CTK Insights

26 May

The Jeweler’s Observation, a look back

Paul Brown, an Australian math teacher and author of Proof, a book that I may characterize as a well-written guided introduction into that most fundamental activity, has brought to my attention a recent post at the Futility Closet blog, The Jeweler’s Observation, which I fully reproduce below:


Prove that every convex polyhedron has at least two faces with the same number of sides.

The solution is initially hidden but becomes available on a button click:

Consider the face with the largest number of sides. If that face has m sides, then it’s surrounded by m faces. Any face must have at least 3 sides. So altogether in this group we have m + 1 faces, and each face must have between 3 and m sides. At least two faces must have the same number of sides.

From Arthur Engel’s Problem-Solving Strategies, Springer Verlag, 1999.

This example provides nice references to two powerful tools in the arsenal of mathematical problem solving: the Extremal and Pigeonhole Principles. However, that was not the reason I decided to write this blog. One of the most (if not the most) important strategy in problem solving, namely, G. Polya's Looking Back step.

So, looking back at the quote from A. Engel's book, what has been actually proved? From the proof we can extract a stronger assertion than that claimed in the problem. This can be formulated in a couple of equivalent ways. But first note that the convexity of the polyhedron was not absolutely essential and that condition can be weakened. For example, the argument appears to work for any polyhedron in which two faces may only share an entire edge (I am not sure even this is necessary). Thus, thinking of such polyhedrons, the original problem can be so generalized:

  • Every polyhedron has a face with two adjacent faces that have the same number of edges.
  • Every polyhedron has two faces with the same number of edges adjacent to the same face.

On a second look back, what is essential for the proof is the requirement that all faces of the polyhedron be convex polygons. If that condition is not satisfied, there are 3D objects with a "skeleton" formed by straight line segments, and "faces" bounded by these edges, with no two faces having the same number of edges. Can you give an example? (That would be a counterexample to the original claim.) I can think of a 3D object that consists of four pieces that have 3,4,5, and 6- sided faces, the ones with 4 and 5 sides not being convex, or even flat.

01 May

A wrapping surprise

Jim Henle's book "The Proof and the Pudding" that I have recently reviewed contains a good deal of surprises. Across several chapters of the book the author looks into the billiard problem (that is very much like the Two Pails puzzle) and its modifications. One of these involves wrapping a ribbon around a 3D box. Start at a corner and move within one of the faces along the diagonal of that corner. When you hit an edge, just bend the ribbon around the edge to the adjacent face, keeping the same angle (45^{\circ}). Here's an example with an incredible box of dimensions \sqrt{2}\times e\times\pi:


Can you figure out how that path will proceed? In a 2D billiard with the crazy dimensions like \sqrt{2} or \pi, the path will circle without end. Most likely, this is what your intuition may suggest happens in the 3D case at hand. My intuition did. So here comes the first surprise:


The path that consists of five legs ends up right at the corner where it started. There is an easy explanation why this is so. As in the study of the regular billiard, we may - instead of bending around an edge - flatten two adjacent faces of the cube and let the path pass on a straight line:


The sequence of five successive face traversals snugly fit into a square of size \sqrt{2}+e+\pi, with the (image of the) path going from one corner of the square to its opposite along the diagonal. Note one face outside the square not touched by the path.

Now, that you may think that you understand and can explain why the path returns to the starting corner, consider wrapping exactly same box but starting in a different face. Before, our first leg lay in a \sqrt{2}\times\pi face. Now, let's start within a \sqrt{2}\times e face. Can you predict what will happen to the path?


As you may surmise, the path will behave - if I may say so - in a more rational way. Given the incommensurate dimensions of the box it was rational to expect an endless path. This is what you get on the second attempt. But there remains a question to ponder: Why was the first path so short? Jim Henle leaves it to his readers to find the answer. So do I, though I'd like to suggest also starting the wrapping in the e\times\pi face.

02 Apr

An impossible building, at least this is how it looks

I had very little time driving through Newark, NJ when I caught the sight of the Panasonic building:


It absolutely appears an impossible structure that reminded me of an old applet Structural Constellation one example of which you can see below:


I'll have to visit this place again to learn what creates the illusion.

01 Apr

Why Learn Mental Math Tricks?

"Why Learn Mental Math Tricks?" is the question author Presh Talwalkar tries to answer in the introduction to his new book "The Best Mental Math Tricks". He gives several reasons, starting with

For one, math skills are needed for regular tasks like calculating the tip in a restaurant or comparison shopping to find the best deal. Second, mental math tricks are one of the few times people enjoy talking about math. Third, mental math methods can help students build confidence with math and numbers.

Mental math tricks are fun to share.

I absolutely agree with all of the above and wish only to remark that his book makes a strong stand against the common meaning of the word "trick" whose connotations include magic, deceit, and disapproval.

The book is not a collection of disparate math facts but rather a textbook of well organized and well explained methods of handling arithmetic operations on classes of numbers grouped by their size, divisibility properties, last digits, etc. A whole section that is devoted to each of the "tricks" contains practice problems with complete solutions to illustrate the method. Every method is accompanied by a mathematical proof that sheds the mystery of why the method works. And this is what places the contents of the book in contraposition with the most likely interpretation of the book's title.

In a recent book "Mathematics without Apologies" Michael Harris devotes a whole chapter to the role played by the so called "math tricks" in the body of mathematics.

While capital-M Mathematics is neatly divided among definitions, axioms, theorems, and proofs, the mathematics of mathematicians blurs taxonomical boundaries. ... A mathematical trick is a notorious crosser of conventional borders.

While it is most unlikely that Michael Harris was ever thinking of tricks described and explained by Presh Talwalkar I have no qualms of making this association. Presh Talwalkar has included in his book

... tricks that are relatively easy to learn, are fun, or have educational value.

I'd go further as I strongly believe that those tricks are more than anything else convey to the early learners the essence of practicing mathematics. Presh Talwalkar's book may also open the eyes of an older generation on what they missed in the early grades.

07 Oct

Review of "Zombies and Calculus" by Colin Adams

Colin Adams, author of the unique book "Zombies and Calculus", opens the book with a warning that "if you are squeamish you should not read the book." I venture an additional warning: if you lack a sense of humor, you should not read it either. As an afterthought, the author considers that, given the title of the book, there's little likelihood a squeamish person wold pick it up, in the first place. In my view, such thinking is a mistake - the first of two mistakes the author has committed. Many a student felt like a zombie in a calculus class. These might have shown interest in the book in the hope of finding an explanation to their experience. To these students I say, no, the book is about hard-core zombies - stiff-legged cannibals, entirely devoid of high mental functionality. The one thing on their mind is feeding on anyone yet alive and thinking. But, unlike in the movies featuring Milla Jovovich, in the book zombies have been observed to be tearing clothes off zombies of the opposite sex - in public and with an obvious intention. Unfortunately, the narrator - a college math professor - being followed by his former students and colleagues who now saw him only as a potential meal had not the time to further dwell on his observations.

One really needs a well developed sense of humor to read about a talented student who, with a chunk of her neck missing and the head at a weird angle, tries to grab her professor of a few minutes ago, and so had to be knocked down. On the other hand, every one would laugh at a delinquent student, oblivious of the surrounding dangers, who hands to the narrator (concerned with escaping a crowd of zombies in hot pursuit around the corner) his late homework. One again needs to summon one's sense of humor to read about the sad fate that befell the unthoughtful student short time after the encounter.

At the outset, I have mentioned that the author committed two mistakes. Here's the second one. Early in the story, several survivors of the initial attack, found themselves locked up in an office, with zombies moaning and banging into(?) each other just outside the door. Naturally there arose a question of active protection, and a girl removed her stockings to be filled with heavy objects that, when swung, could bring a zombie down. The narrator then cuts the stockings with scissors into two legs. However, as every grownup knows, stockings are already a two-piece item. Pantyhose is the one dress item that needs separation.

Stockings Pantyhose

The episode with the stockings serves to develop a conversation (and related mathematical tools) about speed, force, and the strength of human (i.e., zombie's) skull. As the story evolves, other mathematics comes in handy: equations of pursuit, logistic equation, predator-prey model, stationary point of a system of ODEs, Newton's Law of Cooling. Mathematics in the book is impeccable. However, from the incomplete list of topics, one may start suspecting (and justly) that the book should be more properly titled "Zombies and ODE" or "Zombies and ODE Modeling". (The author admits as much in the introduction by pointing to those who already learned calculus as his primary audience.) I do not count this as a third mistake, for this is a common marketing ploy to rely on a sound bite title to attract broader readership. And the book well deserves attention from not squeamish math instructors and a wider audience of intelligent readers, curious of a new literary genre that mixes storytelling with gentle mathematical instruction.

14 Sep

Review of Coffee, Love and Matrix Algebra

I am admittedly a compulsive reader. I either stop reading a book if I do not like it, or I continue reading until finished - with only obligatory interruptions. Gary Davis' book brought me an entirely new and tantalizing experience. Gary used daily tweets on twitter.com to announce new installation at his popular blog Republic of Mathematics (http://www.blog.republicofmath.com/). So there was no other way but to do the reading a chapter a day. Had it been my choice - and I can candidly say that after the fact - I would have gobbled the book in one setting.

The story is about a year long episode in a life of a college math department. Any one, I believe, who ever held a position in an academic department would easily identify the traits of Gary's protagonists as shared by some of their colleagues. The characters were authentic, evolution of events realistic; it took me a while to realize that the book was entirely a work of fiction.

Naturally, while there are similarities, not all math departments are the same; Gary's no different in this respect. It is painted with its own problems and peculiarities. Although a mathematics professor, Gary navigates his story with the skill of a professional writer. He narrates his story that takes several imaginative turns with confidence of a participant and kind humor of life's keen observer. That's a great story, masterfully and engagingly told. Read and enjoy.

01 Aug

Distance to the Horizon on the Fourth of July

I had the luck to celebrate the past 4th of July with our friends in their newly acquired home just above the marina in Atlantic Highlands, NJ. The view from their backyard was absolutely breathtaking. The ambient light that appeared to blur the background made the view even more enchanting.


Here is a map that would help you identify parts of the panorama.

NY-NJ bays

In the middle above the center line, across the Sandy Hook and Low bays the part of the Interstate 278 is the famous Verrazano Bride, past which there are visible Manhattan tall buildings; on the right, that's Brooklyn whose buildings seem taller, but only because of relative proximity.

The sunset was spectacular.

Sunset begins

From the beginning (above) and to the end (below).

NY-NJ bays

With the dusk, came fireworks. We were able to simultaneously see the bursts of illumination over Manhattan (that appeared just above the Verrazano bridge) and Brooklyn. (You can see these too if you click on the photo below.)

NY-NJ bays

And next came the fireworks just above our heads shot from the marina below.

NY-NJ bays

Next day I was downloading my 4th of July photos to my computer from the camera. In the zoomed-in version of the Verrazano bridge I noticed a short upward stick under the bridge which - to my great surprise - happened to be the Stature of Liberty.

NY-NJ bays

To inquire the distance to the stature has now appeared quite natural. This can be estimated with the following map.

NY-NJ bays

It is also possible to use the well known formulas for such estimates, I refer to the wikipedia. The distance is about 22 miles, Verrazano bridge being at about 2/3 of the way. Using the reversed formula it is possible to estimate how high above the marina have I been celebrating the 4th of July this year.

01 May

Making Escher Proud - in Dance

24 Apr

Environmental impact of power lines

This is to simply document my observation which I've been mulling over for a long time until very recently.

A couple of streets that I daily drive over are lined with trees whose branches seem to exhibit strange growth pattern. While their older branches point unremarkably each other way, the younger ones sprout pretty much vertically:

strangely vertical growth pattern

And here is another sample:

Now, at Spring time, the situation became even more salient

I've been preoccupied with this phenomenon until one day I realized that its weirdness was so eye catching that I stopped looking at the more normal trees:

And then all became crystal clear, if not scientifically proven. The vertical growth happened only in the proximity of power lines, all on one side of the street. Same trees on the other - wire-free - side of the street, grew in a natural, sort of a more random way.

So here's my discovery or rather conjecture:

The magnetic field induced by power lines causes the trees in their vicinity to sprout strictly vertically.

More research is necessary to support that conjecture.

17 Feb

Beating a Dead Horse: Mathematics Education Reform - Again

I find the opening paragraph in a recent article by Marcus du Sautoy (The Mathematical Gazette 97 November 2013 No. 540, 386-397) revealing:

During my year as President of The Mathematical Association the government began a review of the curriculum across all subjects taught in school. Given the constant tinkering with the education system by every government, this is probably a sentence that any MA president could write during their tenure.

The same of course holds for every president of the United States and all relevant organizations (NCTM, NEA, MAA, etc.) The fact is that the educational reform is an unceasing undertaking that in mathematical education started at the end of the nineteenth century and that is still going on strong. In the US, there were some discrete moments of new announcements - not to mention the New Math (1960s), there has been a series of standards from NCTM (Curriculum and Evaluation (1989), Professional (1991), Assessment (1995), Principles (2000), Common Core(2010)). It's a given that the effort will not stop any time soon. The CCSS have already gathered a plethora of critics.

There is one person whose name always comes up when I think of the stream of mathematical reforms that come one on heels of another - Diane Ravitch. I never met Diane and all I know about her is that at some point in time she changed her mind about reforms and switched the camps so that one camp praised her courage, the other lamented her betrayal. And when I think of changing one's mind, I am reminded of a curious episode involving a well known Russian physicist Yakov Fraenkel (1894-1952). According to legend, he was shown a slide with the graph depicting the outcome of an experiment. He brought it to light and immediately explained why the experiment went the way suggested by the graph. When one of his students pointed out that he was holding the slide upside down, Fraenkel turned it around and produced an explanation of why the graph had to be that way.

The relevance of my perception of the reform process to Diane Ravitch's story is that I feel that most math educators and after them politicians that make decisions come up with reform ideas viscerally or by the sixth sense. The arguments come later: articles written, committees are set, statistics is collected, theories come forth. All this seldom causes anybody to change one's mind. And I keep wondering why, when it comes to educational reforms, it takes making or changing one's mind, and not some kind of deliberate experimentation.

There were successful experiments. Two are well known and are to my liking: an unorthodox geometry course by Harold P. Fawcett at the Ohio State University starting in the 1930s, and somewhat earlier one by Louis P. Bnzet. The consensus I believe is that a teacher needs to be a Harold Fawcett to manage a course like Fawcett's or W. Eugene Smith who taught that course in the years 1945-1956. Here I believe lies the main reason for the continuous attempts to standardize mathematical education and keep it under control with standardized testing. Other, reasons are usually cited publicly, but I think this is the main one: teachers are not trusted to do the right thing by students on their own.

There is certainly a good reason for mistrust: as in every endeavor, there are excellent, mediocre, and outright bad teachers. For the latter it is easier to follow strict curriculum with testing than to adapt to their class and individual students' progress. Many may not be able to do that.

I strongly believe - what else can I say or how else can I argue - that the only way to find a solution to the real or perceived deficiencies in math education is to admit to that fact. In a country the size of the US, any attempt to teach uniformly, giving every one the same opportunity, so to speak, is simply impossible. Worse, such attempts will always shortchange the better students. As Steven Strogatz wrote in Notices of the AMS,

Though it’s taboo to admit it, I believe there are some kids who have a feel for math.

By extension there are students who do not have that feel, they may be even in majority.

I may be beating a dead horse, but here what Marcus du Sautoy wrote about in the article I mentioned at the beginning:

I often get the feeling that we are still stuck in Napoleonic France, just doing mathematics to serve the state. If only we could find a modem-day Humboldt to take the reins at the Department of Education ....

For the first time in Germany, the study of mathematics formed a major part of the curriculum in the new schools and universities. And mathematicians, freed from the need to model the physical world, began instead to explore mathematical ideas for their own sake. It gave rise to the creation of geometry that lives beyond our three-dimensional universe.

And before that

Just as in English a student isn't meant to grasp the full complexity of a Shakespeare play, we should be prepared to take the risky step of teaching big ideas that a student might not fully comprehend but rather they should be given a way to glimpse something of these great stories. Just as any course in English literature can give just a taste of the great works, a mathematical literacy course would not aim to be complete but to expose students to a sample of what is out there.

I only have one point of disagreement with du Sautoy. It seems to me (I apologize if I am wrong) that he talks of the middle and high school curriculum. I would start earlier by trying to engage little kids with big ideas, and stop ostracizing those who do not get them.

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