*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.)

DATE VARIATION

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:

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

"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

"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

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

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.

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.