## Tuesday, December 22, 2009

### the Lucas Number Triangle

Polygonal numbers have been a recurring topic on this blog (starting with the first post) and although they might eventually exhaust me, I've realized that I'll never exhaust their recreational possibilities.

One interesting way to look at polygonal numbers is to arrange them into number triangles - for a particular $$k \geq 3$$ you can do this by arranging the k-polygonal numbers in diagonals by dimension. If you do this with the triangular numbers, you get Pascal's Triangle. Said another way, the diagonals of Pascal's triangle list  the various dimensional extensions of triangular numbers: the 3rd diagonal gives the usual 'triangular numbers', the 4th diagonal gives the tetrahedrals (now to be referred to as the 'days of Christmas numbers'), the 5th diagonal gives the triangulo-triangular numbers, etc. Each diagonal lists the finite differences of the diagonal below it, so the second diagonal of Pascal's triangle gives what we could call the 'one dimensional triangulars' or the Naturals, and the first diagonal gives us 'zero dimensional triangulars' - the constant sequence of 1s.

The symmetry of Pascal's Triangle allows you to see the various dimensions of the triangular numbers in both the diagonals that slope downward from left to right, and those that slope downward from right to left. However, when you do the same sort of construction with $$k \geq 4$$ you loose this symmetry, and have to pick a direction. When I tried this (described here and here), I chose to make my generalized k-polygonal Pascal Triangles 'left handed', so that the k-polygonal numbers appear in the diagonals sloping downward from right to left. The diagrams below show the left-handed triangles for the square (k = 4) and pentagonal numbers (k = 5).

Using $$\left[\begin{array}{c}n\\r\end{array}\right]^{\mathcal{L}}_{k}$$ to denote the entry in the nth row of the rth diagonal column of the left-handed k-polygonal number triangle, we can express this construction recursively in a way that is very similar to the usual Pascal Triangle definition:

We can express the terms directly using the binomial coefficients:

Instead of making the triangles left-handed, we could instead make them right-handed like these:

Where now all the k-polygonal numbers appear as diagonals sloping downward from left to right. These are almost mirror images of the left-handed variety, and to be honest, I am not sure what the correct value is for the topmost entry. In these formulations, for both the left-handed and right-handed variety, the topmost entry is determined by the value in the far-right column.

Using $$\left[\begin{array}{c}n\\r\end{array}\right]^{\mathcal{R}}_{k}$$ to denote the entry in the nth row of the rth diagonal column of the right-handed k-polygonal number triangle, we can express the construction recursively:

And, just as with the left-handed variety, we can express the terms using binomial coefficients:

For the case where k=3, it is easy to see that both our left-handed and right-handed k-polygonal Pascal Triangles are equal to the usual Pascal Triangle.

It turns out that for the right-handed k = 4 case ('square' polygonals) gives a well-known number triangle called the Lucas Number Triangle.

In the Lucas Number Triangle you find the square numbers, the square-based pyrimidal numbers, and all of the higher-dimensional versions of the same, just as you find the triangular, pyrimidal, and higher-triangular numbers in Pascal's Triangle

In the same way that you can find the Fibonacci numbers by summing left-to-right upward sloping diagonals in the regular Pascal's Triangle, you can find Lucas numbers (see also Wikipedia) by summing  left-to-right upward sloping diagonals in the Lucas Number Triangle. Lucas numbers are a generalization of the Fibonacci sequence - the Fibonacci sequence starts with the numbers $$f_0 = 0$$, $$f_1=1$$, and proceeds according to the rule $$f_n = f_{n-1} + f_{n-2}$$; the Lucas sequence starts with the numbers $$f_0 = 2$$, $$f_1=1$$, and follows the same generating rule as the Fibonacci sequence.

Just as the entries in Pascal's Triangle can be interpreted as coefficients in the expansion of the binomial $$(a+b)^n$$, the entries in the Lucas triangle can be seen to be coefficients in the expansion of the binomial $$(a+2b)(a+b)^{n-1}$$ for $$n \geq 1$$.

In general, right handed k-polygonal Triangles give generalized "Gibonacci Numbers" - number sequences that follow the usual Fibonacci rule but with different first terms, in this case, $$G_0=k-2$$ and $$G_1=1$$. These Gibbonacci Number Triangles have entries whose rows give the coefficients of $$(a+ (k-2)b)(a+b)^{n-1}$$ for $$n \geq 1$$.

A nice overview of the Lucas and Gibonacci Triangles and an interesting combinatorial interpretation of their entries is given in Arthur Benjamin's The Lucas Triangle Recounted.

## Friday, December 18, 2009

### mathematical nature and nurture

Like so many things these days, this post was inspired by a celebrity (well, math celebrity) tweet. This one pointed to a letter published in the October issue of Nature.

The Nature article that du Sautoy referenced is here, and several other related papers by the same group of researchers are here and here. At some point soon I hope to write a bit more about this research, but for now I want to go back to the important issue that I think is raised by du Sauytoy's provocative tweet.

On its most essential level, the question posed by du Sautoy asks if some are born with better mathematical ability than others. This is going perhaps a step further than what was actually asked, but this is a step that many are likely to take, given that a positive answer to this question fits closely with many people's assumptions about mathematics.

Genetic gifts certainly play a role in how our lives turn out, and our ability to do certain kinds of math are not exempt from this. Temperament, attention-span, the gene responsible for holding a pencil properly, and other geneticly influenced factors may equip some more than others for mathematical activiites.

However, belief in the stronger version of this idea, that some are born with math ability while others are not, is one that has plagued the teaching and learning of mathematics for a long time, and may be responsible for excluding vast numbers of people from feeling mathematically competent.

What I believe, and will likely continue to believe until that belief is falsified by convincing evidence, is that mathematics is part of our shared cultural heritage, and is something that we can all lay claim to. Certainly there is much mathematics that I will never understand, but the basics, the fundamentals, and even the spirit that guides the most arcane mathematical research, is available to everyone.

Unfortunately, many feel excluded from the common heritage of mathematics, and part of the reason behind this are beliefs about the nature of mathematics. One belief that causes many to be excluded is precisely the one that states that some are born with the ability to do mathematics and some are not, an assertion that in many cases becomes self-fulfilling prophesy.

Many have written about harmful beliefs about the nature of mathematics - the most harmful being the idea that some are excluded from doing math at birth. Alan H. Schoenfeld has written about how metacognitive concepts like this influence our ability to learn and to teach mathematics. John Mighton has written about the question of math ability being hard-wired in his book The Myth of Ability. In the afterword to his book (online here), Mighton writes how his experience with JUMP tutoring has made him hopeful about challenging these commonly held beliefs - in his Elements of Humanity interview, he calls the belief that only some are born with mathematical ability to be an absurd illusion.

Although the study mentioned by du Sautoy was about measuring individual differences, it is useful to keep in mind how results like these are often misapplied to whole groups of people. In the classic book on group differences and intelligence testing, The Mismeasure of Man, Stephen Jay Gould cautions us:

We pass through this world but once. Few tragedies can be more extensive than the stunting of life, few injustices deeper than the denial of an opportunity to strive or even to hope, by a limit imposed from without, but falsely identified as lying within.

## Tuesday, December 15, 2009

### a Catalan number triangle fractal

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845...
the Catalan Numbers, OEIS: A000108

Catalan numbers occur in a surprising number of counting scenarios, such as counting the ways convex polygons can be cut into triangles, or counting how many ways you can write down a  legal list of brackets (a legal list of brackets is something like this: {{}{}}). In The Book of Numbers, Conway and Guy present at least eight different scenarios that the Catalans count, and Richard Stanley has listed 173 combinatorial interpretations for them (see the chapter and addendum on his Enumerative Combinatorics website). What unites these seemingly different scenarios is their recursive character (polygons contain smaller polygons, and lists of brackets contain smaller lists).

The Catalan numbers can be found in Pascal's Triangle  in a few ways. If you take each 'middle' element and subtract its adjacent entry, you get a Catalan number. Also, if you take a middle element and divide it by its position in the list of middle terms (e.g. divide the 5th middle term by 5), you will get a Catalan number. But Catalans have their very own number triangle too.

Starting, as with Pascal's Triangle, with a 1, the Catalan Triangle is generated using the rule that each element is equal to the one above (in this picture, above on the right slanting diagonal) plus the one to its left (where 'missing' numbers are zero).  The Catalan numbers show up both as row sums and as the last entry in each row.

The rule for creating the Catalan Triangle sounds like a 'slanted' version of the generating rule for the Pascal Triangle (where to produce a number you take the sum of the two above its position), and if you look at the fractal that is created by looking at the Catalan Triangle's values mod 2 (so that even numbers are replaced with 0 and odd numbers are replaced with 1), you get a slanted version of the Sierpinski fractal that you get from doing the same with Pascal's Triangle.

The fractals that sometimes arise from looking at modular values in a number triangle provide a visual clue to the divisibility properties of the numbers in the triangle, and also illustrate some aspects of the generating rule that is used to create the triangle. The Catalan fractal above was made with this Tinkerplots file using this data. The Pascal Traiangle was drawn with Tinkerplots using this file as described here.

## Thursday, December 10, 2009

### unfolding surprises

Here are two videos of some surprising unfoldings.

The first is from New Scientist, posted as part of this article. This shows some unfoldings of a polyhedron with a large number of sides (enough sides so that the polyhedron resembles a sphere).

The second (which may fail to show up in your reader) is from Erik Demain's site. This shows several unfoldings (and refoldings) of a cube.

## Tuesday, December 8, 2009

### more folds

PBS has launched a new Between the Folds website - the online quiz where you have to match the crease pattern to a finished model is neat.

I've mentioned Between the Folds in previous posts (here and here) - if you haven't seen it yet and you get PBS, you really should check it out.

A lot of the origami models featured in the film can also be seen in Origami USA's gallery. If you are looking for something to fold after watching the film, you should check out the diagrams from Origami USA as well.

## Monday, December 7, 2009

### philosophical transactions

TierneyLab at the NYT has a post about a really nice online exhibit celebrating 350 years of the Royal Society.

The exhibit, Trailblazing, is presented as a timeline with links to some key scientific papers that were presented in The Philosophical Transactions of the Royal Society. Mathematics plays a big role in many of the interesting primary sources presented in the exhibit.

Here are a few of the more overtly math-related papers in the exhibit:

Isaac Newton's letter on his theory on light and colours from 1672.

John Hadley's paper describing a new instrument for measuring angles from 1731.

Bayes's posthumus essay on chance and proability of 1763.

Davies Gilbert's paper on the mathematical theory of suspension bridges from 1826.

## Wednesday, December 2, 2009

### a curious population model

Dave Richeson of Division by Zero has recently posted a very nice GeoGebra applet for the interesting discrete logistic equation.

$p(n + 1) = Mp(n)(1 - p(n))$
The discrete logistic equation is a simple model of population growth within a closed environment. p(n) is a decimal between 0 and 1 representing the fraction of the maximum population that has been reached (initial seed value of p(0) = 0.001), and M is the "Malthus factor" a multiplier that represents the fertility of the population (value between 1 and 4). Different values of M determine whether the population dies out, achieves a stable level, or fluctuates.

This simple discrete dynamical system is also easy to implement and explore in Fathom. You only need a slider (for the M value) and a couple attributes (you may want to add others as you explore).

1. In a new Fathom document, create a slider M
2. Add a new Collection, and create an attribute generation, and an attribute population.
3. Provide this formula for generation: caseIndex - 1
4. Provide this formula for population: if (generation=1){0.001, M*prev(population)*(1-prev(population))
5. Add cases in to represent the generations of the population (say 20 or so)
6. Create a graph with population as the y attribute and generations as the x attribute
7. Explore how the graph changes as M varies between 0 and 4.

An example Fathom file is here.

When the population is stable, you have a nice curve like one below.

When the Malthus factor is higher, the population becomes less stable - overcrowding and then dying back.

This population model is mentioned in Mark Haddon's, The Curious Incident of the Dog in the Night-time, whose protagonist is a young mathematician with some behavioural difficulties. See the MAA online review of the book here, which includes pointers to other places to learn about the logistic equation.

## Friday, November 27, 2009

### mathematical lapses

Every man has somewhere in the back of his head the wreck of a thing which he calls his education.
- Stephen Leacock, A Manual of Education

I recently stumbled upon some of Stephen Leacock's writings on wikisource (also available at Project Gutenburg). Stephen Leacock is well known (in Canada at least) as a humourist who wrote in the early part of the 20th century. It is less often noted that a few of his pieces were inspired by (or made use of) school mathematics. Now about a century since they were first published, some of these don't stand up so well, but overall they retain flashes of humor and insight that make them well-worth reading.

The best known of Leacock's school-math inspired writings is A, B, C: The Human Element in Mathematics, republished recently in Mathematics Teacher. Others include Boarding House Geometry, and Aristocratic Education.

What I like the most about Stephen Leacock's math-related humour is that it represents the sort of writing that restores mathematics to its rightful place within the humanities. In the Leacock universe, mathematics is something that every educated person knows about (at least up to a point). It is, or should be, a subject that people have enough facility with that they can find humor in it. Leacock makes it seem completely normal that you would simile knowingly at a mathematical reference just as you might a literary one, and that Euclid would be as familiar a name as Shakespeare.

## Thursday, November 19, 2009

### interesting and uninteresting numbers

In a paper published this week on arXiv, Dann van Berkel mentions the famous proof (or joke, depending on how you view things) that all natural numbers are interesting. Essentially, if there are uninteresting natural numbers, consider the smallest such uninteresting number (which is guaranteed to exist, since the naturals are well-ordered). This number, because it is the smallest uninteresting number, is interesting, providing us with a contradiction.

As Wikipedia points out, Nathaniel Johnson has given a slightly different formulation of the question, that seems more objective, and yet also more paradoxical. And currently, this formulation actually  produces 'uninteresting' numbers. The formulation goes something like this:

A number is interesting if and only if it belongs to an interesting sequence.
The Online Encyclopedia of Integer Sequences (OEIS) contains all interesting sequences.
Consequently, the numbers that do not appear in OEIS are uninteresting.

This gives us (as of last week, according to Johnson) that the number 12407 is currently the smallest uninteresting natural number.

Unfortunately, the sequence of uninteresting numbers seems interesting, and should be considered for inclusion in OEIS... but this would contradict its definition. So far, the sequence has not been included, so it still exists.

Does this formulation give a proof that OEIS cannot contain all interesting sequences? I think there are probably a bunch of statements we can generate with this sort of "interesting" logic, such as:

If an encyclopedia contains all interesting sequences, then every natural number must occur at least once in the encyclopedia.

But then again, you could argue that any number that occurs only once is interesting, and that the sequence of these once-occurring numbers is interesting also, causing them to be included again... In any case, this way of thinking about "interesting numbers" shows a lot of affinity with the Barber paradox, Russell's paradox, and other classics.

But van Berkel was not discussing uninteresting numbers in his paper, but rather a particularly interesting number, 3435.

If you take a natural number and calculate the sum of its digits raised to themselves, you get another natural number. Can this operation ever give you the original number back? Yes. Consider the number 1, which has a single digit, 1, when raised to itself gives 1^1=1. Here are some other calculations:

 n ... sum of digits raised to themselves 1 ... 1^1 = 1 2 ... 2^2 = 4 3 ... 3^3 = 27 4 ... 4^4 = 256 5 ... 5^5 = 3125 6 ... 6^6 = 46656 7 ... 7^7 = 823543 8 ... 8^8 = 16777216 9 ... 9^9 = 387420489 10 ... 1^1 + 0^1 = 2 11 ... 1^1 + 1^1 = 2 12 ... 1^1 + 2^2 = 5 13 ... 1^1 + 3^3 = 28 14 ... 1^1 + 4^4 = 257 15 ... 1^1 + 5^5 = 3126 16 ... 1^1 + 6^6 = 46657 17 ... 1^1 + 7^7 = 823544 18 ... 1^1 + 8^8 = 16777217 19 ... 1^1 + 9^9 = 387420490 20 ... 2^1 + 0^0 = 5 21 ... 2^1 + 1^1 = 5 

If we call this function $\theta(n)$, do we ever get another number than 1 that is a fixed point for $\theta(n)$? Well, yes, and 3435 provides us with the next example:

$3435 = 3^3 + 4^4 + 3^3 + 5^5$
The graph below shows a plot of theta for n up to 5000 - the two fixed points that are found in this range, 1 and 3435, are shown as blue squares (data file here). The huge scale on the y-axis makes the graph look more constant in places than it really is.

Numbers for which $n = \theta(n)$ for a given base $b$ are called Munchausen numbers, and are interesting, according both to common conceptions of what 'interesting' means, and according to Johnson's OEIS-based definition (OEIS entry: A166623, also submitted by van Berkel). Making 3435 even more interesting is the fact that, according to van Berkel's paper, 1 and 3435 are the only Munchausen numbers, base 10.

Munchausen numbers are (presumably) named after the semi-fictional Baron Munchausen, who interestingly enough may not have actually been interesting, but was intent on convincing others that he was. You may know Munchausen from the Terry Gillam movie, and he has also lent his name to the curious Munchausen syndrome, and to the disturbing Munchausen-syndrome-by-proxy. The image at the top of the post is from a 1902 edition of the book, The Adventures of Baron Munchausen.

## Monday, November 9, 2009

### symmetry on TED

To see any pure math lecture on TED is exciting. Why are these things so rare? A recent talk by Marcus du Sautoy, Symmetry, reality's riddle, is now availalbe here. Marcus du Sautoy is a prolific mathematics expositor, and was recently appointed as Oxford's Simonyi professor of the public understanding of science (a post he takes over from Richard Dawkins). I enjoyed his Story of Maths on BBC, and also the one book of his that I've read, The Music of the Primes. His talk touches on topics that he, reportedly, explores more fully in his new book, Finding Moonshine.

For many, the story of Galois that du Sautoy used to launch his talk will be new, but to me it seems to be a tale that mathematics popularizers have stretched a little thin. The story of Galois's duel must rank along side the story of Gauss's defiance of his school master as the top two most popular mathematical biographical capsules of all time. It would be great if someone would go through the popular re-tellings of the Galois story the way that Brian Hayes has for the Gauss story, to see how different authors have embellished and expanded it (perhaps someone has?).

One thing that I found striking and appropriate about du Sautoy's talk is that although it is about group theory it never mentions the phrase "group theory," instead it always stays focused on the motivating notion of symmetry. The mathiest parts of the talk were the examples drawn from the wallpaper groups of Spain's Alhambra fortress (which, along with Galois's duel, are also frequently mentioned in popular literature on symmetry).

Watching this TED lecture made me feel a little depressed that I haven't yet taken the time to really start working through the beautiful and quirky book The Symmetries of Things, by  John Conway, Heidi Burgiel, and Chaim Goodman-Strauss - definitely the next place to go if you are inspired by the lecture.

## Tuesday, November 3, 2009

### The Calculus of Friendship - a review

In The Calculus of Friendship, Steven Strogatz, a Professor of Mathematics at Cornell, gives us a memoir of the friendship he has maintained with his former high school teacher, Don Joffray, as revealed through fragments of their 30 year correspondence. Their friendship was sparked and sustained through their shared enthusiasm for mathematics, and their long correspondence was, almost exclusively, framed by math problems.

The story of the correspondence between these two men is at once charming and subtly powerful.  Strogatz writes directly and honestly, telling the story of a slow-growing friendship that was at once somewhat stilted and yet deep and sustaining. The immediacy and intimacy of Strogatz's writing transform the pleasures and tragedies of normal life into the elements of a compelling narrative, and because the book works so well on this human level, it also very effective in presenting some important lessons about education and about mathematics.

Joffray is presented as the kind of teacher who taught through encouragement and obvious enthusiasm. In many respects, Strogatz's book teaches in a similar manner. Never does Strogatz tell us unequivocally how mathematics should be taught, how we should think about mathematics, or what mathematics is all about. And yet, Strogatz provides us with some distinct impressions on all these questions, so that like the students in Joffray's classes, you leave having learned something important without realizing that you were being taught to.

What does the book tell us about mathematics education? Although Joffray and Strogatz taught and learned in a very privileged environment, the essential lessons of Joffray's teaching seem universal. A good teacher, we learn through Joffray's example, is humble, enthusiastic, respectful, and encouraging. Consider how these qualities reveal themselves in a problem-oriented mathematics curriculum: problems are approached by both teachers and students as co-investigators, the successes of students are celebrated, rich problems are revisited again over time, and the teacher is rarely simply presenting results to passive students. Perhaps the most important quality that Joffray seems to embody is authenticity, and this, more than anything else, seems to have sustained him through his long career and won the admiration of his students. Here again, a subtle lesson - you have to be true to yourself, your instincts, and to what brings you enjoyment if you are going to succeed in education (or in life generally).

What does the book tell us about mathematics? Through the letters of Strogatz and Joffray, mathematics is revealed as a social activity, a language to express excitement and big ideas, and a subject whose growth is fueled by creative (and perhaps somewhat eccentric) enthusiasts who are compelled to do mathematics because of the pleasure it brings them. Strogatz seems always willing to return to the source of his original excitement about mathematics - solving interesting problems and sharing the excitement of discovery, while Joffray never gives up his own explorations and problem solving.

In their letters, Strogatz and Joffray touch on many interesting mathematical topics. Perhaps because he was writing to an enthusiast who was not engaged in "serious" research, much of what Strogatz wrote about in his letters are classics of recreational mathematics. The letters, and the additional references provided at the end of the book, provide many starting points for learning about these topics. Among other things, Stogatz and Joffray discussed pursuit problems (see the MathWorld entry here for an overview and some nice animations) the Monty Hall problem (see this post), and a geometric proof of the irrationality of the square root of two (see "proof 7" at Cut-the-Knot's list of 19 proofs).

One problem that was new to me was the "monk and the mountain" problem, which is related to a whole family of intermediate value and fixed point theorems. Charles Wells has written several interesting posts on the monk and the mountain problem (a recent one here, and another here), explaining how it exemplifies a class of problems that have "naive proofs." The proof that Wells presents is identical to the one that Strogatz describes, and at first it seems too lacking in rigor to be considered truly mathematical. Wells maintains, and I think that Strogatz would agree, that the proof, despite its naiveté, is still a solid mathematical proof. Naive proofs, like this one, have a special place in mathematics education - they can help broaden a learner's perception of what mathematics is and how it can be applied, show that good proofs cut through the unnecessary in order to get to the heart of why something is true, and remind us that mathematics often relies on metaphor in order to communicate its essential ideas.

The Calculus of Friendship is an accessible, engaging, and useful book. Strogatz has taken an engaging personal story and through it has taught some important lessons about friendship, teaching, and mathematics.

The images in this post are from a GSP iteration (file here) that I was reminded of while reading Strogatz's description of a pursuit problem involving 4 dogs chasing each other from the corners of a square.

## Friday, October 30, 2009

This week I had a great origami day - I found Origami For the Connoisseur, by Kunihiko Kasahara and Toshie Takahama at a used book sale (for $3.00CDN), and then when I arrived home I found that my copy of the Between the Folds DVD had arrived. The Between The Folds documentary is great - beautiful, inspiring, informative, and the accompanying short piece, Origametria, would make a great short video to play at a school math department meeting if you want to get teachers fired up about using origami with their students. Between The Folds brings to light much of what draws people to origami, and I think that, for those not familiar with modern paper-folding, it will contain many surprises. The images of origami tessellations are particularly beautiful, and the more representational origami 'sculptures' that are presented are often incredibly detailed. A main focus of the film was the tension between origami technique and artistry. For Robert Lang, there is no conflict between the two - technique sometimes dominates over artistry, but ultimately artistry assimilates and employs technique. At the same time, it seems that Eric Joisel fights an internal struggle to resolve his need for artistic expression and the temptation to focus on technique. Although the conflict, or potential conflict, between the artistic and technical sides of origami is a concern for the practitioners who are at the summit of the art form, I find that neither artistic expression nor technical complexity are important to my own experience of origami. Perhaps that is because I lack both artistic and technical talent. I was encouraged, however, by how Paul Jackson, Erik Demain, and Tom Hull were all manipulating very simple (yet beautiful and interesting) models as they spoke about origami. Paul Jackson contemplated some 'single fold' models, and both Erik Demain and Tom Hull were seen manipulating a hyperbolic paraboloid (instructions for this simple model by Demain are here, and Hull has instructions on page 65 of the handouts for his book, found here). Turning to Origami for the Connoisseur for further inspiration, I found this statement, which was a welcome counterpoint to all the artistry and technical prowess of Between the Folds: Something that is simple and duplicatable too is another origami ideal. A very simple modular origami model for "rotating tetrahedron," designed by Tomoko Fuse, shares the same pages with the quote above. This model is an intriquing toroidal structure of linked tetrahedrons that has been studied by engineers and chemists (see for example Simon Guest's symmetry page and paper). The rotating tetrahedron and hyperbolic paraboloid are examples of the surprising yet simple models that make origami an accessible and beautiful way of exploring mathematics (see this earlier post for other origami-math links). ## Thursday, October 29, 2009 ### JUMP math image from Everyday Number Stories A couple of years ago I saw a presentation by John Mighton of JUMP math. He is an interesting and engaging speaker, and you should check out his short video at MAKE magazine's The Elements of Humanity. An example of how a community organization has been able to implement the JUMP program can be found here, at the Ottawa Centre for Research and Innovation (OCRI) site. ## Tuesday, October 27, 2009 ### Harmonic Denominator Number Triangle This number triangle is made up of the denominators of the Leibniz Harmonic Triangle. From the earlier post about that number triangle, you can see that there are two ways of generating the Harmonic Denominator Triangle: 1. take the reciprocal of the entries in the Leibniz Harmonic Triangle; or 2. multiply the entries of Pascal's Triangle by n+d (here we are using the 'triangular number style' indexing of Pascal's Triangle, rather than the usual 'binomial coefficient style' indexing). The entries of the Harmonic Denominator Triangle ($g^d_n$) are given by: $g^d_n = \frac{1}{h^d_n}$ and $g^d_n = (n+d)t^d_n$ Where$h^d_n$are the entries in the Leibniz Harmonic Triangle and$t^d_n$are the entries in Pascal's Triangle. The connection to the triangular numbers gives us the general formula: $g^d_n = \frac{n(n+1)\cdots(n+d)}{d!}$ I am looking forward to learning more about this number triangle (and the Leibniz Harmonic Triangle too). If you make use of OEIS, you'll see that the The HDT contains a lot of well-known sequences. A quick look at the rows and columns and you'll find the a bunch of well-known sequences: A005430, A002457, A002378, A027480, and A033488, to name a first few. The whole triangle has an entry in OEIS: A003506. One of the comments in the OEIS entry for the HDT points out a neat way to express the relationship between Pascal's Triangle and the Harmonic Denominator Triangle. The entries of the k-th row of the HDT are the coefficients of the first derivative of a polynomial whose coefficients are the entries of the (k+1)-th row of Pascal's Triangle. More specifically, the coefficients of$(x+1)^{k+1}$give you the k+1 row of Pascal's Triangle, while the coefficients of$\frac{d}{dx}(x+1)^{k+1}$give you the k-th row of the Harmonic Denominator Triangle. ## Friday, October 23, 2009 ### Mesopotamian maths On Notation. Whether one believes that mathematics is created or discovered, notation is certainly created. And notation can direct the course of mathematics. - Elisha Peterson, Unshackling Linear Algebra from Linear Notation A common entry point into learning about the mathematics of ancient Iraq is through its notation. The sexagesimal (base 60) system written in cuneiform defines what the mathematics of ancient Iraq means to most people who have been exposed to it through textbooks and short articles. One way to get some exposure to this is to visit Wolfram Alpha, type in a number, and choose "other historical numerals" - you'll be treated to a digitized version of what Babylonian cuniform sexagesimal numbers look like. The Wolfram Alpha implementation of its sexagesimal cuniform translator was not without its problems, but teachers are already using it in their lesson plans. It is worthwhile to think about the long series of abstractions that have lead up to the Wolfram digitized cuneiform. As Eleanor Robson points out in her book Mathematics in Ancient Iraq: A Social History, the earliest 'writing' of numbers were actually the impressions of tokens into the clay tablets - originally the tokens were actually sealed into the clay in order to record counts. Later, it was determined that the tokens themselves could be dispensed with and that the impressions could stand in their place. Later still, the impressions became incised cuneiform markings on the tablets - one of the earlies forms of writing. For a long time since, we have been mimicking these incisions with pen and paper or typeset drawings, and now a cuneiform simulacrum can be generated automatically for us in this new digitized form. While Robson's book tells us a lot about cuneiform writing and the various systems of ancient Iraqi mathematics, it points out that we are severely mistaken if we understand the mathematics of ancient Iraq solely in terms of notational differences between it and our more familiar ways of writing mathematics. The mathematics of ancient Iraq is not merely our familiar mathematics recast with a different base and different writing style, it is fundamentally different in ways that we have trouble appreciating. It seems that when westerners look at the mathematics of another culture or another era, we tend to view it through the fixed lense of our own mathematics, we wonder to what extent did they anticipate our current mathematics, what their "contribution" was, and in what areas they were limited. We tend to ask questions like, did they have a concept of 'zero'?, did they have a 'Pascal's Triangle?' did they know 'Pythagoras's Theorem'? This limited way of viewing the math of others leads us to make a number of untenable assumptions about the history of mathematics. Robson's analysis quickly undermines the simple stories that are often repeated in the margins of math textbooks that place Iraq at the starting point of what became, after elaboration by the Greeks, the "Western" tradition of mathematics. As she notes early on, 'the mathematical culture of ancient Iraq was much richer, more complex, more diverse, and more human than the standard narratives allow." (p. 2) Her book attempts to provide "a new look, and a new perspective" (p. 8) on a subject that has been glossed over far to much, often from an overly simplifying western vantage. In ancient Iraq, numeracy, literature, the mechanics of cuniform, along with the intellectual, state, and educational cultures, grew together to shape each other and to give the mathematics of this period its distinct richness. The 3000 years examined in this book show a a surprisingly diverse history of mathematical practice that begins with emergence of a rudimentary numeracy that developed into applications in accounting and engineering, and that ultimately evolved into a baroque system of numerology and mathematical divination. It is interesting that although the period explored is an ancient one, much of it has only come to light recently. Consequently, much of the non-scholarly writing about the mathematics of ancient Iraq is quite simplistic in its outlook. Robson argues that as examples of cuneiform mathematics came to light in the early and mid 20th century, scholars gravitated towards those texts that fit with their preexisting notions of Greek and Egyptian mathematics (to the exclusion of more representative ones), a process that helped only to reinforce received orientalist ideas about the mathematics of this period. A striking example of how our preconceptions can influence our understanding of how others understand and use mathematics is provided by contrasting two translations of an ancient cuneiform tablet (table 9.1, page 277)- an early translation from the mid 20th century is full of modern mathematical terminology, while a more recent translation attempts to be closer to the original. The translation that uses a modern lens uses the convential terms like adding, subbtracting, multiplying, and dividing numbers to describe the content of the tablet - in sharp contrast, the contextualized reading uses images of lengths "holding" each other, of "turning back" within a calculation, and of "tearing" and "accumulating" surfaces. Robson's approach is rooted in a view of mathematics that is essentially one of social constructivism - one that she admits may not be held by the majority of working mathematicians, but that provides what may be the best perspective for historical analysis. Interestingly, she also argues that a philosophical outlook of mathematical platonism (a view held by many mathematicians and historians of mathematics) inevitably influences people to see more connections between the mathematics of different cultures than can be legitimately said to exist. If we believe that mathematical objects are "real" and "discovered" rather than invented, then we would expect different cultures to somehow discover the same mathematical truths. If we abandon this perspective, differences in mathematical practices and understandings are less surprising. It is interesting to read about the prevelence of educational texts among the surviving cuniform tablets. Some of these tablets mimic "official" documents, but bear the tell-tale signs of being used for the education of scribes. For example, some tablets are identified as educational by "the unrealistic size of the numerical parameters, ... and the lack of credible contextual data" (p56). Creating math problems infused with realistic context has apparently plagued teachers for thousands of years. Robson suggest that there is evidence that scribal training, in some periods, made use of situated learning rather than purely rote schooling (p84). One of the consequences of abandoning a more reductive and simplistic approach to understanding this period and its culture of mathematics is that we loose the "grand narrative" of mathematics that links ancient Mesopotamian mathematics, the Greek mathematics of Pythagoras and Euclid, and what eventually became the western tradition. The simple thread that united these mathematical traditions is cut, in favor of a much more complex weaving of influences. Mathematics in Ancient Iraq is an important reference both for its subject and for its method. For anyone researching the history of mathematics, or this period, it provides an important resource and example. The detail and close readings of original sources and its commitment to situating these within their cultural context, essential for the method that Robson pursues, might, however, prove too much for the reader with only a casual interest in the subject. For this group of readers, hopefully the insights from this text will soon be reflected in more general works and texts. Fortunately there are many good web resources that reflect this emerging understanding of ancient Iraqi mathematics, many of them contributed or maintained by Robson herself (check out the links on her webpage). ## Monday, October 19, 2009 ### whither blogger LaTeX? If you have a Blogger blog, you may be using some$\LaTeX$renderer to make your math look nice. A couple of days ago, the one I was using (from www.watchmath.com) stopped working (hopefully it will be up again by now, or soon...). My limited research has shown me that most Blogger use of$\LaTeX$relies on a small piece of javascript embedded somewhere on the page that calls a larger piece of javascript (possibly located on another server) that in turn invokes some other hosted program that actually does the rendering (possibly located on yet another server): lots of potential points of failure. Another unfortunate aspect of this method is that it doesn't allow the rendered math to show up in readers - you have to look at the page itself to have the script run. If you have lost your math rendering you can learn a little about these things by reading the replacemath.js docs and the mathtex docs. For now, I am using the following bit of script to render the math:  <script src="http://mathcache.s3.amazonaws.com/replacemath.js" type="text/javascript"></script> <script type="text/javascript"> replaceMath( document.body ); </script>  Place this somewhere on your page, and it will invoke the replaceMath function that is defined in 'replacemath.js' - see the documentation about this at the link mentioned above. Although the rendering isn't doing quite as nice a job as the watchmath version did, it's not too bad. I'll look for something better when I have the time... PS: This blog post provides a good overview of the LaTeX rendering options that are out there. ## Thursday, October 15, 2009 ### Math Teachers at Play 17 There is plenty of humor, a few movies, some great activities and explorations, and lots of math in Math Teachers at Play 17. Hope you enjoy reading these as much as I did. Thanks to everyone who submitted posts or answered my requests to include their work. First, a bit of administrativa: Please see this post at Walking Randomly about the rebirth of the Carnival of Mathematics, and some upcommng scheduling changes for Math Teachers at Play. David Richeson at Division By Zero has brought us a great list of activities that are likely to inspire a few budding mathematicians in his post Kindergarten Mathematics. In her post, Elementary Math isn't Easy, Joanne Jacobs directs us to a recent article on the importance of elementary mathematics and elementary math teachers by Hung-Hsi Wu in American Educator. Rachel M encourages us to grow a Counting Garden posted at quirkymomma.com - an activity that helps reinforce basic counting and number recognition. Mathematics is everywhere, particularly at the breakfast table. At Math with My Kids we are reminded of the challenges and rewards of mathematical discovery in Math with Bannanas. Rick Regan suggests an interesting and edible 100 days of school project in his post One Hundred Cheerios in Binary at Exploring Binary. Kakie presents Teach kids how to make$1,000,000 in 30 days with pennies! posted at Bur Bur & Friends: Community Blog. Keeping with the economics theme, Kendra offers Pumpkin Patch: Piggy Bank Math Game posted at Pumpkin Patch.

Much has been written on math blogs lately about how to understand and explain "negative multiplied by a negative is positive." On this fruitful theme, Brent Yorgey presents Minus times minus is plus posted at The Math Less Traveled. Jason Dyer gathers many of the threads that this question has spawned together and offers his own explanation in Negative times negative posted at The Number Warrior.

Denise offers some guidance in How to Solve Math Problems II posted at Let's Play Math!.

Sue Van Hattum offers a way of tackling the problem of math anxiety in her post Math Relax: A Guided Visualization for Overcoming Test Anxiety in Math at her blog Math Mama Writes.

You may have heard of the "Jigsaw" method in collaborative learning, but what about the "Speed dating" method? Kate Nowak introduces this structure in her post Speed Dating at f(t).

John Golden presents a dynamic geometry exploration of the Pythagorean Theorem, considering all triangle types in GeoGebra: Triangle Tuning posted at Math Hombre. See his GeoGebra introduction in an earlier post here. You may also want to see Kate Nowak's instructions for putting your GeoGebra into your blog here.

Kimberly Lightle's post, Dynamic Math and Science Learning With Simulations, at Exemplary Resources for Middle School Math and Science and Maria Andersen's post, Interactive Simulations from PhEt, at Teaching College Math both provide some great links to online simulaiton software. Sue Vanhattum of Math Mama Writes tells us about an early pioneer of learning math through computer interaction in her post Mindstorms: Children, Computers, and Powerful Ideas.

Liz at STEM-ology has pointed us towards a new resource and recruitment program for mathematics teachers, Math for America, in her post X plus Y.

Another math resource has been provided by the Mathematics Department at BYU, which announced that their site When Will I Use Math? has launched. This perrenial question was also answered by Deb Russell in her post Math: When Will I Ever Use This Stuff?, which points to an article on mathematics and 3D animation.

In Pandemics and Their Numbers, Terese Herrera points us to an H1N1 inspired lesson created by The New York Times at the blog Exemplary Resources for Middle School Math and Science.

Humor and mathematics go hand in hand in several recent posts.

Pat Ballew offers a little tongue in cheek math talk in  You Might Be a Mathematician IF... posted at Pat'sBlog.

Xi at 360 presents some great math comics the posts All about A4, Another Math Comic, and How About Another Comic?. Luke Kane points us towards some other recent, and very slick, math comics in his post Comics and Math at Logic Nest.

At Komplexify, we have a bit of calculus poetry in the post Calculus Haiku - The Derivative.

Mistakes, ambiguities, the unexpected, and the impossible - math teachers encounter these, from time to time.

Jakie points out how mistakes often turn into teachable moments in her post When Sketchpad is Wrong at Continuities.

Glowing Face Man presents Ambiguities in Mathematics posted at Glowing Face Man.

Sam Shah shows us some examples of functions behaving badly in his post sin(1/x) at Continuous Everywhere but Differentiable Nowhere.

Vlad Alexeev shows us an impossibly small book of impossible figures in the post Mini Books of Anatoly Konenko at his blog Mathematical Paintings and Sculptures.

John Cook has posted on the math behind musical scales in his posts Circle of fifths and number theory and Circle of fifths and roots of two at his blog The Endeavor.

Alison Blank has put together an inspired and inspiring Prezi presentation, Math is Not Linear, and posted about it on her blog Axioms to Teach By.

Maria H. Andersen of Teaching College Math has put together another impressive prezi presentation  and linked to it from her post How can we measure teaching and learning in mathematics?

The Albany Area Math Circle lets us know about the math documentary Hard Problems: The Road to the World's Toughest Math Competition. To find out more about Math Circles, check out the National Association of Math Circles site.

Green Fuse Films had another math-film announcement on their blog: Between the Folds, their documentary about the math and art or origami, is now available on DVD.

Reidar Mosvold lets us know about what looks like a really neat event: Maths Week in Ireland on his blog Mathematics Education Research Blog.

Marjorie Morgan presents her thoughts on Outdoor Education and Mathematics teaching in her post Lindsay & Sharon - outdoor adventurers at GO! Girls Outdoors.

It surprises some that guessing and measuring can both fall under the mathematical umbrella.

Tom DeRosa presents The Very Exact Science of Guessing posted at I Want to Teach Forever.

Maria Miller presents 10/10 and the Metric Week posted at Homeschool Math Blog.

Staying with a metric theme, Austen Saltz, a senior high school student who is blogging at Talking Science, points us to Nikon's Universcale in his post The Size of the Universe. Universcale is very reminicent of the Powers of 10 film from the 70s, but much flashier (see the film here).

Two approaches to thinking about breaking sticks and making triangles are found at Bill the Lizard and Pat's Blog. Bill takes a simulation-based approach in his posts,  The Broken Stick Experiment and The Broken Stick Revisited, while Pat explains how to tackle the problem using limits in his post, A Limit Approach to a Classic Geometric Probability Problem.

Thanks to everyone. If you submitted an article that was not included, or if you have a post that you think would have made a nice addition, please consider submitting it to the next installment. The next Math Teachers at Play will be up at Math Mama Writes on October 30th.

## Wednesday, October 7, 2009

### Three number triangles, two telescoping series

There are so many relations present [in Pascal's triangle] that when someone finds a new identity, there aren't many people who get excited about it anymore, except the discoverer! - Donald E. Knuth (as quoted by Martin Gardner)

I was inspired by a post on Pat's Blog and by reading A.W.F. Edwards's Pascal's Arithmetical Triangle to look at summing the reciprocals of higher dimensional triangular numbers. It turns out that you can use the same telescoping series technique that allows you to sum the reciprocals of the 2-dimensional (i.e. the usual) triangular numbers, and that the 'telescoping' feature of these sums can be expressed in terms of some nice identities.

The statement of the sum is a nice one. For d > 1 we have:

$\sum^{\infty}_{n=1} \frac{1}{t^d_n} = \frac{d}{d-1}.$
Which gives you a result of 2 for d = 2 as mentioned in Pat's post. In the case where d = 0 or 1, the series does not converge.

In the notes below, none of the identities are new (the newest is, I think, about 400 years old) - the famous quote by Donald Knuth at the top of the post is intended as a caution for anyone who gets carried away and derives scads more.

In all the statements in this post, the index d ranges over the nonnegative integers (0, 1, 2, ...) while n ranges over the natural numbers (1, 2, 3, ...). This may seem somewhat inconsistent, but we like to start with dimension zero (d = 0) and with the first triangular number (n = 1).

The d-triangular numbers, $t^d_n$ (d for "dimension") are defined by $t^0_n = 1$, and

$t^d_n =\sum^{n}_{i=1}t^{d-1}_i \mbox{ for } d > 0.$

The formula for $t^d_n$ can also be expressed as a difference, $t^d_n - t^d_{n-1} = t^{d-1}_n$.

When d = 2 we get the usual triangular numbers, d = 3 gives the pyramidals, and d = 4 gives the triangulo-triangulars, etc. - each dimension up is visualized as a stacking of those below it. Note that this definition has the first triangular number (for all dimensions d) as 1 and not 0, as is sometimes preferred; in this context it makes sense to start at 1.

From this definition, and the Pascal identity, you can establish that

$t^d_n = \left( \begin{array}{c}n+d-1 \\d \end{array} \right).$
If you are familiar with Pascal's Triangle and look carefully at the triangular number definition, you'll see that sum in the definition of the d-triangular numbers is the Pascal Triangle "hockey stick theorem" in disguise. This provides us with a direct formula for the d-triangular numbers:

$t^d_n = \frac{n(n+1)\cdots (n+d-1)}{d!}.$
And this also suggests that we arrange the d-triangular numbers into Pascal's triangle, while remembering that we are not indexing them in the way we usually do for the binomial coefficients.

The formula also gives us the opportunity to generate a bunch of identities, like:

$t^d_n = \frac{(n+d-1)}{d} t^{d-1}_n,$
and
$t^d_n = \frac{n}{d} t^{d-1}_{n+1}.$
From here we flip each entry in the triangle to obtain a new triangle, the Leibniz triangle (so called by Edwards), whose entries are the reciprocals of d-triangular numbers.

There is a nice difference relationship in this triangle too, for > 1,

$\frac{1}{t^d_n} = \frac{d}{d-1} \left( \frac{1}{t^{d-1}_n} -\frac{1}{t^{d-1}_{n+1}} \right)$
To prove this identity, generalize the method that allows you to split up the fraction $\frac{1}{n(n+1)}$ into $\frac{1}{n} - \frac{1}{n+1}$ in the usual telescoping series example.

To get our last number triangle, we divide each entry in the Leibniz Triangle by $(n+d)$, which gives us the Leibniz Harmonic Triangle. In other words, we define $h^d_n = \left(\frac{1}{n+d}\right)\frac{1}{t^d_n}$, and create a new triangle with entries $h^d_n$.

Going back to the formula for $t^d_n$ we can obtain some other identities, like:

$h^d_n = \left( \frac{1}{d+1}\right) \frac{1}{t^{d+1}_n},$
and
$h^d_n = \left( \frac{1}{n}\right) \frac{1}{t^{d}_{n+1}}.$
And, as with the other triangles, there is a nice difference relationship, for d < 0 :

$h^d_n = h^{d-1}_n -h^{d-1}_{n+1} .$
This can be proved following the same ideas as was used to show the difference relationship for the inverse triangulars. It's worth contrasting this identity with the corresponding identity for Pascal's Triangle.

Now, if you've proved these identities, there are two easy ways to find the sum, $\sum^{\infty}_{n=1} \frac{1}{t^d_n}$.

First method: use the Leibniz Harmonic triangle.
Taking advantage of these two identities:

$h^d_n = \left( \frac{1}{d+1}\right) \frac{1}{t^{d+1}_n},$
$h^d_n = h^{d-1}_n -h^{d-1}_{n+1} .$
We can combine them and observe that for d > 1:

$\begin{array}{lll} \sum^{\infty}_{n=1} \frac{1}{t^d_n} &=& d\sum^{\infty}_{n=1} h^{d-1}_n \\ &=& d\sum^{\infty}_{n=1}\left( h^{d-2}_n - h^{d-2}_{n+1} \right) \\ &=& d\left(\sum^{\infty}_{n=1} h^{d-2}_n -\sum^{\infty}_{n=1} h^{d-2}_{n+1} \right)\\ &=& d\left(\sum^{\infty}_{n=1} h^{d-2}_n -\sum^{\infty}_{n=2} h^{d-2}_{n} \right)\\ &=& d\left(\sum^{\infty}_{n=1} h^{d-2}_n -\sum^{\infty}_{n=1} h^{d-2}_{n} + h^{d-2}_{1} \right)\\&=& dh^{d-2}_{1} \\ &=& d\left( \frac{1}{d-1}\right)\\ &=& \frac{d}{d-1}\end{array}$

Incidentally, we also have $\sum^{\infty}_{n=1} h^d_n = \frac{1}{d}$ for d > 0.

Second method: use the reciprocal difference identity for the Leibniz triangle.
Here we see that everything comes directly from the identity:

$\frac{1}{t^d_n} = \frac{d}{d-1} \left( \frac{1}{t^{d-1}_n} -\frac{1}{t^{d-1}_{n+1}} \right).$
We have, for d > 1:
$\begin{array}{lll} \sum^{\infty}_{n=1} \frac{1}{t^d_n} &=& \frac{d}{d-1}\sum^{\infty}_{n=1} \left(\frac{1}{t^{d-1}_n} - \frac{1}{t^{d-1}_{n+1}}\right)\\&=& \frac{d}{d-1}\left(\sum^{\infty}_{n=1} \frac{1}{t^{d-1}_n} - \sum^{\infty}_{n=1}\frac{1}{t^{d-1}_{n+1}}\right)\\ &=& \frac{d}{d-1}\left(\sum^{\infty}_{n=1} \frac{1}{t^{d-1}_n} - \sum^{\infty}_{n=2}\frac{1}{t^{d-1}_{n}}\right)\\ &=& \frac{d}{d-1}\left(\sum^{\infty}_{n=1} \frac{1}{t^{d-1}_n} - \sum^{\infty}_{n=1}\frac{1}{t^{d-1}_{n}} + \frac{1}{t^{d-1}_{1}} \right)\\ &=& \frac{d}{d-1}\left(\frac{1}{t^{d-1}_{1}}\right) \\ &=& \frac{d}{d-1}\end{array}$

After working out all this, I came across the same "reciprocal of generalized triangular numbers" problem at Topological Musings - solution #3 is the same as the one here, but expressed in terms of binomial coefficients (and without extracting some general identities among the terms).

## Monday, October 5, 2009

### beautiful negatives

teacher (emphatically): A double negative makes a positive, but a double positive can never make a negative!
student (lazily, from the back of class): yeah, yeah...

I am not sure where this topic started, but The Number Warrior has collected up pointers to most of the "minus times minus is plus" posts that have been cropping up here.

So far I haven't read about what I thought was the usual visualization (but it has probably been mentioned, burried in some comments): multiplying by -1 is a counter-clockwise rotation by 180 around zero.

The problem with this image is that it takes us off the number-line and has us floating in space for a moment. But this is exactly right - the space we are floating in is actually the field of complex numbers, and seeing "multiplying by -1" as "counter clockwise rotation by 180" is the visualization that corresponds to "the most beautiful equation in the world" $e^{i\pi} = -1$. In the same way, a rotation by 90 degrees corresponds to a multiplication by $i$, and multiplying by $i$ twice gets us to the same place as multiplying by -1, which makes sense, since $i^2=-1$. It has to be counter-clockwise in order to give the imaginary axis its usual direction.

Saying that multiplying by -1 corresponds to a 180 degree rotation does not offer an explanation for why a negative times a negative is positive, but it provides a way of seeing it that is consistent with how we visualize other operations (complex multiplication). If you actually said "this is why a negative times a negative is a positive" someone could easily ask, "why does complex multiplication involve rotation at all?" If anything, this visualization is a way of providing a comfortable introduction to the idea that complex multiplication involves a rotation.

Although "negative times negative is positive" seems obvious once you've gotten used to it, putting double negation in a more general context is interesting. If we always expect not not a to equal a, we are relying on the law of the excluded middle, which not everyone accepts all the time.