"If you have ever used a sundial, you may have noticed that the time registered on the sundial differed slightly from that on your watch."
This is the first page of the section "The Equation of Time." Here Pappas explains how sundials work and how they are related to the shape of earth's orbit.
Here are some excerpts from this page:
"In the 15th century, Johannes Kepler formulated three laws that governed planetary motion. The Sun is located at one of the foci of the ellipse thereby making each sector's area equal for a fixed time interval and the arc lengths of the sectors unequal. This accounts for the variations in the lengths of daylight during different times of the year."
There are two pictures on this page. One of them shows earth's orbit around the sun. Here is this first picture's caption:
"If the time intervals of travel for these elliptical arcs are equal, then the areas of their sectors are equal."
The second picture is of a medieval timepiece:
"A 10th century pocket sundial. There are six months listed on each side. A stick is placed in the hole of the column with the current month."
Pappas will explain more on tomorrow's page. But there are a few things that we notice already. Let's look at a list of sunrise and sunset times in the nearest largest city to me, Los Angeles, for the upcoming month of December:
https://www.timeanddate.com/sun/usa/los-angeles?month=12
December Solstice (Winter Solstice) is on Thursday, December 21, 2017 at 8:27 am in Los Angeles. In terms of daylight, this day is 4 hours, 32 minutes shorter than on June Solstice. In most locations north of Equator, the shortest day of the year is around this date.
Earliest sunset is on December 4 or December 5. Why is the earliest sunset not on Winter Solstice?
And the answer to this last question is related to the current section in Pappas, and so we'll find out all about this in tomorrow's post.
By the way, this is closely related to the idea of Daylight Saving Time. Earlier this month, I wrote about the DST debate -- mainly about Assemblyman Chu's Year-Round DST bill. Some people believe that the DST clock changes would be less annoying if DST weren't nearly eight months out of the year. Why, for example, don't we simply spring forward at the spring equinox and fall back at the fall equinox? Well, the reason for DST is to avoid inconvenient sunrise times -- and sunrise, like sunset, is also dependent on the Equation of Time. Again, all will be revealed in tomorrow's post.
Chapter 10 of George Szpiro's Poincare's Prize is called "Inquisition -- West Coast Style." Here is how it begins:
"The 1980's again saw a flurry of activity. At the time it was not at all clear whether Poincare's Conjecture was true."
Thus in this chapter, some mathematicians will try to prove Poincare false instead of true. One of these mathematicians is Steve Armentrout of Eldorado, Texas. (As a Californian, I don't think of Texas as the "West Coast" mentioned in the chapter title. But we'll be going further west later on in this chapter.)
"He was going to provide a proof of concept, as it were. As tangible evidence of a counterexample's existence this may seem on the light side, but as a mathematical procedure the submission of an existence proof is quite acceptable."
And so Armentrout writes his proof, but not without a warning:
"'It is particularly important to know whether there are mistakes that would invalidate the main result of the paper.'"
You're probably thinking that someone finds a flaw in the proof, but no -- after it is published, we never hear of the proof again. All we know is that a copy of the proof lies in the American Institute of Mathematics in Palo Alto, California, near Stanford. (OK, "West Coast" represent now!)
The next mathematician is Australian Hyam Rubinstein. He devises an algorithm that finds counterexamples to Poincare:
"It could test more and more objects, but even if it never identified a counterexample, this would be no guarantee that one does not lie just around the corner. We will have more to say about Rubinstein's algorithm."
Now Szpiro informs us:
"The next three endeavors do not represent direct attempts at solving the Poincare Conjecture. Rather, they are searches for alternative ways to prove the elusive theorem."
In other words, mathematicians search for theorems that are equivalent to Poincare and are much easier to prove. For example, British mathematician Thomas Thicksun writes:
"'The proposed theorem reads, 'Poincare's Conjecture holds iff every open, irreducible, acyclic 3-manifold, which is a degree one proper image of an open 3-manifold embeddable in S^3, is also embeddable in S^3."
Szpiro explains what iff means -- "if and only if." In other words, Thicksun is attempting to prove a theorem that's equivalent to Poincare. This reminds me of the time I told the students I was tutoring about the abbreviation iff. Their teacher didn't want them to use iff in their proofs.
As it turns out, no one is able to prove the equivalent statement, and so this is another dead end. The next attempt is by David Gillman of UCLA (West Coast represent again!) and Dale Rolfsen from the University of British Columbia. (That's the Canadian West Coast, but nonetheless our coast!) The prove that a conjecture of Christopher Zeeman (mentioned in yesterday's post) is also equivalent to the Poincare Conjecture:
"Well, this same Sir Christopher had proposed a conjecture way back in 1963 that said that every compact, two-dimensional polyhedron that can be shrunk to a single point by moving all of its points along certain paths inside the manifold can be thickened and then collapsed to a single point by triangulating and then removing triangles, one by one, in an orderly fashion."
But this conjecture, just like Thicksun's, is no easier to prove than Poincare itself. The next attempt is by British topologist Colin Rourke and Eduardo Rego. It is publicized by Ian Stewart -- yes, the same Ian Stewart who writes Calculating the Cosmos (mentioned in my August 21st post):
"Since the general public does not generally read Nature, Stewart followed up with an article in the British newspaper The Guardian, and from there, the news item hopped around the world."
But as it turns out, no other mathematicians trust the proof, regarding the two authors as "cranks," or amateurs who make boastful claims about grand proofs:
"However, the listeners never seemed anything other than deeply skeptical, and the atmosphere bordered on the hostile."
Szpiro compares Rourke and Rego to Swedish student Elin Oxenhielm, who thinks she has proved one of the Hilbert problems, only to be mistaken:
"Instead, the bubble burst within days and her budding career was brought to a screeching halt before it had even begun. Ian Stewart has no regret about having been instrumental in bringing the news of Rourke and Rego's purported proof to the public."
With the rise of the Internet, mathematical cranks claiming to have proved Poincare become much more common. For example, the author describes Italian Michelangelo Vaccaro:
"In a few paragraphs, he outlines a purported proof and promised to send hard copies of his twenty-six-page paper to the postal addresses of everyone who was interested."
Of course, Vaccaro fails to deliver. British mathematician Martin Dunwoody, meanwhile, writes his paper, posts it online, learns of an error, rewrites the paper, and repeats the process:
"Anyway, Dunwoody was at version seven of his paper when disaster struck. Colin Rourke had weighed in."
And Rourke locates the fatal error, trying to redeem himself for his earlier follies. Indeed, he and Rego have been working on another algorithm, the RR-algorithm, which searches for possible Poincare counterexamples:
"No input is needed for the RR-algorithm, it is a self-starter and then goes through the list. If a counterexample to Poincare's Conjecture exists, the RR-algorithm."
Combined with Rubinstein's counterexample-hunting algorithm from earlier, the RR-algorithm now becomes the RRR-algorithm. Szpiro compares the RRR-algorithm to the hunt for Higgs bosons:
"What is certain is that running the RRR-algorithm to find counterexamples to Poincare's Conjecture would be immensely cheaper than operating the Large Hadron Collider to find Higgs bosons. But then again, enormous costs are of little consequence to particle physicists."
Of course, those enormous costs eventually paid off, since the Higgs boson has been discovered since Szpiro published his book. The author, meanwhile, writes about one last failed attempt:
"On October 22, 2002, a twenty-one page paper titled 'Proof of the Poincare conjecture' was posted to arXiv.org, an Internet repository for academic papers."
The author is Sergey Nikitin, a Russian mathematician. Just like Dunwoody, Nikitin revises his paper each time a reader spots an error. He writes:
"'Theorem 2.8 was wrong in version 1 and that made the proof incomplete. It was fixed in version 2. The definition of simple connectivity was too strong in version 2. It was fixed in version 3."
But eventually, readers stop correcting Nikitin's paper. Szpiro concludes his chapter by explaining the reason Nikitin is ignored:
"Just three weeks after Nikitin's first posting, in the period between his versions 5 and 6, another Russian mathematician had posted a paper to the arXiv. It would eclipse all previous attempts at proving the Poincare Conjecture."
Normally, I'd be posting today's worksheet, but today is just Day 2 of yesterday's activity. I feel guilty for making a school-year post without a worksheet. But then again, I felt even guiltier for never posting multi-day activities and thereby never giving students an opportunity to continue a worksheet before posting the next one.
So there's no worksheet for me to post today. The students should continue working on yesterday's "Corresponding Parts in Congruent Figures" assignment.
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