## Thursday, December 23, 2010

### functions and CCR

A recent paper on arXiv.org talks about the CCR Instrument, not John Fogerty's guitar, but rather the Calculus Concept Readiness Instrument - a mathematics placement test from Maplesoft. The paper's brief background discussion on what concepts students need to be "calculus ready" is interesting. A large literature is cited that states that the most important concept that students need to be familiar with (for Calculus and for general mathematics education) is the function concept. Unfortunately, the function concept and the concept of function composition have often been cited as weak points for teachers as well as students (see this study by David Meel, for example).

A really nice article to read if you would like to expand your understanding of the function concept and get a sense of how it has mutated over the last few hundred years is Israel Kleiner's Evolution of the Function Concept: A Brief Survey.

As mathematics has advanced, functions have become strange, generalized, and freed from conceptual limitations that threaten to tie them down. Kliener quotes Poincaré in offering an explanation as to why, in contrast, our education proceeds with with simple (antiquated, in Meel's assessment) and well behaved functions.

If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum.

For those of us who are lucky enough to wander through this bizarre museum, the concept of function gets stretched in interesting ways. I seem to remember that learning about functionals in Linear Algebra expanded my appreciation of  functions and their strangeness. Here's an example: consider two sets $A$ and $B$, and suppose that $a$ is an element of $A$. There is a function, call it $\hat{a}$, whose domain is the set of functions from $A$ to $B$, and whose co-domain is the set $B$. This function $\hat{a}$ is defined by the rule $\hat{a}(f) = f(a)$, where $f$ is any function from $A$ to $B$. The first time you see something like this it seems lovely and weird - functions become elements, elements become functions, and the rule that defines the element-become-function $\hat{a}$ looks like a bit of notational sleight of hand. Things can get even stranger of course, such as when arrows (function-like-things) are defined without elements at all.

Concerns about "Calculus Concept Readiness" reminds us how preparation for Calculus is often seen as the ultimate goal for K-12 education. Should this be the case? Discrete/finite mathematics may provide a better goal (convincingly and quickly argued by Arthur Benjamin in his TED talk). It can also be argued that discrete math provides a better setting for learning the fundamentals of the function concept than pre-calculus courses that focus more on functions that are well behaved for the purposes of elementary Real analysis.

## Tuesday, December 14, 2010

### skeptics vs. constructivists

A short while ago, noted skeptic Michael Shermer wrote an interesting article about Steven Hawking's new book "The Grand Design" (another brief review from the Washington Post is here). In Shermer's analysis, Hawking's philosophy of model-dependent realism sounds a lot like radical constructivisim. Skeptical of the extent to which this philosophy should be pushed, Schermer asserts that Hawking's model-dependence has its limits, and that science provides an unprecedented means of overcomming the relativism that model-dependent realism and constructivism seem to lead to.

Coincidentally, there are a number of articles in the most recent issue of Constructivist Foundations (volume 6, number 1) that discuss the same issues that Shermer brings up in his assessment of Hawking's model-dependent realism.

What Shermer and the authors in Constructivist Foundations wrote about, and what seems to trouble people most about model-dependence and radical constructivisim vis-a-vis science, is whether or not science tells us anything about "reality." Shermer answers affirmatively:

"Yes, even though there is no Archimedean point outside of our brains, I believe there is a real reality, and that we can come close to knowing it through the lens of science — despite the indelible imperfection of our brains, our models, and our theories."

Radical constructivists, like Andreas Quale in his contribution to the Constructivist Foundations issue noted above ("Objections to Radical Constructivisim"), doubt that there is any true reality described by theories, even if the theories seem to be getting better:

"...a scientific theory is regarded as a model, constructed to address certain questions that we want to ask, and then imposed on natural phenomena. If the model is successful, fine – but this is then better seen as due to the capabilities of the constructors (scientists) [rather than being closer to reality]."

What does this have to do with math? The debate between scientific realists and relativists maps partially (but perhaps not exactly?) onto the debates between neo-platonists and constructivist/intuitionists and formalists in the philosophy of mathematics (see a few quotes here). Also, after reading a few articles by radical constructivists, I am surprised how many of them are mathematicians and/or mathematics educators (a notable example is Ernst von Glasserfeld, another is Andreas Quale, quoted above).

Few doubt the constructed nature of scientific theories. As Shermer states, the issue is how far you go with constructivist arguments (is there an ultimate reality, or is it turtles all the way down?), and whether ultimately you are a realist, or a relativist. Perhaps asserting the existence of reality requires a Kierkegaardian leap, not of faith, but of rationality - and also marks the reasonable limits of skepticism.

## Tuesday, December 7, 2010

### multiplication table with stars

Here is a printable (2 page pdf) for exploring multiplication patterns with star polygons. Not for memorizing the multiplication table, for exploring.

## Tuesday, November 30, 2010

### this old math blog

This post is an attempt to sum up the contents of mathrecreation. New stuff might get posted at some point, so this might not be a final halmos.

Some of the best posts were (not surprisingly) pointers to other people's stuff - Math Teachers at Play 17 and 23, and the Carnival of Mathematics 63.

Polygonal numbers were the main recreation explored on this blog - most of the posts on these can be found by wandering around.

Looking at polygonal numbers leads to looking at number triangles. In addition to Pascal, others found here include Lucas, Catalan, and the Harmonic Denominator Triangle.

Personally, I liked the book/article reviews - there really were not enough of them. These included Mesopotamian maths, friendly math, math frogs, and mathematical tree climbing.

At one point I worked on developing resources for high school teachers using Fathom, Tinkerplots, and GSP. Posts about these included Monty hall, Rosencrantz and Guildenstern, false positives, pi quilts,  Wallace lines, and the scrambler.

The fun of making mathematically themed objects can't be overstated. Origami is a great vehicle for this, particularly modular origami; but sometimes you might want to glue stuff together.

## Monday, September 27, 2010

### mazes and labyrinths

One of the not-so-subtly-pagen rituals of autumn that seems to be gaining popularity is labyrinth walking. Walking around in circles, preferably by candlelight, has been de rigueur around here for a few years at local fall festivals.

Maze-walking generally stands in stark contrast to this seemingly-closely related labyrinthine activity. For most people, to say that maze-walking is less soothing than labyrinth walking is an under-statement: instead of a growing sense of calm and mental stillness, you get a rising sense of panic and an increase in muttering. When going through a bunch of outdoor mazes a short while ago (right here!) I tried out a suggestion that Marcus du Sautoy casually mentioned in "The Story of Maths" (I think it was in the last episode): if you walk through the maze ensuring that your right hand is always touching a wall, you are sure to complete it (if there is a way to do so). It turns out that this is actually the standard algorithm for traversing a maze (it's in wikipedia, innit?)

The technique worked well, but the kids were a bit disappointed. When you say you have a method of solving a maze, those not used to the ways of math or computer science assume that it will somehow magically allow you to proceed by the shortest route through the maze avoiding every dead-end.

Instead, the right hand rule will lead you into every dead-end in your path, but following it ensures that eventually you worm your way out, and that you do not get lost second-guessing yourself (sounds like your typical computational approach). Following a rule to get through the maze transforms it into a labyrinth - you know that you will reach your goal and you can allow yourself to become lost in the process. Maze-walking in this way provides an example of how being mindful of the opportunities to apply algorithms allows you to become calm and meditative in even the most stress-inducing situations.

On a related note, a nice mathy way to draw a simple labyrinth is to start with a series of concentric circles. Draw a line segment from the innermost circle to a point on the outermost circle, such that the segment is tangent to the innermost circle. Then break each circle on alternate sides of the line segment to form a path.

## Thursday, August 26, 2010

### the new math, 1965

The clip below (you can find it here if it doesn't display) about the new math is from an 1965 broadcast of the CBC television program This Hour has Seven Days.

At the end of the segment, the studio audience provided generally balanced and thoughtful responses to the rather slanted questions posed by Laurier LaPierre - they voiced the concerns and issues that are perpetual companions of public education, and that we still hear today.

If you are wondering about the references to "the parachute man" at the beginning of the clip, you can see just about the full program here (or you can find it as clip #4 on this page).

## Tuesday, August 10, 2010

### mathematical beavers

A recent post at Maxwell's Demon points to the opinion pieces of Doron Zeilberger. Many of Zeilberger's brief posts are sharp and insightful, and all are worth reading. One that caught my eye was "Opinion #95: We need both Birds and Frogs, but most of all we need good Beavers" refering to an essay written by Freeman Dyson (noted here). One theme that is constant through many of these is Zeilberger's unapologetic and enthusiastic championing of the algorithmic/computational approach to mathematics and the  use of computers as creative mathematical tools.

## Tuesday, July 20, 2010

### pi day down under

My maths by mail subscription just notified me that this Thursday is considered pi day in Australia (22/7 rather than 3.14). I am used to celebrating in March, but I am glad to hear that we can justify celebrating twice a year.

Unfortunately, there seems to be only one day that we can celebrate as tau day, and only one day for celebrating e.

## Wednesday, July 7, 2010

### eschering and coxetering

Much has been written about how the work of M.C. Escher was inspired by mathematics and has inspired mathematicians in turn. The relationship between math and art in the work of Escher is still the subject of analysis and discussion: in this month's Notices of the AMS, there is a very interesting article on the mathematical aspects of M.C. Escher's work by Doris Schattschneider.

As Schattschneider describes, the mathematician that alternately inspired and frustrated Escher in much of his work was Donald Coxeter. If you haven't read it yet, you should check out Siobhan Roberts's King of Infinite Space, a popular biography of Coxeter, which also discusses his relationship with Escher.

Way back in 2003, the hyperbolic tilings that Escher and Coxeter corresponded about were used for the Mathematical Awareness Month poster, and Douglas Dunham wrote an accompanying essay to explain how the image for the poster was created.

My own meager contribution: here are two little GSP activities (one here, the other here) inspired by some of Escher's plane tilings.

## Monday, July 5, 2010

### de Morgan's magic square

In The Elements of Arithmetic (1830), Augustus De Morgan (noted in Wikipedia for his peculiarities) has a little fun after explaining how to add multi-digit numbers. On page 20, De Morgan presents this exercise:

Not the most gentle exercise for someone who just learned how to add.

If you divide all the entries by 36, you'll see that this is a magic square of order 11 (magic constant 671).

The Wolfram Mathworld article on magic squares explains various ways of generating magic squares. It looks like  De Morgan used what came to be known as the "Siamese method" starting in the cell (6,7) with an order vector of (1,1) (shown in blue below) and a break vector of (0,2) (shown in red below).

## Monday, June 28, 2010

### football fallacy

Arnold King's short post on Soccer and the Law of Large Numbers made me wonder - when does the law large numbers 'kick in?' I also had to wonder whether King's analysis was an appropriate application of the law of large numbers, or is the expectation that in a high scoring game the lucky scores will even out an example of the law of averages fallacy?

Points in a game of skill are not random, so the law of large numbers applies only to what could be thought of as the occasional 'lucky' or 'unlucky' (i.e. somewhat random) points that disrupt the overall score. To have the law of large numbers apply to these, the amount of random points scored in a game would have to increase dramatically, I would think.

As pointed out by John Allen Paulos in his book Innumeracy, the law of large numbers won't help make a game fair - it is likely that one team will be ahead in 'luck' for the entire course of the game, no matter how long it is played. The unlucky gambler does not get luckier by playing longer.

In a higher scoring game, it may be argued (and perhaps this is what King is really saying) that the points accrued through skill will outweigh those acquired by luck. This reasoning does not suffer from the 'law of averages' fallacy, rather its correctness may depend on how you model the role of luck within a game of skill. Does the number of lucky points stay fixed, or is it proportional to the overall score? If we assume something closer to the former, then skill will eventually dominate luck; if the later is closer to the truth, then bigger scores won't help.

Exploring the law of large numbers and the fallacy of the law of averages is easier with coin flips than with soccer balls (consider the game played by Rozencrantz and Guidenstern).

(BTW, we all know that you can turn a doughnut into a coffee cup, but did you know you can also turn it into a doubly covered soccer-ball?)

## Monday, June 21, 2010

### colored phizz

In "Transitive Decompositions of Graphs and their Links with Geometry and Origami" which appeared in April's American Mathematical Monthly, Geoffrey Pearce described how to edge-color a unit-origami dodecahedron symmetrically using five colors.

Constructing a model based on Pearce's instructions is pretty straight forward if you use Tom Hull's Phizz Unit. Pearce's instructions make use of the usual planar projection of the dodecahedral graph - you need to color your model based on the diagram below (the dark lines represent a single color, you need to repeat the pattern five times with five distinct colors).

If you do this (hopefully with less garish colors), you end up with a model that looks like the one below. In a single picture, you can't really get an appreciation of the symmetry of the coloring, but if you make it yourself and handle it, you'll see that it is indeed a symmetrical, transitive, coloring.

If you want a minimal coloring, rather than a transitive one, you'll only need three colors. A neat way to generate a three-coloring of the dodecahedron is to first find a Hamiltonian circuit for it. To do this, start with two colors - one color to mark an edge-path that touches each vertex only once, and the other color to mark the edges that are not part of the path. Below is a phizz unit dodecahedron with a Hamiltonian circut in yellow (other edges blue). A question before you begin: how many blue edge units and how many yellow edge units should you fold?

Once you have a Hamiltonian circut on your dodecahedron, replace alternate edges on your circut with your third color. Doing this will give you a dodecahedron whose edges are colored such that every vertex has three distinctly colored edges intersecting at it (can you see why?).

Tom Hull describes how to use this method to color phizz buckyballs in his book Project Origami.

If instead of using Phizz units you use Sonobe units, you will obtain models that have the same characteristics, but look more like icosahedrons than dodecahedrons due to the nice duality between these two unit types.

## Thursday, June 17, 2010

### Apollonius, Descartes, Ford, and Farey

A while ago I posted briefly on Ford Circles. I wanted to point out an interesting short post on the Wolfram Blog by Ed Pegg Jr. that shows how to draw these in Mathematica.

If you can, you should also check out the article by Dana Mackenzie on Apollonian gasket circle packing. Both Pegg and Mackenzie connect these packings to Descartes circle theorem.

## Friday, April 2, 2010

### three ring circus

Just for fun, think about what you see when you look at this diagram.  Do you automatically see it as a Venn diagram and start thinking about the principle of inclusion-exclusion? Maybe if you are more concrete minded, you see it as an Euler diagram and start imagining what concepts or things it might describe (see Indexed for examples, like this one).

Three circles of equal radius each circle passing through the center of the others provides a ruler and compass construction for an equilateral triangle (ok, you really only need two of the circles, but the third one balances the diagram nicely).

If you imagine the circles as rings, you probably imagine the famous Borromean rings, but how many (truly) different ring patterns could you imagine? (See answer here).

And if you look at the center of the diagram, you might see a trefoil knot, or triquetra pattern.

Trefoils are chiral and can be presented in many different ways - the picture below is also a trefoil (with an opposite 'handedness' to the one above) made using tiles (described a little here).

Going back to the original picture, you might be tempted to add more circles and make a nice flower pattern. If you have a mystical mind-set, you might see 'the seed of life' in this diagram.

Alternately, you might see a nice way to construct a regular hexagon.

Update: Here is a neat post on origami-blog that you should check out.
Update 2: The intersection of the three circles forms a Reuleaux triangle, as described here.

## 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.