Thursday, April 2, 2015

Section 10-9 Activity: Spring Spheres and Easter Eggs (Day 140)

Today is Maundy Thursday, three days before Easter Sunday. As for the school whose calendar I'm following, even those it's already had its spring break, the school is closed for both Good Friday and Easter Monday. This four-day weekend is commonly observed in other nations such as Canada, and even certain states have a four-day weekend for Easter. California doesn't -- but many districts that take off the week after Easter add Good Friday, and many of those that take off the week before Easter add Easter Monday. The calendar I'm following has a four-day weekend for Easter that's completely separate from the spring break.

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. Last year, Easter was very late -- April 20th, 2014 -- while next year the holiday will be 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:

According to the Bible, Jesus’ death and resurrection occurred around the time of the Jewish Passover, which was celebrated on the first full moon following the vernal equinox.
This soon led to Christians celebrating Easter on different dates. At the end of the 2nd century, some churches celebrated Easter on the day of the Passover, while others celebrated it on the following Sunday.

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.

In 325CE the Council of Nicaea established that Easter would be held on the first Sunday after the first full moon occurring on or after the vernal equinox. From that point forward, the Easter date depended on the ecclesiastical approximation of March 21 for the vernal equinox.
Easter is delayed by 1 week if the full moon is on Sunday, which decreases the chances of it falling on the same day as the Jewish Passover. The council’s ruling is contrary to the Quartodecimans, a group of Christians who celebrated Easter on the day of the full moon, 14 days into the month.

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 is 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. Next 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. Section 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.

Oh -- before I give the sphere surface area, let me comment on yesterday's lesson. It's rare when the Mathematics Calendar of Theoni Pappas gives a problem based on the same lesson that I post on the blog, but that's what happened yesterday. The problem was to give the ratio of a sphere's volume to the that of two cones. But this was slightly different. Although the sphere and the cone's base had the same radius, the height was twice the radius of the sphere (i.e., it was equal to the diameter). Also, the two cones weren't set up as a bicone, but instead had the same base (not vertex). So the ratio ended up being equal to 1 -- and yesterday's date was the 1st.

Both Dr. Franklin Mason and the U of Chicago make the same limiting argument to derive the surface area of a sphere from its volume. This time, the surface area of a sphere is the limit of a union of small polygons -- and these polygons can be taken to be the bases of pyramids, the sum of whose volume is the known volume of the sphere. The volume of a pyramid is V = 1/3 * B * h, and the height of each pyramid is the radius of the sphere. Solving for B, the area of the base, we obtain the equation B = 3 / h * V = 3 / r * V. Adding up all the volumes of the pyramids we obtain the volume of the sphere, so we can plug in V = 4/3 * pi * r^3. And adding up all the bases of the pyramids we obtain the surface area of the sphere. Substituting, we obtain:

B = 3 / r * V
S.A. = 3 / r * 4/3 * pi * r^3
       = 4 * pi * r^2

which is the desired formula.

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 2D 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 (the bases of the pyramids) 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.

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.

And that concludes this post. I wish everyone a happy Easter -- or Passover, or whatever you celebrate this weekend -- and I'll see you on Tuesday, April 7th.

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