Thursday, November 30, 2017

Chapter 6 Review (Day 69)

This is what Theoni Pappas writes on page 30 of her Magic of Mathematics:

"On the other hand, the time intervals of our other clocks are consistent. The difference between a sundial's time and an ordinary clock is referred to as the equation of time."

This is the last page of the section on the equation of time. Actually, Van Brummelen writes a little about the equation of time in the last chapter of his spherical geometry book. But as we found out yesterday, it's the shape of earth's orbit that varies the time of high noon, sunrise, and sunset.

"For example, the chart may look like the one below."

Equation of Time Chart
(The negative and positive numbers indicate the minutes the sundial is slower or faster than an ordinary clock.)

Jan 1       -3
       15    -9
Feb 1      -13
       15    -14
Mar 1     -3
       15    -9
Apr 1      -4
       15    0
May 1    +3
       15    +4
(Pappas doesn't show the chart for June through December.)

Then Pappas claims that if the sundial shows 11:50 on May 15, the actual time is 11:54. This is an error, because the positive +4 indicates that that the sundial is faster than the real time. So if 11:50 is four minutes fast, the actual time is 11:46, not 11:54.

Here's another link to the equation of time:

There are four days per year when the equation of time equals zero. One of them appears in the chart above, about April 15th. The others are June 14th, September 1st, and December 25th. A mnemonic for Americans to remember these days is "Tax. Flag, Labor, Xmas." The minimum value also appears in the chart, February 15th (perhaps a few days earlier), while the maximum is around November 3rd (with a value of +16).

Yesterday, I wrote that the equation of time can be used to determine the optimal dates for setting the clock forward and back for DST. Recall that the goal of DST is to avoid extreme sunrise times. Let's say we pose the following rule:
  • In spring, move clocks forward as soon as sunrise reaches 6 AM, before most people wake up.
  • In fall, move clocks back as soon as sunrise reaches 7 AM, after most people wake up.
This maximizes the number of ideal sunrises, between 6 and 7 AM. OK, so let's attempt this for the nearby city of Los Angeles. Without the equation of time, sunrise ought to be at 6 AM on the equinoxes -- and indeed it would appear as 6 AM on the sundial -- but let's find out when sunrise is once we take the equation into effect:

We notice that even though clocks move forward on March 11th, the sun doesn't rise before 6 AM until a few days later. On St. Patrick's Day it rises at 6 AM and the next day at 5:59. Thus we ought to wait an extra week, until March 18th, before moving clocks forward. And as for the fall:

The sun rises at 7 AM on October 18th, so this is when clocks should be set back to 6 AM. Let's make it easy and say the third Sunday in October.) This is the Sunday we should have set the clock back rather than earlier this month, two weeks late. Those who object to having DST for eight months are correct, but still, we calculated that there should be seven months of DST, from the third Sunday in March until the third Sunday in October.

The equation of time explains why DST should still be more than half the year. At the fall equinox in September, the equation of time is positive. So at sunrise, the sundial will claim that it is 6 AM (Standard Time, or 7 AM with DST in effect), but in reality, it's about 6:40-something. Since sunrise is before 7 AM, the clocks should still be set forward. As September turns into October, the days shorten and the sundial will give sunrise as after 6 AM Standard Time, but with the equation still positive and increasing (towards the early November maximum), sunrise is before 7 AM DST. Not until the third week in October does the shortening day finally "beat" the equation of time -- sunrise is after 7 AM, which should be set back to 6 AM Standard Time.

In the Southern Hemisphere, the seasons are reversed, yet the equation of time isn't. And so while the season change and the equation of time oppose each other in the North in October, they end up reinforcing each other in the South. Sunrise is before 6 AM at the September (or spring) equinox and only gets earlier in October, so clocks actually should be set forward in September -- but not too early that month, since the equation of time is zero on September 1st. This is before the spring equinox, and so sunrise is after 6 AM.

In autumn in the South, when should clocks be set back? Well, in March the equation of time is negative, and so sunrise is after 6 AM Standard Time. In fact, the equation reaches its minimum (or most negative) value in February, and so sunrises may be after 6 AM well before the equinox. In fact, I'd set the clocks back as early as February in the South.

So to conclude, my DST recommendations are:

  • In the Northern Hemisphere, set the clocks forward in March and back in October.
  • In the Southern Hemisphere, set the clocks forward in September and back in February.
Because of the asymmetry of the equation of time, there are seven months of DST in the North and only five months in the South.

Again, the purpose of DST is to avoid extreme sunrise times. It has nothing to do with avoiding extreme sunset times, otherwise we would fall forward and spring back instead of vice versa (but often people suggest falling forward and springing back anyway). Also, DST has nothing to do with temperature at all. I've mentioned that the hottest day of the year is well after the summer solstice, and often approaches the first day of school (cf. the Early Start Calendar debates). Sometimes people suggest springing forward when the temperature rises (which might not be until April or May), and keeping the clocks forward until the temperature falls, as if the purpose of DST is to allow people to enjoy warm, sunlit afternoons outdoors. My DST recommendation works well with temperature lag in the North (where warm days could stretch into October), but not in the South (where warm days may persist well past February).

The third Sunday of the given month could be chosen at the date of the clock change. It might be better to choose the final Sunday of the month instead. This is because Europe changes to DST (or Summer Time) and back on the last Sundays in March and October. Thus that continent doesn't need to change -- instead other countries in the North should adopt Europe's clock change dates.

On the other hand, no country in the South follows the dates I recommend. The closest are New Zealand (which does spring forward the last Sunday in September) and Brazil (which falls back the third Sunday in February).

When we choose either the third or last Sundays of the months listed above as clock changing dates, then both hemispheres have standard time in early March (when the equation of time is negative) and DST in early October (when the equation of time is positive).

Chapter 11 of George Szpiro's Poincare's Prize is called "Watching Things Go 'Pop'." It begins:

"In the late 1970's and early 1980's William Thurston from the University of California, today at Cornell, formulated a spectacular hypothesis."

In this chapter, we prepare for the actual proof of the Poincare Conjecture. Here we learn about all the preliminary material we need in order to understand the framework of the proof. (By the way, the Thurston described by Szpiro died after the publication of this book, in 2012.)

Who is this Thurston? He is a prize-winning mathematician, according to Szpiro:

"Thurston was awarded the Veblen Prize in geometry and a Sloan Foundation Fellowship in 1974 for his work on foliations."

And what exactly is Thurston's work?

"Before investigating how space is shaped, we must verify that all manifolds can be split into basic building blocks or, equivalently, that every manifold can be built up from such blocks."

According to the authors, these building blocks are called "prime manifolds" or "primafolds" and are akin to primes among the natural numbers:

"A bagel is a primafold: No matter where one takes a bite, what remains are the original bagel and the bite."

On the other hand, a figure eight can be divided into two bagels, and a pretzel can be divided into three "bagels," so these aren't primafolds. The categorization of 2D primafolds has implication for our Geometry class:

"The theorem implies that only three types of basic geometries exist in two dimensions. They are the Euclidean, the spherical, and the hyperbolical geometries."

The sphere has no holes, while a plane can be rolled up into a bagel with one hole, and:

"All two-dimensional surfaces with at least two holes -- i.e., with genus two or higher -- belong to the hyperbolic class of objects. In three dimensions things become much more complicated."

For example, a cylinder is formed by "multiplying" the circle with a line:

"In the same manner, three-dimensional cylinders can be built from lower-dimensional building blocks."

Thurston's Conjecture is that there are exactly eight 3D primafolds, but Thurston doesn't prove it -- much to the dismay of John Stallings:

"He explained that these misdeeds 'consist of asserting things that are true if interpreted correctly, without really giving good proofs, thus claiming for themselves whole regions of mathematics and all the theorems therein, depriving the hard workers of well-earned credit."

The next mathematician Szpiro describes is Richard Hamilton. He teaches at several campuses throughout California and Hawaii:

"He no longer chooses his home institution according to the surfing conditions and is now a professor of mathematics at Columbia University."

And he really is still alive, as he'll celebrate his 74th birthday in a few weeks. He also works with Shing-Tung Yau of Hong Kong, whose childhood Szpiro describes as well. When his father dies:

"Only the strong will of his mother and some help by friends and former students of his father allowed Shing-Tung to continue his schooling. Because he could not afford to buy books, Yau spent hours and hours in bookstores to read them there."

Yau eventually works on a problem that has widespread implications in string theory:

"The Calabi Conjecture says that the extra dimensions, beyond the four familiar ones of space-time, are tightly curled up, like the loops in a carpet."

(Pappas also mentions string theory in her book -- see my August 28th post for more info.) The author writes about Yau's collaboration with Hamilton:

"But convinced that this would be the way to prove not only Poincare's but also Thurston's conjecture, Yau encouraged Hamilton to continue on his path."

And "his path" refers to something called "Ricci flow" -- and so Szpiro now digresses to explain who this "Ricci" is. He is an turn of the 20th century Italian mathematician who writes about tensors, which the author describes as a generalization of scalars, vectors, and matrices. At first, tensors have no practical importance:

"Whoever did notice it dismissed it as no more than a technical accomplishment. But a dozen years after its publication, Albert Einstein began utilizing tensors when he reduced gravitation to a geometrical phenomenon."

Ricci flow is related to the curvature of an object. Sometimes in Calculus BC or just beyond, the curvature of a 1D curve is defined as a single number at each point -- tight curves or small circles have large curvature, while nearly straight curves or large circles have small curvature:

"However, for lines that do not just wind along a plane but snake through space, one number does not suffice."

Now let's look at the Ricci flow:

"As envisaged by Richard Hamilton, the Ricci flow does to manifolds what Botox injections do to aging movie stars: They make frown lines, forehead creases, and crow's-feet disappear and render the lady's or gentleman's skin smooth and fresh, as it was when she or he was young."

For example:

"The negative relationship between curvature and scale implies that the more curved a manifold's region, the smaller the scale becomes."

And what's the point of this scale reduction?

"And now comes the icing on the cake: If every beaten-up, kinked, and twisted but simply connected manifold ends up going 'pop' [reducing to a point], Poincare's Conjecture is proven!"

Hamilton comes up with this theory in 1979:

"Three years later, in 1982, he introduced the mathematical world to the Ricci flow. His paper entitled 'Three-manifolds with positive Ricci curvature' was published in the Journal of Differential Geometry and made a big splash."

But not every manifold can be made to go 'pop.' Some of them have strange "singularities":

"Others disappear to a point so quickly that it is hard to see whether their curvature has become round like that of a sphere."

For example, there is a "neckpinch singularity":

"Rather, both the handle and the smaller sphere are simultaneously reduced in size, until only the larger sphere remains, albeit with a small protrusion."

Finally, there is a "cigar singularity," which is like a cigar before the tip has been snipped off. It turns out that this manifold can't be made to go 'pop,' no matter how we slice it:

"But then he got stuck again. The cigar singularity just would not succumb to surgery."

Yes, I know this is difficult to visualize, especially with all of these inexact terms. Be assured that Szpiro describes these terms much better in his book than I can in a short blog post.

Anyway, the author concludes the chapter by describing the day he meets Hamilton in person. He sees that Hamilton is understandably skeptical when a new Poincare proof attempt is announced:

"To his surprise, he realized that this was no hoax. 'This guy may have something,' he told his colleague. But let's not jump the gun, Muffin."

OK, here is the Chapter 6 Review worksheet. I've made changes to the worksheet both three and two years ago, including the activities I give the same week as the worksheet -- and due to these changes, I didn't have much to say about this worksheet last year. Since this week's activity was already given as a multi-day assignment the last two days, there is no activity to give today, so students should just ignore the instruction to perform an activity.

Wednesday, November 29, 2017

Activity: Corresponding Parts in Congruent Figures, Continued (Day 68)

This is what Theoni Pappas writes on page 29 of her Magic of Mathematics:

"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:

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, 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.

Tuesday, November 28, 2017

Lesson 6-7: Corresponding Parts in Congruent Figures (Day 67)

This is what Theoni Pappas writes on page 28 of her Magic of Mathematics:

-- "...movable geodesic skylights. They are designed to optimize the use of solar energy."
-- "Marvelous, but where is the bed?" I asked.

This is the last page of the section "A Mathematical Visit." OK, so now we finally find out what's in Selath's bedroom -- solar skylights shaped like spheres and, most likely, paraboloids (as we saw on page 13 in the November 13th post). OK, let's discover the mathematician's bed:

-- "Just push the button on this wooden cube, and you will see a bed unfold with a head board and two end tables."

Let's skip down to the bathroom mirrors, since this is relevant to our current U of Chicago lesson on mirrors and isometries:

-- To my surprise I saw an infinite number of images of myself repeated. The mirrors were reflecting back and forth into one another ad infinitum.
-- "Now turn around and notice this mirror. What's different about it?" Selath asked.
-- "My part is on the wrong side," I replied.
-- "To the contrary this mirror lets you see yourself as you are really seen by others," Selath explained.

What's going on here? Pappas provides a footnote here:

"Made from two mirrors at right angles to each other. The right-angled mirrors are then positioned so that they will reflect your reflection."

We can use the isometries of Chapter 6 to explain what's happening. In the first case, the mirrors are parallel, and the composite of reflections in parallel mirrors is a translation. Therefore the first reflection image has reverse orientation, and the second has the correct orientation. But you can't see the front of your translation image -- only the back.

The second case provides you with a correct-oriented image of yourself that you can see. The mirrors are perpendicular, and the composite of reflections in intersecting mirrors is a rotation. Indeed, the composite is a rotation of magnitude 180 (twice the angle between the mirrors) and centered where the mirrors intersect. Turning you 180 degrees means that you see the front of your rotation image.

Here's how this section ends -- Selath's two dinner guests arrive, and so the unexpected arrival of the narrator means that there are four for dinner that night:

-- It was hard to conceal my enthusiasm. "But your table is set for three," I blurted.
-- "No problem. With the tangram table I can just rearrange a few parts and we'll have a rectangle.

Tangrams are often mentioned on math teacher blogs as an extra activity to give students. For example, frequent blogger Sarah Carter wrote about tangrams on her blog two weeks ago:

Carter attributes the tangram puzzle to another famous blogger Sara(h) -- van der Werf, to be exact.

Chapter 9 of George Szpiro's Poincare's Prize is called "Voyage to Higher Dimensions." Here is how it begins:

"The situation at the end of the 1960's was miserable: All efforts to prove Poincare's Conjecture had been fruitless."

As the title implies, we read about higher dimensions in this chapter -- and not just the fourth dimension, but 5D, 6D, 7D, and so on. As it turns out, Poincare is easier to prove in these higher dimensions than it is in the third dimension.

Szpiro explains what the analogs of the Poincare Conjecture look like in higher dimensions:

"We also defined the second homotopy group using parachutes. Poincare's Conjecture for four dimensions says that if all loops and all parachutes laid out on a four-dimensional body can be contracted to points, the body can be morphed into a four-dimensional sphere."

As it turns out, Poincare for 5D and above is proved by Steve Smale, though some controversy exists regarding who is actually the first to prove it. Of Smale, the author writes:

"Remember that a biographer of Henry Whitehead's wrote how unspectacular that mathematician's life was, 'with little for the biographer to chronicle'? Smale must be placed at the extreme other end in terms of life stories; there is nearly too much to chronicle."

And so of course I won't write about all of that in this post. (The author tells us that Smale is still alive today -- this summer he celebrated his 87th birthday.) I will say that Smale is educated in a one-room schoolhouse for grades K-8 and goes on to attend the University of Michigan. But he's more interested in politics than in his studies, and applies to grad school in math because he thinks that it's easier than physics:

"Eventually he was warned by the chairman of the math department that he would be kicked out if his grades did not improve."

Fortunately, his grades do improve, and he goes on to earn his master's degree. In fact, he spends time at several other schools, including the U of Chicago (yes, home of our text) and UC Berkeley. He describes what it means to "evert" a sphere:

"This term refers to the turning inside out of a sphere if the skin is permitted to pass through itself, but no holes must be made and no ripping or creasing is allowed."

(Of course, the sphere is evertible only in a higher dimension.) According to Szpiro, Smale also writes papers on chaos theory and the "horseshoe":

"It neatly expressed the idea behind the sensitive dependence on initial conditions, which is one of the catchphrases of chaos theory -- together with the butterfly effect, which, in principle, express the same thing."

(Again, Pappas writes about chaos theory several times in her book as well.) Now Szpiro tells us that Smale earns a Fields Medal, and on the way to Moscow to receive the award, he meets none other than Paul Erdos, the "Man Who Loved Only Numbers," according to Hoffman's book. It's Erdos who tells him that Smale has been subpoenaed by the House Un-American Activities Committee. (As Erdos would put it, "Sam" and "Joe" are at it again!)

"[Smale] had the 'honor' of being summoned by the committee without the hassle of having to be there. The Congress [of the International Mathematical Union] was a usual a grand affair, with thousands of mathematicians in attendance from all over the world."

Later on, Smale receives a grant from the National Science Foundation, but they threaten to take the money back because the mathematician often spends more time at the beach than doing math:

"After all, no government agency wants to be accused of paying for vacations and encouraging laziness."

Just before the turn of the century, Smale formulates a list of problems for mathematicians to work on, similar to those given by David Hilbert at a century earlier:

"Hilbert's problems had determined the direction of much of mathematical research during most of the first half of the twentieth century."

For all his work, Smale receives many prizes, including the Wolf Prize ten years ago. At the ceremony in Israel, he is lauded with the following statement:

"Against mainstream research on scientific computation, which centered on immediate solutions to concrete problems, Smale developed a theory of continuous computation and complexity."

The author now describes Smale's proof of Poincare for dimension five and above. His proof involves a concept called a "cobordism," or boundary, between two manifolds:

"For example, if one manifold is a circle and the other manifold is a pair of disjoint circles, the cobordism would be something resembling a pair of pants: The single circle would be the waist and the pair of circles would be the exits for the legs."

Smale sends his proof to Samuel Eilenberg at Columbia University to look over before submitting it to be published:

"Sammy, as everyone called him, was a Polish Jew who had escaped his fatherland just before the Nazis invaded it and was considered one of the world's leading topologists."

And so Sammy publishes the proof. Meanwhile, others become interested in try to prove Poincare for higher dimensions as well, including Princeton student John Stallings. Szpiro begins with an entertaining story about how Stallings survives lectures by getting the speakers drunk with whiskey:

"Reality television shows and pseudodocumentary films have recently been accused of using the same technique. Another example of mathematics on the cutting edge of culture, one might say.... Stallings had arrived at Princeton at a portentous time when 'topology was God and the Poincare Conjecture was its prophet."

Ultimately, Stallings attends graduate school at Oxford, and he proves Poincare for dimension seven and above. But this is at the same time that Smale is working on the same problem:

"More important, [Smale] had just proved the higher-dimensional Poincare's Conjecture. Now was the time to present his work to a wider audience in Europe."

And so Smale and Stallings meet in West Germany. Smale presents his proof -- and Stallings has a big smile on his face when he spots an error in Smale's proof. And so Smale spends the next four months trying to fix the error:

"In the fall of 1960, Smale's proof was ready to go, and on October 11 the manuscript with the title 'Generalized Poincare Conjecture in Dimensions Greater Than Four' was received by the Annals of Mathematics."

It's a race to who completes the proof first. Szpiro describes how Smale begins publishing his paper first, but Stallings has a finished proof, and to this day it's controversial which mathematician is truly the first to prove it. It's definitely true that Smale's proof works for 5D and up, while Stallings has a proof that begins at 7D. It takes the English mathematician Christopher Zeeman to add 5D and 6D to the Stallings proof.

Szpiro writes a little about Zeeman's early life:

"Just how difficult life must have been for the young boy is apparent from his description of his subsequent service in the Royal Air Force (1943-47) as a breath of freedom."

In other words, to Zeeman, his boarding school is harsher than the military.

Most of the rest of the chapter is about the argument between Smale and the other mathematicians regarding who is the first to prove Poincare for 5D and up. At any rate, it is Michael Freedman, a Southern Californian, who proves the 4D version of Poincare. Szpiro tells us that it's easier to prove Poincare in higher dimensions like 7D, 5D, and 4D than in 3D because there's more "elbow room" to twist the manifolds into spheres.

"Furthermore the only manifold whose parachutes also stretch and shrink to a point is the four-dimensional sphere. Poincare had been vindicated also in dimension four."

Lesson 6-7 of the U of Chicago text covers the Corresponding Parts in Congruent Figures Theorem, which the text abbreviates as CPCF. But a special case of this theorem is more widely known -- corresponding parts in congruent triangles are congruent, or CPCTC.

When I was young, a local PBS station aired a show called Homework Hotline. After school, middle and high school students would call in their homework questions in math and English, and some would be chosen to have their questions answered on the air by special teachers. Even when I was in elementary school, I often followed the geometry proofs that were called in, and more often than not, there were triangle congruence two-column proofs where the Reason for a step was often CPCTC. So this was where I saw the abbreviation CPCTC for the first time. (By the time I reached high school, a few calculus problems were called in to the show. Nowadays, with the advent of the Internet, the show has become obsolete.)

Here's a link to an old LA Times article about Homework Hotline:

When I reached geometry, our text usually either wrote out "corresponding parts in congruent triangles are congruent," or abbreviated as "corr. parts of cong. tri. are cong.," probably with a symbol for congruent and possibly for triangle as well. But our teacher used the abbreviation CPCTC. Now most texts use the abbreviation CPCTC -- except the U of Chicago, that is. It's the only text where I see the abbreviation CPCF instead.

Dr. Franklin Mason, meanwhile, has changed his online text several times. In his latest version, Dr. M uses the abbreviation CPCTE, "corresponding parts of congruent triangles are equal."

Well, I'm going to use CPCTC in my worksheets, despite their being based on a text that uses the abbreviation CPCF instead, because CPCTC is so well known.

Once again, it all goes back to what is most easily understood by the students. Using CPCTC would confuse students if they often had to prove congruence of figures other than triangles. But as we all know, in practice the vast majority of figures to be proved congruent are triangles. In this case, using CPCF is far more confusing. Why should students had to learn the abbreviation CPCF -- especially if they have already seen CPCTC before (possibly by transferring from another class that uses a text with CPCTC, or possibly even in the eighth grade math course) -- for the sole purpose of proving the congruence of non-triangles, which they'd rarely do anyway?

So it's settled. On my worksheet, I only use CPCTC.

Notice that for many texts, CPCTC is a definition -- it's the meaning half of the old definition of congruent polygons (those having all segments and angles congruent). But for us, it's truly a theorem, as it follows from the fact that isometries preserve distance and angle measure.

Another issue that comes up is the definition of the word "corresponding." Notice that by using isometries, it's now plain what "corresponding" parts are. Corresponding parts are the preimage and image of some isometry. Unfortunately, we use the word "corresponding angles" to mean two different things in geometry. When two lines are cut by a transversal and, "corresponding angles" are congruent, the lines are parallel, but when two triangles are congruent, "corresponding angles" (and sides) are congruent as well. The phrase "corresponding angles" has two different meanings here! Of course, one could unify the two definitions by noting that the corresponding angles at a transversal are the preimage and image under some isometry. I tried this earlier, remember? It turns out that the necessary isometry is a translation. This is one of the reasons that I proved the Corresponding Angles Test using translations -- it now becomes obvious what "corresponding angles" really are. I mentioned yesterday, however, that in many ways using translations to prove Corresponding Angles is a bit awkward since it took so much work to avoid circularity. (This is why some authors, like Dr. Hung-Hsi Wu, uses rotations to prove Alternate Interior Angles instead.)

Returning to 2017, today I post the first page from last year's, but not the second page, which is about an unrelated math problem I had on my mind that year.

Instead, I replace it with an activity. Just before Thanksgiving, I wrote about how many teachers often given multi-day activities, but I've never done so on the blog. And so today I post such an activity:

Monday: Lesson 6-6 (Day 66)
Tuesday: Lesson 6-7 and Begin Multi-Day Activity (Day 67)
Wednesday: Finish Multi-Day Activity (Day 68)
Thursday: Review for Chapter 6 Test
Friday: Chapter 6 Test

This activity is based on the Exploration Question in today's text:

a. Find three characteristics that make Figure I not congruent to Figure II.
b. Make up a puzzle like the one in part a, or find such a puzzle in a newspaper or magazine.

Students can perform part a today and part b ("Make up a puzzle") tomorrow. Of course, if they wish, they can do the "find such a puzzle" part tonight. With this plan, we return to having only one day for review -- a formal test review to be given on Thursday.

The following worksheet comes from the website:

Unfortunately, this picture (used as the initial example for part a) doesn't print very well. Then again, most versions I find online are interactive versions that print even worse.

Monday, November 27, 2017

Lesson 6-6: Isometries (Day 66)

This is what Theoni Pappas writes on page 27 of her Magic of Mathematics:

-- "So 24:00 hours would be 30:00 hours, 8:00 would be 10:00, and so on," Selath explained.
-- "Whatever works best for you," I replied, a bit confused.

This is the fourth page of the section "A Mathematical Visit." We're in the middle of the narrator's visit to the mathematician Selath's house, where there are so many strange objects around. On Black Friday, we read about was how water was stored "in" a Klein bottle, but then two pages of this story were blocked by the weekend.

Now the eccentric mathematician is in the middle of explaining that his clock tells time in octal. After all, a shift of work is eight hours, so why not tell time in octal?

Technically, we're about a month away from our annual Calendar Reform post, but some Calendar Reformers do wish to fix the clock as well. Pappas tells us that the narrator's watch reads 5:30 while Selath's clock reads 21:30 -- so the minutes are still apparently in decimal.

[8] (default octal -- Do you remember these change of base signs I explained last summer?)

Or maybe there are only 60 octal minutes in a hour instead of the usual 74. It may seem more logical in an octal world to round 74 up to 100 minutes per hour, but let's think about it:

  • With 30 hours in a day, the length of an hour hasn't changed. This means that we can keep all the old time zones.
  • With 60 minutes in an hour, the minutes are now longer. But then :30 retains its meaning as the half-hour, while the quarter hours are now :14 and :44.
  • There now be 113 seconds in a minute. This number isn't round or convenient, but it allows the second, and all SI units dependent on the second, to retain its value. The hour and second keep their old lengths, with only the minute changing.
The following link gives the time in several different bases, including octal:

It is a pure octal clock, with 100000 "seconds" in a day, each nearly thrice as long as the SI second.

Let's continue on this page:

-- "Now let's go to the master bedroom," [said Selath.]
-- And off we went, passing all sorts of shapes and objects I'd never seen in a home before.
-- "The master bedroom has a small semi-spherical skylight in addition to...."

Well, we won't know what the skylight is in addition to until tomorrow, because this sentence ends on the next page. The only picture on this page is another picture of the interior of the house -- which includes the Klein bottle, but not the skylight in the bedroom.

Today is Cyber Monday. So of course, I had to order something on Amazon today -- and that something is the Pappas Mathematical Calendar for 2018. After she disappoints us in 2017, she does indeed return for 2018. This means that we won't read her Magic of Mathematics in 2018. (But I do notice, in browsing on Amazon, that she wrote Math-a-Day: A Book of Days for Your Mathematical Year, back in 1999. That's the book I probably should have used as a replacement for her calendar this year, instead of Magic. But of course I enjoyed her Magic this year anyway!)

Chapter 10 of George Szpiro's Poincare's Prize is called -- hold on, Chapter 10? Oops, I forgot:

[a] (default decimal)

Chapter 8 of George Szpiro's Poincare's Prize is called "Dead Ends and a Mysterious Disease." Here is how it begins:

"The first person to take a serious crack at the Poincare Conjecture was the Englishman John H. C. Whitehead, who usually went by his middle name Henry."

This chapter is about the earliest attempts to prove the famous conjecture. I've mentioned Henry's uncle Alfred North Whitehead in previous posts, most recently in my October 13th post, as Hoffman mentions him in connection with Erdos. (Oh, and even in decimal base, Lesson 6-6 of the U of Chicago text lines up with Chapter 8 of Szpiro rather than Chapter 6. We read two chapters over Thanksgiving in order to finish his fourteen-chapter book by early next week.)

But today we're reading about his nephew Henry. Szpiro writes that the young Henry -- like several other mathematicians we've seen so far -- is somewhat of a "dren":

"Somewhat careless in his work and not very good at mathematical manipulations, he nevertheless managed to pass the entrance examination to Eton, the most prestigious of England's boys' schools."

During World War II, Whitehead works for Alan Turing -- of Imitation Game fame. Let's pick up the story after the war:

"In 1947 he was named Waynflete Professor of Pure Mathematics at Oxford's Magdalen College. Upon the death of his mother in 1953, Whitehead inherited some cattle from her estate, and he and his wife [Barbara Sheila Carew Smyth] established Manor Farm in the village of Noke, eight kilometers north of Oxford."

Of course, the reason we read about Whitehead is that he works on the Poincare Conjecture:

"At that time it was just one of a host of open problems; nobody knew how fiendishly difficult the proof would be."

At first he believes that he has a solution, but of course he is mistaken:

"The sinking feeling one gets in one's stomach with the realization of a published error is not to be wished on anybody."

The reason his proof is invalid is that it doesn't apply to a certain manifold -- which is now known as the Whitehead manifold:

" proof of Poincare's Conjecture. End of story. So maybe the Whitehead manifold is a counterexample to the Poincare Conjecture. Unfortunately, it also falls short in this regard."

The next mathematician Szpiro mentions in this chapter is Christos Papakyriakopoulos -- known as Papa, for short. The author begins by describing Papa's entanglement in local Greek politics:

"Papa voted openly again the king's return, but the majority went the other way and the king returned to Athens."

And Papa must deal with the invaders during World War II -- and then with the Cold War. He joins a group of Communists as he works at the Institute of Technology in Athens:

"But the atmosphere at the institute was not friendly towards Communist sympathizers. When the professor for whom he had worked as an unpaid assistant was fired, he had to start looking for other pastures."

And those greener pastures are across the sea -- in the United States, at Princeton University:

"During his first ten years in America, Papa produced proofs to three important open problems: the Loop Theorem, Dehn's Lemma, and the Sphere Theorem."

Szpiro tells us that sadly, Papa never returns to his homeland, as he dies of stomach cancer at 62 (and the author adds that many topologists, including Whitehead and Poincare himself, die in their late fifties or early sixties.) He writes:

"The National Technical University in Athens established a prize in Papa's memory, to be awarded every year to an outstanding freshman in mathematics. It is a reflection of his solitude that throughout his career Papa never had even a single coauthor."

So in other words, Papa has an Erdos number of infinity. Now Szpiro returns to Papa's attempts to prove the Poincare Conjecture. The mathematician shows his paper to a young grad student, Bernard Maskit, and his thesis advisor Lipman Bers:

"The bright graduate student was successful, and Bers then told his son-in-law, Leon Ehrenpreis, at New York University, about this."

By now, you probably know the story -- Maskit finds an error in Papa's proof, but not until after the mathematician tries to publish it:

"Why did he not rectify his announcement of March 1962 to the Bulletin [of the American Mathematical Society] if he was already aware of the error in December 1961?"

Szpiro now moves on to Elvira Strasser-Rapaport, who is married to David Rapaport, a Hungarian mathematician and psychoanalyst:

"Strasser-Rapaport had come late to mathematics, obtaining her Ph.D. only at age forty-three after she had raised the couple's two daughters."

As it happens, Papa has reduced the Poincare Conjecture to two sub-conjectures -- prove them both, and Poincare itself is proved. And Strasser-Rapaport is able to prove one of the sub-conjectures. But as it turns out, the second sub-conjecture can be proved false:

"This is exactly James McCool from the University of Toronto did. His paper 'A counterexample to conjectures by Papakyriakopoulos and Swarup' was published in the Proceedings of the American Mathematical Society in 1981."

Szpiro's next subject is born in Oakwood, Texas -- RH Bing. And as the author points out, this is his actual first name:

"Some older readers may remember JR, the eternal villain in the TV soap opera Dallas. Apparently double initials often moonlight as given names in Texas."

Bing becomes a high school math teacher, but then attends UT Austin to earn his MA. He meets Robert L. Moore, who does not like him because he is slightly older than most grad students. As it happens, Moore does not like Jews, women, or northerners either:

"A black student who wanted to join his class recounted that Moore told him, 'Okay, but you start with a grade C and can only go down from there.'"

Oops -- I mentioned race in my second straight post. But I won't give this post the "traditionalists" label, because I'm jumping out of this chapter. To make a long story short, Bing tries to prove -- and later on, disprove -- the Poincare Conjecture, and he fails on both accounts.

Well, we've already seen many "dead ends" as mentioned in this chapter. But is the mysterious disease, suffered by Bing and perhaps other topologists? Is it the disease that causes several other topologists to die young?

"[German Wolfgang Haken] became so obsessed with the problem that he was said to suffer from Poincaritis, an affliction that befell many a good mathematician in the twentieth century."

Haken does ultimately prove the Four-Color Theorem (and in fact, his name is mentioned in the U of Chicago text, Lesson 9-8), but not the Poincare Conjecture.

Szpiro wraps up the chapter thusly:

"But in the meantime let us turn from this rather depressing state of affairs to some more positive developments."

Lesson 6-6 of the U of Chicago text is called "Isometries." In the modern Third Edition, we must backtrack to Lesson 4-7 to learn about isometries.

I didn't write much about Lesson 6-6 two years ago, since I used that day to write about Presidential Consistency and the Common Core. I choose not to repeat that discussion, and so instead I'll reblog what little I wrote about the final isometry -- the glide reflection.

What, exactly, is a glide reflection? Well, here's how the U of Chicago defines it:

Let r be the reflection in line m and T be any translation with nonzero magnitude and direction parallel to m. Then G, the composite of T and r, is a glide reflection.

Just as reflections, rotations, and translations have nicknames -- "flips," "slides," and "turns," respectively -- glide reflections have the nickname "walks." The U of Chicago gives the example of the isometry mapping the right footprint to the left footprint while walking as a glide reflection. Another name for glide reflection is "transflection," since it is the composite of a reflection and a translation.

I once tutored a geometry student who had a worksheet on glide reflections. The student had to use a coordinate plane to perform the glide reflections, which were given as the composite of a reflection and a translation. But the problem was that on the worksheet, the direction of the translation wasn't always parallel to the reflecting line! In fact, in one of the problems the translation was perpendicular to the reflecting line. That would mean that the resulting composite wasn't truly a glide reflection at all, but just a mere reflection!

Friday, November 24, 2017

Black Friday Post: More Songs on the Computer Emulator

Table of Contents:

1. Pappas and Szpiro
2. Fibonacci Music
3. Benchmark Test Song -- Conversion
4. Benchmark Test Song -- Coding
5. Packet Song -- Conversion
6. Packet Song -- Coding
7. Do-Ru-Ma -- Conversion
8. Do-Ru-Ma -- Coding
9. What About 11-/13-Limit Music?
10. Conclusion

Pappas and Szpiro

This is what Theoni Pappas writes on page 24 of her Magic of Mathematics:

"Not quite sure what to expect, I rang the doorbell. A voice asked me to please push the first five terms of the Fibonacci sequence."

This is the first page of the section "A Mathematical Visit." It's a story where the narrator meets a (fictional) renowned mathematician, Selath, and his unusual house.

Here are some excerpts from this story:

"I pushed 1, 1, 2, 3, 5 and the door slowly opened. As I passed through the doorway, I was struck by the catenary stone shaped archway independently suspended at the entrance. In the kitchen we came to a peculiar table with many legs. Selath pulled an equally unusual bottle from the refrigerator. 'As you noticed, this tabel and bottle are not your everyday accessories.'"

The only picture on this page is of Selath's unusual table and bottle.

Chapter 7 of George Szpiro's Poincare's Prize is called "What the Conjecture Is Really All About." I give its opening here:

"Toward the end of the nineteenth century the main thrust of the then-still-young discipline of topology was the classification of bodies and spaces."

As the title implies, in this chapter we finally get to see Poincare's Conjecture itself. Szpiro tells us that the conjecture is a bit tricky for laypeople to understand, and so he spends this chapter giving a few analogies (in lower dimensions) and simple explanations.

So we open the chapter by viewing topology as an attempt to classify different shapes. But how exactly can we do this? Szpiro writes:

"Clearly a single number, such as the number of the body's corners, would not do since a ball and a bagel have zero corners and would thus be classed in the same group even though they are topologically different, while a cube would be classed as different from a pyramid, even though one can be morphed into the other."

Poincare writes a paper about this, and here he makes a bold claim:

"In modern language, this statement would read, 'A three-dimensional manifold that has the same homology groups as the three-dimensional sphere is homeomorphic to it.'"

Szpiro ultimately rephrases this as, Any body that contains no holes and is not twisted can be morphed into a sphere -- though he warns us that the morphing of the 3D body could actually take place in 4D space. But the author compares Poincare's claim to that of Fermat when he wrote about his famous Last Theorem:

"Like Fermat's claim, Poincare's announcement was premature, Unlike Fermat's claim, it was wrong. The somewhat cocky Poincare apparently thought he had a proof in his pocket, but as on numerous occasions, he was mistaken."

His claim is false because he finds a counterexample -- but it's a 3D manifold that is twisted up so much that it can only exist in 4D. As Szpiro explains:

"Even the Klein bottle, which is, after all, nothing but a two-dimensional surface, must be embedded in four-dimensional space so that its features can be fully exhibited."

Hold on a minute here -- let's jump back to Pappas for a second. We've just read page 24, where we see something about a strange bottle on the mathematician Selath's desk. Pappas explains what the bottle is on page 25, but we're skipping it because it's blocked by the non-posting weekend. Well, let's end the suspense and find out what Selath's bottle really is:
"'The water container is known as a Klein bottle -- its inside and outside are one.'"

And there is a picture of a Klein bottle on page 25. According to Pappas, it's unusual in that its inside and outside are not distinct. And here Szpiro gives even more info about what it is -- it's simply a 2D manifold, but the twisting must occur in 4D in order to unite its inside and outside. A mere 3D twisting can't produce a Klein bottle.

Since you can't see the picture of the Klein bottle, let's link to one right here:

Apparently, you can actually order Klein bottles from the above website.

OK, let's get back to Szpiro and Poincare. Notice that the Klein bottle is not the counterexample to Poincare's claim above, since the Klein bottle is a 2D manifold and the claim is on 3D manifolds. At this point, the author continues to describe the difficulties in mentally picturing objects in 4D:

"Some people, such as the physicist Roger Penrose, claim they can think in four dimensions, but most people cannot. However, one can gain an understanding by appealing to analogy: The three-dimensional sphere is to the ball what the surface of the ball is to a disk."

Max Dehn, whom we mentioned in yesterday's post, tries to visualize our 3D counterexample in 4D:

"He suggested extracting a pretzel-like object from the three-dimensional sphere and sewing it back differently."

Another way to obtain this manifold, akin to fellow German Hellmuth Kneser, is to begin with a dodecahedron and glue two of its opposite faces together.

"Now do the same thing with the other five pairs of pentagons, and voila, you have the homology sphere."

Let's return now to Poincare's research:

"Having found a counterexample to his initial claim, Poincare realized that even a series of numbers, such as the torsion coefficients and the Betti numbers, would be unable to furnish a classification of spaces and bodies."

And in his next attempt, Poincare considers sliding rubber bands around a manifold. For example, a bagel might have a band around the outside and another band that loops through the hole:

"These two rubber bands can also be slid about the surface of the bagel, but they will never become aligned next to each other."

At this point, Szpiro suddenly starts writing about group theory. I've mentioned group theory in several posts here on the blog, most recently on October 30th when discussing John Conway, a group theorist who advocates the inclusive definition of trapezoid. (I've also mentioned group theory in connection with other mathematicians like Evariste Galois and Eugenia Cheng.) Szpiro reminds us:

"But not every collection of objects together with an operation represents a mathematical group. For a collection of objects together with an operation to qualify as a group, four conditions, let's call them the groupie requirements, must be satisfied."

Since we've discussed group theory in previous posts, let's just skip to statement of the famous Poincare conjecture:

"Is it possible that the fundamental group of a manifold be trivial and yet the manifold not be homeomorphic to the sphere?"

Poincare's conjecture is that the answer is "no." Here "the fundamental group" refers to the group of all "rubber bands," and this group is trivial if all bands can shrink to the sphere. As we've seen earlier, the fundamental group of a bagel is not trivial. And here "sphere" refers to the 3D sphere, which is a manifold in 4D space.

Of course, Poincare dies without ever proving his conjecture:

"All that was now needed was a proof. And for a hundred years, mathematicians from all over the world searched...."

Fibonacci Music

OK, let's get back to music. The Fibonacci sequence is on my mind today since it's the code to enter Selath's house. So we will start by composing music based on this sequence.

We begin with the 3-limit Waldorf scale. Here's a link to the emulator again:

Type in the following program:

10 DIM S(6)
20 FOR X=0 TO 6
30 READ S(X)
50 DATA 72,64,54,48
60 DATA 108,96,81
70 N=1
80 A=1
90 B=1
100 SOUND 261-N*S(A),4
110 T=A+B
120 IF T>6 THEN T=T-7
130 X=Y
140 Y=T
150 GOTO 100

Remember to click the "Sound" box before typing RUN.

Just like the previous versions of the song, this program reduces the Fibonacci numbers mod 7 and then maps the numbers to the scale, except this time it's the Waldorf scale. Notice that even though there are DATA lines to set up the scale, the song itself doesn't use DATA. Instead, the computer actually calculates the Fibonacci numbers. We see how line 110 adds up two numbers, and line 120 performs the reduction mod 7.

As we know, the Fibonacci numbers mod 7 repeat every 16 terms:


Notice how when we write the numbers in two rows of eight, the numbers in each column add up to seven, except the two zeros in the last column. Since the Waldorf scale is symmetrical around its center note (A in the original scale, D on the emulator), we take advantage of this by assigning 0 to the center note:

4- 5- 6- 0- 1- 2- 3

Then the second half of the song is the mirror image of the first half. This actually reminds me of Lesson 4-8 of the U of Chicago text, Third Edition. This lesson is called "Transformations in Music," which demonstrates how the Common Core transformations can be applied to music. The text refers to this "horizontal reflection" of sorts as an inversion.

It can be proved that in any Fibonacci song (that is, where the numbers are reduced mod m for any m and mapped to notes), the second half is an inversion of the first half, with zero as the line of symmetry. Since songs in the Waldorf scale are supposed to be symmetrical around its central note, we assign zero to it and there we have it -- a true Waldorf-style song:


This program has an infinite loop, so try pressing "Esc" (for "Break") to end the program. As usual, we can switch octaves:

70 N=2

Let's try playing Fibonacci in the major and minor scales. For this, we'll use all ten notes, 1-9 and 0 (to represent 10). To make it easier, change only the following lines:

10 DIM S(9)
20 FOR X=0 TO 9
50 DATA 72,180,160,144,135
60 DATA 120,108,96,90,80
70 N=1
120 IF T>9 THEN T=T-10

Here Note 0 is a perfect tenth, which is D (in the Bb+ major scale). Because the song is symmetrical in Notes 1-9, it might be good to make Note 0 a rest (as some Pi Day musicians do). Then Note 5 also becomes a line of symmetry:

145 IF A=0 THEN FOR I=1 TO 460:NEXT I:GOTO 110

This line, numbered 145 so we can enter it before line 150, starts a small loop where the computer counts from 1 to 460. This takes about one second (or one beat) to accomplish, and so it serves as a quarter rest. As it turns out, the Fibonacci numbers mod 10 take longer to repeat. It takes 60 terms -- in other words, the Pisano period mod 10 is 60.

For the minor scale, make the following changes to the major song:

50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32

Again, the minor scale can be played in one of three octaves: N=1 (high D minor), N=2 (low D minor), and N=3 (low G minor).

Finally, let's play it in the New 7-Limit Scale:

50 DATA 54,105,96,90,84
60 DATA 81,72,70,63,60

There are two possible octaves, N=1 and N=2.

For this scale, I don't like simply making 0 a rest, since we've chosen a special high G note for 0. So instead, I want to erase only the 60th note -- the final 0 before it repeats 0-1-1. This avoids the dissonant semidiminished octave between Notes 0 and 1. Meanwhile, I also want to avoid the sequence 1-6-7 (which does occur in Fibonacci since 1+6=7). This sequence has a dissonant semidiminished fifth between Notes 1 and 6. Indeed, when I hear this sequence, my ear mistakes 1-6 for a perfect fifth, and then when the true perfect fifth 7 is played, it sounds out of place:

145 IF (A=0 OR A=6) AND B=A+1 THEN FOR I=1 TO 460:NEXT I:GOTO 110

This means that if the current note (variable A) is Note 0 or 6 and the next note (variable B) is one more than the current note (that is, Note 1 or 7), then play the quarter rest instead. It turns out that the sequence 1-rest-7 occurs just a few notes before the 1-rest-1 sequence.

By the way, there's another way to refer to the two G's besides G and G+7. The first G is called "white G," while G+7 is called "greenish" G. This is Kite's color notation:

The New 7-Limit Scale, starting on white G:

Key     Sound     Degree     Note     Ratio     Color
0.         45           216          G          1/1         white G
1.         51           210          G+7      36/35     greenish G
2.         69           192          A           9/8        white A
3.         81           180          Bb+      6/5         green Bb
4.         93           168          B7         9/7        red B
5.         99           162          C           4/3        white C
6.         117         144          D           3/2        white D
7.         121         140          D+7      54/35     greenish D
8.         135         126          E7         12/7      red E
9.         141         120          F+         9/5        green F
0.         153         108          G          2/1        white G

In this scale, "white" means Pythagorean, "green" means raised a syntonic comma, and "red" means raised a septimal comma. "Green" and "red" combine to form "greenish." According to Kite, there are also colors "blue" and "yellow," but these are otonal colors -- and as we already know, the EDL computer scale is utonal. So we're stuck with "green" and "red." (How festive -- today is Black Friday which leads to the Christmas season.)

Of course, it's possible to combine green and red in different ways. "Deep" means that there are two factors of the given prime, 5 or 7:

Sound     Degree     Note     Color
36           225           Gb++   deep green Gb
65           196           G#77   deep red G#
86           175           Bb++7 greenish green Bb
16           245           E+77    greenish red E
11           250           Fb+++  triple green Fb

So we can restate our rule to avoid dissonant fifths as, "Avoid playing a white G or D adjacent to a greenish G or D." We also want to avoid dissonant fourths, so we could also say "Avoid playing a greenish G adjacent to white C, or white A adjacent to greenish D." Our Fibonacci song still has a few dissonant fourths, but these are harder to avoid, and so I kept them in the song. On the other hand, all seconds are melodically consonant, and so "greenish G to white A" and "white C to greenish D" don't need to be avoided.

Benchmark Test Song -- Conversion

For the rest of this post, I wish to convert some of my old songs from last year to the new scales. As it is still November, I choose to convert the songs I sang last November (my original songs, not the ones that come from Square One TV).

At the end of the first trimester, I sang the Benchmark Test song:

Benchmark Tests -- by Mr. Walker

Verse 2:
Why do we take Benchmark Tests?
The first trimester is done so let's
See how much we know, know know!

It's some new stuff on Benchmark Tests.
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!

The teacher sees our Benchmark Tests,
Knows what to teach more or less.
That's the way to go, go, go!

When I first created this song, it was in the C major scale, excluding the note B. After I sang it a few times, I instinctively dropped the note F as well, to make the song pentatonic. This suggests that the Waldorf 3-limit scale is a good scale for this song.

Benchmark Test Song -- Coding

Let's start a NEW program:

10 DIM S(7)
15 FOR V=1 TO 3
20 FOR X=1 TO 7
30 READ S(X)
50 DATA 108,96,81
60 DATA 72,64,54,48
70 N=1
80 FOR X=1 TO 4
90 FOR Y=1 TO 7
100 READ A
110 SOUND 261-N*S(A),4
120 NEXT Y
130 FOR I=1 TO 460
140 NEXT I,X
160 DATA 3,4,3,4,3,3,3
170 DATA 4,7,3,7,6,5,4
180 DATA 5,4,5,4,5,5,5
190 DATA 4,1,5,1,2,3,4
200 NEXT V

In this song, I return to labeling the notes 1-7, with the center D as Note 4 rather than Note 0. But the song still follows the Waldorf inversion pattern. Because of this, it's better to repeat lines, as follows:

Why do we take Benchmark Tests?
See how much we know, know know!
The first trimester is done so let's
See how much we know, know know!

It's some new stuff on Benchmark Tests.
'Cause there's still time to grow, grow, grow!
If we don't know it, we take a guess.
'Cause there's still time to grow, grow, grow!

The teacher sees our Benchmark Tests,
That's the way to go, go, go!
Knows what to teach more or less.
That's the way to go, go, go!

Each line is played as seven quarter notes followed by a quarter rest. Each stanza consists of four lines, and then the song repeats for all three stanzas. To accomplish this, we insert a FOR V loop from line 15 to line 200. The DATA lines don't automatically repeat (not even if we place them inside a FOR loop), and so we must RESTORE the data. But RESTORE automatically restores all the data, including lines 50-60 which are used to set up the scale. So we must repeat those lines -- but we can't repeat the DIM line, which is why the outer loop begins at line 15.

Packet Song -- Conversion

The next song we will convert is the Packet Song. Right after Thanksgiving break, the other middle school teachers and I came up with the idea of including all homework and classwork for the week in a packet. This idea fell apart once the administrators insisted that I assign Illinois State homework on the computer instead.

Nonetheless, here is the song:


1st Verse:
Your folks came up to us and said,
"Hey teachers, you can't hack it!
You make the claim you give our kids
Much work, but you can't back it."
So we got together and said,
"Let's stop all this racket!
We'll staple all our good worksheets
To make a ten-page packet."

2nd Verse:
Don't leave your packet behind,
Make sure that you backpack it.
Or if it starts to rain then
Hide it underneath your jacket.
Keep it in a folder so that
You can always track it,
Make sure that you never ever
Leave your ten-page packet.

3rd Verse:
When it's homework time then
Take it out and just attack it.
When there's extra time in class
Then you need to unpack it.
When it's time to turn it in
Make sure that you don't lack it.
To each and every teacher just
Turn in your ten-page packet.

When I first came up with this song, I played it in G major, just like many of my other songs. But in hindsight, I think it's better to play the song in a minor key. Minor keys express sadness -- and of course students are sad that they are assigned so much work in a packet.

So far in my class, the only original songs I played in minor keys are the Mousetrap Song and the first day of school Dren Song. I also wrote the Dren Song so that the second half is an inversion of the first half.

And so let's use our 5-limit just minor scale to create the song.

Packet Song -- Coding

This is a NEW program:

10 DIM S(9)
15 FOR V=1 TO 3
20 FOR X=0 TO 9
30 READ S(X)
50 DATA 30,72,64,60,54
60 DATA 48,45,40,36,32
70 N=1
80 FOR X=1 TO 4
90 FOR Y=1 TO 12
100 READ A
110 SOUND 261-N*S(A),2
120 NEXT Y
130 READ A
140 SOUND 261-N*S(A),4
150 FOR I=1 TO 460
160 NEXT I,X
180 DATA 4,1,4,5,1,1,2,1,3,5,7,8,7
190 DATA 3,3,4,1,5,8,4,7,3,4,7,6,4
200 DATA 4,1,4,5,1,1,2,1,3,5,7,8,7
210 DATA 5,6,3,5,2,3,5,6,6,3,1,3,1
220 NEXT V

In this song, most notes are eighth notes. A longer quarter note appears at the end of every other line ("packet" and its rhymes).

Do-Ru-Ma -- Conversion

There was one more song that I tried to sing last November. It was a special election song, for the day of the presidential election:


If I could, I surely would,
Vote in this election.
Make my choice for president,
Choose the leaders of our land.
Have you registered to vote? No!
Then let's register today.
Soon we'll finally have our say.
So let's all go out and vote, vote, vote!

This is actually a parody of the song "Do-Re-Mi" from Sound of Music. This isn't an original song, but it might be interesting to convert "Do-Re-Mi" to the New 7-Limit Scale.

Hey, that reminds me -- "Do-Re-Mi" is just solfege, and we need to come up with solfege for the New 7-Limit Scale.

Musician Andrew Heathwaite has already invented solfege for many alternate scales. It's helpful to look at EDO scales which represent the 7-limit well, such as 19EDO, or even better, 31EDO:

Heathwaite explains that some of the note names are "grandfathered in" -- that is, they are the old names from the just major scale, such as "do," "re," and "mi." The other names start with the same letters as the grandfathered names but change the vowels. Since the major second "re" starts with R, all seconds start with R. Likewise all thirds start with M, and so on:

Key     Sound     Degree     Note     Ratio     Solfege     Interval (from root note)
1.         51           210          G+7      1/1         do             tonic
2.         69           192          A           35/32     ru             neutral tone
3.         81           180          Bb+       7/6         ma           subminor third
4.         93           168          B7         5/4         mi            major third
5.         99           162          C           35/27     fe             semidiminished fourth
6.         117         144          D           35/24     su             semidiminished fifth
7.         121         140          D+7       3/2         sol           perfect fifth
8.         135         126          E7          5/3         la             major sixth
9.         141         120          F+          7/4         ta             harmonic seventh
0.         153         108          G           35/18      da            semidiminished octave

The last note can't be "do" because that's reserved for the perfect octave. Instead, Heathwaite uses "da" to represent a slightly flattened octave. (If we use the Extended 7-Limit Scale, then Eb+ becomes "lo" and F#7 is a just major seventh, hence it's grandfathered "ti.")

It's also possible to begin the scale on white G. Then greenish G becomes "di," officially a slightly sharpened unison:

The New 7-Limit Scale, starting on white G:

Key     Sound     Degree     Note     Ratio     Solfege     Interval (from root note G)
0.         45           216          G          1/1         do             tonic
1.         51           210          G+7      36/35     di             quarter tone
2.         69           192          A           9/8        re              major tone
3.         81           180          Bb+      6/5         me            minor third
4.         93           168          B7         9/7        mo            supermajor third
5.         99           162          C           4/3        fa              perfect fourth
6.         117         144          D           3/2        sol            perfect fifth
7.         121         140          D+7      54/35     si              semiaugmented fifth
8.         135         126          E7         12/7      li               supermajor sixth
9.         141         120          F+         9/5        te               minor seventh
0.         153         108          G          2/1        do              perfect octave

(In the Extended Scale, Eb+ is "le" and F#7 is "to.") But for our song, we will start it on greenish G, and so we start on "do" and finish on "da."

OK, let's plan the lyrics now for our new song, which will be called "Do-Ru-Ma," the first three notes of the New 7-Limit Scale. We'll keep the original song lines for grandfathered notes:


Do, a deer, a female deer,
Ru, ____________________
Ma, ____________________
Mi, a name I call myself,
Fe, ____________________
Su, ____________________
Sol, a needle pulling thread,
La, a note to follow so,
Ta, ____________________
That will take us up to da, da, da!

OK, we want to enter lines that rhyme -- although not necessarily the rhyme scheme of the original song (as there are now ten lines instead of eight). First of all, the last three notes la, ta, da already rhyme, and so "Ta, a note to follow la," will allow the last two lines to rhyme.

For "fe," I was thinking about fae, or fairies. The line "Fe, a flying little elf," fits, and notice that "elf" happens to rhyme with "myself" in the "mi" line. So now we have:

Do, a deer, a female deer,
Ru, ____________________
Ma, ____________________
Mi, a name I call myself,
Fe, a flying little elf
Su, ____________________
Sol, a needle pulling thread,
La, a note to follow so,
Ta, a note to follow la,
That will take us up to da, da, da!

I'm inspired by the rhyme scheme of yesterday's mouse poem to rhyme the third and sixth lines. For "ma," this sounds like a mother, while "sue" is what you do in court. Let's see:

Do, a deer, a female deer,
Ru, the day that I will fear,
Ma, to mom the babies say.
Mi, a name I call myself,
Fe, a flying little elf
Su, the way I make you pay,
Sol, a needle pulling thread,
La, a note to follow so,
Ta, a note to follow la,
Take us up to da, da, da!

Here "ru" refers to "rue" -- as in "rue the day" that something terrible happens.

Do-Ru-Ma -- Coding

This is a NEW program:

10 DIM S(9)
15 FOR V=1 TO 2
20 FOR X=0 TO 9
30 READ S(X)
50 DATA 54,105,96,90,84
60 DATA 81,72,70,63,60
70 N=1
80 FOR X=1 TO 70
90 SOUND 261-N*S(A),T
110 DATA 1,6,2,2,3,6,1,2,3,4,1,4,3,8
120 DATA 2,6,3,2,4,6,2,2,4,4,2,4,4,8
130 DATA 3,6,4,2,5,2,5,2,4,2,3,2,5,16
140 DATA 4,6,5,2,6,6,4,2,6,4,4,4,6,8
150 DATA 5,6,6,2,7,6,5,2,7,4,5,4,7,8
160 DATA 6,6,7,2,8,2,8,2,7,2,6,2,8,16
170 DATA 7,6,3,2,4,2,5,2,6,2,7,2,8,16
180 DATA 8,6,4,2,5,2,6,2,7,2,8,2,9,16
190 DATA 9,6,5,2,6,2,7,2,8,2,9,2,0,16
200 DATA 0,4,8,4,9,4,7,4,0,4,4,4,1,8
210 NEXT V

Yes, I know this song, but I hope it's worth it.

What About 11-/13-Limit Music?

So far, we haven't written much about the 11- and 13-limits. But some musicians do like to use the 11th and 13th harmonic in music:

  • The undecimal quarter tone is 33/32. When we lower Degree 32 (Sound 229) E by this quarter tone, we obtain Degree 33, Ed (where "d" looks like a half-flat symbol). In Kite's color notation, this is called "amber E."
  • The tridecimal third tone is 27/26. When we raise Degree 27 (Sound 234) G by this third tone, we obtain Degree 26, Gh ("h" is as close I can get in ASCII to this symbol). In Kite's color notation, this is called "ocher G." (Why is it "ocher" and not "orange"? Ask Kite!)
We can keep going with higher and higher primes -- 17, 19, and so on. The largest prime we can play on the computer is the Fermat prime 257. Degree 257 (Sound 4) is E played a small subcomma, 257/256, below white E.

I may consider playing some 11- and 13-limit music. I might consider converting the Dren Song, which is already an inversion song, to 13-limit.


There's so much more I can do with the New 7-Limit scale. One thing about the Extended 7-Limit Scale with Eb+ and F#7 is that these two notes are very close to a just 6/5 minor third apart. (The difference is called a "subcomma," or 225/224.) This allows us to play part of the Extended Scale as a pseudo-harmonic minor scale on red E (where Eb+ sounds like the leading tone D#.)

As Christmas is coming, some holiday songs can be converted to the 7-Limit Scale, especially songs where the song is in G major, but without F# (unless we use the Extended Scale with F#7). For example, on the radio I often here Mannheim Steamroller's Deck the Halls played in the key of F, except sometimes Eb is played instead of E. This version may be converted to the 7-limit scale.

Another possible holiday song to convert is Rise Up Shepherd and Follow, which is lesser known but appears in my Christmas music book. The melody of this song is fully in F Mixolydian, where the note Eb always appears instead of E. Notice that this song is an African-American bluesy spiritual song, and bluesy songs often employ the septimal intervals found in the New 7-Limit Scale.

Oops -- I mentioned race in this post, but hey, this is the bottom of a lengthy holiday post again. Let's put up the "traditionalists" label on this post just to be safe.

And as this is a traditionalists post, let me point out that traditionalist Barry Garelick mentions race in one of his recent posts as well:

Yes, God forbid we should study one subject at a time so we can eventually apply it to other disciplines. Just meld it into one big coloring book activity for teachers to facilitate. And of course, it is understood that mathematics is just white privilege but I’m stepping into other territory so I guess it’s time to stop.

This concludes my post. We'll return to our regular posting schedule on Monday.