In this chapter, Rucker describes the idea of time as the fourth dimension. This idea is a cornerstone of Einstein's Theory of Reality -- instead of the three dimensions of space, we must consider the four dimensions of spacetime, He writes:
Later. Do you hate time? Alarm clocks, sure. Changing the clocks for daylight-saving time [as we just did last week -- dw] is the worst. How can they just take away an hour like that? Remember in 1973 when Nixon took away two hours for the oil companies?
"The older I get, the faster time goes," my [Rucker's -- dw] mother told me."The years just fly by. Every time I turn around, it's Christmas or Thanksgiving."
Let's look at the four puzzles of this chapter:
If we say that the fourth dimension is time, then it is possible to construct a hypersphere in space and time. How?
Take a small spherical balloon. Blow it up and then let the air out. The entire spacetime trail of the balloon's surface and inside is a solid hypersphere. The trail of the surface alone is the hypersurface of the hypersphere.
Commentary: We may draw the temporal definition as if it were a spatial dimension. The resulting diagram is called a Minkowski diagram.
What kind of ideas about the past and future are embodied in this picture, where one thinks of the spacetime solid like a block of ice that melts from the bottom up?
"The melting future" world view corresponds to the notion that future events exist, stored up and waiting for us. A uniform "now" moves forward with the passage of time, and instant after instant is permanently used up. In this viewpoint, past events are totally nonexistent. It is not uncommon for people to feel this way about their lives. Life here becomes a scarce resource that is consumed, and once something is over it doesn't matter at all. This is probably the least rewarding way possible to think about spacetime, as can be seen by thinking about the kind of personal philosophies inherent in the other three world views shown in the figure [only now "exists," only the past and now "exist," and the past, now, and future all "exist" -- dw]. Whenever you cut yourself off from your past, you're in an extremely rootless and vulnerable position. But if you do throw out the past, you might as well throw out the future too, and get totally into the "now."
Commentary: We really get deep into Einstein's Theory of Relativity in this puzzle:
"The relativity of simultaneity" says that differing moving observers will have different opinions about which events are simultaneous. In this problem, we will see how the relativity of simultaneity follows from the two basic assumptions: (1) that moving observers are free to think of themselves as being at rest, and (2) that light always travels at the same speed.
The situation is as follows. A rigid platform is moving to the right at about half the speed of light. On the left end stands Mr. Willy Lee, and on the right end stands Mr. Rye. Mr. Lee sends a flash of light down the platform toward Mr. Rye. Mr. Rye holds a mirror that bounces the light flash back toward Mr. Lee. Mr. Lee receives the return signal. Call these events A, B, and C, respectively. Mr. Lee notes the times of events A and C on his world line. After a little thought he decides which event X on his world line is simultaneous with B. Where does he put X, and why? (Hint: We would place X horizontally from B, but Mr. Lee will not. Simultaneity is relative!)
Mr. Lee will put X halfway between A and C. The reason is that Lee will assume that it takes the light just as long to travel from the other end of the platform as it took it to get there from his end. It is natural for him to think this, in view of the assumptions (1) and (2) mentioned in the puzzle. We, of course, feel that it really takes the light longer to get from A to B than it takes it to get from B to C ... But Mr. Lee will say that we just think that because we're racing past him at half the speed of light!
Commentary: This next puzzle brings up the idea that space is cyclical -- that is, just a circle:
In the figure, we drew a picture [a cylinder -- dw] of a circular space that remains the same size as time goes on. A widely held present-day view of the universe is that our space is an expanding hypersphere, which started out as point-sized about twelve billion years ago. Can you draw a picture of spacetime that represents our space as an expanding circle?
The picture would be a sort of "conical" spacetime, as drawn here. The starting point is known as "the initial singularity," or as the Big Bang. Whether or not our space will eventually contract back to a point is unknown at present. Apparently it depends on how much mass is actually in our universe: if there is enough mass, then the gravitational forces will pull things back together.
Commentary: The idea that gravity will pull things back together is called the "Big Crunch." It is often believed that a Big Crunch will be followed by another Big Bang, and Rucker mentions the idea that time is cyclical -- that is, just a circle. This idea appears in yet another Futurama episode -- "The Late Philip J. Fry."
Let's see -- we haven't had our traditionalist topic this week yet. Once again, today's post is so long that I don't want to make it even longer by having our Andrew Hacker discussion.
Instead, I'll mention that the traditionalist Dr. Barry Garelick is in the news, because he has just published a book, Math Education in the U.S.: Still Crazy After All These Years.
I'll just say that no, Garelick's book won't be the next one I discuss here on the blog after I finish Rucker's book. We can already figure out what Garelick writes in his book -- math education is "crazy" and will remain so until the pedagogy favored by traditionalists is adopted. Again, I repeat that I agree with Garelick and the traditionalists regarding math in the lower grades, but not in the higher grades.
(Actually, come to think of it, perhaps it would be a good idea to purchase Garelick's book and discuss it on the blog, since I devote so many blog posts to traditionalists anyway! Well, in that case, maybe I will buy his book someday.)
I guess it's impossible for me to mention Garelick without bringing up Dr. Katharine Beals, right?
I decided to link to Beals this week because technically she is posting a Geometry problem. Lately she has been focusing on making word problems less wordy. This problem is a classic:
Imagine a rope that runs completely around the Earth’s equator, flat against the ground (assume the Earth is a perfect sphere, without any mountains or valleys). You cut the rope and tie in another piece of rope that is 710 inches long, or just under 60 feet. That increases the total length of the rope by a bit more than the length of a bus, or the height of a 5-story building. Now imagine that the rope is lifted at all points simultaneously, so that it floats above the Earth at the same height all along its length. What is the largest thing that could fit underneath the rope: bacteria, a ladybug, a dog, Einstein, a giraffe, or a space shuttle?
[121 words]Here's a rewrite of the same problem, shortened by about 50%:
Assume the Earth is a perfect sphere and imagine a rope running tightly around the equator. Suppose the rope is lengthened by 710 inches and made to float above the Earth at a uniform height all around. What is the largest thing that could fit underneath: a microbe, a bug, a dog, a person, a giraffe, or a space shuttle?
By the way, the answer is that it's about 710/(2pi) inches, or about 113 inches or 9'5". (Note that 355/113 is a great approximation of pi, which is where the number 710 = 2 * 355 comes from.) So the correct answer is a person (Einstein -- let's keep his name since Rucker talks about him in the chapter that we are now discussing).
This is what I wrote last year about today's lesson:
Today is Good Friday, two days before Easter Sunday. As many readers are aware, I'm often fascinated by calendars, and there's no holiday commonly observed in this country that's more fascinating than Easter. Many people wonder why Easter is late in some years and early in others. Next year, Easter will be very late -- April 16th, 2017 -- while this year the holiday is very early -- March 27th, 2016. It's because of this wide variability that many schools have abandoned tying spring break to Easter. As I mentioned before, early Easters interfere with end-of-quarter exams, while late Easters interfere with the AP exam.
The following link discusses why the Easter date changes so much:
So we see that the Christian Easter is tied to the Jewish Passover. We've mentioned before that the Hebrew calendar is a lunisolar calendar -- that is, it's tied to both the sun and the moon. And we pointed out during our description of the Chinese calendar (another lunisolar calendar) that the phases of the moon and the seasons of the year don't line up exactly. The solar year cannot be divided evenly into lunar months.
So we infer that the Christians wanted an Easter date that is similar to -- yet independent of -- the Hebrew calculation. The date of the full moon was determined by looking it up on a table, rather than depend on the date of Passover. And so this complicated rule of determining Easter was devised.
The link gives a table of the earliest and latest Easters. The earliest Easters between the years 1753 and 2400 according to the table are:
March 22nd: 1761, 1818, 2285, 2353
March 23rd: 1788, 1845, 1856, 1913, 2008, 2160, 2228, 2380
The early Easter of 2008 is still fresh in my memory. A school that took off the week before Easter had March 14th as the last day before spring break, and school resumed on the 24th. But as early as that Easter was, we see that the earliest possible holiday is one day earlier. But Easter hasn't fallen on that date since 1818 -- long before any of us here were born -- and it won't fall on that day again until 2285 -- long after all of us here are dead. What makes March 22nd Easters so rare?
The problem is that there's only one way for Easter to fall on March 22nd -- and that's for there to be a full moon on Saturday, March 21st. If the full moon were a day later, on Sunday, March 22nd, then Easter wouldn't be until the 29th -- since, as the link points out, Easter is delayed by one week if the full moon is on Sunday. And if the full moon were a day earlier, on Friday, March 20th, then Easter wouldn't be until April 19th. This is because, as the link points out, March 21st is considered to be the ecclesiastical first day of spring. So March 20th would still be considered winter, and winter full moons don't count -- only spring full moons do. So we'd have to wait until Saturday, April 18th, the latest possible Paschal Full Moon, which would make the next day Easter.
So we see that if March 22nd were Easter, then the full moon must be exactly March 21st. But surely we shouldn't have to wait nearly 500 years (from 1818 to 2285) for March 21st to be the full moon!
The problem is that these full moons are determined by a table and aren't the dates of the actual full moon (unlike the Chinese calendar, which is based on astronomical dates). Now this table repeats every 19 years -- recall my mention of the Metonic 19-year cycle in earlier posts. So of the 29 dates from March 21st to April 18th, only 19 of those dates are found in the table. The Metonic cycle is not exact, and so the tables are adjusted every century.
What this means is that, in a given century, only 19 of the 29 dates from March 21st to April 18th can be possible full moon dates. If, in a given century, March 21st isn't one of the 19 chosen full moon dates, then March 22nd can't be Easter, since the 22nd isn't Easter unless the 21st is the full moon. As it turns out, the 20th, 21st, and 22nd centuries are all centuries for which March 21st isn't one of the 19 chosen dates. So March 22nd can't be Easter in any of them. And so 19th century was the last time that March 21st was the full moon, and it won't be full moon again that date until the 23rd century.
Now we look at the latest possible Easters:
April 24th: 1791, 1859, 2011, 2095, 2163, 2231, 2383
April 25th: 1886, 1943, 2038
Of course, the late Easter of 2011 is still in recent memory. A school that took off the week after Easter had April 22nd as the last day before spring break, and school resumed on May 2nd. But as late as that Easter was, we see that the latest possible holiday is one day later. This time, Easter will fall on April 25th, 2038. It's possible for me to be alive on that day, as I would be 57 years old -- as opposed to 2285 when I'd be over 300 years old.
Notice that there's also a column at the above link for "Julian calendar." Recall that our current calendar, the Gregorian calendar, was named after a pope -- so why would Eastern Orthodox Christians follow a pope's calendar? To this day, they still follow the predecessor calendar, the Julian calendar, as I explained back in my New Year's Eve post.
This has two affects on the Easter date. First, the equinox date of March 21st on the Julian calendar is actually what the Gregorian calendar calls April 3rd. So if the full moon is too early, it would still be considered winter on the Julian calendar, and Easter must wait until the next full moon. The other effect is that the full moon dates are based on the old table that isn't adjusted every century. So the Julian full moons are themselves later than the Gregorian. In 2015, the full moon on the Gregorian calendar was on April 4th, but the Julian full moon is a few days later. So the Julians must wait an extra week, until April 12th, to celebrate Easter. This year in 2016, the Gregorians celebrate Easter on March 27th, but to the Julians this is still winter. So they must wait an extra month to celebrate Easter, and so they don't celebrate Easter until May 1st.
That's right -- Easter can fall in May on the Julian calendar. According to the link, the earliest Easter in the range of the chart was April 3rd, 1763. (Notice that 1753 was the year that the British converted from the Julian to the Gregorian calendar, which is why the chart begins in 1753.) In recent times, April 4th, 2010 is an early Easter. The latest Julian Easter during the 21st century will be on May 8th, 2078 -- which is Mother's Day in the USA!
In some years, both calendars have the same Easter date. Both calendars agree that April 16th, 2017 will be Easter Sunday. The above link mentions that some people want to reconcile the two Easters by using an astronomical rule, just like the Chinese calendar. In this calendar, not only are March 22nd Easters slightly more likely, but even March 21st becomes a possible Easter date. The following link (near the bottom of the page, with even more discussion on how to calculate Easter throughout the rest of the page) claims that with an astronomical calculation, March 21st, 2877 will be Easter:
OK, that's enough about the Easter date. Let's get on with the lesson. Lesson 10-9 of the U of Chicago text is on the surface area of a sphere. This is the final formula of Chapter 10, and the only one that doesn't appear in any of our Pre-algebra texts.
Now today is an activity day. Considering how we found the surface area of a cone earlier this week -- by cutting out a circle sector that can be wrapped around the lateral area of the cone -- it would be elegant if we could take four great circles, each with area pi * r^2, and cover the surface area of the circle with these four circles.
But this is impossible. We can't cut out any figures on a flat paper and expect them to cover the surface area of a sphere accurately. This is known as the Mapmaker's Dilemma -- it's impossible to cover areas on the surface of the spherical earth on a flat 2-D map. The Mapmaker's Dilemma implies that any trick of using the areas of flat figures to find the surface area of a sphere is doomed.
Here's my idea of an activity: we take the idea of dividing the surface of a sphere into figures that are nearly polygons and run with it. Now Dr. M divides his surface into triangles, but here we will use square Post-it notes instead. After all, we measure areas in square units, not "triangular units." The task directs students to estimate how many Post-it notes it takes to cover the surface of a sphere before they actually try it.
(Note: I did make one change to this activity from last year. I changed the order of the lessons so that the students haven't learned the volume of a sphere yet. Students should estimate the volume of the sphere by multiplying the number of Post-its by the area of a Post-it, so they don't need to know the volume of anything to complete the activity.)
I have decided to name this activity "Spring Spheres." The name actually refers to an an incident a few years ago (it was 2011 -- the year when Easter was very late) where a volunteer in a classroom was not allowed to bring Easter eggs to school because they were religious. So she decided to bring the students "spring spheres" instead. Here I twist the use of that name around -- it's springtime and we're finding the surface area of a sphere -- hence the name "Spring Spheres."
This activity is long and requires that there are several balls in the classroom -- and even if we divide the class into groups and ask some students to bring balls to school, they may simply play with the balls rather than complete the activity. So here are some other activities that I am posting today:
-- From the U of Chicago text: calculate the surface area of the earth. Then compare the area of the United States and other countries to that of the entire earth.
-- Let's balance out "Spring Spheres" with a question about Easter -- specifically the Easter date. Even though Easter is determined by a table, the table can be calculated using a formula. The following link gives a link to what is known as the Conway Doomsday Algorithm -- and that's Conway as in John Horton Conway, the mathematician who also argued for the inclusive definition of trapezoid. In fact, Doomsday is used to determine the day of the week -- and that's part of calculating Easter, since we need to know when Sunday is. The link also describes how to calculate the Jewish holidays of Passover and Rosh Hashanah, but these are more complicated than calculating Easter. A neat trick is to verify that the Easter calculation works this year, then calculate when it falls next year. Notice that this is a math lesson, but if your school is similar to the Washington state school where Easter eggs have to be called "Spring Spheres," then just stick to the Spring Spheres lesson in the first place.
(Remember that I posted about Conway's Doomsday Algorithm back on Leap Day.) By the way, according to the link, we read:
If you pay attention to the dates of Easter and Passover from year to year, you will notice that although they usually fall within a week or so of each other, on occasion Passover falls about a month after (Gregorian) Easter. At the present time, this happens in in the 3rd, 11th, and 14th years of the Metonoic Cycle (i.e., when the Golden Number equals 3, 11, or 14). The reason for this discrepancy is the fact that although the Metonic Cycle is very good, it is not perfect (as we've seen in this course). In particular, it is a little off if you use it to predict the length of the tropical year. So, over the centuries the date of the vernal equinox, as predicted by the Metonic Cycle, has been drifting to later and later dates. So, the rule for Passover, which was originally intended to track the vernal equinox, has gotten a few days off. In ancient times this was never a problem since Passover was set by actual observations of the Moon and of the vernal equinox. However, after Hillel II standardized the Hebrew calendar in the 4th century, actual observations of celestial events no longer played a part in the determination of the date of Passover. The Gregorian calendar reform of 1582 brought the Western Church back into conformity with astronomical events, hence the discrepancy.
Similarly, you will notice that in many years Gregorian Easter (the one marked on all calendars) differs from Julian (Orthodox) Easter, sometimes by a week, sometimes by a month. Again, this is due to the different rules of calculation. A major difference is that Orthodox Easter uses the old Julian calendar for calculation, and the date of the Vernal Equinox is slipping later and later on the Julian calendar relative to the Gregorian calendar (and to astronomical fact). Also, the date of Paschal Full Moon for the Julian calculation is about 4 days later than that for the Gregorian calculation. At present, in 5 out of 19 years in the Metonic Cycle--the years when the Golden Number equals 3, 8, 11, 14 and 19--Orthodox Easter occurs a month after Gregorian Easter. In three of these years, Passover also falls a month after Gregorian Easter (see above).
As it turns out, 2016 has a Golden Number of 3, so both the Jewish Passover and Orthodox Easter are actually a month away. Indeed, the Jews are now celebrating Purim, not Passover. Of course, I already mentioned how Jehovah's Witnesses already celebrated their annual Memorial this week (on Adar II 14th, not Nisan 14th). Also, ABC is still airing The Ten Commandments this weekend, even though Passover (the celebration of Moses and the Ten Commandments) isn't until next month.
Yes, every time I turn around, it's Christmas or Thanksgiving -- or Easter, that is. Yes, the Long March is over, and it's now spring break. My next regular school post will be on Monday, April 4th.
And so this concludes my last post before spring break. Once again, I plan on making one or two posts next week, during spring break itself. One of the posts may finally get to the Andrew Hacker article that I've been wanting to discuss for a month. And the other may be about Lesson 9-5 of the text -- a lesson that we'll skip as it's not important to PARCC or SBAC, but it's interesting to me. The Common Core standards devote much time to reflections and other transformations of the plane, and now Lesson 9-5 discusses what happens with reflections in 3-D space. After the break, we will continue with pyramids and cones. I wish everyone a happy Easter -- or Purim, or whatever you celebrate this weekend.