Sunday, October 8, 2017

Another Look At Triangles With Area 1 And Perimeter 8

Here again are all the triangles with area 1 and perimeter 8, this time shown with their circumcircles.

The values of the minimum and maximum radii involve cube roots; they can be found by extremizing xy(8 − xy) subject to the constraint (x − 4)(y − 4)(x + y − 4) = ¼.

Wednesday, October 4, 2017

Some Notes On The Wisconsin Gerrymandering Case

I started following Gill v. Whitford because it's such a high-stakes case. Even recognizing that politics is a dirty game, I think we should worry when a minority of the voters can secure a large majority in the statehouse election after election. I wish we could treat politicians like the toddlers they are, and apply the rule of "you cut, I choose." But apparently it doesn't work that way.

Oral arguments happened on Tuesday. I've read contradictory tea-leaves analysis (1, 2, 3). I do wonder, though, if Roberts on Tuesday viewed Kennedy as a lost cause and sought to use the questioning time for purposes of damage control. Roberts emphasized the policy risks of a decision for the plaintiffs…might this convince Kennedy of the need to write as narrowly as possible in their favor? Or perhaps the difficulty of setting a manageable standard will deter Kennedy one last time from striking down a legislative map on the basis of partisanship—even a map like Wisconsin's that all nine justices would probably stipulate is corrupt. The Supreme Court might well end up adopting the view of the dissenting judge in the district court, who wrote, "I am … unable to conclude that Act 43's passage was anything other than the kind of “politics as usual” that courts have routinely either tolerated or acquiesced in."

Some reflections on the controversy.

  • News outlets usually describe the Wisconsin case as a dispute between Republicans and Democrats. However, the parties in Gill v. Whitford aren't the Republicans and the Democrats; the parties are the State and the citizens who are seeking relief. When Wisconsin legislators drew the challenged congressional districts, they clearly had party in mind; yet they weren't on party business. They were enacting law with the force of state government. Likewise, though the plaintiffs in the case belong to the party out of power, they bring their case as citizens alleging that the State has infringed their individual Constitutional rights. The more you can ignore the media's horse-race coverage, the better you will understand the arguments in the case.

  • Kennedy is interested in gerrymandering as a First Amendment issue. To a layperson like me who tends to think of the speech part of the First Amendment as a "free-to-be-you-and-me" guarantee, this has been confusing. However, by putting together bits and pieces of Kennedy's previous decisions (as quoted in the district opinion), it seems that the First Amendment is supposed to function not only as a liberty guarantee, but also as an essential mechanism for democracy. It's as if the important speech from the Constitutional perspective is political speech. The give-and-take of public debate on the candidates and issues of the day is how we come to agree on what ought to be done or not done by government and who should represent us in that activity. And similarly, the kind of "free association" that matters for our system of government is particularly the getting together to have such discussions and debates. Perhaps this explains why Kennedy believes that corrupt districting is a First Amendment issue: it threatens to render coming together pointless, and threatens to remove from the occupiers of the legislature the burden of accounting for their behavior to the people at large in the public forum.

Amendment I
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.

Some notes on the district court decision.

A key passage in the district court opinion (p. 56) summarizes a long process of sifting through relevant Supreme Court decisions, in which there was often little consensus. The takeaway is a proposed standard for deciding political gerrymandering cases:

"the First Amendment and the Equal Protection clause prohibit a redistricting scheme which (1) is intended to place a severe impediment on the effectiveness of the votes of individual citizens on the basis of their political affiliation, (2) has that effect, and (3) cannot be justified on other, legitimate legislative grounds."

Amplifying the notion of intent in (1), a footnote says it is "an intent to make the political system systematically unresponsive to a particular segment of the voters based on their political preference", for example by creating safe seats and/or by making for an entrenched minority party.

For (2), the efficiency gap (EG) is used as evidence, not as proof or a standard; and not as the only evidence: it is also important that two elections have happened with the map in question. (The "entrenching" of power highlighted by Kennedy in previous decisions has something to do with duration. Also important to avoid hypotheticals.)

For (3), it is noted that that the maps went through many drafts, each one consistent with legitimate principles, yet each one better for the party in power than the one before—making legitimate principles no justification for the final map. Also, while the defense of the state's "natural Republican geography" is qualitatively valid, it is not quantitatively enough to justify the map.

(A footnote…In Zimblog-related news, Judge Ripple's written decision makes correct use of the abbreviations i.e. and e.g. However, I still hold to my argument that even correct usage of these abbreviations is unwise.)

Sunday, October 1, 2017


This year I watched all four of the major King Kong movies:
I'll compare the films in two ways.
  • First are my Oscar picks: this is the way I would hand out selected Oscars if the competition were only open to King Kong movies. 
  • After revealing the Oscar picks, I present my "King Kong Scorecard." Think of this as the way the Oscars might have been organized had they only ever been open to King Kong movies.
Three of the Kong movies are really good. Perhaps it's hard to make a really bad King Kong movie, because Kong is mythic, and the power of a myth is roughly independent of the particular telling. (C.S. Lewis: "The story of Orpheus strikes and strikes deep, of itself; the fact that Virgil and others have told it in good poetry is irrelevant.")

On with the show:


The award for Best Visual Effects goes to 2017's Kong: Skull Island. The effects in this movie were uniformly excellent. In fact, the final fight in Skull Island might be the best monster battle in the history of monster movies.

(Last place goes to the 1976 King Kong, basically a movie about a guy in a gorilla suit.)

Suave, right??
Best Actor in a Leading Role. Duh—it's Kong! But which one? There are good arguments for the 2005 Kong (utterly convincing as a primate) and for the 2017 Kong (utterly convincing as a monster). But I give the award to 1933 Kong, because even with primitive special effects he manages to evoke our sympathy.

Best Actor in a Supporting Role. Samuel L. Jackson gave a committed performance in Kong: Skull Island, as did John C. Reilly. But the Oscar goes to Charles Grodin in the 1976 King Kong, because he was the only actor in the film who seemed to be in on the joke. Always great to watch Charles Grodin camping it up.

Best Actress. Fay Wray (1907–2004) was a Hollywood icon (an obituary is here). She starred in many films, but the role of Ann Darrow made her a legend. In his review of the 1933 King Kong, Roger Ebert recounts an amusing anecdote:

"At a Hollywood party in 1972, I saw Hugh Hefner introduced to Fay Wray. 'I loved your movie,' he told her. 'Which one?' she asked."

There's only one Fay Wray. But the award for Best Actress in a King Kong movie goes to…

Naomi Watts. Watts is the 2005 film's center of gravity. She rightly has top billing in the film. She honors the historic role while making it hers. Watts has skills that few actors in any era possess. Her physicality and facial acting deliver the film's beauty-and-beast narrative, which is primary in the 2005 retelling. (Watts also has a good partner in Kong himself, played naturalistically by Andy Serkis via technology.)

Naomi Watts in King Kong (2005)

Best Supporting Actress. There is no supporting actress in a King Kong movie.

Best Cinematography. The 2005 King Kong.

Best Director. The directors in 2017 and 1976 didn't seem sure of what they were trying to do. The award goes to Peter Jackson, whose vision is fully realized in the 2005 King Kong.

From "Jungle Jungle" (2001), by Rico Gatson
Best Original Song. The award goes to Max Steiner (1888–1971) for "Aboriginal Sacrifice Dance," the music that accompanies the Skull Island marriage ritual in the 1933 King Kong. Steiner was a Hollywood musical legend who scored countless films including CasablancaGone with the Wind, and The Searchers. Steiner's Kong-song is both completely inauthentic and completely groovy, a complicated verdict that artist Rico Gatson might agree with: his 2001 video installation "Jungle Jungle" draws upon footage from the 1933 ritual sequence. (I watched the loop many times when I came across it playing in an art museum a dozen years ago.) Peter Jackson added a nice touch to his 2005 film when he used Steiner's music during the "Eighth Wonder of the World" sequence in New York.

Best Original Story. Naturally this has to be the 1933 King Kong. Most movies that get made and re-made over the decades tell stories from literature: Dracula, Frankenstein, Hamlet, Pride and Prejudice, Romeo and Juliet. King Kong is the only perennial movie I can think of that tells a story that was originally developed specifically for the screen.

Best Picture. This isn't close. Peter Jackson's 2005 King Kong is not only the best Kong movie, it's a wonderful movie in any genre. Jackson's King Kong is a vintage pleasure that can only be described with the language of a 1930s movie poster. It's got romance! Adventure! Thrill and chills! I watched it last of the four movies, because freeing up three-plus hours to watch it wasn't easy. But once I settled in, I appreciated its luxurious storytelling. (Admittedly, some of the set-pieces went on a bit long.)

A few highlights that show various elements of film-making working together at a high level:

  • The transporting Art Deco aesthetic in the opening titles, New York sets, costumes, and of course the Empire State Building itself. 
  • The steady current of homage to the original 1933 movie, including a conversation in a taxi in which director Carl Denham asks if Fay Wray is available to star in his movie; a restored "insect pit" sequence (the sequence along these lines was cut from the 1933 movie when it horrified audiences); and the Max Steiner music noted earlier.
  • The nightmarish encounter with the Skull Islanders. And here let's pause to note that the basic Kong myth, um, raises issues. In 2017, the makers of Kong: Skull Island moved to table such questions, by their handling of the Skull Islanders and by heavily de-emphasizing the beauty-and-the-beast arc. But the 2005 King Kong handles the touchiest issues by doubling down on them. For example, King Kong's "thing for blondes" is handled with tongue in cheek while Kong searches for Ann in the streets of New York. And the Skull Islanders in the 2005 film are the furthest thing you could imagine from a politically correct ideal of aboriginal peoples. There was no cliché, however; I was shocked, and gripped, by Jackson's nightmare vision of an ecstatically murderous society clinging to a landscape of barren crags. (Do they know that the other side of the island looks like Malibu?)
  • The lovely Central Park "ice-skating" sequence, an inspired addition to the traditional story. 
  • The Depression-era framing, which created atmosphere and added depth to Ann's character. 
  • The winsome sequence when Ann Darrow charms Kong on Skull Island.

King Kong Scorecard!

1933 1976 2005 2017
Pre-island buildup / sense of going on a voyage   A B A A
Mysterious island / pulp-era delights & horrors A C A A
"Beauty and the Beast" theme/pathos B B A C
New York mayhem A A A F
Special effects quality C C A A

Tuesday, September 26, 2017

Tone It Down (How Cliches Weaken Your Case)

     Surely: the adverb of a man without an argument.

                —From Bad News, by Edward St. Aubyn

In politics, I think people are best served by plain writing that is dense with testable claims and publicly verifiable facts. Writers do try to "energize the base," and in campaigns there is a place for that. But few stylists are capable of elevated prose about live political issues. The more energetic a piece of political writing, the worse that piece of writing probably is.

And the less convincing it probably is. Clichés can make a writer sound like somebody on a rant—like a loud TV you just want to turn off. Extra words subtract gravity from what you're saying. Stock phrases cover up the fact that no argument is being made.

Enough preamble—let's see if the following examples speak for themselves. In every case, I think that striking words would have made for a more effective statement. Sources for the original statements are listed at the end.

Weaker:  I have no swastika or Third Reich related tattoos. PERIOD.

Stronger:  I have no swastika or Third Reich related tattoos.

Weaker:  But there wasn’t a shred of evidence that any insurer had “abused” the boy or his mom.

Stronger:  But there was no evidence that any insurer had abused the boy or his mom.

Weaker:  Jessie Ford, who’s like this great sociologist at NYU, …

Stronger:  Jessie Ford, a great sociologist at NYU, …

Weaker:  If we recognize men and women who identify with the genders they were assigned at birth (cisgender) and we recognize men and women who do not identify with their assigned gender (transgender), then surely we agree this difference is worth recording. 

Stronger:  If we recognize men and women who identify with the genders they were assigned at birth (cisgender) and we recognize men and women who do not identify with their assigned gender (transgender), then this difference is worth recording. 

I may gradually add more examples to the list over time. Here are the sources of the original statements, in order:

Sunday, September 17, 2017

Answers to Amy's Multiplication Puzzle

From an earlier post:

Amy multiplied an eight-syllable number by an eight-syllable number and obtained a four-syllable number. What could her numbers have been?

(These are whole numbers we're talking about.)

To solve this, I first reflected that an eight-syllable number is necessarily pretty large…therefore the four-syllable number must be pretty large…but what kinds of large numbers only have four syllables? Probably something like "twenty trillion." A number like that ends with a lot of zeros, which means that it contains a lot of factors of 10, or in other words a lot of 2s and 5s.

So, I tried putting powers of 2 against powers of 5, and pretty soon I hit upon the following fact:

128 × 125 = 16,000

This is (6-syllable) × (6-syllable) = (4-syllable). Not far from Amy's problem! To patch it up, I added "thousand" to both factors; doing so increases the syllable count in each factor by two, but doesn't change the syllable count of the answer (because "sixteen thousand" becomes "sixteen trillion").

128,000 × 125,000 = 16,000,000,000

Now we have (8-syllable) × (8-syllable) = (4-syllable), as desired. Amy's numbers could have been 128,000 and 125,000.


Using a computer, I also found some interesting cases:

1) The smallest instance of Amy's numbers that I could find was 1,120 × 6,250 = 7,000,000.

2) Some extreme versions of the puzzle, with answers.

Amy multiplied a 22-syllable number by a 22-syllable number and obtained a 4-syllable number. What could her numbers have been?

6,103,515,625 × 1,474,560,000,000,000,000,000 = 9 nonillion

Amy multiplied a 26-syllable number by a 69-syllable number and obtained a 4-syllable number. What could her numbers have been?

2,147,483,648 × 1,396,983,861,923,217,773,437,500 = 3 decillion

3) The four-syllable number 9 septillion can be written as a product of two 13-syllable numbers in three different ways:

9 septillion = 156,250 × 57 quintillion, 600 quadrillion

9 septillion = 7,812,500 × 1 quintillion, 152 quadrillion

9 septillion = 39,062,500 × 230 quadrillion, 400 trillion

4) The four-syllable number 1,000,000,002 can be written as a product of two 11-syllable numbers in two different ways:

1,000,000,002 = 11,829 × 84,538

1,000,000,002 = 23,658 × 42,269

That's it for Amy's puzzle! In a later post, I'll share my code for generating number names. Once you have the number name, counting syllables is easy; but converting a digit string into a string of words is harder. UPDATE 9/21/2017: Instead of sharing my code, here is a webpage with number-naming code in many programming languages.

Saturday, September 16, 2017

Answer to Eyeballing Rates Of Change

Watch carefully and you'll see the straight lines both glowing green at the moment when the blue area leads the red area by the greatest amount. Were you in the ballpark?

Here is a video in MP4 format. And here is one-page walkthrough of the Calculus for the general case where ¼ is replaced by a general parameter λ.

Tuesday, September 12, 2017

Eyeballing Rates Of Change

At about which point in this movie would you say that the numerical difference between the blue and red areas is greatest?

A video file that you can play, pause, rewind, and download is here. One way to share an answer would be to pause the video at the moment of your best guess, take a screenshot, and email it to me at

I sometimes wonder if studying Calculus or working in a Calculus-heavy field makes people any better at perception tasks like this. I do believe that teaching kinematics for so many years improved my direct perception of acceleration; being able to see a couple of derivatives down has helped me to avoid some erratic drivers over the years.

In case any Calculus-trained readers would like to analyze the problem symbolically, I'll give the formulas I used to make the animation:

  • For any value of m ≥ 0, the blue region is defined by x ≥ 0, y ≥ 0, ym x, and yx ex
  • For any value of m ≥ 0, the red region is defined by x ≤ 0, y ≤ 0, y ≥ ¼ m x, and y ≥ x ex.

You should find a simple answer for mbest, the value of m that maximizes the difference blue areared area.

For some extra symbol pushing, replace y = ¼ m x with y = λ m x, where 0 < λ < 1. You should find \(m_{\rm best} = \lambda^{{\sqrt{\lambda}}/(1-\sqrt{\lambda})}\). The limit of mbest as λ approaches 1 is interesting.