CTK Insights

I have recently posted a constructive solution by Dan Shved to the problem of finding a line through a given point ($D)$ that halves the area of a given triangle $(\Delta ABC).$

There is a calculus based proof of existence (see, e.g., Honsberger.) For $D$ outside $\Delta ABC$, there are two rays emanating from D that form the smallest angle containing the triangle. The rays from $D$ in this angle cross and divide the triangle into two parts, say, left and right (looking from $D).$ If $\alpha$ is the angle form one of the extreme rays that describes the position of a ray through D, then the difference in the "left" and "right" areas is a function $f(\alpha)$ of $\alpha.$ The function is continuous and is negative at one extreme location and is positive at the other one. By the Bolzano Theorem, $f$ vanishes somewhere between the two extremes. For this angle, i.e., for this position of the ray, the left and right areas cut off by the ray from the triangle are equal - and this is exactly what is needed.

For point $D$ within the triangle, the proof changes only slightly. Lines through $D$ divide the triangle into "left" and "right" parts. By choosing one of the lines and rotating it $180^{\circ}$ degrees either direction, we get the same line but with the "left" and "right" parts interchange such that the difference in the two areas flips the sign, implying - again by the Bolzano Theorem - that somewhere along the way the two areas were equal.

There is of course a huge conceptual difference between knowing a problem has a solution and being able to actually produce that solution. However the generality of the foregoing proof has its advantages. For example, it is clear that - slightly modified - it will solve the problem of establishing the existence of a perimeter splitter through a given point.

It is then meaningful to ask whether a line could be simultaneously an area and a perimeter splitter, and seek a characterization of such lines if they exist. As a matter of fact, they do, and there may be $1,$ $2,$ or $3$ such lines. This was shown by A. Todd a senior at university. In truth, the problem has an interesting history.

In 1994, A. Shen published in The Mathematical Intelligencer a list of the so-called coffin problems - elementary looking problems with hard solutions - that were offered to Jewish applicants at the entrance exams at the Mathematics Department of the Moscow State University in the 1970s and 1980s. Problem 5 in Shen's list requires to Draw a straight line that halves the area and perimeter of a triangle. I. Vardi published solutions to those problems in a postscript file. (Unfortunately, my attempts to install a postscript reader led to my computer infestation that much diminished my interest in Vardi's solutions.)

It is not hard to establish that any line that halves both the perimeter and the area of a triangle has to pass through its incenter, see, for example, [Honsberger, Kodokostas, Kung, Todd]. The proof below is based on the area splitter construction. It makes use (like probably all other proofs) of the fact that the area $S$ of a triangle can be found from $S=rs,$ where $r$ is the inradius, $s$ the semiperimeter of the triangle.

Assume that $FG$ is the area splitter. Then $r(BF+BG)/2$ is the sum of the areas of triangles $BIF$ and $BIG$ which differs from the area of $\Delta BFG$ unless the incenter $I$ lies on $FG.$

It is as easily established that passing through the incenter is a necessary and sufficient condition of a perimeter splitter to be an area splitter, and the statements are reversible.

The following diagram shows Dan Shved's solution adapted to the case where $D=I$.

The diagram depicts the configuration where two solutions $FG$ and $F'G'$ cross sides $AB$ and $BC$. These are symmetric in the bisector of angle at $B.$

In a thorough investigation, D. Kodokostas tabulated all the possible combinations of the angles of $\Delta ABC$, proving there are cases when the number of solutions is exactly $1,$ $2$, or $3.$

References

1. R. Honsberger, Mathematical Delights, MAA, 2004, 71-74
2. D. Kodokostas, Triangle Equalizers, Mathematics Magazine 83, No. 2 (April, 2010) 141–146; available at http://dx.doi.org/10.4169/002557010X482916
3. S. Kung, Proof Without Words: A Line through the Incenter of a Triangle, Mathematics Magazine 75, No. 3 (Jun., 2002) 214
4. A. Shen, Entrance Examinations to the Mekh-mat, The Mathematical Intelligencer, Vol. 16, No.4 (1994) 6–10
5. A. Todd, Bisecting a Triangle, Pi Mu Epsilon Journal, Vol. 11, No. 1 (1999) 31–37
6. A. Todd, New and Letters, Mathematics Magazine 84 (2011) 396
7. I. Vardi, Mekh-mat entrance examination problems

A wooden cube - after being painted all over - has been cut into $3\times 3\times 3$ smaller cubes. These were thoroughly mixed in a bag, from which one was produced and tossed. What is the probability that a painted side turned up?

On impulse, one would approach the problem in a more or less standard way. There are $8$ corner cubes with $3$ painted sides, $12$ mid-edge cube with $2$ painted sides, and $6$ face-central cubes with only $1$ side painted. The total probability is then

$\displaystyle\frac{8}{27}\cdot\frac{3}{6}+\frac{12}{27}\cdot\frac{2}{6}+\frac{6}{27}\cdot\frac{1}{6}=\frac{1}{3}.$

Now generalize: cut the cube into $n\times n\times n$, $n\gt 1$ smaller cubes and ask the same question. The problem is not awfully difficult but needs some figuring out. Following the foregoing pattern, we eventually arrive at

$\displaystyle\frac{8}{n^3}\cdot\frac{3}{6}+\frac{12\cdot (n-2)}{n^3}\cdot\frac{2}{6}+\frac{6\cdot (n-2)^2}{n^3}\cdot\frac{1}{6}$

This expression simplifies, as you can verify, to $\displaystyle\frac{1}{n},$ which, by the way, confirms the answer for $n=3.$

The above is a useful and not too difficult exercise, but there is a delightful shortcut that avoids most of the counting of cubes and their sides.

$n^3$ cubes have a total of $6\cdot n^3$ sides. Of these, $6\cdot n^2$ are painted. All sides have the same probability of turning up, therefore, a painted side will turn up with the probability $\displaystyle\frac{6n^2}{6n^3}=\frac{1}{n}.$

References

1. R. Honsberger, Mathematical Delights, MAA, 2004, pp. 77-78

Subitizing is the ability to discern the number of object on a group without actually counting the objects. Even babies and animals do subitizing with small groups.

The photos below present an opportunity for subitizing in increasing order of difficulty.

See, you do not miss a distant bird that is seen flying between two branches of the right trunk.

Here, I can count fast if I split the birds into three groups.

When shooting in proximity to a puddle, the reflection of an object comes out sharp if the focus taken is as if you were shooting the object itself.

This becomes quite clear when you compare the two pictures below. In the first one the focus is on the reflection, in the second it's on the surface of the puddle.

Last week I got ahold of the new book by Daniel Dennett - Intuition Pumps And Other Tools for Thinking. I knew of Dennett - a prolific author and a noted philosopher at Tufts University - from his earlier books, Brainstorms, The Mind's I, Consciousness Explained.

I've been recently spending time in physical therapy where, while lying face down and getting my neck and back massaged, I've been reading Moby Dick. So, when I came across a New York Times article that featured Dennett, my interest was immediately picked by Dennett's quote

Philosophers can seldom put their knowledge to practical use. But if you’re a sailor, you can.

Herman Melville would likely agree. Getting a Kindle edition was a matter of seconds; the last two sessions in Physical therapy this is what I was reading. I also did some reading in between, but not much. The bottom line is I did not intend to write about the book before I went through most of it, if at all, and certainly not so soon. I would not do that now if it were not for an accidental tweet from @PrincetonUPress that informed the reading public of the new book Change They Can't Believe In: The Tea Party and Reactionary Politics in America by Christopher S. Parker & Matt A. Barreto. All I have to say about this book I garnered from the publisher's blurb which caught my attention because it served a perfect ground to apply some of Dennett's teaching.

The title - Intuition Pumps And Other Tools for Thinking - may suggest that the book belongs to a self-help variety but it is not so. Dennett tells the reader "This book shows what I've been up to all these years." There are definite borrowings from the three books I own, Brainstorms (1978), The Mind's I (1981) and Consciousness Explained (1992). (It looks like a while that I last bought Dennett's book which explains in part why I was open to a new purchase.) The book features chapters "Tools for thinking about meaning or content", "Tools for thinking about evolution", "Tools for thinking about consciousness", "Tools for thinking about free will", and more. I did not yet get that far, but I got appreciation for the second chapter "A dozen general thinking tools". If your perception of the word "thinking" -like mine - connotes purposeful activity whose meaning is more accurately expressed as "reasoning" or "reflection", you could agree that "Mental Tools" or "Tools for Polemics" would better reflect the content (and probably the intent of the author) that "Tools for Thinking". For example, Dennett draws reader's attention to the device he calls Rathering, which he describes as

a way of sliding you swiftly and gently past a false dichotomy. The general form of a rathering is "It is not the case that blahblahblah, as orthodoxy would have you believe; it is rather that suchandsuchandsuch - which is radically different.

So this is not exactly a tool one would apply to arrive at a conclusion but rather to mislead or just convince somebody - a reader, a listener, an interlocutor. Dennett does say that some ratherings are just fine, and I hope that mine is. To qualify a rathering needs to juxtapose inescapable dichotomies. Those that do not I would classify as "a tool for demagoguery", not "tools for thinking".

There are also ratherings that don't use the word "rather". A terser version may be of the form "_____, not ______." Regardless of the form, the demagogic ratherings exploit the common implication of the word "rather" that "there is an important incompatibility between the claims flanking it."

Reading Dennett's book made me sharply attuned to this kind of argument such that I could not miss the introductory sentences of the blurb for the Princeton U book I mentioned above:

Are Tea Party supporters merely a group of conservative citizens concerned about government spending? Or are they racists who refuse to accept Barack Obama as their president because he's not white? Change They Can't Believe In offers an alternative argument--that the Tea Party is driven by the reemergence of a reactionary movement in American politics which is fueled by a fear that America has changed for the worse.

The juxtapositions - and there are multiple - are false. Can't I say that Tea Party supporters are conservative citizens concerned about government spending who are (naturally, reactionary) racists fueled by a fear that America has changed for the worse? Neither syntactically nor semantically there is anything wrong with such an assertion! I may only wonder at the purpose of bringing up those labels - creating a straw man, Dennett would call it - only to follow up with a make-belief refutation in the next sentence. Being semantically covered by various flavors of "reactionary movements" the labels stick on, if not elucidate the essence of the Tea Party's reactionism.

With this introduction I can't believe the book is worth reading.

Now, back to Dennett's, "rathering" is not exactly a "tool for thinking", although being aware of the trick may strengthen your defenses against demagoguery. But there are very real thinking tools valuable in any kind of reasoning, math problem solving, in particular.

One such tool - I think it may be more properly called a metatool - was invented by Douglas Hofstadter and described yet in The Mind's I. Consider a tool or an argument as having many settings, each of which can be modified by turning a knob. Turn a knob, see the effect that transpires, evaluate the importance and relevance to the whole of a particular detail.

What role is being played in the blurb by the word "merely": "Are Tea Party supporters merely a group of conservative citizens..."? Think of it as a knob. Turn it hard till the word "merely" disappears. This reduces the stringency of the implicit finger pointing. The question becomes rather innocent, "Are these conservative citizens concerned about government spending?" The word "merely" makes us suspect that this is not quite so, it begs an elucidation that apparently comes in the next two sentences, linking them into a single damaging claim that to all purposes has been presented as three possible alternatives. "Merely" makes all three into one.

Another knob: President's skin color. Turn it all the way. May a citizen reject President's policies without being suspect of racism in case the said president isn't white? I believe the authors would do well to discuss separately whether there are racists in the Tea Party and how many of its members object to Obama's policies.

I'll stop here. I think that I said enough about the book that I am not going to read. I said probably too little about the book I am reading and enjoying. This is a good refresher for those who used to read Dennett's books, and a wonderful introduction into the scope of fascinating philosophical ideas, for those who never heard of him.

Robert Taplin's work is on display at the Ground of Sculpture, Hamilton, NJ.

To add the magic, these had to be viewed at night as in

And an extra one from the permanent collection

Several photos that I took on a deck of my house and on a short trip to a park next to the Raritan river.

Some reflections are surprisingly accurate; some are disturbed with ripples in the water. The ripples may be caused by wind, or by fowl.

I have several books in my review stack, so that when the book by Lance Fortnow arrived the day before yesterday, I opened it for cursory inspection, just to get a first impression. Yesterday I finished reading it. It's a tremendously good read, entertaining and informative.

"P vs NP" is one of the seven millennium problems selected by the Clay Mathematical Institute as the most important challenges inherited unsolved from the twentieth century. One of these - the Poincaré Conjecture - has already been solved, the others remain open.

Lance Fortnow tells the story of the "P vs NP" problem - its emergence, attempts to solve, the significance of a solution, its repercussions whichever way it comes out. Of the seven problems, a solution to "P vs NP" will have the most profound, probably even evolutionary, effects on the human society.

"N" stands for a class of problems that we know how to solve (efficiently); "NP" describes a class of problems that we are interested in solving, but only know how to check whether a specific solution is correct or not. Colloquially speaking, "N" denotes what we know, "NP" what we want to know. The "P vs NP" problem asks whether the two classes are (potentially) the same. If P = NP, we get (and in a pretty swift order) everything we want (even if later on we may regret getting that). If P ≠ NP, solving difficult problems may require growing hardships, fresh insights and new technologies. It is reassuring to know that, by now, there is probably no one who believes that P = NP, but the fact is that no one knows for sure. Much of what is going on on the internet concerning security depends on the assumption that P ≠ NP. Many things will go awry if a proof is found to P = NP. However, in this case, the proof could not be of mere existence. To solve the problem one must supply an efficient algorithm for solving one of what's known as NP-complete problems, and prove that the algorithm will work regardless of the underlying data. Luckily, the NP-complete problems are among the hardest.

This is like solving Sudoku problems. There is a huge amount of them, each depending on the initial set of clues. To solve the "P vs NP" problem, an algorithm should be able to solve any of particular Sudoku problems, regardless of the initial clues, and regardless of their size; the algorithm should come with a proof that this is indeed so.

The "P vs NP" problem was conceived during the Cold War, independently on the two sides of the Iron Curtain. The book draws a picture of dramatic and unexpected developments "here" and "there"; the juxtaposition reinforces the view that it would be better for everyone if the two classes of problems were different.

There is (p. 81) a compelling anecdote how the famous Russian mathematician Andrey Kolmogorov saved the study of Probability Theory in Russia. According to Marxist philosophy, there are laws that govern every phenomenon in the world such that there could not be possibly independent events. Kolmogorov came up with an example of a priest who, during a drought, prayed for rain. Next day it rained. For the Marxist philosophers it was impossible to admit that a prayer might have helped, forcing them to accept that the prayer and the next day rain were two independent events.

For a problem being hard does not imply that even in special cases it may not have an efficient solution. There are ways and ways to tackle hard problems: solving a problem approximately, finding and adapting a solution to a close but different problem, speeding up computers, even inventing altogether new computing technology. Thus the book describes the novel field of research - quantum computing. Awfully exciting.

Many approaches have been tried to solve either way the "P vs NP" problem. Nonetheless, here is what the author had to say about the progress made so far (p. 121):

We are further away from proving P ≠ NP than we ever were. Not literally, but in the sense that there is no loner any obvious path, no known line of reasoning that could lead to a proof in the near future.

The only known serious approach to the N versus NP problem today is due to Ketan Mulmuley from t he University of Chicago. He has shown that solving some difficult problems in a mathematical field called algebraic geometry (considerably more complex than high school algebra and geometry) may lead to a proof that P ≠ NP. But resolving these algebraic geometry problems may require mathematical techniques far beyond what we have available today.

This may be so; still, it is reassuring that another branch of abstract mathematics seems to find bearing on a problem of practical importance. Talk of The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

I just made an amazing discovery while reading a biography of Vito Volterra.

Naturally, the book mentions many other mathematicians and describes their work. What arose my interest was a mention of a paper by Giuseppe Peano "Il principio delle aree e la storia di un gatto" (The principle of areas and the story of a cat.) The paper appeared in Rivista di matematica.

The article referred to the discussion that had taken place at the Académie des sciences in Paris about why a cat, no matter how it falls, always lands on its feet. The question has arisen after a series of photographs had provided documentation of the dynamics of the fall.

It's not clear from the book whether, and if so then how, the Académie des sciences in Paris has settled the question such that Peano's solution might well be original. I do not know if you ever tried to find an explanation to the notorious ability of the feline species to land on their paws. Last year I had a chance to ponder that question while enjoying Mark Levi's book which bore the question as the title:

Curiously, Peano's explanation was wrong and Mark Levi gave a reason why it was so. Peano's brought an engagingly plausible argument:

this animal left to its own devices, describes with its tail a circle in the plane perpendicular to the axis of its body. In consequence, by the principle of areas, the rest of its body must rotate in the direction opposite to that of the motion of its tail; and when it has rotated by the desired amount, it stops its tail and with that arrests at the same time the rotational motion, saving in that way both itself and the principle of the areas.

Levi blithely refutes this kind of argument:

Some people say that the cat does that by spinning the tail. On close inspection this turns out to be false. As an experimental fact, tailless cats are just as good as tailed ones in flipping over. Alternatively, a theoretical argument shows that to accomplish a 180° flip in a fraction of a second, the cat would have to spin its tail so fast that its tip would have to break the sound barrier, or to come close. This would create a sonic boom, or a loud whistling at the very least. And enormous centrifugal force would cause a part of the tail to tear off and become a deadly projectile, almost like a bullet. So the "tail" theory quickly flunks the sanity test.

The question appears to be of broad interest. There even are two wikipedia pages that shed light on the subject Cat righting reflex and Falling cat problem both of which answer the question in agreement with Levi's book.

Moreover, advances in high-speed photography allowed National Geographic to produce a film that leaves no doubt of the veracity of the later-day answer. Admittedly, the French Academy was at a great disadvantage trying to tackle the problem some 125 years ago.

So, what is the right answer? As seen in the movie, the cat first bends in the middle, then twists so that its halves spin in different directions (likely in conformity with Peano's principle of areas, but also simply preserving the momentum), and finally straightens out.

Now, the thing that surprised me most in the course of the investigation was a wikipedia reference to the 1969 article by T. Kane and M. P. Scher "A dynamical explanation of the falling cat phenomenon" (International Journal of Solids and Structures 5 (7): 663–670. doi:10.1016/0020-7683(69)90086-9), as the solution as "originally due to (Kane & Scher 1969)." This appears to imply that the problem remained (officially, at least) unsolved for about 80 long years - quite on a par with the, say, better known Poincaré conjecture. But think of it, most probably the members of the Académie des sciences in Paris were not the first to ponder the question, which leads to a conclusion that the question has a much longer history. Hmm, I would never guess.

Wikipedia notes that

The cat righting reflex is a cat's innate ability to orient itself as it falls in order to land on its feet. The righting reflex begins to appear at 3–4 weeks of age, and is perfected at 7 weeks.

This discovery is reported as late as 19 December 2011! (Disparagingly, the link to the research is not publicly available.) What is unclear is by which method the researcher came up with the "3–4 weeks of age" estimate and how big was the observed feline population.