Sunday, June 25, 2017

Alice's Adventures In Wonderland

The most valuable book I own is an 1868 edition of Alice's Adventures in Wonderland that I bought in England twenty-five years ago and presented to my mother as a gift. She loved the book and displayed it in a casual and elegant way on a Shaker table in her living room. It's worth something today, even with the smoke damage. But I'm not inclined to part with it.

For everyday purposes I keep a pretty edition from 1992 that was given to me as a gift. Recently I read the book to my kids, and the experience reminded me what a slog Alice can be.

In praise of the book one can say a lot. How many writers, besides Shakespeare, have contributed permanent figures of speech to English? We could try to count them, but let's not go down that rabbit hole. The anarchy of the book recognizes and respects something deep that Carroll understood about childhood. And the topsy-turvy setting, in which the laws of logic and physics lose their grip, must have been influential in the genesis of some notable twentieth-century art, such as Dada, theater of the absurd, and (who knows?) magical realism.

(Before you ask: I left Borges off the list because, although Borges was very fond of Carroll's work and called it "authentic fantasy," nevertheless in Borges's actual writings, the citations to Carroll were mostly not to Alice. Is there a single specific allusion to Alice's Adventures in Wonderland in Borges's stories or essays?)

Alice's Adventures in Wonderland has effective passages, the best in my view being those involving the Duchess, pictured above. The scene in Chapter VI enacted by the Duchess, her pig/baby, and the violent maid might well be performed today by an avant-garde theater company. A later scene involving the Duchess combines some of the best antics in the book (the Duchess's absurd moralizing) with a claustrophobic creepiness.
[Alice] had quite forgotten the Duchess by this time, and was a little startled when she heard her voice close to her ear. "You're thinking about something, my dear, and that makes you forget to talk. I can't tell you just now what the moral of that is, but I shall remember it in a bit."

"Perhaps it hasn't one," Alice ventured to remark.

"Tut, tut, child!" said the Duchess. "Everything's got a moral, if only you can find it." And she squeezed herself up closer to Alice's side as she spoke.

Alice did not much like her keeping so close to her: first, because the Duchess was very ugly, and secondly, because she was exactly the right height to rest her chin on Alice's shoulder, and it was an uncomfortably sharp chin. However, she did not like to be rude, so she bore it as well as she could.

"I daresay you're wondering why I don't put my arm round your waist," said the Duchess after a pause…
Real literary moments in the book are rare, however—as is fine prose. For purposes of comparison, consider these passages of strong and lovely prose from, respectively, Wind in the WillowsAbel's Island, and Charlotte's Web:

He thought his happiness was complete when, as he meandered aimlessly along, suddenly he stood by the edge of a full-fed river. Never in his life had he seen a river before—this sleek, sinuous, full-bodied animal, chasing and chuckling, gripping things with a gurgle and leaving them with a laugh, to fling itself on fresh playmates that shook themselves free, and were caught and held again.

He became somnolent in his cold cocoon. In his moments of dim-eyed wakefulness he had no idea how much time had passed since he was last awake—whether an hour, a day, or a week. He was cold, but he know he was as warm as he could get. The water in his clay pot was frozen solid. His mind was frozen. It began to seem it had always been winter and that there was nothing else, just a vague awareness to make note of the fact. The universe was a dreary place, asleep, cold all the way to infinity, and the wind was a separate thing, not part of the winter, but a lost, unloved soul, screaming and moaning and rushing about looking for a place to rest and reckon up its woes.

Then came a quiet morning when Mr. Zuckerman opened a door on the north side. A warm draft of rising air blew softly through the barn cellar. The air smelled of the damp earth, of the spruce woods, of the sweet springtime. The baby spiders felt the warm updraft. One spider climbed to the top of the fence. Then it did something that came as a great surprise to Wilbur. The spider stood on its head, pointed its spinnerets in the air, and let loose a cloud of fine silk. The silk formed a balloon. As Wilbur watched, the spider let go of the fence and rose into the air.

Nowhere does the writing in Alice compare. Here is a typical passage:

Down, down, down. There was nothing else to do, so Alice soon began talking again. "Dinah'll miss me very much to-night, I should think!" (Dinah was the cat.) "I hope they'll remember her saucer of milk at tea-time. Dinah, my dear! I wish you were down here with me! There are no mice in the air, I'm afraid, but you might catch a bat, and that's very like a mouse, you know. But do cats eat bats, I wonder?" And here Alice began to get rather sleepy, and went on saying to herself, in a dreamy sort of way, "Do cats eat bats? Do cats eat bats?" And sometimes, "Do bats eat cats?" for, you see, as she couldn't answer either question, it didn't much matter which way she put it. She felt that she was dozing off, and had just begun to dream that she was walking hand in hand with Dinah, and was saying to her very earnestly, "Now, Dinah, tell me the truth: did you ever eat a bat?" when suddenly, thump! thump! down she came upon a heap of sticks and dry leaves, and the fall was over.

The passage aims to create a sense of amazement or absurdity at the idea of a fall through the air so lengthy that all of the above can take place. But couldn't the fall have been made to seem long by some means other than boring the reader? This passage illustrates how the enormous intellectual merits of Carroll's creation strain against its aesthetic flaws. The place where the book shines aesthetically is in the illustrations by John Tenniel, which are the best reason to own a copy of the book.

Carroll elevated his prose in the book's epilogue, and in particular the book's final two sentences, each about a hundred words long. By this time in the story, Alice herself has gone away but her sister remains on the bank, reflecting dreamily on what Alice had told her:

So she sat on, with closed eyes, and half believed herself in Wonderland, though she knew she had but to open them again and all would change to dull reality—the grass would be only rustling in the wind, and the pool rippling to the waving of the reeds—the rattling teacups would change to tinkling sheep-bells, and the Queen's shrill cries to the voice of the shepherd boy—and the sneeze of the baby, the shriek of the Gryphon, and all the other queer noises, would change (she knew) to the confused clamour of the busy farm-yard—while the lowing of the cattle in the distance would take the place of the Mock Turtle's heavy sobs.

This, the penultimate sentence in the book, is quite a contraption but it's fairly well put together. So far so good, but then sentiment intrudes unforgivably in the final sentence:

Lastly, she pictured to herself how this same little sister of hers would, in the after-time, be herself a grown woman; and how she would keep, through all her riper years, the simple and loving heart of her childhood: and how she would gather about her other little children, and make their eyes bright and eager with many a strange tale, perhaps even with the dream of Wonderland of long-ago: and how she would feel with all their simple sorrows, and find a pleasure in all their simple joys, remembering her own child-life, and the happy summer days.

The reader rejects this not so much because it reads as a maudlin message from Lewis Carroll to Alice Liddell, but rather because within the world of the book its sentiment seems unearned. There's not much evidence in the book that Alice really has a loving character. She's not cold. But she's no Fern Arable, who devotedly stood vigil by the runt whose life she herself saved; she's no Jim Hawkins or Bilbo Baggins, who both acted heroically despite fears. Polite, considerate, and impatient at the way everybody bosses her around, Alice is an Everychild, which is to say she isn't very interesting.

Nor is curious or queer automatically interesting—especially when rendered in the and-then-and-then-and-then format of a bedtime story. Charlotte's Web, The Hobbit, Abel's Island, and newer classics like the Harry Potter books have generous, rounded plots, and they take up enormous themes: the facts of life and death, self-reliance, the emergence from childhood into experience, and society and its sins. Alice is an intermittently delightful fantasy, the work of a highly original mind; but despite its status as a classic of children's literature, I'm not sure it is literature at all.

Saturday, June 24, 2017

Movie Review: Logan

From Logan we learn a good rule of thumb: never allude to the Western in a comic-book movie. It only makes the viewer realize that he ought to have gone to see a great Western film instead of going to see whatever comic-book story he is watching now. To be fair, the effect of Logan's allusions to Shane wasn't as dismaying as, say, the way Limp Bizkit's cover of "Faith" makes you want to drop what you're doing and listen to George Michael's "Faith." But it's probably at least as bad as the way U2's cover of "Paint It Black" makes you want to put on some Rolling Stones. Westerns and comic books are both beloved of boys, but the difference between them is that nobody has made a comic book movie yet that ranks with the great Westerns like The Searchers, Once Upon a Time in the West, or The Man Who Shot Liberty Vance. Every adult male I know has one or two favorite Westerns, but only teenagers (and the artistically-still-teenagers) will exercise themselves to any degree with the problem of ranking comic book movies.

If this is a rant, it's meant to be a subtle one. There's some great film making in comic book movies, especially in their best opening sequences:

To these I'd add the notorious pencil scene from The Dark Knight and the sequence in Unbreakable when young Elijah Price receives a gift from his mother. Meanwhile in Logan, the half-dozen interesting moments were buried in plot clichés and stale story lines. The film's best feature is probably its affecting treatment throughout of Charles Xavier's senescent frailty, which works so well because of Patrick Stewart's committed performance. Unfortunately, that is the film's only excellent acting.

James Mangold, the director of Logan, also directed Girl, InterruptedKate and Leopold3:10 to Yuma; and Walk the Line—films that together earned nine Oscar nominations and two Oscar wins. His 2013 film The Wolverine unfolded like a blazing comic book on the screen: it was top-grade movie making in the genre. But Logan doesn't work, despite its director's resume and notwithstanding the film's nearly unanimous positive reviews. Logan's attempt at a grown-up comic-book movie only proves that comic book movies don't grow up.

Thursday, June 22, 2017

The Whimsical Clock Maker


A whimsical clock maker fashioned both of the hands on his clock to be the same length. Here are some pictures of the clock. What time is it in each picture?







Is the clock maker's clock display ever ambiguous? If so, at what time(s)?

Monday, June 19, 2017

Harmonic Half-Pipe

When a skateboarder skates back and forth in a half-pipe, executing the motion the same way every time, the motion is periodic in both the horizontal sense and the vertical sense. We can make a simple version of this problem by allowing a bead to slide without resistance on a wire under the influence of gravity:

For most wire shapes, solving for the motion of the bead will require numerical methods. In any case, the specific time dependence of the bead's motion will depend on the shape of the wire. I wondered what wire shape would cause the bead's vertical motion to be the same as if the bead were simply bobbing on a spring.

I did a minute or two of searching and didn't see the problem solved anywhere online, so I worked it out. Here's an animation of the solution:

At all times, the height of the "skateboarder" is the same as the height of the mass that bobs on the spring. (This is called "simple harmonic motion.")

The shape of the wire is given by the graph of \(x = \pm p(y)\), where \[p(y) = \sqrt{y_0\,y-y^2}+y_0\sin^{-1}\!\!\sqrt{y/y_0}\,.\] Here, the constant \(y_0 =\frac{2g}{\omega^2}\) is the height of the half-pipe, which is also double the amplitude of the spring motion; and \(\omega\) is the angular frequency of the simple harmonic motion you want to produce. (There are also more complicated solutions that don't turn vertical at the ends.)


To find this solution, I worked in the first quadrant with the wire given by \(x = p(y)\) and \(p(y)\) unknown. Let the bead begin from rest at \((x,y) = (p(y_0), y_0)\). From energy conservation,

\[\frac{1}{2}mv^2 + mgy = mgy_0\,.\]

Express the speed as \(v^2 = ((p^\prime)^2 + 1)\dot{y}^2\). Now if we specify the vertical motion as \(y(t) = y_0\cos^2\left(\frac{\omega t}{2}\right)\), then we can express the time derivative in terms of the coordinate itself: \(\dot{y}^2 = \omega^2(y_0\,y - y^2)\). This eventually leads to a separable first-order nonlinear differential equation for \(p(y)\), \[\frac{dp}{dy} = \sqrt{\gamma\frac{y_0}{y}-1}\,\] where \(\gamma = \frac{2g}{\omega^2y_0}\). Note that \(\gamma \geq 1\), because the vertical acceleration can't possibly exceed \(g\), so that \(\frac{1}{2}\omega^2y_0 \leq g\) or in other words \(\gamma \geq 1\). Also note that \(\frac{dp}{dy}\rightarrow\infty\) as \(y\rightarrow 0\), which makes sense because the half-pipe must be horizontal where the vertical motion reverses direction.

Integrating with the boundary condition \(p(0) = 0\) so that the half-pipe goes through the origin, we get, with the help of Dwight 194.21 and 192.11, \[p = \int_0^y{\sqrt{\gamma \frac{y_0}{\xi}-1}\,d\xi} =  \sqrt{\gamma\,y\,y_0 - y^2} + \gamma y_0 \left(\frac{\pi}{2}-\tan^{-1}\!\!\sqrt{\gamma\frac{y_0}{y}-1}\right)\,.\] If we take \(y_0 = \frac{2g}{\omega^2}\) then \(\gamma = 1\) and this simplifies to \[p = \sqrt{y\,y_0-y^2} + y_0\left(\frac{\pi}{2}-\tan^{-1}\!\!\sqrt{\frac{y_0}{y}-1}\right)\,.\] The arctangent can be expressed more simply as \(\cos^{-1}\!\!\sqrt{\frac{y}{y_0}}\) (to see this, draw a right triangle with legs \(y\) and \(\sqrt{y\,y_0-y^2}\)), whereupon we also notice that \(\frac{\pi}{2} - \cos^{-1}(\cdots) = \sin^{-1}(\cdots)\), so in the end \[p(y) = \sqrt{y_0\,y-y^2}+y_0\sin^{-1}\!\!\sqrt{y/y_0}\] as above.

Monday, June 12, 2017

Ninth Circuit: The Law Forbids Setting Immigration Policy Via Press Release

To date, President Trump's travel bans have generated a whole catalog of district court and circuit court opinions. Reading some of these opinions, I've been surprised to see how often the judges have bypassed the statutory interpretation issues in the case and reached the constitutional questions pertaining to the Establishment Clause of the First Amendment and the Due Process clause of the Fifth Amendment.

It's surprising because the accepted practice in Federal courts is that you only reach constitutional questions if you must. When there's a way to decide a case short of deciding a constitutional question, you should take that route. And here it seems clear that there's at least a statutory question to be settled, because there are two specific statutes in the Immigration and Naturalization Act (INA) that contradict one another:

1182(f). Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate.

1152(a)(1)(A). [N]o person shall receive any preference or priority or be discriminated against in the issuance of an immigrant visa because of the person’s race, sex, nationality, place of birth, or place of residence.
So which is it? Can the President ban everyone from Country X, or may the President do no such thing? Or is there no contradiction here after all, because 1182(f) refers to "entry" while 1152(A)(1)(a) refers to 'issuance of a visa'?

Recently, when the Fourth Circuit upheld a temporary restraining order on the travel ban, it too bypassed this statutory question and reached constitutional questions, finding against the Government on First Amendment grounds. However, one of the concurring judges, Barbara Milano Keenan, did say that she would have arrived at the same conclusion by the simpler expedient of statutory interpretation. Keenan's argument wasn't that 1152(a)(1)(A) overrules 1182(f); indeed, she indicated that thought the two were compatible on their face. Instead, her statutory reading was confined to 1182(f) itself, and in particular the burdens it places on the President by its text. Here is the crux of Keenan's argument for why the travel ban exceeds the powers granted to the President by Congress in the Immigration and Naturalization Act:
        The plain language of Section 1182(f) permits a president to act only if he "finds" that entry of the aliens in question "would be detrimental to the interests of the United States" (emphasis added). In my view, an unsupported conclusion will not satisfy this "finding" requirement. Otherwise, a president could act in total disregard of other material provisions of the INA, thereby effectively nullifying that complex body of law enacted by Congress.
        Here, the President's "finding" in Section 2(c) is, in essence, a non sequitur because the "finding" does not follow from the four corners of the Order’s text. In particular, the text fails to articulate a basis for the President’s conclusion that entry by any of the approximately 180 million individuals subject to the ban "would be detrimental to the interests of the United States."
In other words, a finding must be more than a bare assertion—and what's more, the Executive Order, read carefully, doesn't even make a bare assertion. It includes some rhetoric about the countries and how dangerous those countries are, but it never even claims that the aliens are dangerous, let alone finds them to be so, with any of the due diligence or rationality that is connoted by the notion of a "finding" in law.

Keenan's argument appears to have been influential beyond her own circuit, because today, in upholding yet again the temporary restraining order against the Government, the Ninth Circuit court backed away from its prior constitutional reasoning and embraced Kennan's approach. Like Keenan, the Ninth Circuit held that (here I give a bunch of excerpts):
the President did not meet the essential precondition to exercising his delegated authority: The President must make a sufficient finding that the entry of these classes of people would be "detrimental to the interests of the United States."

There is no finding that present vetting standards are inadequate, and no finding that absent the improved vetting procedures there likely will be harm to our national interests. 

The Order makes no finding that nationality alone renders entry of this broad class of individuals a heightened security risk to the United States

The Order does not tie these nationals in any way to terrorist organizations within the six designated countries. It does not identify these nationals as contributors to active conflict or as those responsible for insecure country conditions. It does not provide any link between an individual's nationality and their propensity to commit terrorism or their inherent dangerousness.

the Order does not provide a rationale explaining why permitting entry of nationals from the six designated countries under current protocols would be detrimental to the interests of the United States.

The Order's discussion of country conditions fails to bridge the gap. Indeed, its use of nationality as the sole basis for suspending entry means that nationals without significant ties to the six designated countries, such as those who left as children or those whose nationality is based on parentage alone, should be suspended from entry. Yet, nationals of other countries who do have meaningful ties to the six designated countries—and may be contributing to the very country conditions discussed—fall outside the scope of Section 2(c). Consequently, EO2’s focus on nationality "could have the paradoxical effect of barring entry by a Syrian national who has lived in Switzerland for decades, but not a Swiss national who has immigrated to Syria during its civil war."

 the Order specifically avoids making any finding that the current screening processes are inadequate. As the law stands, a visa applicant bears the burden of showing that the applicant is eligible to receive a visa or other document for entry and is not inadmissible. See 8 U.S.C. § 1361. The Government already can exclude individuals who do not meet that burden. See id. The Order offers no further reason explaining how this individualized adjudication process is flawed such that permitting entry of an entire class of nationals is injurious to the interests of the United States.

Former Presidents have invoked § 1182(f) under non-exigent circumstances to address compromised security conditions abroad but have tied exclusions to the culpable conduct of barred aliens, such as aliens who contributed to a country's situation in a specified way or were members of particular narrowly defined and/or dangerous groups. 

President Obama's Executive Order 13726, for example, suspended the entry into the United States of persons who were responsible or complicit in particular actions or policies that threaten the stability of Libya

In two instances, former Presidents have distinguished classes of aliens on the basis of nationality. But these distinctions were made not because of a particular concern that entry of the individuals themselves would be detrimental, but rather, as retaliatory diplomatic measures responsive to government conduct directed at the United States. For example, President Carter's proclamation barring the future entry of Iranians occurred during the exigent circumstance of the Iranian hostage crisis. 

President Reagan's suspension of entry of certain Cuban nationals as immigrants came as a response to the Cuban government's own suspension of "all types of procedures regarding the execution" of an immigration agreement between the United States and Cuba, which had "disrupt[ed] normal migration procedures between the two countries."

Section 1182(f) requires that the President exercise his authority only after meeting the precondition of finding that entry of an alien or class of aliens would be detrimental to the interests of the United States. Here, the President has not done so.


Logically speaking, I don't think the Ninth Circuit's decision hinges on resolving the contradiction between 1182(f) and 1152(a)(1)(A). But drawing a line between what Trump did and what Carter, Reagan, and Obama did becomes easier for the Ninth Circuit if 1152(a)(1)(A) limits 1182(f), and this unsurprisingly is what the Ninth Circuit finds. By applying several canons of statutory interpretations, the opinion imposes this reading on them:
In prohibiting nationality-based discrimination in the issuance of immigrant visas, Congress also in effect prohibited nationality-based discrimination in the admission of aliens.
The argument, basically, is that in a reasonable set of immigration rules, the rules for entry can't be meaningfully different from the rules for visas. Remember, though, that Judge Keenan opined differently in her concurrence for the Fourth Circuit. On the other hand, her fellow Judge Stephanie Thacker didn't join that part of her opinion…so it seems the basic meaning of the INA as a whole remains somewhat unsettled.


Having found the Executive Order to be in conflict with immigration law as written, the Ninth Circuit reminds everyone that if the legislative and executive branches don't like the law, they're perfectly entitled to come together and rewrite the law to their mutual satisfaction:
We have based our decision holding the entry ban unlawful on statutory considerations, and nothing said herein precludes Congress and the President from reaching a new understanding and confirming it by statute. 
Indeed, it is Congress's inability (or unwillingness) to write legislation over the past ten years that has encouraged Presidents to write so much policy via Executive Order. In this trend Trump represents a new low, because thanks to his administration's overall incompetence, his Executive Orders aren't much better than press releases. Another word for policy-by-press-release is authoritarianism.


In a series of footnotes, the Ninth Circuit did make several down-payments on constitutional questions, in case those questions arise later in the Supreme Court. Probably the juiciest footnote in the opinion is the one that goes out of its way to rule Trump's tweets in-bounds according to the Federal Rules of Evidence:
See Donald J. Trump (@realDonaldTrump), Twitter (June 5, 2017, 6:20 PM), ("That’s right, we need a TRAVEL BAN for certain DANGEROUS countries, not some politically correct term that won’t help us protect our people!") (emphasis in original); see also Elizabeth Landers, White House: Trump's tweets are "official statements", CNN (June 6, 2017, 4:37 PM), (reporting the White House Press Secretary's confirmation that the President's tweets are "considered official statements by the President of the United States"). We take judicial notice of President Trump’s statement as the veracity of this statement "can be accurately and readily determined from sources whose accuracy cannot reasonably be questioned." Fed. R. Evid. 201(b)(2).
I had to laugh when I read that parenthetical "emphasis in original." Translation: GRANDPA WRITES IN ALL-CAPS.

Friday, June 9, 2017

i.e. and e.g.: I'm Done

Instead of trying to explain the difference between i.e. and e.g., I'm going to propose today that we, all of us, strike these abbreviations from our writing entirely.

Reading a Supreme Court decision last week, I was amazed to see that e.g. was used where i.e. was meant.

Unless I'm mistaken on the substance—always possible, since I'm not a lawyer—Justice Sotomayor wanted the sense of in other words here; but she wrote e.g., which has the sense of for example.

Folks, written English doesn't get any higher-status than a published decision of the U.S. Supreme Court. If a distinction isn't observed there, then it's hard to maintain that the distinction exists.

This week at work, I was sent a memo that used i.e. for e.g., or maybe e.g. for i.e., or maybe both (it happened more than once). The writer was a Harvard graduate.

I'm not criticizing Harvard here, or Yale for that matter (Sonia Sotomayor's alma mater). Rather, what I take from these two examples is that i.e. and e.g. are strictly meaningless

From now on, if I want to say "for example," then that's what I'll write. Likewise for "that is," or "in other words." 

Abbreviations mar prose anyway. I hereby purge i.e. and e.g. from my written lexicon.

Tuesday, May 9, 2017

1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13, ...

Sequences are ordered progressions of numbers or other objects, as in the examples

1, 2, 4, 8, 16, …


A, B, AA, BB, AAA, BBB, …. 

In school mathematics, sequences are often called "patterns." I don't like that usage, because it's terribly limiting. Mathematical patterns can be found in visual designs, crystals, the multiplication table, or some totality of facts…patterns aren't just about sequences! You'll often hear people say, "Mathematics is the study of patterns." They don't mean, "Mathematics is the study of sequences."

My colleague William McCallum has a useful dictum: "Patterns are a tool, not a topic." For example, patterns in the multiplication table could be used as a tool for teaching about the properties of operations; patterns in the sequence 1, 2, 4, 8, 16 could be used as a tool for teaching about exponential functions. Whatever work is done with sequences at a given grade ought to transcend 'patternology' to intentionally build up students' strengths in the most important mathematical topics at each grade level.


Lately in idle moments I'll try to pass the time by thinking up a sequence of numbers that isn't yet included in the Online Encyclopedia of Integer Sequences (OEIS). It's a fun game—like trying to think of a notable topic that isn't in Wikipedia.

This was my first attempt:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 30, 41, 50, 61, 70, 81, 90, 111, 200, ...

The rule of this sequence is that, beginning with zero, each successive number must share no digits with the previous number—and be as small as possible subject to that constraint. For example, the number after 200 will be 311, because 311 is the smallest number greater than 200 that doesn't have 2 or 0 as a digit.

This is sequence A030283 in OEIS.

To create the sequence whose initial terms are in the title of this post, I considered a sequence of regular polygons with sides of unit length (triangle, square, pentagon, hexagon, etc.). Choosing a standard position for the polygons (centered at the origin), and also choosing a standard orientation (with the top edge horizontal), I counted how many grid points (m, n) were inside the polygon or on its boundary. (Here m and n are integers.)

Here are some diagrams showing the first eleven values 1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13:

My sequence doesn't appear to be in OEIS. There's a partial match with A219844, but the two sequences aren't the same.


When you look at the numbers 1, 1, 1, 3, 5, 5, 7, 9, 9, 9, 13, what do you notice? Will the feature you noticed hold true for the entire sequence, out to infinity?

Feel free to write any observations, conjectures or arguments down in the comments. Also, feel free to put a sequence of your own in the comments!


My daughter and I talked a bit about sequences during Saturday School a few weeks ago. That morning, my daughter was writing in a workbook she'd brought home from school. Looking across the table, I noticed that whoever had written the workbook had misunderstood 4.OA.5, a standard that involves sequences. In an effort to rescue the math, I quickly sketched the following problem:

Start with 256. At each step, divide by 2. Repeat forever!

a) Show 4 steps of the pattern (sequence)

b) Prove or disprove: every number in this pattern (sequence) is a whole number.

(As you can see, I took this opportunity to reinforce fractions and calculation of quotients. Also, note that the rule for the sequence is given. This is a math problem, not an  IQ test.)

By making the starting number a large power of 2, I had hoped to lure her into believing that all of the numbers in the sequence would be whole numbers. That gambit didn't work, however—she saw from the outset that the values would eventually drop below 1, and she proceeded to show this by generating sufficiently many values. (Actually, more than enough values! I then drew her attention to the second of the two arguments below.)

My daughter had fun solving this, and we enjoyed talking about it. What's cooler than infinity? Done right, I think 4.OA.5 can prompt some very good mathematics. I expect it has also helped publishers and school systems hang onto old-style, non-aligned "pattern" work in the curriculum.

Thursday, May 4, 2017

My Own Struggles With Grammar

Articles on grammar seldom feature the author's own mistakes or shortcomings. The genre typically delivers advice, not confession. But in order to balance my other posts on the topic, I thought I'd list some areas where I tend to struggle.

Words That Give Me Trouble
  • Embarrass is hard for me to spell (one r or two?).
  • There's a dead neuron in my brain where the proper spelling of maybe is supposed to be stored: sometimes I write mabey the first time!
  • Are dubious and doubtful synonyms? I can't tell.
  • I only recently learned the distinction between masterful (done in a domineering way) and masterly (done in the way an expert would do it). Almost nobody marks you down for using the words interchangeably, and the third definition of masterful in the American Heritage 4th Edition makes masterful and masterly synonymous. However, the conceptual distinction itself is real, and I generally favor making distinctions so I'll give this one a try.
  • The word publicly just looks wrong to me! We don't write mysticly, franticly, or graphicly…. It should just be publically, a spelling I see in online writing from time to time.

Rules I Refuse To Follow 

  • Some of the classical rules (or "rules") of grammar. I split infinitives, occasionally end sentences with prepositions, and often use who for whom. (On these and some other matters I'm with Steven Pinker.) 
  • The rule about capitalizing words in titles. It's absurdly intricate…there's an entire website devoted to figuring it out! It also leads to typographic ugliness, as in

In the early years of this blog, I tried to respect the classical rule; an example would be "Reverie on the Principle of Equivalence" from 2007. Later I gave up and adopted the simple rule of capitalizing every word in the title. See for example the recent post "In Honor Of William Wootters On The Occasion Of His Retirement." This might be wrong, but even so I prefer the simplicity of my rule.

Emerging Usage I Have Embraced

  • Internet abbreviations. In informal email I'll use IIRC, BRB, nvm, and others.
  • They as singular pronoun. A few years ago I began losing patience with the construction "he or she." In email, I started abbreviating this to "s/he." Finally I gave up and started allowing myself to use they as a singular pronoun. (If I can easily revise it away then I do, but I don't work as hard at that anymore.) The big news in copy editing so far in 2017 is that the Associated Press style guide will now allow for limited use of the singular theyHere is Grammar Girl with the story.

Tuesday, May 2, 2017

Revisiting My Case Against Trump

In my post just before the election, "Why I'm With Her," I offered four reasons to vote for Hillary Clinton. I don't do a lot of overt politicking on this blog—my own politics are complicated anyway—but in this case, I was definitely writing to persuade. I was hoping to convert just one voter from non-Clinton to Clinton, and I was hoping to convince just one wavering Clinton voter to show up on election day. I forwarded the post to some friends and relatives, which led to some good conversations.

I wasn't enthusiastic about voting for Clinton, but I swallowed my reservations and titled the post with affirmative language, rather than more grudging language along the lines of "Why We Mustn't Elect Donald Trump." An anti-Trump title would have described my feelings better, but I didn't think that a negative message would bring a dissatisfied person out to vote or persuade a third-party voter to pull the lever for Clinton instead.

In writing the piece, I worked to craft a set of points that would be almost inarguable. I wanted there to be no good comeback to these items. I judged, for example, that even a Clinton hater would have to grant that Clinton's temperament is preferable to Donald Trump's.

After the election, I showed my list of reasons to a friend—a Republican who didn't vote for Trump or Clinton. My friend agreed with all of the points except one, the one about the alt-right vs. the Marxist Left. If my audience had been Republicans in particular, then I wouldn't have included that point.

Of course, one can agree with all of the individual points and still not believe that they add up to a yes vote for Clinton. I tried to make important points, and briefly underline their importance. Still, a person might have reasons of their own that they feel are more important. Some of the most persuasive essays I read during the campaign aired pro-Trump arguments and countered them directly. (Here was an example.)

Nobody can know how Hillary Clinton would be governing today, so there's no way to know whether my arguments were "right." Conversely, documenting the evidence against Trump in each of these categories would be a big job that I can't undertake. But below I list the reasons once again. They still seem right to me—perhaps even more than they did a hundred-odd days ago.

One writer I follow appears to be rethinking things. Andrew Sullivan recently published a thought-provoking piece titled "Maybe America Wasn't Crazy to Elect Donald Trump." Sullivan is a longtime Clinton critic who voted for her anyway and was as attuned as anybody to the downsides of Donald Trump. He argues now however that it might have been worse had Clinton been elected:
I still would never have risked putting this menacing clown into the Oval Office. But in the long run, if catastrophe doesn’t strike, it might even be better for the future health of our politics that Clinton is not president. 
Read Sullivan's case here.


1. She has a better temperament. The President's job is stressful, so I think temperament matters. Donald Trump seems excitable and impulsive, which in a President makes me nervous. Hillary Clinton seems like a calmer, more deliberate person.

2. Her base is more manageable. The alt-right is nasty enough as a fringe movement. If Trump wins, the alt-right will have a seat at the table in policy discussions.

Bad things would also happen if the Marxist left were to grow in power, but the situation isn't symmetrical. The relationship between Clinton and progressive activists isn't nearly as cozy as the relationship between Trump and the alt-right. And the Republicans in Congress won't roll over for the Marxist left the way they will for the alt-right.

3. She is more fact-based and can listen better. I don't like all of Hillary's policies, and I especially don't like her interventionist instincts. But I think she is capable of weighing the arguments of critics and capable of adjusting her plans in response to facts. Donald Trump can't take any criticism, and facts don't inform his plans at any stage.

4. Criticism of her seems overblown, while criticism of him is just. Matt Yglesias's take on the email thing is pretty much how I look at it, and for every bad trait of Hillary's, Trump has the same trait except worse (12). The record of Trump's bad character spans decades. He is a small, insecure person with a cruel streak and a terrible lusting after power for its own sake. Lots of people are trying to explain how it happened—but however it happened, this year the Republican party nominated a candidate who deserves to lose.

Sunday, April 30, 2017

In Honor Of William Wootters On The Occasion Of His Retirement

William Wootters, my earliest mentor in theoretical physics, will retire from Williams College this year. Wootters is a pioneer of quantum information theory: his groundbreaking paper "A single quantum cannot be cloned," coauthored with Wojciech Zurek in 1982, has been cited over 4,000 times.

Bill decided against having a Festschrift conference, so I won't be able to give a talk to celebrate his retirement. However, the alumni association invited Bill's former students to share reflections, and I wanted to share mine here as well.


The story of my physics career at Williams begins and ends with Bill Wootters. In my freshman year, Bill co-taught a course with Karen Kwitter about the major advances in physics and astrophysics during the 20th century. It wasn’t the typical starting point for physics majors, but for an under-prepared student like myself it was a perfect introduction to the subject. The fascinating course material, the lively teaching, and the professors’ abundant kindness all gave me courage to move ahead in the subject.

By the time of my senior year, I was completing my astrophysics major and writing an honors thesis in physics with Bill. It was a busy year for me, in some ways a tumultuous and overcommitted year. Bill’s kindness, charity, and steady professionalism kept my spirit calm and my mind on my tasks. Through his research supervision, Bill exposed me to new frontiers of physics and inculcated in me his own high standards for scholarship. He exemplifies the golden intersection of two of the most noble roles a person can have: on the one hand, scientist; on the other hand, educator.

When I arrived at Berkeley to study for a doctorate in physics, I couldn’t help noticing that my classmates who’d been undergraduates at large universities tended to know more than I did about particular topics—whether it be the Standard Model, or plasmas, or what have you. That first semester with its oral qualifying exams and its dozens of impressive classmates (Gino Segré! Nima Arkani-Hamed!) was once again an intimidating start. But I soon discovered that Williams had prepared me well. In truth, not all of my classmates understood the fundamentals as deeply as one needed to. I might not know everything, I often thought; but what I know I truly know. It was a gift to be given such a firm foundation on which to stand, and I thank Bill and the entire Williams physics faculty for that gift.

Tuesday, April 25, 2017

An Efficient Connector: Solution For the L Shape

In this post I'll provide the solution to the problem I posed recently:

For the L-shaped set shown here (1 unit tall and 2 units wide), find two points of the L such that, if the two points are joined by a straight line segment, then the average distance between points in the set decreases by the greatest possible amount.

(Measure point-to-point distances along the segments; that is, all "travel" remains within the L and/or the connector.)

Note, in the problem as phrased here, when you calculate the average distance between points in the "after" picture, you must consider not only distances between points in the L, but also distances between points in the L and points in the connector, and between points in the connector and other points in the connector. Call this the homogeneous form of the problem (the connector becomes part of the network being analyzed, as opposed to its being thought of as made of different stuff or serving a different purpose). One could also consider the inhomogeneous form of the problem, in which the points of the connector aren't considered when calculating the average for the "after" picture. ...

Here's my result for the homogeneous problem, which I haven't checked thoroughly (details here):

The optimal connector attaches to the vertical about 63% of the way up, and it attaches to the horizontal about 36% of the way over. Adding this connector lowers the average distance between points by 12.8%.

To solve the problem, I generalized the dimensions of the L and derived a general expression D(abcde) for the average distance between points in the network.

(See the PDF for the actual expression.) Next I specialized to the problem at hand by expressing a, d, and e in terms of b and c. (For example, d = 2 − c.) The result was a messy function of two variables, D(b, c).

Complexity of the expression aside, the nature of the problem guarantees that the behavior of this function will be pretty simple, with just one basin to find. Here's a contour plot of D(b, c):

The minimum value D0 = 0.8719… occurs at (b0 = 0.3725…, c0 = 0.7186…).

Mathematica can express D0, b0, and c0 in exact form, but the expressions are lengthy; for example, D0 is the fifth real root of a certain 18th-degree polynomial whose constant term is


That's it for the homogenous problem, although the foregoing should be thought of as provisional given my lack of stress-testing. 

By the way, having a general expression for D also makes it easy to solve cases where the L has a width:height ratio different from 2:1. For example, to analyze a square L, we can put d = 1 − c instead of d = 2 − c. Then the optimal connector looks like this:

The triangle in the upper-left corner has horizontal and vertical legs of equal lengths, about 0.364. The optimal connector for the square L decreases the average distance between points by about 15%.

Now if one wished to add another segment to decrease the average distance further….

Monday, April 24, 2017

Packing Books Into Boxes

As frequently as I've moved apartments in my life, I ought to be better than I am at packing up books. Taping down the flaps on a box, I'm apt to think: Might I have gotten more books in here if I'd done it differently? After so many years, why don't I have a system for this?

Apparently I'm not the only person who feels this way. Here's a moving company that offers advice for packing books—from the comments online, you can tell that book-packing vexes people.

"A constant puzzle with every box": that actually describes pretty well the abstract version of this problem, known as the three-dimensional bin-packing problem. A research article on the problem says that "The problem is strongly NP-hard and extremely difficult to solve in practice." Even the one-dimensional version of the problem is NP-hard.

To give a flavor of the subject, here's a snippet from the research article giving an approximate algorithm:

Right—why didn't I think of that?

For a much friendlier tour of the problem, click here.

I wonder if researchers have ever studied the book packing problem—that is, bin packing in the case where the bins are all 'book-shaped.' Conversely, perhaps the mathematicians who study bin packing might be able to learn something by talking to people who pack books for a living! Anyway here are a couple of videos for packing books. In common with some of the mathematical approaches I saw, both videos use strategies of pre-sorting the books and recursively creating layers. Check back the next time you need to pack up a library.

Saturday, April 22, 2017

An Efficient Connector: A Warmup Problem

(Updated below.) I haven't had much time to dig into the problem posed in my last post, that of adding an efficient connector to this L shape:

As a warmup however, I did calculate the average distance between points on the boundary of an equilateral triangle with unit side lengths (measuring distances along the boundary, as if the sides of the triangle were roads).

The average point-to-point distance is ¾, which is less than the length of a side. For details, see this two-pager.


Waking up fresh this morning, I think the mean distance is easy to calculate as follows. Once the first point is chosen, the second point is equally likely to be closer when measured clockwise or closer when measured counter-clockwise. If closer measured clockwise, all distances up to 3/2 are equally likely; if closer measured counter-clockwise, all distances up to 3/2 are equally likely. Therefore once the first point is chosen, all distances up to 3/2 are equally likely, and the mean distance is
\[\int_0^{\frac{3}{2}}{v\cdot\left(\frac{2}{3}\,dv\right)} = \frac{3}{4}\,.\]
This reasoning generalizes to any triangle if we replace 3/2 by the semiperimeter s:
\[{\rm mean\ distance} = \int_0^s{v\cdot\left(\frac{1}{s}\,dv\right)} = \frac{1}{2}s = \frac{1}{4}p\]
where p is the perimeter of the triangle.

Friday, April 14, 2017

An Efficient Connector: The Solution And Some Additional Questions

Problem from the previous post:

Airport authorities would like to build a connector road (dashed line) so that maintenance vehicles can drive from one runway to the other without having to go all the way back to the airport terminal.

Given that the connector will be vertical, how far from the terminal should it be?

In my approach to the problem, I neglect the startup cost of building the connector road. My goal instead is to minimize the average distance that a maintenance vehicle would have to travel in order to get from the top runway to the bottom runway.

With the coordinates shown, the problem is to minimize, with respect to s, the value of the definite integral
\[ \frac{1}{c^2}\int_0^c{\int_0^c{\left(h + 2(b/c)s+ |x-s|+|y-s|\right)\,dxdy}}\,. \]
Before giving the result, let's make a couple of intuitive observations. First, if the two runways were parallel, then the optimal location for the connector would pretty clearly be at the midpoint. Second, since the runways aren't parallel, putting the connector at the far right looks worse than putting it at the far left…in both cases, the vehicles have to trundle the entire length of the runway to connect, but when the connector is at the far right, the vehicles also have to cover a long distance to get from one runway to the other. From these observations alone, we might expect the optimal location of the runway to lie to the left of the midpoint.

Evaluating the integral and minimizing the resulting function of s, we obtain

sbest = ½(cb).

This result shows how the parameter b pulls the optimal location leftward.


We could generalize this problem very far by asking something like the following: Given a closed, bounded, non-convex set L in a Euclidean space, find two points of L such that, if the two points are joined by a straight line segment, then the average distance between points in the set decreases by the greatest possible amount. In this form (if it is well posed), the problem might be difficult; even without the complication of identifying an optimal pair of points, just calculating the average distance between points in a convex set is apparently not easy. (This 2009 paper has some promising references if you're interested. Another relevant paper is this one from 2012.)

For the sake of having something to chew on, let's ask the following:

For the L-shaped set shown here (1 unit tall and 2 units wide), find two points of the L such that, if the two points are joined by a straight line segment, then the average distance between points in the set decreases by the greatest possible amount.

(Measure point-to-point distances along the segments; that is, all "travel" remains within the L and/or the connector.)

Note, in the problem as phrased here, when you calculate the average distance between points in the "after" picture, you must consider not only distances between points in the L, but also distances between points in the L and points in the connector, and between points in the connector and other points in the connector. Call this the homogeneous form of the problem (the connector becomes part of the network being analyzed, as opposed to its being thought of as made of different stuff or serving a different purpose). One could also consider the inhomogeneous form of the problem, in which the points of the connector aren't considered when calculating the average for the "after" picture. That's more like the runway problem.

I haven't done the problem in either the homogeneous or inhomogenous form; feel free to beat me to it!

I chose the L because it seemed like the easiest possible asymmetrical case. (Update 4/14: maybe an easier case would be a pair of parallel line segments of different lengths…but let's stick with the L.) One could try the problem for other non-convex sets, such as a polygon boundary, a circle, or some other plane curve. Two-dimensional sets could also be considered, such as two square regions given in position with respect to one another (a situation considered in this paper, though not with the goal of finding an efficient connector).


After you add a segment to a given set, you could repeat the problem with the resulting set, and so on. (If there were ever a choice between several equally good pairs of points, that would have to be handled somehow, either by choosing one of the pairs randomly or by choosing all of them at once.) I wonder what the limiting set looks like! Some kind of lacy fractal? And how well can one do, in the end? That is, given an initial network, does there exist an infimum value for the average distance between points attainable by adding connectors ad infinitum?

Monday, March 27, 2017

An Efficient Connector

Airport authorities would like to build a connector road (dashed line) so that maintenance vehicles can drive from one runway to the other without having to go all the way back to the airport terminal.

Given that the connector will be vertical, how far from the terminal should it be?

In my approach to the problem, I neglect the startup cost of building the connector road. My goal instead is to minimize the average distance that a maintenance vehicle would have to travel in order to get from the top runway to the bottom runway.

For example, with reference to the next figure, suppose that the vehicle is initially on the top runway, at the point indicated by the gray circle. To get to the indicated point on the bottom runway, the vehicle must drive along the magenta path to the other gray circle.

One way to begin the problem is to create a formula for the distance between a given initial location and a given final location. Then, one can use integration to average the distances over all initial and final locations. Because the average will depend on the location of the connector, the possibility arises of choosing the location of the connector that minimizes the average.

I did the problem for the dimensions shown:

In case anyone wants to solve it for themselves, I'll post my answer at a later time.

If you don't know calculus, you can try using your intuition to place the connector in an efficient location. Do you think that the most efficient connector will be halfway across, or less than halfway, or more than halfway?

Follow-up questions: Was it safe to assume that the most efficient connector is vertical? What if the runways are asymmetrical, as in this configuration?

Sunday, March 26, 2017

On Having A Favorite

To enroll in a frequent-flyer program online, I had to answer half a dozen security questions, including "What is your favorite kind of vacation?" and "What is your favorite cold-weather activity?" Who has ready answers to these kinds of questions?

"I like red the best!" says the toddler, as if his outfit didn't already make that clear. For grownups as well, having a favorite is for people who are at the toddler stage in their appreciation of something. I have a favorite bourbon, and that should tell you that I don't know much about bourbon. A good way to know that somebody isn't much of a reader is if they have a favorite book.

Now it is true that when I eat in a familiar restaurant, I almost always order the same thing. Always the pork curry at the Thai place in my neighborhood, always the chicken tikka at the Indian place. There's more to Indian food than chicken tikka, of course, but that's why God created other Indian restaurants. And if I didn't want a Shackburger and a strawberry shake, then I wouldn't be at Shake Shack, now would I?

This is not to say that Shake Shack hamburgers are the only hamburgers I like. I also like an occasional double quarter-pounder, or a "Mexican style" hamburger with Jack cheese, salsa, and avocado. Can't go wrong with a barbecue burger either—that bacon and zippy sauce!

Also, crumbled blue cheese is excellent on a thick hamburger.

Yet there are people out there who say things like, "My favorite hamburger is In-n-Out." Hearing this always makes me sad, not because I object to In-n-Out burgers in themselves, but because having a favorite hamburger just seems like a sad way to live.

To have a favorite X is to care too little about X's to arrange your life in such a way as to be surrounded only by wonderful X's. As few ties as I own, I can't say I have a favorite one. I like them all, or else why would I have them? Do you really want to put on a tie and think, "Eh, not my favorite"?

On the other hand, there are costs to not having a favorite. I can spend a long time in the morning choosing a tie to wear. A trip to the bookstore can take me hours and lead to no firm decisions; likewise for a trip to my Netflix queue. If nothing else, having favorites is efficient: a fact well known to parents of toddlers.

Saturday, March 25, 2017

Piece Out

Back for a few more notes on grammar and spelling:

1. Please know that the past tense of lead is led.

LeBron James lead the Cavaliers through a relatively stress-free fourth quarter on the way to the win. (Sports Daily, December 2016)
LeBron James led the Cavaliers through a relatively stress-free fourth quarter on the way to the win.

2. The preferred past tense of plead is pleaded.

When arraigned Friday morning Goff plead not guilty and was assigned a public defender. Shortly before 1:30, through a second public defender, he plead guilty and was sentenced to 12 months in the Arkansas Department of Correction. (Booneville Democrat, March 2017)
When arraigned Friday morning Goff pleaded not guilty and was assigned a public defender. Shortly before 1:30, through a second public defender, he pleaded guilty and was sentenced to 12 months in the Arkansas Department of Correction.
My American Heritage Fourth Edition (2000) also lists pled as an acceptable spelling of the past tense.

Unfortunately, I see from the online version that the Fifth Edition now lists plead is an acceptable spelling of the past tense. My advice though is not to use that spelling. For one thing, it's inconsiderate writing—you'll trip up some fraction of your readers when they mistakenly read plead as present-tense the first time through.

I do think pled is fine, though according to the American Heritage entry linked above, the usage panel for the Fifth Edition prefers pleaded to pled by a wide margin.

And I see that the usage note doesn't even address plead as a past-tense spelling. I suspect that's because zero percent of the panel would find this spelling acceptable.

Although plead as a past-tense spelling is apparently widespread enough to be listed in the dictionary, I don't think it is used very often in high-status writing. Anecdotally, I find several thousand hits for a google search of "He pleaded guilty" on, but only about a hundred hits for "He pled guilty" or "He plead guilty." A quick scan suggests that the instances of "He plead guilty" tend to come from complicated constructions (" was required that he plead guilty...."), or from direct quotes of speech, or from internet comments.

Bottom line: use pleaded, or use pled if you prefer, but don't use plead because you might trip up your readers and/or come across like an internet commenter.

3. Don't hand-wave with around.

The committee developed a set of guidelines around ethics.
The committee developed a set of ethics guidelines for members.

The use of around, in the sense of the first sentence, is almost always a sign of vagueness. In the second sentence, we better serve the reader by stating more clearly what is going on—the guidelines are for members. Even if we don't give any additional information beyond the first sentence, we can cut out flab by writing
The committee developed a set of ethics guidelines.

Around, in the sense of the first sentence, is gaining currency. In addition to saving the writer the effort of being clear or concise, it appeals to those writers who worry that more common prepositions just don't sound smart enough.

4. Know your own verbal tics. Is piece one of them?

With the tic:
We haven't figured out the professional development piece.
We haven't figured out professional development.

Normally I don't comment on speech, just writing, but piece is a prominent verbal tic among knowledge workers. Their jobs require them to analyze systems into parts; it's natural to think of the parts as "pieces," and to speak accordingly. But I have yet to hear a sentence that wouldn't be improved by just not doing that.

Because it's unpretentious, I would even prefer
We haven't figured out the professional development thing.

Sunday, March 19, 2017

The Sliding Ladder Problem

Yesterday a colleague mentioned a problem that professors often assign in their Calculus courses. The problem went something like this:
A 5m ladder is leaning against a wall. If the bottom of the ladder is pulled along the ground away from the wall at a constant rate of 0.40m/s, how fast will the top of the ladder be moving down the wall when its bottom is 3m away from the wall?
(This is a version of the problem I found today at math.stackexchange.)

The problem is straightforward as stated (the answer is 0.3 m/s), but my colleague wanted to know what I thought, as a physicist, about the fact that the speed of the top point of the ladder exceeds any finite value as this point nears the ground. I can see from a different math.stackexchange page that this is a frequently asked question, so I thought I'd post some thoughts about it here.

Before I get to that, however, I ought to substantiate the point my colleague was making about the ladder's speed. The following animation helps to visualize the motion:

I added a reversed phase of the motion just because I think it aids the eye to see the "bounce."

The equations I used to generate this figure were

x(t) = t
y(t) = (1 − t2)½
0 ≤ t ≤ 1

where point F = (x(t), 0) is the location of the ladder's point of attachment to the floor at time t, and point W = (0, y(t)) is the location of the ladder's point of attachment to the wall at time t. At time t = 0, the ladder is vertical; at time t = 1, the ladder is horizontal.

These equations are the non-dimensionalized versions of

X(T) = vT
Y(T) = (L2 - v2T2)½
0 ≤ TL/v

where v is the constant speed of the ladder's foot, and L is the length of the ladder. (That is, x = X/L, y = Y/L, and t = vT/L.)

The functions x(t) and y(t) are differentiable on the interval (0, 1), which means that points F and W both have well-defined speeds during this interval. However, y(t) has no derivative at t = 1, not even a left-sided one; this means that the speed of point W doesn't exist at the moment of time when the ladder is horizontal.

People usually say that the speed of point W "becomes infinite" at t = 1…but if "the speed of point W" means |dy/dt|, then saying that the speed is infinite is saying that dy/dt = ∞, and I used to discourage my beginning Calculus students from writing such things, because I wanted them to appreciate that dy/dt is defined by a limit, and limits are numbers that either exist or don't.

(It's certainly true that the speed of point W exceeds any finite value as t approaches 1, and that's enough to justify asking for a physicist's take.)

To think about the ladder's motion physically, concentrate on the motion of the ladder's center of mass point. (The motion of the center of mass point tells you about the net force acting on the ladder.) Here's an animation showing the motion of the center-of-mass point.

The coordinates of the CM point are ½(x(t), y(t)). As the ladder moves, the CM point traces a circle of radius ½. This isn't uniform circular motion, however, because the CM point isn't moving at constant speed.

The next animation shows the horizontal and vertical locations of the CM point.

The CM point has zero horizontal acceleration during the motion. But the CM point has a downward vertical acceleration that increases without bound throughout the motion.

From the Second Law, Fnet, cm = Mtotacm, we conclude that during the time when the ladder is dropping, there is zero net horizontal force on the ladder, while the net vertical force on the ladder is downward, increasing without bound as the ladder reaches the ground.

So I think the main takeaway from physics is that no finite force is capable of making a ladder move this way.

To analyze the problem any further, I would want to specify the coupling between the ladder and the wall and floor. For instance, on the wall there could be a freely sliding ring, allowing point W to move vertically without resistance, but constraining point W so that it can't move horizontally. That version of the problem is considered in this article, again leading to the conclusion that the constraint forces cannot be finite. This had to happen, given the motion of the CM point during the stipulated motion.


The ladder problem reminds me of a class of problems treated by philosophers of science, called supertasks. A supertask is an infinite sequence of operations conducted in a finite amount of time. One question is whether supertasks ever happen, or are ever possible. Zeno's Paradox could be thought of as a question about whether motion itself is a supertask. You can read about supertasks on Wikipedia.

After Zeno, the most famous of a supertask is the example of Thomson's lamp; here is a paper about that paradox by John Earman and John D. Norton. I learned about some of Norton's work while writing my paper on the Law of Inertia and determinism in Newton's Laws.