FOCUS is EVERYTHING

A few weeks ago I mentioned the book "Ghost Map" about Dr. John Snow's dedication in discovering the cause of Cholera in the London suburb of Chelsea.

Last night as I watched a BBC special on the miraculous rescue of the 33 miners in Chili, I couldn't stop thinking about the incredible amount of resources that had been brought together from around the world, and about the last headline I read on my internet news reader.... "Haiti cholera death toll passes 250 "....

As I watched caravans of trucks that brought water across a desert to keep the drill cool (the first one, that never made it to the miners) I kept thinking about the 250 and growing death toll in Haiti.... We know what killed them... it's the water... and yet..... there seems not to be any great need on the part of the world to rush to Haiti and drill water wells, much easier than a hole to extract miners from a cave deep in the earth... and saving tens to hundreds of times as many people...

I loved the rescue, the beautiful technology applied, and then, we turn our head... or maybe not..

Donate here


Water For People helps people in developing countries improve quality of life by supporting the development of locally sustainable drinking water resources, sanitation facilities, and hygiene education programs. Around the world, 884 million people do not have access to safe drinking water and 2.6 billion are without adequate sanitation facilities. Every day, nearly 6,000 people who share our planet die from water-related illnesses, and the vast majority are children. But the real failures are all the broken pumps, filled latrines, and solutions that aren't. We want to change all that. The solution? Programs that last and examine entire districts and regions rather than purely households and villages. Create solutions that last, and not only do people benefit for a long period, but organizations don't have to expend time and energy going back again and again to the same location.

A Little Math Humor

Shecky, who blogs at Math Frolic had a link to a site with some math jokes... This was the one that gave me the best chuckle.

Addendum: My Apologies to Steven Colyer whose blog "Multiplication by Infinity" is the blog I linked to above. Sorry not to have credited Steven in the first place... just must have been a busy day. Mia Culpa

Physics and Poker and NPR, Oh My!

The folks at "All Things Considered" are talking about how physics may make you a better poker player.

A few years ago, physicist Jeff Harvey invited Eduard Antonyan to a game of poker at a friend's house. Antonyan was a graduate student of Harvey's at the time, in the physics department at the University of Chicago.

"I invited Eduard to play because we're always looking for new victims," Harvey tells NPR's Guy Raz. "But it didn't exactly work out that well."

They Found a Way to Measure How Long it Takes to Fall in Love?

NO fooling.... the header reads,

Falling in love can elicit not only the same euphoric feeling as using cocaine, but also affects intellectual areas of the brain. Falling in love only takes about a fifth of a second. The findings raise the question: "Does the heart fall in love, or the brain?"

Here is the link on Science Daily...see if you can figure out what they are measuring, or how they measured it... Keep in mind you have to put two people who are not in love in a situation in which they fall in love, and be measuring love (somehow) when it happens.

More Harmonic Thoughts

I got a note after my last post that reminded me how little Julian Havel seems to be known in the US. Denise, of Let's Play Math, related that her local library had NO books by him. Impossible... No, I don't mean that it is impossible they have no books by him (it should be)...but that is one of his really good books.



Havel is (or was) a headmaster at a British College (sort of like the last year of our HS in the US).

In his book, Impossible, he has a chapter on the Harmonic Series in which he mentions a couple of more interesting notes I might have added for students.

For example, it is quite easy to see that H₁= 1, and H₂= 3/2. H₆= 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 2 9/20... and those three are the only ones in the entire infinite series that end in a terminating decimal. Havel states that, "it can be proved" but does not... anyone know an tractable proof for that to approach Pre-calc students?

Another quick point to mention is the extremely slow rate at which the function diverges.... Ralph Boas Jr was co-finder of the smallest n for which H_n> 100. It turns out it takes more than 15 10⁴² terms to reach 100. Which means it is adding less than .01 each step... and yet... as Borovik cautions us... It Diverges..

A Nice Presentation of the Harmonic Series

Here is a nice video here about the Harmonic series. I would love for all my Pre-calc and calc students to see this video just to see the really nice proof by contradiction that the series is infinite. The paradox itself is just a bonus, and the fact that the squares add up to a value involving pi is a beautiful story that ought not to be popped out as an aside... sometimes we need to take the time to dig a little deeper into what Euler has given us... but a nice presentation all the same. Enjoy, but remember the adomonition that Borovik gave to his students,

" Today I said to the calculus students, "I know, you're looking at this series and you don't see what I'm warning you about. you look at it and you think, 'I trust this series. I would take candy from this series. I would get in a car with this series.' But I'm going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.

For more about the harmonic series, here and here and one more here, but surely there are more...use the search function.

DaVinci Was Right

HPO MVI 0043 from U of T Engineering on Vimeo.

I subscribe to eGFI, a site that is there to encourage future students to go into Engineering. I recommend it to students who ask about or seem to want to go into Engineering. Today they had this on the site, with the video above..


Human-Powered Plane Flies Like A Bird

"For the past four years, Todd Reichert, an engineering student at the University of Toronto, has been working to perfect one of Leonardo Da Vinci’s greatest concepts – an ornithopter.

An ornithopter is a human-powered aircraft that flies by flapping its wings, and with the help of 30 other students, as well as $200,000, Reichert made history by building such a vehicle and piloting a sustained flight.

Called the Snowbird, the aircraft is made of carbon fiber, foam and balsa wood, and weighs less than 93 pounds. Its wingspan is 104 feet, which is comparable to that of a Boeing 737. The vehicle works by pumping a set of pedals attached to pulleys and lines that bring down the wings in an elegant flapping motion.

Leonardo Da Vinci conceived the ornithopter back in 1485. For much of his life, Da Vinci was fascinated by the phenomenon of flight and produced many sketches of flying machines; however, he never actually made one."

The Bead by Mamikon's Visual Caclulus

Over the last few days I came upon a way to explain the volume of a bead with an (almost) non-calculus approach. My solution was inspired by a comment posted by Arjen Dijksman. I admitted I didn't understand his suggestion to use Mamikon's visual calculus, so he did what good teachers do. He posted a very clear explanation that even I could not miss. It's a beautiful blog, pretty math... check it out..

and Thanks loads Arjen.

A Non-calculus Explanation for the Volume of a Bead

A few days ago I wrote about the "surprising constant" that a bead formed by drilling a hole through the center of a sphere had a volume that was dependent only on the height of the remaining bead. At the time I admitted I could not think of a good "non-calculus" explanation for why the volume for a given bead height was a constant.

Afterward Arjen Dijksman,(see his blog) who regularly provides helpful insights, suggested that the volume could be explained using the ideas of Mamikon Mnatsakanian that I discussed here. While I was not able to see Arjen's solution, it did lead me to realize that a "non-calculus" proof existed (with some allowances).

Thinking of Mamikon's annular sweep led me to realize that every cross-section through the bead perpendicular to the axis of the drilled hole would be an annulus. In fact, using R as the original sphere's radius, r as the radius of the hole, and h as the distance above the center of the sphere where the cross section was taken, the area of the annulus would be Pi times (R² - r²-h²), which is just the area of a circle.

Now by Cavalieri's Principal, "If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal."

Several candidates exist with circular cross-sections; cylinders, cones, and spheres are examples. The cross-sections of the bead went from zero at the top, to a maximum at the center, so the cylinder is excluded. We can exclude the cone since their radii decrease linearly from top to bottom and our bead does not. And since we are already tempted with the idea that the volume of the bead is the same as a bead with a radius equal to half the total height of the bead, why not try to compare those?

To make the notation easier, I decided to call the height of the bead 2H (letting H² equal the value (R² - r²). So the cross sectional area of the sphere with raidus H at any height, h, above the center will be Pi(H² - h²). As we have shown above, the cross-sectional area of the bead at the same height h is Pi (R² - r²-h ²). and since H² is equal to (R² - r²) these two are equal.

So we have proven that the volume of the bead is the same as the volume of a sphere with the same height;... but did we use calculus? I fear that the proof of Cavalieri's Principal is founded in the calculus, and thus we still have a "calculus-based" explanation...but it's the best I have to offer to date.

Folium

I realized in Caculus class the other day that I didn't have a link for folium on my Mathwords webpage, so I decided to write one here and just put a link...
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Folium is the Latin word for leaf. Student’s frequently are introduced to them in learning to graph polar curves such as r=sin(3*theta) which is a trifoil, or three leaf, sometimes called a “rose” (shown above).

The one that came up in my Calculus class a few days ago was one of the more widely known mathematical folia, the “Folium of Des Cartes”, x³ + y³=3xy. A nice chance to use the newfound skill of implicit differentiation. This curve was created by Des Cartes to challenge the methods of Fermat in finding extrema in the early days of the emerging calculus. I should mention that both men found the extrema without the calculus of limits as we now know it. I mentioned Des Cartes method, as extended by Hudde, in an earlier posting here.



The Folium can be rendered in parametric form by using the substitution y=tx to give

x³ + (tx)³ = 3tx² and dividing out x² we get to

x + t³ x = 3t... This can be factored for x to get x(1+t³)= 3t.... or

x= (3t) / (1+t³) and thus by y=xt we get y= (3t²)/ (1+t³)

I recently came across this neat stamp image with a picture of Des Cartes and his Folium.

An Interesting Triangle Property

I'm considering this one for my "Almost Pythagorean" file. I came across this recently and found it interesting. The Pythagorean Theorem is actually a property about triangles with 90 degree angles; but this one is a property of all triangles that contain a 60 degree angle at vertex A.

The "Then" implied by the above is that

The derivation is not too difficult and might well be presented to a good pre-calc student as a challenge. If you don't want me to spoil it, stop reading now until you have tried it.... It makes me wonder what we could come up with if we started with a thirty-degree angle.

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Surprisingly Constant

Some things never change... and some of the things that don’t change are pretty remarkable. I was recently reminded of two that always seemed non-intuitive, and therefore quite pleasing. Both of these can be proved toward the end of a first year calculus course, and at least one can be explained with a little hand-waving, to a good pre-calc student.

The first is the simple fact that if you take a slice a sphere with a pair of parallel planes that are some distance, d, apart; the surface area of the part of the sphere between those planes is the same no matter where the slices are made. Near the “poles” or near the “equator”, the area is the same. It is nice to be able to tell students that this incredibly interesting fact was known to Archimedes in the first Century of the modern era when he proved that the surface area of a sphere is the same as the lateral area of a cylinder that contains the sphere.

This image is part of an explanation provided at a math site at the University of Regina

The essence of the explanation uses nothing beyond similar triangles. If we approximate the sphere’s surface between the slicing planes as a frustum of a cone, then the slant height (AP) can be shown to be in the same proportion to the height (AB)as the radius of the frustum (QP) is to the radius of the sphere(KP)). This image is also from the U Regina site mentioned above.Another nice thing about this relation is that when you set up the rather fearsome looking integral for the surface area, it reduces through simple algebra to a constant.




The second similarly interesting idea is about the volume of a shape I call a bead, for lack of a more precise term. The bead is the solid remaining when you drill through the center of a sphere. The unexpected constant here is that for any radius sphere you start with, the volume of the bead depends only on the height. And perhaps even less expected, is that the volume is the same as the volume of an un-drilled sphere with the same height (i.e., when 2r = h).

I have not found a simple explanation for pre-calc students that will explain this one. If you have one in your pocket, drop me a note.

RIP Benoit Mandelbrot

Benoît B. Mandelbrot, a maverick mathematician who developed an innovative theory of roughness and applied it to physics, biology, finance and many other fields, died on Thursday in Cambridge, Mass. He was 85. Here is a recent Ted Talk he gave about his work in Mathematics. Enjoy...

More about Timid Testers

I recently wrote about my disappointment with a very small part of Kaiser Fung's new book, "How Numbers Rule Your World." I want to point out again that I have really enjoyed the book. One of the things that makes it really interesting is the little asides of historical note that are just the kind of detail I love...
Case in point:
In the same chapter I spoke of in the last blog, Kaiser explains a little about the history of the lie detector.

William Marston, A Harvard-trained psychologist who was the first to relate truth telling and blood pressure variations, failed to popularize the concept in the early twentieth century, but he ultimately achieved immortality by creating the comic book heroine Wonder Woman, who not coincidentally wieled a Magic Lasso that "makes all who are encircled in it tell the truth".

Timid Testers and Magic Lassos

Kaiser Fung, of Junkcharts, has written a really nice book about statistical ideas and applications called "Numbers Rule Your World" which I read about half of early Sunday Morning while my beautiful sweetheart was sleeping in to recover from her trans-Atlantic flight. I really like the book, but came across a short passage where I was distrubed by the seemed support for bad statistics.

The issue in question was the use of lie detectors to acquire confessions. "In the United States it is legal to obtain confessions via the reporting of false evidence, which means the police are free to tell a suspect he or she failed a lie detector test no matter what happened." Then he goes on to give an example of when the method had been effective...In the case of the murder of Angela Correa of Peekskill, New York in 1989. There was a 16 year old classmate suspect who met the psychological profile the police had for the killer. There was a problem, though. Physical evidence was either absent, or contradictory... Hair samples did not match the accused, and there were none of his fingerprints on any of the evidence found near the body. But most damning was the fact that the DNA evidence from the semen samples inside the raped victim clearly excluded the suspect. But after all this comes the line, "The police knew they had the right guy..".
so they talked the 16 year old into taking a lie detector test, the least accurate investigation tool (other than a DA looking for a conviction) in the police arsenal. After "eight hours in a ten foot by ten foot room, facing in turns Detective McIntyre and Investigatior Stephens." In the end, the young man made a confession. Fung concludes the section with the sentence, "Without the polygrpaph exam, there would have been no confession and thus no conviction."

I was so troubled by the story, that I went to look up the case, and it turned out that shortly after Fung stopped following the case, (he comments on the young man's release in September of 2006) in November of 2006 the physical evidence led to another confession:

"A convicted killer was indicted yesterday in the killing of a Peekskill teenager, a crime for which another man spent 16 years in prison after being wrongfully convicted. Steven Cunningham pleaded not guilty yesterday in State Supreme Court to charges of second-degree murder and first-degree rape. Mr. Cunningham, who is already serving 20 years to life in state prison for strangling his girlfriend’s sister in 1994, is accused of murdering Angela Correa, 15, on Nov. 15, 1989. He was linked to the crime after new testing matched his DNA to semen found in the girl’s body, prosecutors said. Westchester District Attorney Janet DiFiore convened a panel last week to review the circumstances that led to the wrongful arrest and conviction of Jeffrey Deskovic, one of Miss Correa’s classmates, who was released in September.

Prerequisites

A dialogue on the AP Calculus EDG about whether homework should be graded or not (or even given) led Dan Teague to contribute the following short poem by Ralph Boas, who was a professor at Northwestern for thirty or so years. Having just finished grading a pre-calc test in which several students had decided that (-3)/(-4) = -(3/4) and many, many more had given me the so common as to be beyond surprising simplification of (x-3)^2 = x^2 + 9 may have colored my identification with the poem.

Prerequisites, by Ralph Boas

How could you be a cowhand
If you couldn't ride a horse?
If you yearn to cook for gourmets
You'll need some food, of course.
You can master many subjects
If you only have the will;
But how do you hope to cope with calculus If your algebra is nil?

How could you sing in opera
If you haven't any voice?
If music seems too difficult
There is another choice.
Rewards in Math are plenty
But this obstacle looms big:
How can you shine in calculus
If you won't learn any trig?

Ralph Boas is often remembered for another literary/math creation, "A Contribution to the Mathematical Theory of Big Game Hunting", published in the American Mathematical Monthly.

Research on Brain Science is......confusing.

Just looked at a page from Science Digest... here are the headline leads to the stories. See if you can figure out the truth about education and Alzheimer's.

If you don't have a college degree, you're at greater risk of developing memory problems or even Alzheimer's. Education plays a key role in lifelong memory performance and risk for dementia, and it's well documented that those with a college degree possess a cognitive advantage over their less educated counterparts in middle and old age.

Education Does Not Protect Against Age-Related Memory Loss, Say Researchers (Jan. 10, 2007) — Adults over 70 with higher levels of education forgot words at a greater rate than those with less education,

Brain Exercises May Slow Cognitive Decline Initially, but Speed Up Dementia Later (Sep. 1, 2010)

Education Helps Against Dementia, Swedish Study Finds (June 1, 2010) — Researchers have discovered that education not only delays the early symptoms of dementia, but can also slow down the development of the disease

And now you know.... Sorry Paul Harvey

A Math Book for ????

I'm thinking about connections, and suddenly I come across,

"In 1972, the physicist Freeman Dyson wrote an article called “Missed Opportunities.” In it, he describes how relativity could have been discovered many years before Einstein announced his findings if mathematicians in places like Göttingen had spoken to physicists who were poring over Maxwell’s equations describing electromagnetism. The ingredients were there in 1865 to make the breakthrough—only announced by Einstein some 40 years later. "

A while back I got a free copy of "The Mystery of the Prime Numbers"
by Matthew Watkins, which is lavishly illustrated with drawings by Matt Tweed. I don't know either guy, but Matthew somehow stumbled across my blog and thought I would be interested... I was, but more on that later.
So then I came across a link (sorry whoever) to an article in Seed Magazine by Marcus du Sautoy, who spends a little time with primes himself. The quote above is the opening paragraph.
Here are a couple of clips that tingled my head...

"In their search for patterns, mathematicians have uncovered unlikely connections between prime numbers and quantum physics. Will the subatomic world help reveal the elusive nature of the primes?"

Ok, Primes are mysterious and hard to figure out, and so is the nature of the universe, and everyone knows some quote about physics being written in the language of math... but linking Primes to String Theory???
So they laid a foundation:

This unexpected connection with physics has given us a glimpse of the mathematics that might, ultimately, reveal the secret of these enigmatic numbers. At first the link seemed rather tenuous. But the important role played by the number 42 has recently persuaded even the deepest skeptics that the subatomic world might hold the key to one of the greatest unsolved problems in mathematics."

Ok, so now we have physics, math, and the Hitchhiker's Guide to "the meaning of life, the universe, and everything". Ok, I know that this is not the first time 42 shows up in math and physics, "The angle at which light reflects off of water to create a rainbow is 42 degrees." But this is seriously heavy stuff... read the article..

Ok, so we are back to the idea of connections, and how to make them happen, and that brings me back to "The Mystery of the Prime Numbers". THIS is the book you give non-mathematicians who don't think math has any "soul". I really believe that this book would be a wonderful gift for a clever 12 year old, and I may be shortchanging clever 10 year olds in that statement (Think Christmas). This is a book about mathematical ideas that seriously tries to avoid the barrier of mathematical symbols, and that is where Matt Tweed's creative illustrations come in....... but don't be confused, this is not "baby math" or watered down math, this is Primes straight on without barriers...or at least with some of the confusing barriers removed. Ok, let me be blunt..this is the book I give my granddaughter, and the book I give my math colleges.

" By "number system" I just mean the entire sequence of counting numbers together with the usual rules for adding and multiplying them. We're going to treat this as a single entity, almost like an organism, and look it it properties, its anatomy, if you like."

They say there will be two more... Can't wait...

Studies on the Gender Problem in Math

Several new reports on the reasons for the limited numbers of women in mathematics... Here are an eclectic collection of issues that ought to be good reading for teachers, and parents of female students.
Mostly they differ by who they blame:
Blame the teachers...
This first one has been around for awhile, and points to the need to provide strong math teachers for elementary students.
Believing Stereotype Undermines Girls' Math Performance: Elementary School Women Teachers Transfer Their Fear of Doing Math to Girls, Study Finds (Jan. 26, 2010) —

Blame the Parents: "Dads have a major impact on the degree of interest their daughters develop in math. That's one of the findings of a long-term University of Michigan study that has traced the sources of the continuing gender gap in math and science performance."

Blame the culture... "Girls around the world are not worse at math than boys, even though boys are more confident in their math abilities, and girls from countries where gender equity is more prevalent are more likely to perform better on mathematics assessment tests, according to a new analysis of international research."


I am reminded of an old Earth Day Poster from Walt Kelly,

Rhythm and Reasoning Go Together?

An interesting story out of the Karolinski Institute in Sweden indicates another connection between music and intelligence, sort of.....

"Researchers at the medical university Karolinska Institutet and Umeå University have now demonstrated a correlation between general intelligence and the ability to tap out a simple regular rhythm. They stress that the task subjects performed had nothing to do with any musical rhythmic sense but simply measured the capacity for rhythmic accuracy. Those who scored highest on intelligence tests also had least variation in the regular rhythm they tapped out in the experiment."

Here is another quote about the study that seems to go against some of what we have been told about the relation between brain structure and intelligence,

"They also demonstrated a correlation between high intelligence, a good ability to keep time, and a high volume of white matter in the parts of the brain's frontal lobes involved in problem solving, planning and managing time.

"All in all, this suggests that a factor of what we call intelligence has a biological basis in the number of nerve fibres in the prefrontal lobe and the stability of neuronal activity that this provides."

The study appears in 'Intelligence and variability in a simple timing task share neural substrates in the prefrontal white matter', Fredrik Ullén, Lea Forsman, Örjan Blom, Anke Karabanov and Guy Madison, The Journal of Neuroscience, 16 April 2008. But I got my information from an article at Science Daily.

Halmos, Full Stop

As long as I'm mentioning things on this date, I should add that this was the date in 2006 that Paul Halmos died.

Halmos always protested the common view that held mathematicians were "number crunchers" and proposed that mathematics was much closer to art.

His description of how to do (and learn) mathematics is still the best advice I could give my students.

" Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"

He was also the originator of IFF for If and Only If... and the use of the tombstone symbol to indicate the end of a proof.

"The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like ▯, and is used to indicate an end, usually the end of a proof. It is most frequently called the 'tombstone', but at least one generous author referred to it as the 'halmos' ".", Paul R. Halmos, I Want to Be a Mathematician: An Automathography, 1985, p. 403.

Happy Birthday, Bud, You're Still on First

October 2nd was the birth date of showbiz legend Bud Abbot, who should always be remembered as the great straight man in one of the great showbiz acts of all time... How could we ever get to the point where this isn't funny.. enjoy.