student is the perspective that algebra alone can be a powerful approach to optimization.

Wednesday, October 9, 2019

Updating The Family Movie Night List

Just added The Day the Earth Stood Still (1951) to the list of classic films for family movie night. We liked it! Kids steeped in contemporary alien-encounter movies will have to adjust to some understated special effects. The film is absorbing and still distinctive for the way its humanity-scale drama plays out on a human scale.

Sunday, September 8, 2019

Work/Life Balance

Work/life balance is a continual struggle, but over the past couple of years I think I've gotten better at it. Partly this is because I've adjusted my ways of working. For example, instead of attending to my phone all weekend, getting stressed out by its pinging sounds, I'll often go on airplane mode until Monday morning. Weekend work and checking email still happen, but they usually happen at my desk, at times when I choose; and I observe a no-work-on-Saturdays rule pretty faithfully. This has forced me to be more efficient on the other six days of the week. Another step I've taken is to limit my weekend travel to seven days per year. (I did the math on this once and calculated that seven weekend days amounted to around 20% of my annual quality time with family; this was as much as I thought I ought to be willing to give away.) Having a finite budget for these days makes me choosier about saying yes to events that take me away from home on Saturdays or Sundays.

Selfie in a hotel 
When I do travel for work, I have a tradition of recording a nightly voice memo for the kids, in which I read aloud one of Aesop's fables; my wife can play it for the kids at bedtime. I believe this probably bores them, but that's why I chose fables (most of which take less than a minute to read). Some of the fables are funny, and sometimes a moral from the fables will prove useful in a practical situation.

The "Moon Tree" in Sacramento, CA
(from a seed that orbited on Apollo 14)
I try to enjoy the places I travel to for work. Maybe the city I'm in has a famous barbecue place or a famous tree; maybe instead of taking Uber to my meeting, I can walk through a historic district. A trip of multiple days often has an evening free, and small cities often have nice museums: in recent years I've been able to dip into the De Young, the Briscoe, the Frist, and the Arizona-Sonora Desert Museum.

From this exhibition
However, the most important factor in improving my work/life balance over the past couple of years hasn't been working smarter; it's been living better independently of work. I'm a member at some museums in the city, and a museum membership is a good value compared to many other forms of entertainment. When a museum sends notices of future exhibitions and one of them looks good, my wife and I choose a specific weekend when we'll see it, and then we put the item on our calendars. We try to have something booked every other month or so.

(My go-to recipe is here.)
I've resumed old pastimes—reading, cooking, and exercising. The temptation to work on a project on a Sunday afternoon is now in healthy competition with the temptation to finish a great book I'm partway through. A good chunk of my Saturday is typically spent shopping for ingredients, cooking, and washing up. As for exercise, while it can feel like an obligation, and I'm no longer the athlete I once was, fundamentally I still enjoy breaking a sweat. (Sometimes I even get good ideas on the elliptical machine.) Exercise also makes me hungrier, so I better enjoy what I eat.

Everything ebbs and flows, and sometimes work trumps all of this. Sometimes I'm just too lazy to maximize the denominator in the work/life fraction. And nothing is ever finally figured out. At best I may continue to learn how to live, year upon year.

Thursday, September 5, 2019

Approximating Pi By Dividing N-Digit Numbers

While on vacation I added a new sequence to the Online Encyclopedia of Integer Sequences. The sequence is about approximations to π:

     π ≈ 3/1

     π ≈ 44/14

     π ≈ 355/113

     π ≈ 3,195/1,017

     π ≈ 99,733/31,746

     π ≈ 833,719/265,381

     π ≈ 5,419,351/1,725,033

     π ≈ 80,143,857/25,510,582

     π ≈ 657,408,909/209,259,755


The nth term of the sequence is the best approximation for π that can be given by a quotient of two n-digit numbers. In a case like 3195/1017 = 3550/1130 where two or more quotients of n-digit numbers each give the best approximation of π, we take the fraction with the smallest numerator and denominator. [Fixed 10/4/19—a tip of the hat to reader Andy for noticing that my original statement of the rule was ambiguous!]

Here are the corresponding decimals. In each case, the last digit shown is the first digit that deviates from the decimal expansion of π.

     3/1 = 3.0…

     44/14 = 3.142… (note this equals the more familiar 22/7)

     355/113 = 3.1415929…

     3195/1017 = 3.1415929… (note 3195/1017 equals 355/113)

     99733/31746 = 3.14159264…

     833719/265381 = 3.141592653581…

     5419351/1725033 = 3.1415926535898…

     80143857/25510582 = 3.141592653589792…

     657408909/209259755 = 3.14159265358979322…

When I thought of this sequence, I searched for it in OEIS and was surprised not to find it. Admittedly this sequence seems unimportant, but it also seems like a natural idea. Anyway, not finding the sequence in OEIS, I googled a few terms and discovered that a Russian computer programmer named Oleg Zelenyak had earlier thought of the sequence as an example in a book on computer algorithms. Zelenyak gives eight terms, so to add value I calculated the 9th term, 657408909/209259755. Finding the 9th term took twelve-plus hours on my laptop, using a pure brute force approach.

You can see some of these details by going to sequences A327360 and/or A327361. The first sequence gives the numerators of the fractions, while the second sequence gives the denominators. (OEIS is an encyclopedia of integer sequences, so it represents fraction sequences by pairs of integer sequences.)

Some other fraction approximations for π can be found at Wikipedia's article on "Milü." Wikipedia's more general article on approximations to π contains some amazing results.

Friday, July 12, 2019

A Math Problem From My Notes

Been thinking about this one lately while exercising:

Triangle ABC lies within the closed unit disk. Angle A measures θ degrees. In terms of θ, what is the greatest possible area of triangle ABC?

This is one instance of a variety of problems one could pose, along the following lines. "Shape 1 lies within Shape 2. Something is known about Shape 1. What else can be known about Shape 1?"

Monday, June 17, 2019

My Conjecture For 4 Points In A Circle

(I promised I wouldn't solve this problem, but I can still make a guess, can't I?)

Recall the Points in a Circle Problem:

Given N obstacle points X1, …, XN in the closed unit disk, let
aN(X1, …, XN)
denote the greatest possible area of a triangle in the disk that doesn't strictly contain any of the Xi. Let
aN* = minimum possible value of aN(X1, …, XN).
Then for each given value of N, our problem is to calculate aN* and find a minimal set of obstacles, that is, N points Xi* for which aN(X1*, …, XN*) = aN*. 
  • a1* = 1 and a1(X) = 1 if and only if X = O, the center of the disk.
  • a2* = 1 and a2(X1, X2) = 1 if X1 = O or X2 = O.
  • a3* = 0.8286369… and a3(X1, X2, X3) = 0.8286369… if and only if {X1, X2, X3} = {O1, O2, O3} in some orientation.
The points Oi are equally spaced around a circle of radius r0 = 0.320963…. The value of r0 is given exactly by the real root of the cubic polynomial 5x3 − 4x2 + 7x − 2, namely

and the value of a3* is given exactly by positive root of the polynomial 160000x6 − 265824x4 +  412857x2 − 209952, namely

Here is my guess for N = 4:

The four obstacle points lie on the vertices of a square that is concentric with the disk. The obstacle points have coordinates given by (±c, ±c) where c0.3966 0.246 is given exactly by the positive root of the polynomial −1 − 2x + 13x2 + 40x3 + 34x4 − 8x5 − 20x6 + 8x8.

For these four obstacle points, the greatest area I have found for a viable triangle is approximately 0.731, or in exact terms the positive root of the polynomial −1024 + 2496x − 509x2 − 2596x3 + 1468x4 + 736x5 − 560x6 −  64x7 + 64x8.

I'm not sure if I've found the maximum possible area of a triangle, given the above obstacles, and I'm not sure the above obstacles are optimal either.

Friday, June 7, 2019

What Is Your Favorite Sea Animal?

Recently I had to reset my security questions on an airline website. (See also: On Having A Favorite.) One question was, "What is your favorite type of reading?" To this, my answer would be something like: "Yes." That however was not one of the choices. A person could answer that 'my favorite type of reading is online forums,' and a person could answer that 'my favorite type of reading is cook books.' But I was amazed to see that no option was available to say that one's favorite type of reading is fiction. Here is the entirety of the dropdown menu on the website:

I wonder if the people who created this menu have ever been on a plane. In the Library of the Sky, people mostly read thrillers. A reading survey of frequent flyers would have to be like,

Which is your favorite type of reading?
A woman is missing, and the narrator is unreliable
Somebody has unwisely pissed off the protagonist
Southern Gothic, Nordic Gothic, or Vampire Gothic
An old vicarage detective story but with guns/bombs
Courtroom stuff
Could be a serial killer on the loose
Ennui, and prose, but also death

Looking at the screen shot, I see that I've passed over the previous security question, "What is your favorite sea animal?" This question must be intended for passengers under 6. "Shark! Rawr!" (Maybe it should've asked "What is your favorite stuffed animal.") A subsequent question asked, "What is your least favorite vegetable?" This I found hard to answer, because I've attained the age at which vegetables are no longer a threat. "N/A" wasn't an option however, so I cast my mind back to childhood—to the time when I was four years old, and my mother had put me in a preschool. This was a well intentioned effort on her part to get me out of the yard so I could make some friends. The session only lasted six weeks, but I remember a lot about it: the daily nausea as the hour approached to leave the house, the having to lie on a plastic mat and close my eyes at naptime even though I wasn't tired; how only the most aggressive boys ever got to build with the cardboard bricks; and how lame the daily snack was. Every day, the same thing. I could tell you what it was, but that would be a security risk.

Monday, June 3, 2019

Lawn Roller Smash Problem - Some Animations

Here's what it looks like when a lawn roller of width 4 units follows the blue curve y = x2:

Inside the orange quasi-triangle, a bug would get smashed twice. And just below that region, there's another sort of quasi-triangular region in which a bug would get smashed three times. Here's a view in which the triple-smash is easier to notice; watch the y-axis between y = ½ and y = 2.

By the way, the orange curve is the graph of

16x6 + x4(16y2 − 40y − 191) − x2(32y3 + 96y2 − 30y − 688) + 16y4 − 136y3 + 225y2 + 544y − 1156 = 0.

I derived this polynomial by creating time-parametrized expressions for the endpoints of the segment, and then eliminating the time variable. (I needed Mathematica to do the elimination step, and even Mathematica crashed at first; then I thought to make a certain algebraic substitution, and Mathematica found the answer quickly.)

The boundary of the triple-smash region is composed of the orange arc that passes through y = 2 together with two envelopes, which are part of the graph of the polynomial

(y/31/6)3 = (x/4)2


y = ½ + ¾(2x).

I derived this by calculating the velocity of the segment points at each distance from the center, and then setting the local, instantaneous velocity equal to zero to obtain the locus where the segment "rolls without slipping" along the envelope. Interestingly, for a segment of width greater than or equal to 1 unit, the form of the envelope is independent of the width of the segment; it's just that the envelope extends further from the y-axis if the segment is wider.

Anyway, I first thought of this situation in an odd way. I was thinking not about steamrollers, but about the fact that any time we draw a mathematical curve on a piece of paper, we're actually painting a thick streak of ink on the page, and this must inevitably introduce some inaccuracies. If the thickness of the streak is small enough compared to—I suppose—the radius of curvature, then little harm is done. But if we paint with a very broad brush, then we distort the target curve qualitatively as in the case illustrated above.

Saturday, June 1, 2019

Steamroller Smash Problem (Actually, It's A Lawn Roller)

A lawn roller 4 feet wide moves along a curve that has been traced on the ground.

Lawn roller. Image: Dave Thompson,

The roller is following the curve y = x2, where x and y are measured in feet. Here is a picture of the curve:

If a bug is standing on the curve, the bug will get smashed. But some bugs in the path of the roller will have insult added to injury: they will get smashed twice.

Are there any bugs that will get smashed more than twice?

By the way, here is what it means for the roller to "follow the curve." As the roller moves, visualize the points of contact between the roller and the ground as being a 4-foot-wide line segment; then the roller moves in such a way that this segment is always perpendicular to the curve, and bisected by it. (I think this is the most natural sense of the phrase "following the curve" in this context.)

Sunday, May 19, 2019

Book Reviews: Nonfiction by Amis, Malcolm, and Yang

Capsule reviews of three nonfiction books

Martin Amis, The Rub of Time: Bellow, Nabokov, Hitchens, Travolta, Trump: Essays and Reportage, 1994–2017. A less learned collection than Amis's The War Against Cliché, but on average more fun. Had the book been half as long, it would have been twice as good. Some of the features on pop-culture were weak, and some of the literary essays went over my head. But in the middle zone were a number excellent pieces and many great sentences. One that I'll remember was this opening sentence in a reported piece on poker: "If for some reason you were confined to a single adjective to describe Las Vegas, then you would have to settle for the following: un-Islamic." Although some of the pieces are over twenty years old, the collection still feels current because it takes up questions that have only become more lively today, such as what to make of great art made by flawed men.

Janet Malcolm, Nobody's Looking at You: Essays. I recommend this book, especially if you haven't read most of it before. While browsing in a bookstore I bought it on impulse, on the strength of the author's name alone, because I've so enjoyed reading her in the past—and was disappointed to discover when I got home and opened the book that I'd already read almost everything in it. Still, the pieces were worth rereading and I'm glad to have the book in my library. I do wish the book had included more essays; too much of it consists of magazine profiles. The profiles are first-rate examples of the genre, but they aren't essays. From this writer especially, I'd hoped for more examples of her analysis, argument, and insight.

Wesley Yang, The Souls of Yellow Folk: Essays. The five essays here are worth reading, but unfortunately they make up less than a third of the book. About two-thirds of it consists of profiles, feature articles, and book reviewing. All of that is good reading, too—but when I bought this book, I thought I was getting, you know, essays.

If the part of the title that comes after the colon is misleading, the part that comes before the colon is a troll. Yang is an incisive thinker, and he may be the best writer about the Asian (male) experience in America; but a collection of articles is just a collection of articles. The Souls of Black Folk by W.E.B. DuBois took up its questions. Echoing DuBois's title was not only tasteless but also unwise: it made a very worthwhile book look trifling by comparison. I look forward to what Yang does next.

Friday, May 17, 2019

Where Is The Abortion Debate Heading?

I think the pro-life movement is about to make some major gains. This is due to some smart strategy by conservatives—but it's also due to the underlying fact that the pro-choice movement hasn't shifted public opinion far enough during the forty-six years since Roe was decided. Have a look at this graph showing the past eighteen years of polling data about what we might call basic team affiliation:

Pro-choice affiliation is flat over two decades, and even the youngest cohort barely cracks 50%.

For a deeper dive into 2018 views, this graph of Gallup data is illuminating:

The bottom category is the most telling. Fewer than half of Americans today want abortion on demand to be legal during the first trimester.

All of this comes as a surprise to me. I guess I've been in a bubble on the abortion issue. I used to wonder how the pro-choice movement could stand by and allow conservatives to pass increasingly tight restrictions on abortion even for pregnancies in the earliest stages. Now I think I understand why the pro-choice movement has strategically avoided calling the question. The reserves of popular support are too low to cash in.

Today I did a google search on "abortion amendment." Setting aside the news of the day (Alabama, etc.), I found that the vast majority of the resulting webpages described proposed constitutional amendments to outlaw abortion (or to recognize fetuses as persons, etc.). Almost none of the webpages described constitutional amendments about guaranteeing a woman's right to choose.

It's illuminating to try to draft such an amendment. Here's a first attempt, call it Version A:

Section 1. The right of an adult woman to abort her pregnancy for any reason during the first trimester shall not be denied or infringed by the United States or by any State.
Section 2. The Congress shall have power to enforce this article by appropriate legislation.

This merely codifies the key holding of Roe, and yet the 45% data point from the 2018 graphic suggests that such an amendment could never pass. (I do know that about two-thirds of Americans want Roe v. Wade to stand; that isn't enough to make a new constitutional amendment feasible however, because amending the Constitution requires three-fourths of the states.)

What would a constitutional amendment look like that could pass? Maybe this, call it Version B:

Section 1. The right of an adult woman to abort her pregnancy for reasons of rape, incest, or endangerment to her life, during the first trimester, shall not be denied or infringed by the United States or by any State.
Section 2. The Congress shall have power to enforce this article by appropriate legislation.

The data suggest that this amendment could pass; see in particular the 77% and 83% data points from the 2018 graphic. But you can see how far this text retreats from the goals of the pro-choice movement, which include at least the right to abortion on demand during the first trimester (as first guaranteed by Roe). It's understandable, then, why the amendment strategy never emerged. 

It's conceivable to me that in its next major abortion case, the Supreme Court could effectively establish Version B as the meaning of the Constitution regarding abortion. However, that's assuming they uphold stare decisis to any extent at all. If not, I could imagine the Court reading abortion out of the Constitution entirely. 

However the Court decides, or even if the Court doesn't take up a major abortion case, we're likely to see red states and blue states sorting themselves on abortion policy even more than they do now. Then Alabamians and Vermonters, Californians and Floridians, plus everybody in between, will get no more freedoms than those which they vote for and demand. 

Sunday, April 28, 2019

Overrated/Underrated, Cont'd


Snow White and the Seven Dwarfs (1937). Sergei Eisenstein, who directed Battleship Potemkin, and who should have known better, once referred to Snow White (a boring animated feature) as the greatest film ever made—enough evidence all by itself to make Snow White overrated. Roger Ebert includes Snow White in his "Great Movie" category; here's how he concludes the review: "The word genius is easily used and has been cheapened, but when it is used to describe Walt Disney, reflect that he conceived of this film, in all of its length, revolutionary style and invention, when there was no other like it—and that to one degree or another, every animated feature made since owes it something." Fair enough, but that doesn't make it a fun movie to watch.

Tacos. Tacos break when you try to eat them! What the fuck.

Fried chicken. I’m talking about deep-fried, bone-in chicken parts (like in the image at left). Although a sandwich of boneless, fried chicken breast is great, I have to say that with all due apologies to the South, deep-frying bone-in chicken parts is a mistake. It's a holy mess to eat, and a lot of the meat is hard to get at. The coating on fried chicken is usually either soggy and greasy, or else absurdly thick and crunchy. Now maybe you have the perfect home recipe for fried chicken parts, and maybe you execute it beautifully every time. But how does that help me?

Glamorous exclusivity awaits you.
Airport lounges (like Admirals’ Club, Sky Club, etc.) The lounge is better than the gate area for long layovers, but it isn't as much better as it's supposed to be. First of all, lounges admit too many people and are often more crowded than the gate area. Recently in a Sky Club I gave my chair to a woman who was far along in her pregnancy and who later said that she’d been wandering for ten minutes looking for a place to sit. Imagine a pregnant woman dragging a suitcase behind her, looking for a seat and not being immediately offered one! Alas that image is consistent with the general atmosphere of greed that suffuses the airport lounge—where people print emails for the pure perk-lust of doing it and then swarm the buffet like freeloaders at a wedding. The food and wine in the lounge aren’t even as good as what you can get at a wedding. Free is good, but there are apparently fees to "belong" to these lounges, and anyhow free isn't the issue given that most people in a lounge are traveling alone on the company’s dime, not feeding a family of four out of the vacation budget. I'm thinking now of the man in suit and tie who sat in a club chair beside mine recently, crouched over his cardboard cup of smelly vegetable soup and eating it with a plastic spoon. Elegant! Give me a ShackBurger in the terminal instead.

You know that food they serve in chafing dishes during your business meetings?
Why not eat exactly the same thing on your way there and back, too?


Alexander Sinardo/Wikipedia
Jennifer Lopez. Specifically, she's underrated in the movies. I'll rewatch Out of Sight any time, and everybody was wrong about Maid in Manhattan (yeah I said it). Actually, Lopez covers all the bases: she's underrated (in the movies), correctly rated (in dance), and overrated (in music).

Dishwashing machines. This was a topic on Twitter recently, initiated by a thread from everywhereist. I think she's a funny writer, but I don't understand the premise of perennial dishwashing-machine jokes. At my house the dishes come out of the dishwasher perfectly clean almost every time, with no rinsing beforehand. I can put basically everything in our dishwasher, except for heavy/large pans and wooden things. Over the course of a year the dishwasher saves me a ton of premium evening hours at home. Maybe don't blame the dishwashing machine for your failure to consult Consumer Reports?

Thursday, April 18, 2019

Cutting A Corner Saves 15% To 30% Of The Distance

Sometimes when I'm out walking and I cut a corner to save time, I wonder how much time (or distance) I saved. Today I finally did a bit of math on this. I assumed a right triangle with sides in a given ratio, and I calculated the percent savings from walking the hypotenuse. For side ratios between 1:1 and 1:5, the percent savings ranges from 15 to 30 percent.

When the ratio of side lengths is greater than about 5, taking a shortcut isn't very tempting and I'm likely to just walk around the corner. So in practice, when I do bother cutting a corner I think I'm typically saving around 15 to 30 percent on distance.

Math notes: if the side length ratio is r, then the percent savings is given by

\[p(r) = 1-\frac{1}{\sqrt{1+\frac{2}{r+\frac{1}{r}}}}\,.\]

Here is a graph of p(r) over the range r = 1/5 to r = 5:

The formula looks simpler if you express it in terms of one of the angles of the triangle, but I find angles hard to estimate visually in a real-life corner-cutting situation; estimating a side ratio seems easier. Anyway, I'd say that the upshot here is that the detailed behavior isn't very important; rather, over a pretty wide range of cases, the savings are well described as being in the 15-to-30 percent range.

Wednesday, March 27, 2019

Five Historical Questions About Determinism And Differential Equations

Not as fun as a Buzzfeed list, I know! But ten or fifteen years ago I wrote this list of questions and I thought I'd just put them out there now in case any history experts have thoughts.

The topic is the history of determinism in relation to the history of uniqueness proofs in mathematics. I'm puzzled by the relationship between these two histories.

Here are the questions, followed by a bit of discussion.

  1. Why was Laplace in 1795–1814 so committed to the idea of Newtonian mechanics as a deterministic theory, decades before there was any mathematical proof that it was?
  2. Prior to 1890–1894 (when the mathematical question was settled), did anyone attempt to dispute Laplacian determinism by invoking the potential for mathematical indeterminacy in Newtonian mechanics?
  3. Can Peano in 1886–1890 really have been the first to produce an example of an initial value problem with continuous flow and multiple solutions?
  4. Why couldn't Newton himself have produced indeterministic examples in the 1600s?
  5. Is there anything in the historical record to suggest that Newton ever made mathematical investigations into non-uniqueness?

Regarding Questions 1 and 2, here is Laplace's famously elegant statement of the deterministic thesis, from his 1795–1814 Essai Philosophique des Probabilités:1
We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it—an intelligence sufficiently vast to submit these data to analysis—it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.
Although metaphysical determinism predates the scientific revolution, in this passage Laplace's fanciful Intelligence appears to conduct a specifically Newtonian analysis. Laplace must have conceived of Newtonian mechanics as a mathematical framework with certain properties of uniqueness of solutions. However, uniqueness theorems powerful enough to justify a deterministic conception of Newtonian mechanics weren't proved until the mid- to late-1800s, culminating in the Picard-Lindelöf theorem of 1890–1894.2

Why was Laplace in 1795–1814 so committed to the idea of Newtonian mechanics as a deterministic theory, decades before there was any mathematical proof that it was?

The deterministic worldview has a certain majesty, but also a certain horror. Prior to 1890–1894, did anyone attempt to dispute Laplacian determinism (for example, in order to defend free will) by invoking the potential for mathematical indeterminacy in Newtonian mechanics?

Regarding questions 3 through 5, in 1890 Giuseppe Peano published the example dx/dt = 3x2/3, x(0) = 0 to demonstrate that solutions to initial value problems need not be unique.3 But can Peano possibly have been the first to produce such an example? (Shouldn't Cauchy have produced numerous such examples as part of his own investigations in the 1820s?)

Why, for that matter, couldn't Newton have produced indeterministic examples in the 1600s? Newton regarded his Laws of Motion partly as a means of discovering force laws from observed motions; the Principia presents complex examples of this method, such as the Kepler problem and the less important but still impressive Proposition 10, Problem 3.4 In comparison with such problems, it's trivial to consider a hypothetical motion such as x(t) = t4 and to inquire as to the forces that might produce this motion. At time t = 0, the particle is at rest (x'(0) = 0), and no force acts on it (F(0) = mx"(0) = 0). Therefore Newton would have said that the particle should remain at rest. And yet the particle moves.

Newton would easily have obtained the required force law in this peculiar example, namely F(x) ~ x½. Given the initial condition x(0) = x'(0) = 0, this force law permits the particle to remain at rest…or to execute the given motion…or to wait an arbitrary length of time and then spontaneously begin the given motion. Newton might well have cast the example aside as being unimportant for physics, but he couldn't fail to observe mathematically that the Second Law differential equation sometimes admits multiple solutions to the initial value problem. Is there anything in the historical record to suggest that Newton ever made mathematical investigations into non-uniqueness?

I find it hard to accept that the careers of Newton, Leibniz, the Bernoullis, Euler, Laplace, and Cauchy all passed before anybody discovered the problem dx/dt = 3x2/3 (x(0) = 0) or looked closely at a motion like x(t) = t4. Maybe my premises are mistaken. Any thoughts would be welcome.


1. Pierre-Simon de Laplace, A Philosophical Essay on Probabilities, trans. F.W. Truscott and F.L. Emory (Mineola: Dover Publications, 1996), p. 4. Translation of Pierre-Simon de Laplace, Essai philosophique sur les probabilités (6th ed.), from 15-volume collected works (Paris: Gauthier-Villars, n.d). First edition published by Courcier, Paris, 1814.

2. Tom Archibald, "Differential equations: A historical overview to circa 1900," in H.N. Jahnke (Ed.), A History of Analysis (History of Mathematics Vol. 24) (American Mathematical Society, 2003).

3. Giuseppe Peano, "Démonstration de l'intègrabilité des équations diffèrentielles ordinaries," Mathematische Annalen 37, 1890, pp. 182-229.

4. Isaac Newton, Mathematical Principles of Natural Philosophy, trans. I. Bernard Cohen and Anne Whitman, assisted by Julia Budenz (Berkeley: University of California Press, 1999), pp. 655–664; see also pp. 167–177.

See also my "Inertia and Determinism," Brit. J. Phil. Sci. (2008), 1–12.