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 clich├ęs 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.