## Tuesday, March 30, 2010

A post today on Revolutions called Scientists misusing Statistics and one posted last week on bit-player called Statistical Error both look at an article in Science News called Odds are, it's Wrong. While you are reading these, you might also want to read an older post from the Endeavor, Most published research results are false.

## Tuesday, March 23, 2010

### antiquated math word of the day

Today's OED word of the day (subscribe here) is sagitta.

A doubly-antiquated word (an old term for the old term 'versed sine'), it exhibits some interesting etymological connections between arrows, arches, fish-ears, sponges, and geometric constructions. I haven't yet parsed through the mathematical descriptions offered - it might be interesting to see how they connect to each other. The most surprising thing I learned from this (so far) is that there used to be a name for the middle horizontal line in an epsilon (an application of the geometric description).

sagitta
(sdt)
[L.,lit. an arrow.]
1. Astr. A northern constellation lying between Hercules and Delphinus: = ARROW n. 4. 1704 in J.HARRIS Lex. Techn. I. [And in mod. Dicts.]

2.  Geom. a. The versed sine of an arc: = ARROW n. 6. [1594: see ARROW n. 6.] 1704 in J.HARRIS Lex. Techn. I. 1726 LEONI Alberti's Archit. I. 9/2 The..Line..from the middle Point of the Chord up to the Arch, leaving equal Angles on each Side, is call'd the Sagitta. 1853 SIR H. DOUGLAS Milit. Bridges (ed. 3) 32 The sagitta, or versed sine, of the curvature being about one fifth of the side of the triangle.
b. In extended sense: The abscissa of a curve. Obs. rare01727-41 in CHAMBERS Cycl.

3. Arch. The key-stone of an arch. 1703 R. NEVE Builder's Dict. (1736). 1823 P.NICHOLSON Pract. Build. 592. 1849-50 WEALE Dict. Terms.

4. The middle horizontal stroke in the Greek letter . [App. an application of sense 2.]
1864 ELLICOTT Pastoral Ep. (ed. 3) 103 The thickened extremity of the sagitta of .
1881 Dublin Rev. VI. 134 The disputed line is really the sagitta of an epsilon.

5. Anat. ‘The sagittal suture’ (Cent. Dict. 1891).

6. Zool. a. One of the otoliths of a fish's ear. 1888 ROLLESTON & JACKSON Anim. Life 86 There are [in the ear of the perch] generally two large otoliths, a sagitta in the sacculus, an asteriscus in the recessus cochleae 1897 PARKER & HASWELL Text-bk. Zool. II. 199. b. One of the components of certain sponge-spicules: see quot. 1898 SEDGWICK Text-bk. Zool. I. 83 The Triæne consists of the rhabdome, or shaft, and the cladome, which consists of the three cladi, a straight line joining the ends of the two cladi is the chord. The sagitta is a perpendicular from the origin of the cladome to the chord.

## Friday, March 19, 2010

### structuralism comix

A few posts back I suggested that Logicomix take on the intellectual history of structuralism - its story seems to parallel the one they told in their first graphic novel on the search for the foundations of mathematics.

I guess they had already thought of it - check out their new comic about one of structuralism's founders, Claude Levi-Strauss. Is there a comic on Ferdinand de Saussure (pictured above) forthcoming?

BTW: While you're browsing, don't forget to check out Math Teachers at Play 24, over at Let's Play Math. :)

## Wednesday, March 10, 2010

### polygonal wanderings

About three years ago I sent in a manuscript to Mathematics Teacher called "Triangulating Polygonal Numbers" - and it has finally made its way into the magazine's March issue. Phew!

Since writing the first draft of that article I've continued to wander along the polygonal number trail - usually recording something about them on this blog. To celebrate that old article finally getting published, I thought I would try to collect together a few of the neat things I've stumbled upon while wandering through this topic.

I can't exactly remember why I first started looking at polygonal numbers, but it may have been while trying to find examples of interesting diagrams to draw in Fathom. Using a bit of recursion, it turns out to be pretty easy to create a Fathom document that can draw nice diagrams where you can control the number of sides and the length of the sides of the polygonal numbers that are drawn.

(I haven't yet implemented the "polygonal number diagram maker" in a more open or free platform - but it would be a nice project.)

I found out later that another fun way to diagram these numbers is to put them on a quadratic number-spiral. The images below are for the triangular, pentagonal, 12-agonal, and 13-agonal numbers - the square numbers look very uninteresting when you plot them on this spiral. The image at the top of this post shows both the triangular and the hexagonal numbers plotted on the same spiral. (Update: see Mike Croucher's Mathematica and Python implementations for drawing polygonal number spirals over at Walking Randomly - the implementations are straight forward and the images look great.)

The MT article looks at generalizations of the familiar $s_n = t_n +t_{n-1}$ identity, which tells us that a square number is the sum of two triangular numbers. From a geometric point of view, its obvious that you can split a polygon into triangles, but I thought it was interesting that you could also split a polygonal number into triangular numbers. A nice outcome of this geometric point of view is that it provides some nice "proofs without words" (the diagram below illustrates a relationship between hexagonal numbers and triangular numbers, $h_n = t_{2n-1}$).

Another interesting way to generalize things is to look at "higher dimensional" polygonal numbers. If you look at three dimensional polygonal numbers (visualized as stacked pyramids of spheres with different polygon bases), the familiar $s_n = t_n +t_{n-1}$ shows up in the standard $n \times n$ multiplication table. It turns out (surprisingly) that the upward sloping diagonals of the standard multiplication table sum to tetrahedral numbers, and of course, the main downward diagonal is the sum of all the squares in the table. So, the sum of the entries in the main upward sloping diagonal and the one above it is equal to the sum of the entries in the main downward sloping diagonal (shown in the 4x4 multiplication table below).

These identities among the higher-dimensional triangular numbers come in handy when you try to sum their reciprocals.

A different way of generalizing this same idea is to to look at splitting up higher powers into higher-dimensional triangular numbers. We know that $n^2 = t_n +t_{n-1}$, but what aboutn $n^3$, and $n^4$, and so on? Exploring this question leads you to another very interesting set of numbers, the Eulerian numbers, which show up as coefficients in the equations below (the 'exponent' on the $t$ is just an index indicating its dimension - $t^d_n$ is the n-th d-dimensional triangular number).

"Higher-dimensional triangular numbers" is a bit too fancy sounding - these things are much more recognizable to most people as the diagonals in Pascal's Triangle (2-dimensional triangular numbers are highlighted in grey in the image below) - the "dimension" corresponds to the diagonal column number (starting with index zero, for the "zero dimensional triangular numbers" which are just the constant sequence of 1's).

The other polygonal numbers (square, pentagonal, etc.) also occur in (less) well-known number triangles - the Lucas and Gibonacci Triangles.

If you look at the polygonal numbers for any length of time, you begin to appreciate that there are  many formulas for them. One surprising formula for the higher dimensional triangular numbers is their ordinary power series generating function. I found this formula surprising because it illustrates an interesting relationship between the rows of Pascal's Triangle and its diagonals. It shows that if you take the reciprocal of a particular expression whose coefficients are taken from a row in Pascal's Triangle, you get a formal power series whose coefficients are the entries in a corresponding diagonal column of Pascal's Triangle. Well, I was surprised, at least.

Those are the main highlights of my tour of the polygonal numbers, for now. Mathematicians (and idlers) have been exploring them since (at least) the time of Pythagoras, so I'm confident they'll still be around when I have time to look at them again.

## Monday, March 8, 2010

### formulas for fidelity and infidelity

I am sure someone could put together a better post on this topic. I really don't have the time, but when I stumbled on these two items on same day I couldn't resist putting something down.

First, take a look at the mathematical formula for the perfect wife. Please forgive the sexist slant to this, particularly today. I suppose a better title would have been 'mathematical formula for the perfect life partner,' but a silly article deserves a silly title, I suppose.

Next, take a look at how this has been adapted to be the formula for the perfect affair.

It is left as an exercise to try to rehabilitate the above into something that actually could be said to make sense.

These 'formulas' make some explicit assumptions about the genders involved, and hinge on the fact that they are restricting our attention to pairings that involve a single male and female. This restriction tells me right away that no real mathematicians could have been involved, since the first thing that they would have done is generalize the result.

In either case, whether you are looking for happiness with the person you've publicly pledged yourself to, or the person that you are secretly breaking that pledge with, a key factor determining happiness (of the male, I assume) is that the woman is more intelligent than the man. Probably not too hard to arrange. Since this seems all very male-centric anyway, maybe the only truth that we can get out of this is that dumb men are happier in all cases.

The original research paper that prompted all this looks like it would also provide some giggles. It appears in the European Journal of Operational Research, Vol 202, Issue 2 (April, 2010). Here is the abstract:

Research shows that the success of marriages and other intimate partnerships depends on objective attributes such as differences in age, cultural background, and educational level. This article proposes a mathematical approach to optimizing marriage by allocating spouses in such a way as to reduce the likelihood of divorce or separation. To produce our optimization model, we use the assumption of a central “agency” that would coordinate the matching of couples. Based on a representative and longitudinal sample of 1074 cohabiting and married couples living in Switzerland, we estimate various objective functions corresponding to age, education, ethnicity, and prior divorce concerning every possible combination of men and women. Our results show that the current state of marriages or partnerships is well below the social optimum. We reallocate approximately 68% of individuals (7 out of 10) to a new couple that we posit has a higher likelihood of survival. From this selection of new partners, we obtain our final “optimal” solutions, with a 21% reduction in the objective function.

Notice how they helpfully translate the 68% to 7 out of 10 for us. Thanks guys!

The vision of society that someone must hold in order to propose, even in a silly model, a central agency for allocating marriages is in keeping with one that would assume that someone who stays in a bad or even abusive relationship is more successful than someone who leaves.

In any case, here's an example of really dumb research giving way to even more ridiculousness as it gets picked up by the press (and, mea culpa, probably giving way to something even worse when people start blogging about it).

## Thursday, March 4, 2010

### Carnival of Mathematics 63

Welcome to the 63rd Carnival of Mathematics. Here's a property of the number 63, courtesy of Number Gossip:
Consider two functions $f$ and $g$ where
- $f$ maps a natural number onto its Roman Numeral (a string of letters);
- $g$ maps a word onto the sum of the numerical values of its letters (the value of a letter is its position in the alphabet).
It turns out that our number 63 is a fixed point of the endomorphism $gf$, ($63 \mapsto LXIII \mapsto 12+24+9+9+9 = 63$). Number Gossip assures us that there are not too many fixed points of this map.

What is a math(s) carnival? Please see this post by Mike Croucher at Walking Randomly. Want to see more mathematics carnivals? Check out the list of past incarnations of Carnival of Mathematics here, along with past Math Teachers at Play carnivals here (if nothing else, be sure to check out the most recent MT@P!).

The images appearing in this edition of the blog carnival are courtesy of Jeff Miller's mathematicians (and mathematics) on postage page.

Bryan of Soul Physics, presents two recent posts: How to time-reverse a quantum system and Unitary operators and spacetime symmetries.

John Baez and Mike Stay have both blogged about their recent paper Algorithmic Thermodynamics at the n-Category Cafe and reperiendi, respectively.

At bit-player, Brian Hayes has written an interesting post about someone who may have been the first proto-blogger on computational topics (circa 1970) in his post Gruenberger’s prime path.

Charles Siegel proves a big theorem in his post Monodromy and Moduli posted at Rigorous Trivialities.

Martin wonders where zero might be pointing in Zero's signs posted at Enigmania

Peter Rowlett's post Truchet, Braille and Euler from Travels in a Mathematical World provides us with a whole new way of seeing a familiar identity.

Fëanor presents Common Errors (31): Pythagoras posted at New at LacusCurtius & Livius.

International Women's Day is coming up next Monday (March 8th). Suzane Smith reminds us of 15 Female Scientists Who Changed the World posted at EKG Classes. Reading through this list reminded me of an excellent article, Not good with numbers, posted by Izabella Laba at The Accidental Mathematician.

On his new math blog, Puzzle Zapper Blog, Alexandre Muñiz presents Holy Hyperbolic Heptagons!. The neat hyperbolic diagrams there reminded me of M.C. Esher's hyperbolic tilings.

Coincidentally, Chaim Goodman-Strauss of the Math Factor recently blogged (and pod-casted) about the largest Escher exhibit ever.

A Twitter discussion about the relative merits of MathType and Latex has prompted Robert Talbert of Casting out Nines to blog about Five reasons you should use LaTeX and five tips for teaching it.

Inspired by the same tweets, Kate Nowak gives us a demonstration of how well MathType can work for those who've mastered their keyboards in MathType Challenge at f(t).

Keeping with the 'tools of the trade' theme, Mike Croucher proves that Mathematica does not suck by teaching us about Integrating Abs(x) with Mathematica posted at Walking Randomly.

And now, some introductions...

Jamie introduces his two new math blogs with What is Daily Mathercise? posted at Daily Mathercise, and What is Math Factoid? posted at Math Factoid.

Nancy Goroff introduces us to George Hart's work at Make Magazine's Math Mondays with Make: Online : Math Monday: Sierpinski tetrahedron posted at MAKE Magazine.

Following John Allen Paulos, students from the University of Leicester comment on mathematics in the media on their blog, Math Students Read the Newspaper.

John Cook presents his latest post on laws of numbers - large, small, and now medium in The Law of Medium Numbers — The Endeavour posted at The Endeavour.

Inspired by a recent essay by Micha Gromov, T of Meteoroids from Mindspace takes us on a tour of mathematical platonism in a context for Gromov's program.

Is there really such a thing as a coincidence in mathematics? By chance or by design, Pat Ballew has written about A Serendipitous Coincidence? The First-Ever Pursuit Problem. posted at Pat'sBlog.

Jason Dyer reprises a worthy, and surprisingly heated, topic in Multiplication is Not Repeated Addition? Revisited posted at The Number Warrior.

Jason's post prompted Sue VanHattum to ask What is Multiplication? posted at Math Mama Writes....

All the ensuing discussion on both blogs reminded me of 360's classic series of posts on the various ways we can multiply (warning for those with strong feelings: repeated addition is on the list).

Thanks to everyone who submitted, and to everyone who has visited. Be sure to submit your preK-12 math posts to Math Teachers at Play, and your everything-but-the-kitchen-sink math posts to the next Carnival of Mathematics. Be on the look out for the 64th edition of CoM, coming next month at Teaching College Math.

### teachers - born or built?

A very interesting article in the NYT Magazine, Building a Better Teacher, by Elizabeth Green came out earlier this week. On page 8 of the article, Green writes:
Working with Hyman Bass, a mathematician at the University of Michigan, Ball began to theorize that while teaching math obviously required subject knowledge, the knowledge seemed to be something distinct from what she had learned in math class. It’s one thing to know that 307 minus 168 equals 139; it is another thing to be able understand why a third grader might think that 261 is the right answer. Mathematicians need to understand a problem only for themselves; math teachers need both to know the math and to know how 30 different minds might understand (or misunderstand) it. Then they need to take each mind from not getting it to mastery. And they need to do this in 45 minutes or less.
And these are not even the most interesting observations that the article makes.