Did Newton Invent the Calculus?

Many times, at social gatherings, dinner parties and such, people ask me: “Did Newton really invent the calculus?” I always answer: “Yes! He absolutely did.”

How can I be so sure? What benchmark has to be met to be considered “inventor?”

My understanding is that while there was an existing corpus of knowledge concerning geometry and algebra stretching at least as far back as 312 B.C. when Archimedes demonstrated his “Method of Exhaustion” for determining the area of geometrical figures, especially circles, no progress had or could have been made in applying algebraic methods to geometric problems, this had to wait for Descartes in the early 1600s. Archimedes showed the area of a circle could be determined by the easier calculation of the area of an inscribed regular polygon with an ever increasing number of sides until the tiny edges pretty much coincided with the perimeter of the circle. This concept could have led directly to the calculus if the ancients had a sufficiently well developed algebra whose equations could represent geometrical shapes and curves that could be exploited to accommodate an ever increasing number of smaller and smaller polygonal edges generated by the Method of Exhaustion, but this was something not available nor even obvious to the ancients and European mathematicians of the Middle Ages right up to 1665. Newton’s first claim to “Inventor” is he realized this relationship which he called “analysis” and then identified a process for calculation of the area under a curve called “integration.” The inverse of integration would give the instantaneous motion of a particle along the curve, the “velocity” (“particles” and “motion” come from the fact Newton was studying what today would be called theoretical physics for the dynamics of moving particles--generally planetary motion--in his case). Newton’s second claim was that he was the very first person to understand and state clearly the “Fundamental Theorem of Calculus: “...the fundamental theorem is understood as a statement that the two central operations of calculus, differentiation [velocity] and integration, are inverse operations...Newton discovered the fundamental theorem in 1665.” (Niccolo Guicciardini, “Isaac Newton on Mathematical Certainty and Method,” 2009, p. 182-183).

The justification for Newton’s claim of primacy in inventing the calculus is that he simply announced the fundamental theorem. 

What makes this sweet for me is 20 years ago on one pleasant afternoon in the main library of the University of the Pacific I happened upon the complete 8 volume set of Derek Whiteside’s edited “The Mathematical Papers of Isaac Newton.” 

In volume 1 I found a table composed by Newton in 1665 (at the age of 23!) and copied it down. I could see instantly that he had written down the same differential and corresponding integral equations used and taught to this very day (vol. 1, page 305 “Calculus becomes an algorithm”). Here’s the table, see if you don’t agree:

So there you have it, incontrovertible proof that Newton was onto the calculus: a table giving the differentials and the corresponding integrals of equations for the first time in the history of mathematics; before anyone else.


Note that in 1665 a young Leibniz was still involved in law and diplomacy and didn’t even meet a real physicist, Huygens, until 1672. In later decades he became Newton’s adversary, claiming he invented the calculus first, but I think the above table and the Fundamental Theorem statement completely vindicates Newton’s claim.

Now what he did with this powerful tool is another story. Newton’s one major publication was the “Principia Mathematica.” Written in Latin it remains impenetrable even today with a modern translation. The mathematics are nothing one could learn without spending an inordinate and intolerable amount of time re-learning Newton’s method of presenting his results in geometric fashion. Why so cryptic? Well, 320 years ago people we would now call scientists were still developing ways of interacting with each other and sometimes old ways were held over for reasons we now find obscure, byzantine and prolix. To illustrate the modern view of Newton’s motivations I present a condensed paragraph from a book review in the journal “Historia Mathematica:”

“Did Newton use the calculus to find out “most of the Propositions” in his “Principia” as he once claimed? Or was his natural philosophy calculated as it was presented, in the established Archimedean tradition of geometrical physics? With a few exceptions, Newton did not work out the propositions of the “Principia” using his method of infinite series and fluxions [that is to say, using his calculus]. Newton did work out a solution to the notorious inverse problem analytically [another synonym for the calculus], and, if one combines Newton’s binomial expansion with the Taylor series expansion and takes into account his analytic expression of Taylor series from the 1690s, one arrives at the statement that the force is proportional to the second fluxion of displacement [or Newton’s famous Second Law: F = ma, where “a” is the second differential of the position with respect to time, the “second fluxion of displacement”]. Newton saw his synthetic geometric methods as continuous with ancient geometry. Classically, algebra, although useful for invention, could not be used in demonstration, which must exhibit the construction which solves the problem. Also, Newton saw geometry as grounded in mechanics; as such it fitted seamlessly with his natural philosophy. Indeed, he often based his demonstrations on physical insights which analysis [calculus] would only obscure. Analytic techniques did not constitute a unifying method for Newton’s natural philosophy, so he did not allocate them a prominent role. Yet, neither are Newton’s demonstrations in keeping with the strictures of classical geometry. The majority depend on his innovative method of first and ultimate ratios the geometrical limit procedures that constitute Newton’s “synthetic method of fluxions.”

(The preceeding paragraph was condensed from “Reading the Principia” by Niccolo Guicciardini, a book review by Richard Arthur, “Historia Mathematica” 28 (2001), 54-57.)

So while Newton had invented the calculus and could use it effectively he was more likely to resort to a geometrical method of analysis to present his results because it was in line with the accepted methods practiced by the ancient geometers like Euclid and Archimedes. Basically, Newton was caught with one foot in the past while the other foot was stepping gingerly into the future. It took about another 100 years before the calculus as we know it today was finally ready and geometric analysis ceased to be the paradigm for physics research.

And here is the complete page from Whiteside's "Mathematical Papers".

More better confusion--Quote

This is just too good not to copy and publish on my own blog; oh that we all could have such successes!

Scientific confusion

by gavin, 16 November 2011, at

http://www.realclimate.org/index.php/archives/2011/11/scientific-confusion/

“We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things.”

Fermi Problems, Introduction

“Fermi Problems” is one of the neatest things you will learn from taking physics: you can calculate anything, simply and efficiently. This is worth the price of admission. The only tools needed are your imagination the analytical power of your brain and a pencil and paper. You guess data, estimate sizes and make simple assumptions for your calculations. The title of one of the books recommended below pretty much gives away the whole trick: “Assume a spherical cow.”

Fermi Problems is a standard problem solving technique commonly taught in high school and college physics classes. It involves  essentially nothing more than solving word problems without having problem-relevant data.

For example how many ping-pong balls will fit into a suitcase, where the suitcase and ping-pong ball sizes are left undefined. Sometimes a hint of data can be gotten like from watching pieces of torn paper float in the wind.  The point is to concentrate on the physics of a situation and not waste time worrying about exact numerical answers (some people will say that’s the engineer’s job anyway).

In my days of yore (slide rule times) my physics professors had a standing rule that as long the answer was within half an order of magnitude it was correct; nobody cared about the numbers, just the thinking.  

This concept has even made its way into employment interviews where it’s been reported interviewees are asked to give a rough estimate, using only their brain, to some seemingly bizarre problem—like how many dump trucks are needed to haul away Mt. Fuji (hint: it’s a fairly well defined cone)—to show they are capable of rational analysis and reasonable problem solving abilities.  

Fermi problems are a good way to sharpen your intuition, but who was this Fermi guy and why did he get an eponymous genera of physics problems named after him?

Fermi was an Italian physicist, both theoretical and experimental (eponymous “Fermi level” anyone?), who was experimenting on why atoms disintegrated when bombarded with alpha particles. He emigrated to America in the late 1930s; built the first nuclear reactor; demonstrated the uranium fission chain reaction; and worked at Los Alamos. One early morning in 1945 he found himself standing on a desert in New Mexico gazing south into the darkness wondering if the first atomic bomb was going to work or not. He was a crafty applied physicist too and since there was only minimal instrumentation available to measure the yield of the “gadget” he thought of a way to make his own measurements while waiting for the official results.  The bomb goes off as planned with a big fire ball and a shockwave starts propagating away from the blast. Fermi waits until it gets to him, drops some tiny pieces of paper he had ripped up from shoulder height and watches to see how far the shock wave takes them before they hit the ground.  He estimates the distance they traveled and looks this number up on a pre-calculated table to find the yield was at least 10 kilotons of TNT.  The rest is history (the actual yield from instrumentation was 20 kilo tons, but at least Fermi knew things had gone per theory).

This story give the outlines of doing Fermi problems: collect or even just assume some simple data; use a simple model to analyze the data; critique the answer for plausibility.

Fermi problems have a serious side too. Physicist find it useful to check their work against simple models and calculations to make sure they are on the right track often even making a rough hand drawn graph of the data to make sure an experiment is performing properly.

Over the years a few books have been written by researchers to illustrate the techniques of quickly calculating results. The following is a short bibliography of some of these books.

“Used Math For the First Two Years of College Science,” Clifford E. Swartz, p.4. 1993, American Assoc. of Physics Teachers. This book also contains a splendid review of all the math one needs in college.

“Guesstimation Solving the World’s Problems on the Back of a Cocktail Napkin,” Lawrence Weinstein and John A. Adam, Princeton U. Press, 2008. 73 Fermi questions posed and answered.

“The Cosmological Milkshake A Semi-Serious Look at the Size of Things,” Robert Ehrlich, 1994. 135 somewhat off-beat, but important Fermi Questions.

“Consider a Spherical Cow A Course in Environmental Problem Solving” and “Consider a Cylindrical Cow More Adventures in Environmental Problem Solving,” John Harte, 1988 and 2001 University Science Books. These are more focused questions with deeper computations that show how Fermi Questions can be made to give real quantitative insight.

“The Flying Circus of Physics With Answers,” Jearl Walker, 1977, John Wiley & Sons. Interesting questions most demanding only an intuitive answer, so perhaps the simplest Fermi Problems, but the answers require real physical insight.

Other references and historical information can be found at:

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

Finally, what can go wrong? Well, plenty, if one is not careful in assuming data for the calculation. Ludwig Prandtl a founder of the science of aerodynamics was at a fancy dinner party early in the 20th century. A guest asked how it was that a bumble bee can fly. He immediately made some calculations on the back of a menu and concluded by announcing that he had just proven that it is not theoretically possible for bumble bees to fly! So one must choose input data wisely for the Fermi Problem to work! Oh, and yes, performing Prandtl’s calculation with more accurate values from biology shows it is perfectly acceptable for bumble bees to fly.

Note added Nov. 15th 2011.

It has recently come to my attention that Tom Murphy, Professor of Physics at UC San Diego, has a blog called “Do the Math,” somewhat focused on the environment, but other things as well, where he uses Fermi-type computations to get a handle on challenges in our future. This is a very interesting blog, highly recommended!

http://physics.ucsd.edu/do-the-math/

Subsequent post will illustrate a couple of Fermi Problem calculations I have made concerning measuring the earth’s axial tilt and the size of rocks in the rings of Saturn. Stay tuned!

Bring Back Drafting!

Back when I was in high school we had the option to take “Mechanical Drafting.” This was a very rigorous class devoted to producing manufacturing drawings for industry using pencil, paper and a whole slew of mechanical aids to generate circles, lines and what not encountered in every day machinery.

Mechanical drafting is understood to be the creation of what is known as descriptive geometry by French mathematician Gaspard Monge while working for the military in approximately 1766. Certainly descriptive drawings have existed for many centuries previous to Monge, for instance Leonardo da Vinci made many wonderful drawings of mechanisms, sculpture, human anatomy and buildings, many of which have been turned into patent office-worthy working models by aficionados over the years. But until Monge made a critical discovery, all previous drawings have been limited in the amount of useful information available to a user of a drawing hoping to make a useful product from it. The reason for this limited usefulness was that, as Monge realized, at least three orthogonal views are required to adequately describe an object and as an added benefit these views can have features labeled with the appropriate dimensions without having to interpolate them from oblique or foreshortened views. Sacrebleu!

This subject has in large measure been discontinued in high school because, I think, our society lost the desire to make things. Plus software called CAD, or Computer Aided Design, came on stream and I guess folks just figured computers would make all the drawings hereafter. Well, yes and no, computers make wonderful drawings in the same sense the T-squares and pencils and plastic triangles of yore made great drawings too. What hasn’t changed or been supplanted by computers is the ability to visualize in one’s mind, assisted by sketches on paper, the desired end product. Once this has been visualized the CAD people can take over and make a drawing in days.

My point is everybody involved in STEM needs the ability to visualize something. And there is an old axiom that says: One does not understand something until one can draw it. So not teaching drafting is counter-productive to understanding.

Now, here’s the exciting thing. I am not suggesting we go back to drafting boards, but instead teach a freehand version using the rules of drafting to produce sketches that convey information and insight both to the sketcher and to others down stream who may need input! Yes, a small class could be added to a school curriculum for science majors to learn how to draw real objects in three-space! The drudgery of T-square mentality is gone leaving behind the shear beauty and usefulness of a sketch. And because well-known rules are involved, a syllabus can be easily developed and grading is a snap while the subject can be taught in an entertaining manner.

I give you two motivations for this approach.

First, in order to patent something, one generally needs to document the invention in a notebook which includes clear and detailed drawings of the invention. QED.

Second, While making the drawing the draftsperson / scientist sees new possibilities open up on the paper not fully realized in the imagination and conversely one also sees problems or impossibilities develop not previously recognized thereby saving time from following dead ends. The drawing has a deeper meaning than the conveyance of beauty from artist to viewer.

Art in service of technology...exciting, no?

The Atomic Solar System or Five Things Dirac Wants You to Know About the Electron

The troubles begin in tenth grade biology. It is announced that all life and it’s diversity in the biosphere is due to variations in a DNA molecule. So what’s a “molecule?” A simple explanation is provided involving a large helix made of different colored balls, but the details are left for next year’s chemistry class. Once in chemistry class it’s revealed that molecules are the result of various atoms binding to each other in certain specific ways. So what’s an “atom?” For that we’re told to wait until physics class next year. Finally, as a senior, you know the moment of truth is at hand if for no other reason than that there are no more classes to be had. Then after months of discussing Newton’s Laws, in the last days before graduation, the teacher casually tells you atoms are miniature solar systems with a heavy, tiny nucleus sun in the center and nearly weightless electrons orbiting like planets. Mystery explained, off to graduation and a lifetime of believing in tiny atomic solar systems! As Einstein would say: “Oy vey!”

A few of us managed to stumble into university physics classes where we started hearing about how electrons were sometimes particles, sometimes waves, depending on how you interacted with them. And in atoms they were more like clouds than planets. We may have even heard they had spin, but a spin unlike anything we knew from before! But I’m willing to bet that even those of us lucky few who graduated in physics still see an atomic solar system in our mind’s eye when we hear the word “atom.” And so, for the vast majority of people, atomic solar system it is. But can we do better?

Yes and no. When quantum mechanics began in the early years of the 20th Century there was some justification early on for a solar system model. It was the very first conceptualization made by the scientists who were trying to be develop a quantitative understanding. Niels Bohr was one of the founders of quantum mechanics who was able to mathematically analyze a solar system model to made some useful predictions. This is his sketch drawn in 1910 of a hydrogen atom clearly showing a little “x” for the electron and a circular arrow for its orbit in the hydrogen atom and below the dihydrogen molecule where now the orbit circles the line connecting the two individual atoms.

But by 1925 it had become clear that little balls of negative electricity whizzing around a heavy, inert, nearly invisible, positively charged nucleus was not a sophisticated enough model to explain the intricate physics of atoms. Yet this mental image of little balls orbiting a tiny sun remains, firmly rooted in our educational consciousness ever since, to be handed down to generations of students just because it is so easy to visualize. To do better we have to read and understand the works of the great physicists like P.A.M. Dirac who finally solved the mysteries of atomic physics making it one of the most well analyzed fields of all physics.

A few years ago in an article in SIAM News recounting the life and work of Paul Dirac (SIAM News, Volume 36, Number 2, March 2003 “Spinning into Posterity,” by Dana Mackenzie) included a list, page 2, that gave the five things Dirac wants you and all posterity to know about atomic electrons. The list:

“Why Spin Does Not Equal Rotation

“Seventy-five years after Dirac’s breakthrough, nearly every popular account of electron spin still describes electrons as if they were rotating billiard balls. And they are all equally wrong. Here are five reasons that this “mental picture,” as Dirac would call it, does not conform to reality:

“Electron spin is quantized; the angular momentum of a classical billiard ball is not. Nothing can gradually “slow down” or “speed up” an electron’s spin.

“The electron’s spin “axis” is completely reassigned by any attempt to measure it. That is, a spin ½ electron will, if measured, also have spin ½ or –½ around the x-axis, spin ½ or –½ around the y-axis, and spin ½ or –½ around the z-axis. (These measurements cannot be performed simultaneously.) By contrast, a billiard ball’s axis of rotation is independent of (and may be oblique to) any axis chosen by an experimenter.

“The electron’s magnetic moment is two times too large for a spinning ball of charge. (Or its spin is two times too small for its magnetic moment.)

“If an electron were a spinning ball, the linear velocity of its surface would exceed the speed of light.

“Quantum physicists know that an electron does not orbit a nucleus in the same way a planet orbits the Sun. So why should the electron rotate like a planet?”

I think the lesson is that just because something is easy to see it is not necessarily right; and if something is complicated there must be a lot of interesting things going on!

As a post script to this blog entry one may infer from the first paragraph that one way to enhance the educational process might be to teach physics first in ninth grade, then chemistry and finally senior biology since each subject builds upon the logical foundation provided by the previous one. And this in fact is presently being proposed and debated. Check the American Association of Physics Teachers web site for “Physics First” to read about the benefits of a logical progression, the reverse of what I described above, in learning science.

The Aha! Moment and the Joys of Mathematica

I should start this entry with a disclaimer like Wikipedia: “Caution! This article is written like an advertisement.” So it may seem, but what I am advertising can be readily had for a few hundred dollars from software company Mathematica and after reading this you may really want it.

First, I’m sure everyone reading this has had at least one of Martin Gardener’s “Aha” moments: that split second when hours of study have all clicked and the answer suddenly becomes clear. I can remember my first time. It was late at night and the college library was empty and closing. I had to retreat to a vacant classroom (left open all night in a small Ohio college town) to keep trying to figure out what all this epsilon/delta stuff in the calculus book was all about. When it finally hit I felt a brain-rush like I’d never experienced before. It was an intellectual high when your brain goes: “Aha, so that’s how it works!” After that, like a true addict, one can’t stop trying to experience it over and over again and, for some reason, mathematics seems to give more “Aha”s than many other endeavors.

Imagine then if one could have something that could give Aha-moments on demand? In mid-1988 software called “Mathematica” (a name recommended by Steve Jobs, as recently divulged by Stephen Wolfram the developer of Mathematica on the occasion of Steve Jobs passing) blossomed in the fledgling computer world with a program initially written to take advantage of Apple computer graphics capabilities. Suddenly almost every mathematical function known to man could be had for the cost of a few key strokes (albeit sometimes the syntax could be a bit obtuse) for unfailing execution, the results plotted in 3-D if so desired. The software was successful and the product grew and matured until today it stands at version 8.0. The reason why it offers ‘Aha’ moments is for the two-fold thrill of deciphering the commands to calculate your result and the production of the result itself. A quick look at the Mathematica web site will easily convince you of the extreme power and versatility. Today undergrads at MIT solve quantum mechanics problems of undreamed of complexity, in class, with their laptops running Mathematica. And the benefits reach all the way down to grade school. With version 8.0, Mathematica offers a web site of thousands of “Demonstrations” where some phenomena or other, from simple arithmetic to laser physics, is given on a screen with slide buttons: the effects of manually changing variables can be seen instantly. High school teachers love the ability to demonstrate a physics (or chemistry, math, astronomy...anything using math) principle projected on a white board with the ability to write on the board as well to help explain the concept under consideration.

Buy this software today!

And now for the odd connection, or not, again I have no answer, just conjecture. For quite a while, I don’t know when he started, but up to the day after the Challenger Space Shuttle disaster on January 28, 1986, Richard Feynman used to teach physics to Hughes employees (like me; I happened to have an office next door to the auditorium he used). Every other Wednesday afternoon he would hold forth in the auditorium on the first floor of building R1 on Imperial Blvd. in El Segundo; at that time it was all Hughes Aircraft. You can verify this astounding fact in his book “Surely You’re Joking, Mr. Feynman?” on p.330 (Actually, in the book, he says he “…used to teach…” And while the book was published in 1985 he was still teaching in 1986; I wonder if this wasn’t one of his famous ploys to keep the celebrity-curious from interfering with his enjoyment of lecturing on physics)?

OK, so I had Feynman as a teacher for 2 hours every other week. So, what didn’t I ask him? Well, I didn’t ask him what he saw in his mind’s eye when someone said “electron.” Wouldn’t that have been keen to hear? I asked him to draw me a copy of the very first Feynman Diagram he ever drew. He couldn’t; he didn’t remember what it was! I also didn’t ask him what his relationship was to Mathematica. But why would I think he had anything to do with Mathematica?

Now for another one of my completely unsubstantiated conjectures. In 1985 physicist and science historian Silvan Schweber published a paper in the “Reviews of Modern Physics,” (vol. 58, p. 452), discussing Feynman’s contributions to quantum mechanics. He mentions that while Feynman was still in high school in 1933 he assigned every symbol provided in the top row of keys on his typewriter a mathematical definition. This allowed him to type complex number equations using a sort of non-standard English language syntax unlike anything in use in mathematics at the time. The illustration provided looks sort of like what Mathematica uses today for entering functions. For instance in Mathematica you can type: “AgeOfUniverse” then a space to signify multiply and “SpeedOfLight.” Hit enter and you get 10 to the 26.1489 power meters; the distance light has traveled since the beginning of the universe. And Stephen Wolfram, the developer of Mathematica worked with Feynman at Caltech for about ten years right before it was released. And there is a famous photo of the two of them huddled together, Feynman writing, Wolfram watching, taken about the time Wolfram was developing Mathematica. And I can almost read Feynman’s lips saying: “Listen Wolfram, I think you need computer software that can be programmed to solve big complex problems by simply writing down the names of functions and giving the range of variables to evaluate!” And, you know what, that’s pretty much what we have!

Oh, yeah, and maybe this isn’t completely unsubstantiated. In 2005 Wolfram wrote: “We [Feynman] talked a lot about how it should work. He was keen to explain his methodologies for solving problems: for doing integrals, for notation, for organizing his work. I even managed to get him a little interested in the problem of language design. Though I don't think there's anything directly from Feynman that has survived in Mathematica.” Yet it’s hard not to think: use Mathematica and get a little help from Feynman!

And, if any further advertising were required to encourage you to try Mathematica, I give you John Forbes Nash (remember the movie “A Beautiful Mind?”): “I spoke on how I had been using MATHEMATICA in my work on the game models…I used the talk materials as prepared for my talk earlier in 2003 in Napoli (which was the latest lecture on the topic of game theory and economic interest). (It happened that progress since then had been slow, because of difficulties in actually finding solutions by computational means.)…were included to illustrate the key topic of the applicability of the…MATHEMATICA software "to complex problems in game theory"”. (July 25, 2003).

Buy this software and get Feynman and Jobs!

How Einstein Didn’t Flunk High School Science

How many times have we been at a party and someone asks what we do for a living and finding out we’re in science they say: “Wow! Science! That’s hard, even Einstein flunked high school science (or math)!” As we fight to stifle a scream we look at our shoes and mumble: “Well…gee…uh...” Because we really don’t know the story of how he didn’t flunk we’re sort of stuck agreeing from ignorance. Well, the real story is one of the best April Fools-type jokes ever!

It turns out Einstein left his German high school early without his diploma, but with a letter of recommendation from his teachers to the ETH in Zurich, Switzerland, saying that he was an outstanding mathematician and should be allowed to take their entrance exam a year early. Although he was still two years younger than the minimum entrance age he passed the science and math parts with such high grades the head of the physics department, Professor Weber, requested Einstein be allowed entrance immediately! However, due to some rather more mundane grades in history and foreign languages, and lack of a high school diploma, it was decided he should take a senior year of Swiss high school just to even out his total intellectual ‘package’ and pick up the diploma required for entrance by the ETH.

His first semester at the Aargau Kantonsschule high school was accomplished with a GPA of 2. Not too good right? Wrong! They were using a grading system of “1” as superior through “6,” a total failure. So he was doing B-level work (although his math and physics grades were “A”s his foreign language and liberal arts were still “C” level).

Now it gets interesting. After the first semester the Swiss educational bureaucracy issued a fiat that for the next semester the grading system was going to be turned on its head: 6 for superior and 1 for the less advantaged. I’m sure you can see where this is going!

Of course! The second semester he failed everything miserably by the grading standards of the previous semester! Although his second semester GPA was 5.4 out of 6, a B+ level. So he actually got smarter the second semester.

And at the end of the year he was welcomed into the ETH with open arms and as they say: “The rest is history.”

However, as time passed and the hoary mists of history enveloped the finer details of a foreign secondary school system, all that was remembered was that, somehow, inexplicably, because even the great Einstein failed his high school science, science must be really, really hard, so why study it?

Well, now you know how he didn’t fail. I wonder if Einstein ever thought about the irony of his senior year high school grades? And the nameless Swiss bureaucrats who made the switch would have pulled it off without notice if they hadn’t had one certain brilliant young man in their school system that year: 105 years later Time magazine’s “Man of the Century.”

All the above information can be easily had by reading “The Collected Papers of Albert Einstein,” volume 1, page 17.

The lesson learned is that if you are going to use a system (e.g. alphabet letters, numbers, emoticons, etc.) to indicate goodness it should be set-up to leverage seemingly pre-conceived human expectations of a top score (first letters, biggest numbers, smiley face). In this respect the American tradition of assigning “A”s coupled with 4 GPs is a stroke of genius combining the best of both of the Swiss grading systems that bedeviled Einstein!

It can be reasonably inferred after reviewing the standard Einsteinian biographies that Einstein himself never gave his high school grades a second thought; that’s what publishing articles in Annalen der Physik will do!

CERN LHC Luminosity Milepost

What with all the newsy swirl the last few days this went completely unreported:
“Last night, [October 3, 2011] while most of the collaboration was sleeping, LHC reached the 2011 milestone of delivering one inverse femtobarn of luminosity to LHCb.” http://www.quantumdiaries.org/2011/10/03/one-inverse-femtobarn-lhcb/

The Fermata Joke

A high school neighborhood friend recently gave me what has to be the most esoteric joke I’ve ever seen, this cartoon created by her friend Thomas Stumpf:

But my knowledge of music is legendary for its nonexistence so I had to ask Linda, an accomplished pianist and piano teacher to explain the in-joke:

“'fermata' in Italian, means 'stop'; stop signs in Italy look just like ours, but say 'fermata'! In music, it means to just wait on the note, for as long as seems to be the best length, in context of the music. The fermata sign is the top half a circle with a dot inside. The fermata is over a rest,(which is silence), which would mean to be silent for more time than the time value of the rest, which is 2 beats.”

So, there are no notes on the score which means there is no sound and the fermata above means that one should keep doing what is written below for as long as necessary, which is nothing or: “shut up.”

Columnar Basalt

Just about anybody who lives in California, and many others who don’t, know about Devil’s Post Pile Park located on the west side of Mammoth Mountain ski area. It is a spectacular west-facing wall of basalt columns several feet wide rising 60 feet straight up. When I visited there in 1999 one scary fact impressed on the casual trekker by a Park Service sign was that the whole thing was earlier on considered a nuisance and a prime source of building material for a dam on the adjacent river for gold mining purposes or some such. The whole display of the post piles is large, but well within the purview of the visitor, so obviously this site could have been easily destroyed. As much as I wanted a rock sample I realized it would be totally inappropriate to take a piece of this magnificent structure! Not to mention that the very smallest samples would have weighed in at a significant fraction of a ton!

Much to my surprise and delight then, while shopping for flagstone pavers at the local rock and brick construction supply yard, I spotted a box labeled: “Small columns,” and inside were fragmented pieces of columnar basalt, supposedly from Washington State. I was able to find a nice piece for my collection, a few inches long, see photo. Notice it has four well developed faces that could be extrapolated out to give six faces total, although since columnar basalt is not a single crystal, the number of faces are not determined by crystal structure considerations and it could possibly have only 5 faces. Plus one end has been broken off showing the basic mineral structure and the other end has been roughly sawed showing a cross section view.

Faster than light expert opinion

The experts are starting to publish opinions on the neutrino-speed of light issue. Here is Sean Carroll, Caltech professor of theoretical physics:

“Faster-Than-Light Neutrinos?
by Sean
Probably not. But maybe! Or in other words: science as usual.
For the three of you reading this who haven’t yet heard about it, the OPERA experiment in Italy recently announced a genuinely surprising result. They create a beam of muon neutrinos at CERN in Geneva, point them under the Alps (through which they zip largely unimpeded, because that’s what neutrinos do), and then detect a few of them in the Gran Sasso underground laboratory 732 kilometers away. The whole thing is timed by stopwatch (or the modern high-tech version thereof, using GPS-synchronized clocks), and you solve for the velocity by dividing distance by time. And the answer they get is: just a teensy bit faster than the speed of light, by about a factor of 10-5. Here’s the technical paper, which already lists 20 links to blogs and news reports.
The things you need to know about this result are:
It’s enormously interesting if it’s right.
It’s probably not right.
By the latter point I don’t mean to impugn the abilities or honesty of the experimenters, who are by all accounts top-notch people trying to do something very difficult. It’s just a very difficult experiment, and given that the result is so completely contrary to our expectations, it’s much easier at this point to believe there is a hidden glitch than to take it at face value. All that would instantly change, of course, if it were independently verified by another experiment; at that point the gleeful jumping up and down will justifiably commence.
This isn’t one of those annoying “three-sigma” results that sits at the tantalizing boundary of statistical significance. The OPERA folks are claiming a six-sigma deviation from the speed of light. But that doesn’t mean it’s overwhelmingly likely that the result is real; it just means it’s overwhelmingly unlikely that the result is simply a statistical fluctuation. There is another looming source of possible error: a “systematic effect,” i.e. some unknown miscalibration somewhere in the experiment or analysis pipeline. (If you are measuring something incorrectly, it doesn’t matter that you measure it very carefully.) In particular, the mismatch between the expected and observed timing amounts to tens of nanoseconds; but any individual “event” takes the form of a pulse that is spread out over thousands of nanoseconds. Extracting the signal is a matter of using statistics over many such events — a tricky business.
The experimenters and their colleagues at other experiments know this perfectly well, of course. As Adrian Cho reports in Science, OPERA’s spokesperson Antonio Ereditato is quick to deny that they have overturned Einstein. “I would never say that… We are forced to say something. We could not sweep it under the carpet because that would be dishonest.” Now there’s a careful and honest scientist for you, I wish we were all so precise and candid. Cho also quotes Chang Kee Jung, a physicist not on the experiment, as saying, “I wouldn’t bet my wife and kids because they’d get mad. But I’d bet my house.” A careful and honest husband and father.
Scientists do difficult experiments all the time, of course, and yet we believe their results. That’s simply because it’s proper to be extra skeptical when the results fly in the face of our expectations: extraordinary claims require extraordinary evidence, as someone once paraphrased Bayes’s Theorem. When the supernova results in 1998 suggested that the universe is accelerating, most cosmologists hopped on board fairly quickly, both because we had a simple theoretical model in hand (the cosmological constant) and because the result helped explain several other nagging observational problems (such as the age of the universe). Here that’s not quite true, although we should at least mention that Fermilab’s MINOS experiment also saw evidence for faster-than-light neutrinos, albeit at a woefully insignificant level. More relevant is the fact that we have completely independent indications that neutrinos do travel at the speed of light, from Supernova 1987A. If the OPERA results are naively taken at face value, the SN 87A should have arrived a couple of years before we saw the explosion using good old-fashioned photons. But perhaps we should resist being naive; the SN 87A events were electron neutrinos, not muon neutrinos, and they were at substantially lower energies. If neutrinos do violate the light barrier, it’s completely consistent to imagine that they do so in an energy-dependent way, so the comparison is subtle.
Which brings up a crucial point: if this result is true (which is always a possibility), it is much more surprising than the acceleration of the universe, but it’s not as if we don’t already have ways to explain it. The most straightforward idea is to violate Lorentz invariance, a strategy of which I’m quite personally fond (although I’ve never applied the idea to neutrino physics). Lorentz invariance says that everyone measures the speed of light to be the same; if you violate it, it’s easy enough to imagine that someone (like, say, a neutrino) measures something different. Once you buy into that idea, neutrinos are an interesting place to apply the idea, since our constraints on their properties are relatively weak. It’s an interesting enough topic that there are review articles, and even a Wikipedia page on the idea.
And there are more way-out possibilities. Graininess in spacetime from quantum gravity might affect the propagation of nearly-massless particles; extra dimensions might provide a shortcut through space. This experimental result will probably give a boost to theorists thinking about these kinds of things, as well it should — there’s nothing disreputable about trying to come up with models that fit new data. But it’s still a long shot at this time. I hate to keep saying it over and over in this era of tantalizing-but-not-yet-definitive experimental results, but: stay tuned.”
http://blogs.discovermagazine.com/cosmicvariance/2011/09/23/faster-than-light-neutrinos/

Speed of Light Exceeded?

"A total of 15,000 beams of neutrinos -- tiny particles that pervade the cosmos -- were fired over a period of 3 years from CERN towards Gran Sasso 730 (500 miles) km away, where they were picked up by giant detectors.

"Light would have covered the distance in around 2.4 thousandths of a second, but the neutrinos took 60 nanoseconds -- or 60 billionths of a second -- less than light beams would have taken.

"It is a tiny difference," said Ereditato, who also works at Berne University in Switzerland, "but conceptually it is incredibly important. The finding is so startling that, for the moment, everybody should be very prudent."

Obviously this overturns Einstein's axiom that the speed of light is the maximum allowable velocity in the universe, one of the foundational laws of modern physics.

Number Series the GLAT and the SAT

Number series

At some time or other we were all challenged in grade school with an aptitude test that included a question requiring the next number in a series. You remember: “What’s the next number in the series 1, 2, 3, _?” “4.” Right? Wrong! It’s “2”! The numbers in the series are the largest primes dividing the natural numbers starting with one! Who knew?

At the risk of raising bad memories, especially after the school informed your parents, based on such ‘scientific’ testing, that you were a chronic underachiever I would like to submit that number series are interesting and not too terribly intellectually taxing by providing two examples and finally, at the end, show you a tool to crack any number series!

The Fibonacci numbers is a very simple series to generate. As with all number series one has to start somewhere. So start with the most basic numbers of all: 0 and 1. To find the third Fibonacci number, F3, simply add the preceding two numbers, 0+1, getting 1, boring, but it gets better. The fourth number, F4 is 2, then F5 is 3 and the series is off and running: 0,1,1,2,3,5,8,13,21…

Still, so what? Well, the usefulness of Fibonacci numbers almost exceeds the imagination! In third grade science we saw how flower petals and leaves and seed pods spiral out from the center in beautiful arcs to maximize accessibility to sunlight and rain. Maybe we counted the number of clockwise turning and counter-clockwise turning spirals while looking at the stem end of a closed pine cone or a pineapple and found 5 and 8 spirals; two adjacent Fibonacci numbers. Then in tenth grade art we saw how the beautiful Parthenon in Athens had linear height and width dimensions that were again a ratio of adjacent Fibonacci numbers. Finally in senior year algebra it was explained that the ratio of adjacent Fibonacci numbers was always the same number, the golden ratio, phi, 1.618… Wikipedia does a fine job with all the details and uses of these numbers (including stock trading).

But there is one thing they can do for you while you are driving that long stretch of I5 to San Francisco: amaze your family by converting miles to kilometers with ease. Here’s how. A moment of reflection shows that the numerical value for phi and the number of kilometers per mile are almost identical values, within half a percent! If you are going 65 miles per hour you decompose 65 into Fibonacci numbers: 55+8+2. Increase these to the next highest ones: 89+13+3 which adds to 105 kilometers per hour. To convert to miles, use the next lower numbers. I know, it’s a lame trick making minimal use of Fibonacci numbers, but still…well, I’ll bet you’ll try it anyway.

The next number series, with no particular usefulness what-so-ever, was made famous because it was problem number 20 in the GLAT (Google Labs Aptitude Test for perspective employees) booklet that Google inserted inside several popular magazines in 2004. Google “GLAT” for all the details and answers from Mathematica and others.

This integer series is interesting because it appears intimidating and shows how the human mind can be easily fooled.

Question 20. “What number comes next in the sequence: 10, 9, 60, 90, 70, 66, …?

A) 96
B) 1000000000000000000000000000000000\
0000000000000000000000000000000000\
000000000000000000000000000000000
C) Either of the above
D) None of the above”

Notice that answer “B)” is without a doubt intimidating and probably causes your subconscious inner-voice to say something like: “Well, let’s skip this one and come back later.” Or at least you realize there must be some kind of strange algorithm at work here to cause this number to be even considered as an answer.

But the key to this series is simply the largest natural number requiring exactly n letters in English. Since “10” is larger than 1, 2 or 6 and all four have three letters in their names the series starts with 10. So you just have to count the number of letters in the name of each number to figure out the next one. The only real mathematical knowledge you need is to recognize that answer B) has it’s own name, it’s called a “googolplex,” 10 letters! Ha, ha, fooled you! But the lesson to be learned, especially if you are a student interested in taking the SAT is: don’t panic! Look for some sort of conversion process going on like substituting names for numbers, etc.

Finally, the tool promised at the beginning is an absolute marvel of what the human brain can produce. The “On-line Encyclopedia of Integer Sequences,” http://oeis.org/, can without a doubt crack any sequence: simply type the first few numbers of interest in the opening page box and the Encyclopedia returns the full sequence with generating formula, historical notes and references to similar sequences. Take a look; show it to a student facing one of those SAT tests. What a great way to gain familiarity with and confidence in solving number series!

My favorite sequence is: “An optimal n-mark Golomb ruler [which] has the smallest possible length (distance between the two end marks) for an n-mark ruler.” OEIS reference number A003022.

Finally, Google has many entries concerning methods of solving for the next number in a series which can be of great help to the test taker.