Papert is Wrong

I usually find Papert right on top of things...and mostly agree with his views...but today at "The Daily Pappert" the quote was:

“The institution of School, with its daily lesson plans, fixed curriculum, standardized tests, and other such paraphernalia, tends constantly to reduce learning to a series of technical acts and the teacher to the role of a technician.”

I know that is not true.  I deal with wonderful kids every day who have not let their education be reduced "to a series of technical acts".  I work with teachers every day who refuse to be reduced to "the role of a technician."

Yesterday a student who might have been focused on the AP Calculus test just one month away had asked me about Euler's proof that the solution of the sums of the reciprocals of the squares, the famous Basel Problem, was pi^2 / 6.   I had to admit I didn't know off the top of my head how to do that..So this morning I brought him one of my Julian Havel books that discussed the proof, and he walked out with it this afternoon.  I don't know, but I bet he will be reading it this weekend rather than studying for my test...which I can live with. That may not constitute master teaching, but just seizing the moment when I could feed a student's hunger with more math.  But not being a "technician" either, Joe wasn't that hungry to learn when I met him a few years ago.  I am proud to believe that being in my classes has been one of the factors that fanned the flames of curiosity. 

Yes, I teach the curriculum, but every kid who ever walked through my class knows that there are those precious moments when we walk away from the syllabus to talk about the things that get Mr. Ballew excited.... anything about Euler....about Geometry ("darn it kids, you gotta' learn geometry to know math")  ... about vectors.. and the history of math.  OK, we go off on a lot of tangents, but come the standardized tests and the AP tests and all those things that they use to measure kids and schools and teachers, my kids make me proud.  Proud of them, proud of the teachers I work with, and proud of the little contributions I have made to enrich their learning.

Maybe what Papert says is too often true in education, but it doesn't have to be that way.  If you are a  student,take charge of your own education.  Let it be driven by YOUR curiosity. And if you are a teacher, find moments to teach your passion.  Kids WANT to know that learning is satisfying. Talk to them about what YOU are learning, about what you are struggling to learn.

What Do Math People Do?

Steven Colyer wrote recently to comment on people who were known for other things, but were, or had been mathemticians (too me, it's like being a marine, once you are,  you always are...  "Hooo raaah"... except I'm not, and never was a marine).  He mentioned a couple of folks that I knew, and reminded me of a blog post I had started once and never gotten finished..... So here it is....Thanks Steven
-------------------------------------------
I was a big fan (still am, I guess) of James Burke's series, "Connections", which are still popular on you tube it seems..and was just reminded of a "Burke-like" connection..

It started when I had one of those, "What can you do with math?" questions this morning. Not from a kid trying to avoid learning, but one who was curious because they couldn't imagine what they might do with a math education except teach (as if they know what people do with a history major).
I mentioned my "other life" expereince working in reliability and quality control, and I describe things some of my ex-students do working in non-educational fields in a general cross section of applied mathematical/statistical/actuarial jobs in government and business. 

And when this question comes up I always mention people who have math/sci majors who do/did "other" stuff. Terri Hatcher was a math/engineering student before she went to a casting call to keep a nervous friend company (and found fame). Art Garfunkle earned a Masters from Columbia in mathematics. And of course I mention the Simpsons and Futurama creator, Matt Groening who supposedly had a math or science degree from Evergreen State College in Washington.   (**see more  below) 

Today I happened to mention to the questioning young lady that Hedy Lamar, the famous (she thought she might have heard of her) actress had a patent for a submarine guidance device.
I wasn't sure what she studied in school, but knew that she was regarded as one of the "brainy" beauties of Hollywood. Still, I remembered it so poorly (I blame my age) that I looked back at some notes and was pleased to be reminded of an interesting "connection" . Hedy Lamarr had come up with her navigation device with another artist, the pianist/composer George Antheil.

http://www.wired.com/science/discoveries/news/2008/08/dayintech_0811
Together they conceived and patented a frequency-hopping torpedo guidance system: Lamarr contributed the knowledge of torpedo control gained from her husband and Antheil a method of controlling the spread spectrum sequences using a player-piano mechanism similar to those used in the "Ballet Méchanique".


**So I found a list of people who had a math based educations up to some point... Here are a few of them:
Basketball player Michael Jordan started out as a math major, but switched majors his junior year. (journalism if I recall)

San Antonio Spurs center David Robinson received a BS in mathematics from the United States Naval Academy

Tennis player Virginia Wade, who won the singles title at Wimbledon in 1977, received a bachelors degree in mathematics and physics from Sussex University

Teri Hatcher (Lois Lane on "Lois and Clark") was a mathematics and engineering major at DeAnza Junior College.  (The story I heard is that she went to a screening with a friend and they promplty hired her instead)

Danica McKellar (Winnie Cooper on "The Wonder Years") was a mathematics major at UCLA and graduated summa cum laude.  (you have all heard aobut her "Math Doesn't Suck" books)

Retired supreme court justice Harry Blackmun received a degree in mathematics from Harvard.

Corazon Aquino, ex-president of the Philippines, was a mathematics minor at the College of Mt. St. Vincent.

and of course I have mentioned that Lewis Carroll, author of Alice in Wonderland, was in real life a lecturer at Oxford in mathematics, and is famous for his book on Logic.
One of my favorite Lewis Carroll stories is about his gift of a book to Queen Victoria.   Here is the version as it is told on the Mathworld page.

Several accounts state that Lewis Carroll (Charles Dodgson ) sent Queen Victoria a copy of one of his mathematical works, in one account, An Elementary Treatise on Determinants. Heath (1974) states, "A well-known story tells how Queen Victoria, charmed by Alice in Wonderland, expressed a desire to receive the author's next work, and was presented, in due course, with a loyally inscribed copy of An Elementary Treatise on Determinants," while Gattegno (1974) asserts "Queen Victoria, having enjoyed Alice so much, made known her wish to receive the author's other books, and was sent one of Dodgson's mathematical works." However, in Symbolic Logic (1896), Carroll stated, "I take this opportunity of giving what publicity I can to my contradiction of a silly story, which has been going the round of the papers, about my having presented certain books to Her Majesty the Queen. It is so constantly repeated, and is such absolute fiction, that I think it worth while to state, once for all, that it is utterly false in every particular: nothing even resembling it has occurred" (Mikkelson and Mikkelson).

Christopher adds in a comment, "I know that John Astin from the Addam's Family has a mathematics degree from Johns Hopkins University. It doesn't say that he graduated with a math degree on Wikipedia, but I remember from the playbill for "Once Apon a Midnight" that he has a masters degree in math."  

Strange Connections

I recently came across a note that the Peter Mark Roget whose name is associated so closely with the thesaurus was a scientific type.  More particularly, in 1815 he invented the log-log scale (the logarithm of the logarithm of the number on the C and D scales... ) on the slide rule which facilitated finding powers and roots of numbers. 

I looked into his history and found that his education was in medicine and his work on the thesaurus was part of a lifelong coping mechanism to fight depression.  But before he actually got around to publishing his great book, he not only invented the slide rule scale, but following the death of Sir John Hershell, Roget was secretary of the Royal Society from 1827 to 1848. He also made observations in sight and persistance of vision that influenced the development of movies. 

From"Cheshire Antiquities,© Craig Thornber, Cheshire, England, UK.
"His observations were based initially on looking at the world through a series of slits such as one might have in a vertical Venetian blind or pallisade. A rotating cartwheel viewed through such as system gives an optical illusion. The spokes at the top and bottom appear straight but those at the sides appear to bend downwards. Roget worked out the path of the light to show how this happened. He went on to explain a phenomenon that often perplexes devotees of Westerns, a hundred years before the invention of film. At certain speeds, the cartwheel appears to stop or go backwards. Roget's observations were made by viewing through vertical slits but he showed the position of each spoke in the wheel at each glimpse and how this could lead to the optical illusion of stasis or backward motion. The same phenomenon is observed when a film is made with a cine camera. In 1820, Roget worked with Michael Faraday and Joseph Plateau in a series of experiments on vision leading to Roget's paper to the Royal Society on the Persistence of Vision. Roget's work showed that an image persists in human perception for about one sixteenth of a second and this forms the basis on which animations, film and television are based. "
(Wikipedia)

On 9 December 1824, Roget presented a paper entitled Explanation of an optical deception in the appearance of the spokes of a wheel when seen through vertical apertures. ...
While Roget's explanation of the illusion was probably wrong, his consideration of the illusion of motion was an important point in the history of film, and probably influenced the development of the Thaumatrope, the Phenakistiscope and the Zoetrope.

He also was involved in the creation of University of London and the precursor to the Royal Medical Society.    A busy Guy...

While I was searching this out, I came across the fact that the invention of the Ln scale (for finding e^x) was by an 11th grade high school student. 
From a post by Robert Adams:

The Ln scale was invented by a high school student, Stephen B. Cohen, in 1958. The original intent was to allow the user to select an exponent (in the range 0 to 2.3) on the Ln scale and read e^x on the C (or D) scale and e^(-x) on the CI (or DI) scale. Pickett and Eckel were given exclusive rights to the scale in the early sixties. Later, Stephen Cohen created a set of "marks" on the Ln scale to extend the range beyond the 2.3 limit, but Pickett never incorporated these marks on any of their slide rules.

And just one more footnote, I always found it a little quirky that the log scale on the slide rule was linear.

Phone Bill for Adam Ries

Found this old file on Reuters news service just in time for the commemoration of the death of German mathematician Adam Ries (Ries died on March 30th in 1559):

BERLIN – A German mathematician who died 450 years ago has been sent a letter demanding that he pay long-overdue television licence fees, residents at his former address said today. Germany's GEZ broadcast fee collection office sent the bill to the last home address of Adam Ries, an algebra expert who bought the house in 1525. A club in his honour was set up at the property four centuries later.
"We received a letter saying 'To Mr Adam Ries' on it, with the request to pay his television and radio fees," said Annegret Muench, who now heads the club.
Muench returned the letter to the GEZ with a note explaining the request had come too late because Ries had died in 1559, centuries before the invention of television and radio. She nonetheless received a reminder a few weeks later.

And from My files on Ries (or Riese):

 
When the early Italian mathematicians began to develop the foundations of algebra using the new Arabic numerals, they used the Latin word cosa, unknown, to represent the unknown quantity in a problem. In the 15th and 16th century with the emergence of the German Rechenmeisters, or master reckoners, the word coss was used in the same role. One of the most influential of these master reckoners was Adam Riese, who played a significant role in the movement toward the use of "Hindu-arabic" numerals and away from the counting boards and Roman numerals that still persisted. His Die Coss in 1524, and his other arithmetic primers were so well received that "nach Adam Riese" is still used in Germany to describe mathematical accuracy and precision. One of the reasons for his popularity was that Riese wrote in German rather than Latin which was more commonly used for math books.  This allowed for much greater access to his work.  The image at right appeared on a German stamp on the 400th anniversary of his death. It is from a woodcut that appeared on the cover of Rechenung nach der lenge, auff den Linihen vnd Feder .  The word "cossist" became a term for algebraist throughout Europe during this period. Many of the symbols we now use were developed during this time, including the signs "+" and "-" for addition and subtraction, and the first use of decimal fractions.

---

Happy Birthday Karl Pearson

Sunday was Karl Pearson's birthdate, and I let it slip being busy with other things... But David Bee posted a reminder, as he often does with statistical birthdays, to the AP Stats EDG.  With grateful appreciation, I will steal his entire post:  (the picture above is the same, I believe, as the cover of Ted Porter's biography of Pearson.  )
---------

Karl Pearson, FRS, 27 March 1857 - 27 April 1935

Karl Pearson, who according to author and statistician David Salsburg, started the/a “statistical revolution” in the 1890s, founding the world’s first academic statistics department in 1911 at University College London, was born in London. Brought up in an upper-middle-class family, he was home-schooled until age 9, then sent to University College London, studying there until he was 16, at which time he left because of illness.

He was home-taught by a private tutor for two years, leading him to Take the Cambridge Scholarship Examinations in 1875. He came in Second on the Exams, winning a scholarship to King’s College.

Pearson graduated from Cambridge University in 1879 [the same year Einstein was born], ranking high in Mathematics. He then traveled to Germany to study at the University of Heidelberg, studying both physics and metaphysics. He returned to England in 1880, going to Cambridge and for a while studying Law. During 1882-84 he lectured around London on a wide variety of topics, also writing essays, articles, and reviews as well as substituting for professors of mathematics at KC and UCL, leading to him appointed to the Chair of Applied Mathematics in 1885.

Pearson was highly productive during the next 5-10 years. One book he completed was The Grammar of Science (1892), which prior to World War I was considered one of the great books on the nature of science and math, with perhaps the primary reason being it accessibility to practically anyone.(Another reason was its anticipating some of the ideas of Einstein’s Theory of Relativity.)

 

Pearson was married in 1890, with son Egon Pearson, who was to become another big contributor to Statistics, born in 1985.

Pearson was heavily influenced by Galton’s 1889 book Natural Inheritance, becoming interested in developing mathematical models for studying the processes of heredity and evolution. His 18 papers entitled Mathematical Contributions to the Theory of Evolution, written from 1893 to 1912, contain his most valuable work, including contributions to correlation and regression analysis and including the Chi-Square Test (1900), thus making it the oldest inference procedure still in use. (It has been said the chi-square test was produced in an attempt to remove the normal distribution from its central position.)

In 1900, Pearson, with W.F. Raphael Weldon and Galton, in response to biologists at the time not accepting biological conclusions based on mathematical analysis, founded the journal Biometrika, with Pearson coining its name and being its first Editor from the first issue, which appeared in October 1901, until his death almost 35 years later.

  

In 1911, Pearson started the Department of Applied Statistics at UCL, which was the result of Galton combining two other units to form it.

With World War I beginning in 1914, another sort of war had not yet begun. In September 1914, Pearson received a manuscript from a 24-year-old person by the name Ronald Fisher. The two corresponded in a friendly manner during the next couple of years until May 1917 when Pearson and some of staff criticized it unfairly, misunderstanding assumptions of Fisher’s maximum-likelihood method. By this time Fisher sensed that Pearson had been responsible for the rejection of many of his earlier papers, and the Pearson-Fisher War was on. (Pearson stopped considering any paper by Fisher to appear in Biometrika. As a matter of fact, the Fisher’s “maximum likelihood” didn’t appear in print (elsewhere) until 1922. Also, Fisher turned down an offer as Chief Statistician in 1919 because he would have to work under Pearson. The dispute between them continued to grow in intensity year after year, with each taking every opportunity to attack the other’s views.) (*** see addendum at the bottom)

Pearson became a Fellow of the Royal Society in 1896. His son Egon became head of a split Department of Statistics at UCL in 1933.

The following are many of the terms in Statistics due to Karl Pearson (along with the year first used, arranged chronologically):

moment (1893)

standard deviation (and use of sigma as, 1894) mode (1895) skewness (1895) correlation coefficient (1896; note: Galton first used "correlation" in 1888) chi-squared (1900) goodness-of-fit (1900) homoscedastic (1905) multiple correlation (1908)

 

For Pearson's connection to Einstein and much more, read the bio at:

http://en.wikipedia.org/wiki/Karl_Pearson

Thanks Dave

*** A short while after I posted this, I came across a historical note about the Fisher-Pearson disagreements on a post about the Chi-square tests from Dan Teague of the North Carolina School of Science and Mathematics...

A Song for Wisconsin, and Workers Everywhere

 Got a note from Jonathon in New York that included an old Pete Seeger tune that he has been sharing during the recent show of support for teachers and workers in Wisconsin, Indiana and the several other states where conservative forces have been trying to gut the unions... Since lots of my AP students also take AP History and Government classes, thought I might post it here where they may see it...




Maybe we all need to revisit history once in awhile to remember how it has been, and how it could be again. Liberty is balanced more on a razor's edge than the divergent limits I spoke of yesterday...
Maybe our students learn a little too much about the Revolutionary war and too little about the 20th century..

Hanging it Up

I was thinking again about the bent rail problem from a couple of days ago, and got to wondering what the height would be if it was an inverted catenary arc instead of a circular arc.  I was led to this idea by one of my students who also had suggested that it might NOT be circular....and I had no way to assure him it would be... I have no reason to assume it is a catenary either, but it was a curve I felt I needed to know more about so I used it as a self-teaching exercise.

At that point I realized I didn't know much about the catenary curve in general.  (what I mostly knew is about its history and can be found here).  So I set out to see if I could learn a little more and figure out how to solve the question of how high would the inverted catenary reach.  Below is what I have attempted to do... the answer is close enough to the circular amount of bowing that I wonder if it can be right, so if you are up on this stuff, please comment.

I knew that the catenary curve y= a cosh(x/a) has a minimum at (0,a).  For students who haven't seen "cosh" that's just a hyperbolic cosine function (much like the circular cosine function in many ways, and often pronounced to rhyme with "gosh"... see an explanation of hyperbolic functions here).  It can be replaced with an alternate form that may be more palatable to your eyes:

Of course it is easy to see that the vertex is at (0,a).  What I had never known, and find particularly wonderful about the curve, is that for any catenary, the area under the curve from x=c to x=d  divided by the constant a of the equation is the length of the curve.....A good calculus student can show this is true pretty easily with the exponential form since the derivative of cosh(x) is sinh(x).... and for hyperbolas we have cosh^2 - sinh^2 = 1 instead of the addition. That makes the integral for arc length the same as the integral for area between the curve and the x-axis.

So the length of the catenary from the vertex to any other value of x is given by a* Sinh(x/a)  (OK, you can guess that is the hyperbolic Sin, usually pronounced like "cinch" and it is the same as the hyperbolic cosine except the two exponentials are joined by subtraction rather than addition..)

I worked the problem out upside down because I wasn't smart enough to turn it rightside up... ... I assumed that the rod was hanging under a secant line.  For our hanging (and now very flexible) rod, the arc length is 5281 feet, and the secant (parallel to the x-axis) is 5280 feet.  We can position the end points so that they are symmetric around the y-axis so that our hanging rod will have it's vertex on the y-axis. 

So we know that the vertex is at (0,a) and we know that the length of the cable from the vertex to x=2640 is going to be 2640.5 which must equal a*sinh(x/a).    This seemed to give us enough to write an equation that we could solve for a, and my answer came out close to 82,809... which seems like a really big constant.  Buty the effect of a on the catenary is to adjust both the distance from the origin, but also how fast it turns up.  Here is a graph showing catenary curves for a=1 (red) and a=2(blue). 

So we want a very wide catenary, we need a really big number... maybe 82,809 will work.  So our equation for this curve is y = 82,809 * cosh(x/82809).   Now we can use the equation for the curve to find the height of the curve at the point where x= 2640.  This turns out to be just a smidge over 82,851 or almost exactly 42 feet above the y-intercept of 82,809.  So if we flipped it over and set it on the ground, the one mile wide arch would be only 42 feet high.   Very near to the 45 foot answer we got from the circular arc. 

Recently I wrote about Willima Whewell's role in the creation of the word scientist. I just found that there is also a Whewell equation for the catenary which gives the slope of the tangent of the catenary as a function of the distance from the origin, s. Tan(T)= s/a. That means in the catenary version of our inverted rail, the angle between the curve and the horizontal secant is just over .03 radians.... I think I did that correctly.

As a comparison, the St. Louis arch is 630 feet high, and has a width from side to side of 630 feet.  It is not a true catenary because it is adjusted for weight, called a "weighted" or "flattened" catenary. If we assume the difference is small (I don't know for sure) then the information above should allow you to find the constant "a" for the arch, and then find the length of the arch... Good luck... 

The Sharp Edge Between Infinity and "Finity"

After my problem on the steel rail got me in trouble, I have decided to stick to a purely theoretical problem that has no messy reality features to get in the way...


I recently came across two limit problems that got me to thinking about how narrow the line between infinity and the finite can be....

Try the following two limits.....

Both can be simplified by the old "multiply by one" trick ... letting the "one" in this case be the conjugate of the binomial ...For the first, it works like this...

Ok, sorry, I got those last two images reversed... the middle one is the simplest........
Which leads to the conclusion that the limit is 1/2... Ok, not terribly difficult, so what...???

 But if we do the same sort of thing with the second, we get :

Yikes, now the Limit has shot off to infinity... The difference between a one and a two for that leading coefficient has made a huge difference. At first we might think that we can find some number a such that :

has a finite limite between 1/2 and infinity, But when you try it.... if a is greater than 1, the limit is infinity, and if a is less than one, the limit is negative infinity, and in between, balanced on a razor's edge is a=1, which has a limit of 1/2...

Just goes to show that 1/2 is the average of Plus and minus infinity,

Fractal Language

Shecky, at Math Frolic, posted a cute joke today (yesterday?) .. enjoy...


"Do you know what the "B" in Benoit B. Mandelbrot stands for?
.
.
.

answer: "Benoit B. Mandelbrot"....


For those who are stuck...here is an explanation of the mathematical idea.

Will It or Won't It

In my last blog I tried to solve a problem about a rigid steel rail a mile plus one foot long that was forced into a space one mile long.  I came up with an answer of about a 45 foot bulge that agreed with other mathematical answers, but then I got a comment from Jonathon who suggested, "I think you need to take the compression of the steel into account. A mile of steel rail would be pretty heavy and at that small angle (if it could be held perfectly in a line above the ground) it would probably compress itself until almost flat... " 

My first thought was that he was probably right.  I even commented that I thought so... and then... I begin to wonder... Would the steel flatten?  I imagine a fine steel wire rod, or maybe even a hollow tube.. let's say 1 inch in diameter and the given length....Steel has pretty remarkable strength... Would it bend or would it compress along its length and sag to the ground?

If I take a wooden yardstick and bend it to fit a 35.5 inch distance (I know, not the same scale, but the same idea) it will bulge up quite well, and I think of wood as being equally strong under lateral compression... but I know that when we scale up the length, the masses will scale by the cube of that ratio.. so ???

ANYONE, ANYONE, .....Bueller?

  So if anyone out there knows how to decide with appropriate technical engineering skill how much such a rod would sag... chime in here...  And what would you do if you wanted it to arch supported only at the extermes and reach a height of 45 feet... Can that be done with one piece of steel.  I guess I could scale the problem down, but at lengths I could manage, the height to the top of the bend would be pretty small.  If I scaled down the one mile to 52.8 feet then the extra length would be .1 feet or about an inch and a quarter, and the height of the bend would be .45 feet or just under 5.5 inches.   However I can't afford a 53 foot lenght of metal pipe, and another scale of ten reduces the problem to about 1/2 an inch of bow...

HELP!

Monkey Around

Steven Colyer  posts some really interesting stuff at Mutliplication by Infinity... Today's is a good chuckle..Go there!

Oops, Not Quite to Specification

If you enjoy sharing problems with your math classes, here is one that will challenge the thinking skills of your upper level students.   Suppose a perfectly flat piece of ground is prepared that is one mile long.  A thin rail of steel is ordered to go across the flat surface.  Unfortunatly, the bid went to the lowest bidder, and they were not very good with a ruler, so the rail was 5281 feet long instead of 5280.  The engineers decided to squeeze it into place and let the middle bow up as much as necessary.  And the question is.... How much did it bow up? 

The intuitive approach is to estimate whether the hight above the ground in the center of the rail would be

a)  just enough to slide a piece of paper under

b)  just enough to for a small dog to walk under

c)  just enough for the average man to walk under

d) just enough for the average car to drive under

e)  just enoug for a London double-decker bus to drive under

And of course the tougher challenge would be to calculate the height exactly (or at least some reasonable accuracy)..

Stop  now if you don't want to see the solution...   or read on for the solution and a related question....

S
P

O

I

L
E
R

If we assume that the bowed rail makes a circular arc, as shown, then we can apply some of the geometry we have learned to get a pretty good (I hope) approximation... but first we might try to find a back of the envelope aproximation that would answer the multiple choice portion: 

If we tried to create the right triangle shown at left we can hope that with some reasonable estimates we could come up with a (very) rough value for the bowed distance DE.  DC is 1/2 mile or 2640 feet, and the Arc from C to E is 1/2 a foot longer, or 2640.5 feet.  So what should we say about CE, the hypotenuse of the right triangle.  Can we suggest 2640.25 as a compromise length.  If so, then by the Pythagorean theorem we can obtain DE² = 2640.25² - 2640² . From this we can estimate that the  height of the bend is about 36 feet... We could get slightly different estimates for the height for different estimates of CE, but even if we assume it were only 2640.1the height to the bend is over 20 feet. 

The biggest problem in a more exact solution is that we come up with a couple of messy equations in two unknowns (or at least I do)...  I will call the angle CAD simply angle A for typing ease, and we have from two common formulas from analytic geometry that the arc length is equal to r times the central angle.  For our problem, the arc CE will equal A*r.  We can also use the right triangle ADC to show that the Sin (A) = 2640/r... or  r * Sin(A) = 2640...

 Now that is pretty messy to try to graph on a calculator, and it turns out that Wolfram alpha folded on the problem....

At this point they have a perfect place to apply the series approximations to Sin(x) that you made them learn... ie that Sin(x) =  x - x³  /6 ... and if the angle A is even less than 1 radian, we can assume our error is less than 1/120 radians... not a terrible estimate. 

That allows us to write   an expression for Sin(A)/ A that will simplify into something tractable.

From this we can conclued that A must be about .0337... and from that we can conclude that the radius must be about 2640.5/.0337 or about 78,350  feet .  Now we can subtract the distance AD which is r Cos(A) or 78,305 to get a (hopefully) close approximation of the height of the bowed rail of  45 feet... room for that double-decker bus for sure...

And the follow up question... What does that make the actual length of that hypotenuse we were estimating earlier... and is there a pretty pattern in there somewhere about the length of a chord... no time to think...kids coming in  the door..."Run away.. ...run away!"   

-------------addendum---------------

Jeffo has pointed out that if you used the equation Sin(A)/A above instead of the system of equations then Wolfram alpha will solve the value of a, and gets the same value, but with much greater accuracy....

A ~~ ±0.03370775880988154948897645696380541681505...

Thanks,  Jeffo.

Addendum two::: The history of this problem has been well documented by David Singmaster.  This is from his notes

A railway rail of length  L  and ends fixed expands to length  L + ΔL.  Assuming the rail makes two hypotenuses, the middle rises by a height,  H,  satisfying  H² = {(L+ΔL)/2}² ‑ (L/2)²,  hence  H @ Ö(LΔL/2).

          However, one might assume the rail buckled into an arc of a circle of radius  r.  If we let the angle of the arc be  2θ,  then we have to solve   rθ  = (L + ΔL)/2;  r sin θ = L/2.  Taking  sin θ @ θ - θ³/6,  we get  r² @ (L + ΔL)³/ 24 ΔL.  We have  H = r (1 - cos θ) @ rθ²/2  and combining this with earlier equations leads to  H @ Ö{3(L+ΔL)ΔL/8}  which is about  Ö3 / 2 = .866...  as big as the estimate in the linear case.

The Home Book of Quizzes, Games and Jokes.  Op. cit. in 4.B.1, 1941.  P. 149, prob. 12.  L = 1 mile,  ΔL = 1 ft or 2 ft -- text is not clear.  "Answer: More than 54 ft."  However, in the linear case,  ΔL = 1 ft  gives  H = 51.38 ft  and  ΔL = 2 ft  gives  H = 72.67 ft,  while the exact answers in the circular case are  44.50 ft  and  62.95  ft.

Sullivan.  Unusual.  1943.  Prob. 15: Workin' on the railroad.  L = 1 mile,  ΔL = 2 ft.  Answer:  about  73  ft.

Robert Ripley.  Mammoth Believe It or Not.  Stanley Paul, London, 1956.  If a railroad rail a mile long is raised  200  feet in the centre, how much closer would it bring the two ends?  I.e.  L = 1 mile,  H = 200 ft.  Answer is:  "less than  6  inches".  I am unable to figure out what Ripley intended.

Jonathan Always.  More Puzzles to Puzzle You.  Tandem, London, 1967.  Gives the same question as Ripley with answer "approximately  15  feet".  The exact answer is  15.1733.. feet  or  15 feet 2.08 inches. 

David Singmaster, submitter.  Gleaning:  Diverging lines.  MG 69 (No. 448) (Jun 1985) 126.  Quotes from Ripley and Always.

David Singmaster.  Off the rails.  The Weekend Telegraph (18 Feb 1989) xxiii  &  (25 Feb 1989) xxiii.  Gives the Ripley and Always results and asks which is correct and whether the wrong one can be corrected -- cf Ripley above.

Phiip Cheung.  Bowed rail problem.  M500 161 (?? 1998) 9.  ??NYS.  Paul Terry, Martin S. Evans, Peter Fletcher, solvers and commentators.  M500 163 (Aug 1998)  10-11.  L = 1 mile,  ΔL = 1 ft.  Terry treats the bowed rail as circular and gets  H = 44.49845 ft.  Evans takes  L = 1 nautical mile of 6000 ft and gets almost exactly  H = 50 ft.  Fletcher says it took 15 people to lift a 60ft length of rail, so if someone lifted the 1 mile rail to insert the extra foot, it would need about 1320 people to do the lifting.

Standing Up

A few days (March 13)  back I wrote about the discovery of Uranus by the great amateur astronomer William Herschel and his sister Caroline.  In that same blog I mentioned that
a)  "Amazingly it was the same date  that science took a setback in the state of Tennessee when the Senate passed the Butler Act in 1925 (Yeas: 24; Nays: 6).  The law prohibited public school teachers to teach anything which denied the Biblical account of man’s origin. "The law also prevented the teaching of the evolution of man from what it referred to as lower orders of animals in place of the Biblical account."  Within two months this will lead to charges being filed against John Scopes,  " 

and also that

b)  "And the question of evolution in education is still under attack on March 13 of 2011. From the Memphis Flyer..
As you probably know, House Bill 368—which allows teachers to critique such "controversial" theories as the theory of evolution—is coming up for a vote on March 16 in the general subcommittee of the House Education committee.
Although the bill also attacks global warming and human cloning, the primary aim of the bill's sponsor (Bill Dunn) is to gut the teaching of evolution in public schools."


So a few days ago when I was asked by Jonathon to post something about the Solidarity Blog Day in support of teachers under attack I was glad to participate.  I should make a disclaimer.  I'm not that political anymore.  I'm near the end of my career and tired of fighting I guess, and I'm not even that much in favor of unions, or any large bureaucracy....... BUT... I'm totally opposed to the gross abuses of power that happen without them,,, and seem to be about to occur even with them.

So Today, I'll wear red,......for all those kids I talked into teaching when they could make so much more money elsewhere.....       I don't think there is a fence on this one...

Go ahead, teachers, rattle the bars of your cage... use your voice while you still have one....

Here is a clip from a letter by Lynne Winderbaum that was posted this morning on JD2718 that I really liked:

So you ask me, “Why did you need a union? You had a good reputation as a teacher, good  reports from your supervisors, loved your job, and thanks to social networking, have students who keep in touch with you from a career that spanned nearly forty years. Couldn’t this have happened without your union?” No, never.
What we want from our existence is very simple really. We want to be able to support our families. We want to be able to take care of their health needs. We want to be able to protect them if we die or are too disabled to work. We want respect in the workplace. We want to be able to provide our children with an education that will give them a life better than the one we led. Women want to the right to raise their children and return to their jobs. We want to be able to live out our old age with independence and dignity. Couldn’t this happen without my union? No, never.

And sign in on the facebook page...

Springtime and Daffodils

Spring has come to East Anglia, and the Daffodils are popping up everywere.  Each time I see a patch I hope for a little breeze that will demonstrate their beautiful geometric construction.  I first learned about this a few  years ago right around Easter, with snow on the ground... Hoping that doesn't happen this year, but  here is that blog again anyway:
-----------------------------------

I’ve been thinking about geometry a lot lately. Partly that is due to the fact that I’m going through Trig and Vectors in my Pre-calc class. Partly it is probably because it has popped up in science stories I have been reading lately. On the same day I wrote about the Daffodils in the Snow, I read a note from a researcher on why they respond differently to wind than other similar flowers. The short answer is geometry.

When you watch Tulips, for instance, they will lean away from the wind. Daffodils, on the other hand, remain almost completely erect, but turn their tilted heads away from the wind. William Wordsworth must have had this in mind when he wrote,

“ Ten thousand saw I at a glance
Tossing their heads in sprightly dance.”


The reason the daffodil twists like a weather vane and the tulip bends more in the wind is the geometry of the cross-section of the stem. A tulip stem is nearly round, and so it can bend, but not twist. The same effect causes your garden hose to crimp up and cut off the flow of water when it can’t twist. They are very good at bending, very poor at twisting. The engineering types call the twisting motion torsion, and it is related to the cumulative sum of the fourth power of the distances from the center to the edge of the stem. Circular things are far away in all directions. But a Daffodil has a cross section that is more elliptical. If you pick it up you can see the difference in the long and short axis easily with the naked eye. Since the sum of the fourth powers of the distances is lower, it is more able to turn away from the wind. Scientists studied one type of daffodil and reported that up to about 22 mph, the stems stayed essentially erect, and the trumpets all turned away from the wind. After that, the flowers both turned and bent some, but they cannot bend as low as a tulip in any wind… all geometry.


The geometry of sharks popped up too. It seems that shark geometry may be to thank for the engineering advances that will result in a few more swimming records falling at the Olympics in China this year, although I assume the swimmers will want some of the credit. It seems a shark has dimples on its scales that breaks up the flow of water, reducing the drag so it uses less energy. Speedo reckons it can make better swimsuits making its swimmers go faster using something similar. The new body suits are already in the pools and having an impact.

Another bit of research answers the question, with such a huge blue ocean to wander around in, how exactly do marine predators like sharks find their next meal? Yeah…. You guessed it, geometry. It has been known that many land animals search for food much like a shopper in the super market searches for a particular item. The math term is called a Levy walk (actually a Levy flight), a fractal type structure from geometry where the small parts are self-similar. Ok, the actual rules are a little technical but in essence it means that the animal undergoes lots of short-distance journeys interspersed with fewer longer-distance journeys. Just as you go to an area of the store where you think the item is located, then circle around in that area looking for it. If you don’t find it, you go off to another area and begin a close search there.


Geometry, making your life better… See, because if two sides and the included angle of one triangle….

Gauss and Constuctable Polygons

By special request, for Steven Colyer who wrote, "I WILL point out one thing I wish you'd mentioned, and that was Gauss' accomplishment with the number 17. "

I assume he is talking about  Gauss' discovery that the heptadecagon was constructable with the classic tools of Greek Geometry.... hope that's it anyway. 

So here is my version of that story, with many clips from Wolfram and Wikipedia...

One of the great problems of antiquity, along with doubling the cube and squaring the circle, was construction of a  heptagon, a seven-sided polygon, with a straight edge and compass.  Euclid had demonstrated the construction of the pentagon, the hexagon was easy, and so the early geometricians focused on the "next one". The early geometers knew that if an n-gon could be done, a 2n-gon was easy, so the octagon was easy, the decagon was easy, butt the nonagon also caused problems.  Over the years it became an open question in mathematics, "Which polygons are constructable with straight edge and compass?"

 On April 30, 1796, Gauss, at the age of nineteen, proved the constructability of the regular 17-gon (heptadecagon).  He did this by showing, (I think) that it could be factored into equations involving no more than quadratics.    I have been looking on line for a copy of Disquisitiones Arithmeticae  in my price range (literally-free) but have not found one to see how he explains.  Within five years he had developed a theory that described exactly which polygons were, and which were not, constructable. He didn't actually construct one, but he proved it could be constructed. The proof relies on the property of irreducible polynomial equations that roots composed of a finite number of square root extractions only exist when the order of the equation is the product of powers of two and Fermat Primes.  Gauss showed that the 17-gon is constructable since the sine and cosine of  ^2π⁄₁₇ can be expressed with basic arithmetic and square roots alone (which can be formed with a compass and straightedge).   The first actual method of construction was devised by Johannes Erchinger, a few years after Disquisitiones Arithmeticae was published. In the book Gauss supposedly writes cos(2pi/17) as an expansion of square roots which would look like this in modern notation

 

(thank you wikipedia)

(wikipedia)


Guass was so  proud of this, among all his great discoveries, that he told his close friend, Farkas  Bolyai, that the regular 17-gon should adorn his tombstone, but this was not done. There is a 17 pointed star on the base of a monument to him in Brunswick because the stonemason felt everyone would mistake the 17-gon for a circle.

 Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructability of regular polygons:

A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes.

Gauss stated without proof that this condition was also necessary, but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837.

Detailed results by Gauss' theory

Only five Fermat primes are known:

F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537 (sequence A019434 in OEIS)

The next twenty-eight Fermat numbers, F₅ through F₃₂, are known to be composite.
Thus an n-gon is constructable if

n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, … (sequence A003401 in OEIS),

while an n-gon is not constructable with compass and straightedge if

n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, … (sequence A004169 in OEIS).

Connection to Pascal's triangle

There are 31 known numbers that are multiples of distinct Fermat primes, which correspond to the 31 odd-sided regular polygons that are known to be constructable. These are 3, 5, 15, 17, 51, 85, 255, 257, … , 4294967295 (sequence A001317 in OEIS). As John Conway commented in The Book of Numbers, these numbers, when written in binary, are equal to the first 32 rows of the modulo-2 Pascal's triangle (see below), minus the top row. This pattern breaks down after there, as the 6th Fermat number is composite, so the following rows do not correspond to constructable polygons. It is unknown whether any more Fermat primes exist, and is therefore unknown how many odd-sided constructable polygons exist. In general, if there are x Fermat primes, then there are 2^x−1 odd-sided constructable polygons.

Pascal's triangle mod-2
1  1   = 3
1  0  1  = 5
1  1  1  1  = 15
1  0  0  0   1 = 17
1  1  0  0    1   1  = 51
1  0  1  0   1   0  1 = 85
1  1  1  1  1    1   1  1  = 255   etc...


This biography of Gauss, by far the most comprehensive in English, is the work of a professor of German, G. Waldo Dunnington, who devoted most of his scholarly career to studying the life of Germany's greatest mathematician. The author was inspired to pursue this project at the age of twelve when he learned from his teacher in Missouri that no full biography of Gauss existed at the time. His teacher was Gauss's great granddaughter, Minna Waldeck Gauss. Long out of print and almost impossible to find on the used book market, this valuable piece of scholarship is being reissued in an augmented form with introductory remarks, an expanded and updated bibliography, and a commentary on Gauss's mathematical diary, by the eminent British mathematical historian, Jeremy Gray.