This news story made the rounds many months ago, but I didn’t read of it until I was paring down my Inbox this week (1388 messages. It was getting a little overwhelming) and found it in a news digest. In late March, the Japan Aerospace Exploration Agency accepted a proposal of a project led by Shinji Suzuki to make origami spacecrafts and launch them from the International Space Station. How cool is that?

One worry is that they would burn up because friction from entering the atmosphere tends to make things rather hot, but it’s possible/likely that they won’t both because of their shape and because they will be traveling so slowly through the atmosphere (plus the paper, made from sugar canes, is heat resistant). When I first read this I envisioned Giant Origami Planes, but they’re actually small: the shuttles will only be 8 inches by 4 inches after folding, and weigh just over an ounce.

Another worry is that there is no way of controlling where they land, or even how to track them. This is a much bigger deal, and perhaps one reason they’re not going with the Giant Origami I’d envisioned (can you imagine if one of those swooped down onto your lawn?). But don’t go looking too soon: the grant they received is for 3 years of feasibility studies.

While you’re waiting, you can learn how to make an Origami Rocket.

I read this story in many places, but got most of the info for here from Discovery News.

My cousin Taimi is a linguist, and at our Big Family Reunion last month she told me that some people tried to develop a mathematical symbolism for language (language in general, regardless of the actual language spoken) and it worked well for speech that was meant to be informative. For speech patterns that were social and culturally based, though (when and how to thank someone, for example), the math language of language fell apart. It just couldn’t be applied universally correctly.

So when we returned to New York I looked up what this might be and found myself, well, confused. The closest example that I could find (which may or may not be what Taimi was referring to) was in a Keith Devlin column from 1996. He used an example from X-bar theory. (Doesn’t X-bar sound like a drinking establishment? In fact, there is such a place in Los Angeles, although it looks geared more towards Gen X than mathematicians).

In linguistics, X-bar theory seems to be a way of describing all sorts of phrases in a recursive fashion. So if you want to talk about the noun cat, you might add on some descriptions like gray or fabulous. The grammatical rules allow you to change X (a noun, a verb, a proposition, an adjective, etc.) into something that’s modified with things called complements and adjuncts, and the result is called X-bar and should be written but is often written just as X’ because it’s hard to typeset the whole “bar” thing. So a noun-bar (N’) would be fabulous gray cat (instead of just cat) or food bowl (instead of just bowl)

Then you can move up to X Phrases. An X Phrase is an optional specifier, followed by an X’, and then maybe some Y Phrases. This is written along the lines of
XP→(specifier)X’YP*

As an example a noun phrase (NP) would be something like “the fabulous gray cat” where the is the specifier and fabulous gray cat is N’. Another noun phrase is “the food bowl” where the is a specifier and food bowl is N’.

From this you can form a verb phrase (VP) “sees the food bowl”. Here the V’ is just the single verb sees but it’s followed by the noun phrase the food bowl. In other words, VP→(specifier)V’NP*. This is where the recursive part comes in, using Y phrases to build X phrases.

Presumably the next step would be to combine “the fabulous gray cat” and “sees the food bowl” into an actual sentence, but building sentences involves a whole other set of rules.

In many ways this reminds me of diagramming sentences, except that instead of starting with the sentence and breaking it down, the rules have to be developed in such a way that they can be described regardless of the actual sentence or even actual language.  Because the beauty and complication of this is that all of the X-bar rules apply regardless of whether your noun phrase is “the fabulous gray cat” or “el gato gris fabuloso” or “η μυθική γκρίζα γάτα”.

How big is an acre? It is, of course, 4046.9 square meters, although that 0.9 is shorthand for 0.8726… if you’re surveying in the USA and shorthand for 0.8564… if you’re not. But how big is that? If the answer of “four roods” doesn’t help you, then a popular comparison is that it is about 3/4 of a football field:

Created by Xyzzy n and posted under GNU Free Documentation License

It was in searching for a visual description of an acre that I ran across the great site How Much is Inside? (with translations into Arabic, Dutch, German, Spanish, and Swedish). Rob Cockerham and his crack team of reporters tackle all sorts of questions about non-standard measurement. For example,

• How much gold is inside a bottle of Goldschläger?
• How many CDs can be labeled with a Sharpie® before it runs out of ink? (more than you think!)
• How much actual granola is in a box of granola bars?
• And, of course, How much is inside an acre?

As a bonus, there’s a Science Club which does fun things like drop toast to see if it usually lands butter side down and dissect Hot Pockets. Hooray for science!

There was an interesting article in The Economist a few weeks ago about how to get an accurate estimate of a not-necessarily-well-known quantity (e.g. How many people were on board the RMS Titanic on its premiere voyage?). One way, which was already examined by James Surowiecki is to ask a bunch of people rather than just one. In general, the too-high guesses and too-low guesses start to cancel out, and the average number is a pretty good estimate, or at least more accurate in general than only asking one person.

But what if you don’t have a crowd? Then Edward Vul and Harold Pashler said that you can just answer twice. Indeed, whether you make two guesses one right after the other or two guesses several weeks apart, the average will (on average) be more accurate than the original guess. Better accuracy comes when the guesses are several weeks apart, but several commenters note that this may be because people looked up the answer in the meantime.

Let’s simulate this. I asked four random people to make two guesses as to the number of people who sailed on the Titanic . For comparison, Wikipedia says there were 2223 people aboard that night.

Person #1 First Guess: 900
Person #1 Second Guess: 2300
(Average: 1600, closer than the first guess!)

Person #2 First Guess: 2000
Person #2 Second Guess: 5000
(Average: 3500, a lot worse than the first guess)

Person #3 First Guess: 3200
Person #3 First Guess: 1400
(Average: 2300, nearly exact!)

Person #4 First Guess: 4000
Person #4 First Guess: 5000
(Average: 4500, worse than the first guess)

The average of the first guesses was 2525, which was more accurate than every first guess except the guess of 2000. This illustrates the idea that the average over crowds tends to be more accurate than an individual.

Now for the purposes of illustrating this article, the average guesses should be a little more accurate than the initial guesses. This happened for two people, but not for the other two: the average Average guess was 2975, which is less accurate than 2525. Dang, real data is so uncooperative. Of course, Vul and Pashler asked 438 people instead of 4, so maybe if I had asked more people I’d have gotten a result similar to V&P.

[You can see what seems to be the original article from Psychological Sciences here.]

Jean-Robert Argand was born on July 18, 1768. He was a bookkeeper and amateur mathematician, and is remembered for having introduced a geometric interpretation of the complex numbers as points in the Cartesian plane (a discovery that had been anticipated by Caspar Wessel in a paper published in 1799). Argand’s treatment of the subject appeared in a self-published anonymous monograph in 1806, which found its way to Legendre and eventually Jacques Français, who published an article about the idea and some of its ramifications, and asked for help in identifying the originator of the idea. Argand came forward, and a debate ensued in print as to the validity of working with complex numbers as geometric quantities, not merely algebraic objects (Argand and Français arguing in favor of geometry, François-Joseph Servois arguing against). This exchange served to solidify the association between Argand’s name and the geometric representation, and henceforth the name has stuck.

Curiously enough, Servois and Argand nearly share a birthday.

Mathematicians with birth or death anniversaries during the week of July 13 - 19:

July 13: Birthday of John Dee [1527]

July 14: Birthday of Laurence C. Young [1905] (generalized curves); deaths of Lorenzo Mascheroni [1800] (analysis) , Augustin Fresnel [1827] (wave theory of light), and Benjamin Gompertz [1865] (mortality rates)

July 15: Birthdays of Adolph Pavlovich Yushkevich [1906] (historian of mathematics) and Stephen Smale [1930] (differential geometry, Field’s Medalist 1966)

July 16: Birthday of Siegfried Aronhold [1819] (theory of invariants)

July 17: Birthday of Wilhelm Lexis [1837] (time series); deaths of Henri Poincaré [1912] (algebraic topology, polymath) and Adolph Pavlovich Yushkevich [1993]

July 18: Birthdays of Robert Hooke [1635] and Jean-Robert Argand [1768]; death of Abraham Sharp [1742] (calculator, mathematical and astronomical tables)

July 19: Birthdays of François-Joseph Servois [1768] (operator theory) and Charles Briot [1817] (theory of functions)

Source: MacTutor

Now is the time on Sprockets 360 when we plagiarize share some of the fine things going on around the Internet.

Are you looking for some fun math problems to do? You can try out the bi-weekly Monday Math Madness. MMM #10 at Blinkdagger is a probability problem with a twist: if you know the probability of a group winning the lottery over 25 years, what is the probability of someone in the group winning the lottery at least once in a 5-year period? Solutions are due by Monday night/Tuesday morning at midnight. MMM #11 will appear the following Monday on Wild About Math.

There’s also a new regular math problem being posted. Walking Randomly has started posting an Integral of the Week. Integral #1 is . (Incidentally, if you google “Walking Randomly” then the program suggests that you mean “Working Randomly”, which feels rather like my summer.)

Finally, why was 6 afraid of 7? Because 7 8 9. (Thank you, thank you, I’ll be here all week.) The Barenaked Ladies said it much better on this YouTube video, which I actually found here at Wiskundermeisjes, a site created by Ionica Smeets and Jeanine Daems. (This isn’t the first time I’ve stolen borrowed from them, either — I originally found the idea for Godzilla’s Sierpinski Cookies from Evil Mad Scientist Laboratories on their site. Indeed, their site really makes me wish I could speak Dutch.)

The Carnival of Mathematics took a little break, but it’s back with Carnival of Mathematics #36 over at Rigorous Trivialities, a blog of mathematics and other neat things (Congrats on your recent engagement!) hosted by Charles at the University of Pennsylvania. There are quite a few offerings here — from polygons, cats, and negative dimension all the way to GL_n. Lots to read on (what is here, back in Rochester) a rainy Friday!

Have any of your read Jodi Picoult? I confess, I hadn’t heard of her until just a couple months ago, when I noticed a slew of her books in the store and, reading the backs, thought they sounded interesting. Then I discovered that she’d written Nineteen Minutes (a fictional account of a school shooting), which I had in fact heard of but hadn’t read. So I read one of her books. And then another. And then another.

In other words, I’ve found that I like her books a lot: they’re a mixture of what I think of as summer reading [books that you can spend days reading while lounging out on the porch, or inside on the couch if it's not actually summer] and genuine ethical conflict. And to my surprise, I’ve encountered quite a bit of math in passing. Nothing central to the plot, but when was the last time I found a novel that mentioned cuisenaire rods? Or tangrams? Or algebra jokes?

Way to go Jodi!

The San Francisco Chronicle carries Sudoku by the comics, but it turns out that it has an additional Sudoku-like puzzle hidden in the classifieds. And by Sudoku-like, I mean not really at all like Sudoku except that it involves digits being put into boxes. Actually, it’s more like Kakuro now that I think about it, except you can have the same digit appear more than once in a row or column. So, in fact, it’s not really like either one.

The idea is that there is a 4×4 grid with one digit in each box. The sum of each row is given on the right (shown below in bold), the sum of each column is given below the column (shown here in bold), and the sum of the diagonals is also given. In addition, some of the entries are already given.

In theory, these could be solved logically, but it turns out that (at least for the puzzles in the Chronicle) there are multiple solutions, and the practice of making a guess, checking how far off you are, and then adjusting the guess works pretty well for these.

Aside: My brother-in-law Scott, who brought these to our attention (thanks Scott!) said he’d looked around on the internet for info on this puzzle but couldn’t really find anything. This may be due to the fact that “Cross Numbers” seems to refer to more than one game: I also found a game that was more like Crossword puzzles, with teach Across or Down clue leading to a number, one digit per square. You can see an example here.

This is the Impossible Triangle, also known as the tribar or as the Penrose Triangle.  It was invented by Roger Penrose Oscar Reutersvärd in 1934, and even though he didn’t get his name attached to it, he did get a postage stamp in his honor.  

Even though this figure is impossible, it’s been constructed several times.  Sort of.  Brian MacKay and Ahmad Abas designed one for the Claisebrook roundabout in East Perth, Western Australia:

Photo by Greg O’Beirne, published under GNU Free Documentation License.

The triangle also appears at the Deutsches Technikmuseum in Berlin, Germany.

Photo by EmptyTerms, published under GNU Free Documentation License.

Of course, this is how it looks from another angle.

Photo by EmptyTerms, published under GNU Free Documentation License.

If you want to create your own, here is a one page printable pdf with instructions for a paper model, from Cool Optical Illusions.  

I’ll leave you with a final video showing yet another possible impossible version.

Edited (7/7) to add:  The video I really wanted to include was this one: