Lesson 9-5: Reflections in Space

Chapter 10 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality is called "Time Travel and Telepathy." These are, of course, two of the best-known science fiction tropes.

He writes:

Why does travel have to be so hard? The perfect vehicle is easy to imagine: a sort of automobile with some special buttons on the dash. Get in, punch the code numbers of where and when you want to be, turn the ignition key, and -- presto -- there you are in 1920's Paris, on the Great Plains before the pioneers, on the moon, or even in another galaxy.

Actually, such a vehicle as Rucker describes does exist -- it's called the DeLorean! Notice that Rucker first wrote this book in 1984, just barely before the world's most famous time travel movie, Back to the Future, was released -- otherwise he probably would have mentioned it here. (On the other hand, the movie E.T. was released in time for Rucker to mention it in this chapter.)

The movie Back to the Future II was mentioned in the news about five months ago, mainly because the date to which its main character, Marty McFly, travels happens to be October 21st, 2015. So there was much lighthearted discussion about whether any of the predictions made by the movie about the year 2015 had come true. (One notable prediction that failed was that the Chicago Cubs would win the 2015 World Series -- ironically, they were eliminated from the playoffs on October 21st! On the other hand, the movie correctly predicted the proliferation of flat-screen TV's and drones.)

Rucker's chapter, meanwhile, is more about time as the fourth dimension as well as the many paradoxes associated with time travel. (Yes, I know that David Kung's lectures were all about paradoxes, but he doesn't mention time travel.) The most famous time travel paradox is the Grandfather Paradox -- Rucker describes a version of this paradox where you attempt to kill your own past self.

There are six puzzles in this chapter. Throughout this blog post, I will make several comparisons to my own favorite sci-fi series, Futurama. (This show has its own resolution of the Grandfather Paradox -- the main character Fry goes back in time to become his own grandfather!)

Puzzle 10.1:
If you were to have complete freedom in moving forward and backward in time, then you could duplicate most of the feats of a hyperbeing who can move ana and kata at will. How could you use time travel to enter a sealed room? How could you use it to remove someone's dinner from his stomach without disturbing him?

Answer 10.1:
To get into the sealed room, travel far, far into the future until you reach a time where the room's walls have crumbled. Step into the room's space and then travel back in time. Note that this is exactly analogous to moving ana, then moving across the location where the wall would be, and then coming back kata into the room's space.

Getting the food out of someone's stomach is a little trickier, since you yourself won't fit in the stomach.... Take a little scoop and set it right on the bed where his stomach was. Now send the scoop back a few hours, to when he was there, and then bring it back to the empty bed....

Puzzle 10.2:
Special relativity says that it's impossible to permanently label any given space location. "Right here last week," in other words, has no absolute meaning. How would the existence of a time machine go against this assumption?

Answer 10.2:
If we assume that the time machine goes straight backward in time, then "right here one week ago" is wherever the time machine appears when I send it one week back. One would not expect the Earth to still be "right here" a week ago, so traveling backward in time might well land one in empty space. Science fiction writers usually deal with this problem by somehow setting their time machine to track along the Earth's spacetime path.

Commentary: In this next question, Rucker explains how faster-than-light (FTL) travel is, in fact, equivalent to time travel.

Puzzle 10.3:
Here is a picture explaining how FTL travel can lead to time travel to one's own past. The traveler goes from A to B to C, B being an event on the world line of a distant galaxy that is moving away from Earth at half the speed of light. Explain how the paths AB and BC both can be regarded as pure FTL trips.

Commentary: Here is a link I found of an image that's similar to Rucker's. The left image is the one relevant to this puzzle (though Rucker alludes to the right image elsewhere). The particle begins in the upper left corner (A) and travels "superluminally" (i.e., FTL) to the upper right corner (B), and then travels FTL again to the lower left corner (C), which is in A's own past.

Answer 10.3:
The point is that "instantaneous" is a relative concept. Relative to Earth, B happens at the same time as A, so one can travel instantaneously from A to B. Relative to the distant galaxy that moves away from us, C is simultaneous with B, so one can travel instantaneously from B to C. Combining the two trips take one from A to C, and thus into one's own past.

Puzzle 10.4:
Not only does FTL travel lead to time travel, the converse is true as well. Time travel leads to FTL travel. Given a rocket and a time machine, how could you send a probe all the way around the galaxy, yet have it get back the same day?

Answer 10.4:
Equip the rocket with a good robot brain so that after its hundreds-of-thousands-of-years-long journey, it can use the time machine to jump hundreds of thousands of years back in time and find the Earth. Once it finds the Earth again, it makes a small time jump to the right day (the day of launch) and lands.

Commentary: The TV show Futurama avoids the problems with FTL travel to distant galaxies by declaring that in a few centuries, the speed of light will have increased!

Puzzle 10.5:
If time itself were bent into a vast circle, then one might hope to reach the past by traveling "around" time. But thinking about a universe in which time is a vast circle leads to some strange problems. Say, for instance, that you build a very durable radio beacon and set it afloat in space near the Earth. Is it possible that this beacon can last all the way around time. If it does, then once you set one afloat, how many more should you be able to detect? What if you decide to set your beacon afloat if and only if you detect no beacons out there before the launch?

Answer 10.5:
The idea of a beacon that lasts all the way around time would lead to difficulties. If it drifted away from Earth never to return, then there would be, it seems, endlessly many of them out there, as a consequence of the one launch! This seems nonsensical. The situation is particularly vicious if we suppose that B_1 sends out a signal that can inhabit the launch of B_0. A yes-and-no paradox!

...It is, in other words, impossible to build a truly indestructible object if time is circular! For anything you build must in time disintegrate into pieces so that you can "again" build it.

Commentary: As I mentioned in my last post, one particular Futurama episode, "The Late Phillip J. Fry," explores the idea that time is circular. In this episode, Fry, the robot Bender, and the Professor travel in a time machine that can only travel to the future, not the past. So in order to return home, the crew must travel all the way around time. So the time machine itself is B_0, an object that travels the entire wheel of time.

Here's how the episode avoids the paradox -- upon arriving, the time machine B_0 lands right on top of time machine B_1, thereby destroying it. (Actually, it's B_2, not B_1 -- the crew accidentally overshoots its home year and must make a second lap around time.) With B_2 destroyed, we no longer have to worry about there being infinitely many time machines around.

Puzzle 10.6:
We discussed a number of paradoxes that arise from an ability to travel to the past. But just being able to communicate with the past leads to paradoxes as well. Suppose, for instance, that I have a magic telephone with the following properties: Whenever I pick up the receiver and dial "1," the magic telephone rings an hour earlier. I am thus able to telephone my past self. If I happen to hear the phone ringing, and pick up the receiver, I can expect to hear the voice of my future self. By now, what if at 9:00 I decide I will dial "1" at 11:00 unless I have received a call at 10:00?

Answer 10.6:
We have a yes-and-no paradox here. I dial "1" at 11:00 if and only if I don't get a call at 10:00, but I get a call at 10:00 if and only if I dial "1" at 11:00. In other words, I get a call at 10:00 if and only if I don't get a call at 10:00. This particular paradox was first raised in a paper by G. Benford, D. Book, and W. Newcomb, "The Tachyonic Antitelephone" (1970). Gregory Benford, is, by the way, a science fiction writer, as well as being a physicist. The "tachyons" his paper refers to are hypothetical particles that, unlike ordinary mass particles, always go faster than light. Benford's paper argues that since tachyons could be used to send messages into one's own past, it must be, even in principle, impossible to detect them. If, then, tachyons are real, they fill a sort of undetectable ghost universe whose time direction is in a sense perpendicular to our own time direction.

Commentary: Futurama also has a method of avoiding time paradoxes. In the Futurama movie Bender's Big Score, some of the characters go back in time to tell their past selves not to travel back in time. This is a paradox as now there are now two copies of the character. The Professor explains that the universe corrects the paradox by destroying one of the time copies. This follows Rucker when he writes, "Or maybe the cosmos would, in the interest of self-preservation, strike dead anyone about to perform a paradoxical time travel experiment!"

This is wrote last year about Lesson 9-5 of the U of Chicago text, a special spring break lesson:

We already discussed translations and rotations for Common Core Geometry, but those transformations applied to the two-dimensional plane. The physical world has three dimensions, and so it's the 3-D transformations that actually apply to physics.

Lesson 9-5 of the U of Chicago text is on reflections in space. I find this to be an interesting topic, but since it doesn't appear on the PARCC, it would be a waste of time to cover it in class. There will be no worksheet for this lesson, since it's not intended to be taught in class. This is why I waited until spring break to blog about this topic.

The text begins by defining what it means for a plane in 3-D, rather than a line in 2-D, to be a perpendicular bisector:

"In general, a plane M is the perpendicular bisector of a segment AB if and only if M is perpendicular to AB and M contains the midpoint of AB."

Now we can definition 3-D reflections almost exactly the same way we define 2-D reflections:

"For a point A which is not on a plane M, the reflection image of A over M is the point B if and only if M is the perpendicular bisector of AB. For a point A on a point M, the reflection image of A over M is A itself."

If you think about it, when you (a 3-D figure!) look at yourself in a mirror, the mirror itself isn't a line, but rather a plane. Mirrors in 2-D are lines, while mirrors in 3-D are planes. So now we can define what it means for two 3-D figures to be congruent:

"Two figures F and G in space are congruent figures if and only if G is the image of F under a reflection or composite of reflections."

Most texts don't actually define what it means for two 3-D figures to be congruent. We know that the traditional textbook definition, that congruent figures have corresponding sides and angles congruent, only applies to polygons. It doesn't even apply to circles, much less 3-D figures. But we can simply use the Common Core definition -- two figures are congruent if and only if there exists an isometry (i.e., a composite of reflections) mapping one to the other -- and it instantly applies to all figures, polygons, circles, and 3-D figures.

In 2-D there are only four isometries -- reflections, translations, rotations, and glide reflections. An interesting question is, how many isometries are there in 3-D?

Well, for starters, translations and rotations exist in 3-D. We can define both of these exactly the same way that we do in 2-D -- a translation is the composite of two reflections in parallel planes, while a rotation is the composite of two reflections in intersecting planes.

Notice that every 2-D rotation has a center -- the point of intersection of the reflecting lines. The same thing happens in 3-D, except that the intersection of two planes isn't a point, but a line. Thus, a 3-D rotation has an entire line as its center -- every point on this line is a fixed point of the rotation. But usually, instead of calling the line the center of the rotation, we call it the axis of the rotation. One 3-D object that famously rotates is the earth, and this rotation has an axis -- the line that passes through the North and South Poles. Confusingly, the mirror of a 2-D reflection is often called an axis -- but in some ways, these two definitions are related. One can perform a 2-D reflection by taking a 2-D figure and rotating it 180 degrees about the axis in 3-D (and recall that A Cube has reflected A Square in exactly this manner).

Glide reflections also exist in 3-D -- although these are often called glide planes in 3-D. A glide reflection is the composite of a reflection and a nontrivial translation parallel to the mirror. Notice that there are infinitely many directions to choose from for our translation in 3-D, but if this were a 2-D glide reflection there are only two possible directions for a translation that's parallel to the mirror.

But are there any other isometries in 3-D? Well, we notice that a glide reflection is the composite of two other known isometries, a reflection and a translation. So the next natural possibility to consider is, what if we find the composite of the other two combinations? What is the composite of a reflection and a rotation, or a translation and a rotation?

In 2-D, the composite of a translation and a nontrivial rotation is another rotation. This is possible to prove, as follows: let T be a translation and R be a nontrivial rotation, and G be the composite of T following R. Both translations and rotations preserve orientation, and so their composite G must preserve orientation as well. In 2-D, three mirrors suffice -- that is, every isometry is the composite of at most three reflections. Since G preserves orientation, it must be the composite of an even number of rotations. Therefore G is the composite of two reflections (or the identity transformation -- the transformation that maps every point to itself), and so G is either a translation or a rotation.

Let's try an indirect proof -- assume that G is a translation. That is:

T o R = G

Since T is a translation, it has a translation vector t, and as we're assuming that G is a translation, it must also have a translation vector g. Now let U be the inverse translation of T -- that is, the translation whose vector is -t, the additive inverse of t. We now compose U on both sides:

U o T o R = U o G

Now since U and T are inverses, U o T must be I, the identity transformation:

I o R = U o G

Since I is the identity transformation, I o R must be R:

R = U o G

Notice that G and U are both translations, whose vectors are g and -t, respectively. Then their composite must be another translation V whose vector is g - t. So now we have:

R = V

that is, a rotation equals a translation. Now no rotation can equal a translation (as the former has a fixed point, while the latter has no fixed point) unless both are the identity -- which contradicts the assumption that R is a nontrivial rotation (i.e., not the identity). Therefore, the composite of a translation and a nontrivial rotation isn't a translation, so it must be a rotation. QED

But this proof is invalid in 3-D. This is because the proof uses a step that only works in 2-D -- namely that three mirrors suffice. We must show how many mirrors suffice in 3-D.

Let's recall why mirrors suffice in 2-D. Let G be any 2-D isometry, and let A, B, and C be three noncollinear points whose images under G are A', B', and C'. The first mirror maps A to A', the second mirror fixes A' and maps B to B'. It could be that the image of C under both mirrors is already C', otherwise a third mirror maps it to C'. Notice that the proof of the existence of these mirrors is nontrivial and depends on theorems such as the Perpendicular Bisector Theorem, since reflections are defined using perpendicular bisectors.

As it turns out, four mirrors suffice in 3-D. To prove this, we let G be any 3-D isometry, and let A, B, C, and D be four noncoplanar points. The first mirror maps A to A', the second mirror fixes A' and maps B to B', the third mirror fixes A' and B' and maps C to C', and the fourth, if necessary, fixes A', B', and C' and maps D to D'.

And so this opens the door for there to be a new transformation in 3-D, one that is the composite of a translation and a rotation as well as the composite of four reflections. We can imagine twisting an object like a screw. A screwdriver rotates the screw about its axis, but then it's being translated into the wall in the same direction as that axis. And because of this, this new transformation is often called a screw motion.

We still have one last combination, the composite of a reflection and a rotation. It is subtle why the composite of a reflection and a rotation in 2-D is usually a glide reflection -- why should the composite of a reflection and a rotation equal the composite of a (different) reflection and a translation? And it's even subtler why the composite of a reflection and a rotation may be a new transformation in 3-D.

But the simplest example of this roto-reflection is the inversion map. In 3-D coordinates, we map the point (x, y, z) to its opposite point (-x, -y, -z). This map is the composite of three reflections -- the mirrors are the three coordinate planes (xy, xz, and yz). As the composite of an odd number of reflections, it must reverse orientation. Yet it can't be a reflection, since it has only a single fixed point (0, 0, 0) and not an entire plane. Similarly, it can't be a glide plane because glide planes don't have any fixed points at all.

Roto-reflections are formed when the axis of the rotation intersects the reflecting plane in a single point -- and of course, this single point is the only fixed point of the roto-reflection.

So now we have six isometries in 3-D -- reflections, translations, rotations, glide planes, screw motions, and roto-reflections. Are there any others? As it turns out, these six are all of them -- and the proof depends on the fact that four mirrors suffice in 3-D.

Returning to the U of Chicago text, we have the definition of a reflection-symmetric figure:

"A space figure is F is a reflection-symmetric figure if and only if there is a plane M such that the reflection of F in M is F."

Similarly, a figure can be rotation-symmetric, as well as roto-reflection-symmetric. In the text, figures such as the right cylinder have reflection, rotation, and roto-reflection symmetry.

But a figure can't have translation symmetry unless it's infinite, as translations lack fixed points. So likewise, figures that have glide reflection or screw symmetry must also be infinite, as these transformations are based on translations.

Of course, there exist dilations in 3-D space as well. There is little discussion of similarity in 3-D, except to compare the surface areas and volumes of 3-D figures.

Now that we are reading about the fourth dimension, the next thing we wonder is, what would a reflection in 4-D look like? We follow the same pattern that we followed for 2-D and 3-D -- we begin with the perpendicular bisector of a segment. In 2-D the perpendicular bisector is a 1-D line, in 3-D it's a 2-D plane, and so in 4-D it's a 3-D hyperplane. This hyperplane M is the mirror of a 4-D reflection, mapping each point A to its image B if and only if M is the perpendicular bisector of AB
(and of course mapping each point on the hyperplane to itself).

We know that the 3-D isometries include all of the 2-D isometries (translations, rotations, reflections, and glide reflections) plus two new isometries (screws and roto-reflections). So we expect the 4-D isometries to include all the 3-D isometries plus two new ones.

One of these new isometries is called a double rotation, as it is the composite of two rotations. Notice that the center of a 2-D rotation is a 0-D point, and the center (axis) of a 3-D rotation is a 1-D line, and so the axis of a 4-D rotation is a 2-D plane. The double rotation is the composite of two rotations whose respective axes (planes) intersect in a single point. A point inversion is a double rotation.

The other new 4-D isometry is a roto-glide reflection. It is the composite of a roto-reflection (which requires three dimensions) and a translation in the fourth dimension. Just as three mirrors suffice in 2-D and four mirrors suffice in 3-D, we see that five mirrors suffice in 4-D.

But since today we read Rucker's chapter on time travel, we may ask, how do 4-D reflections work if we take time to be the fourth dimension? That is, suppose the mirror hyperplane is a single instant in time (that is, it contains only the three spatial dimensions), then what would a reflection look like?

In today's Rucker chapter, the author writes that a small particle called a positron may in fact be the time-reflection image of an electron:

If we look at it in terms of the "moving Now" viewpoint, it seems a little surprising that the electron and positron manage so neatly to appear and disappear together. But according to physicist Richard Feynman, one can take a spacetime viewpoint and regard the positron as an electron that is traveling backward in time. From this standpoint we simply have a nice little closed causal loop.

Rucker then follows this with a short story he wrote, called "A New Experiment With Time," in which an entire person undergoes a time-reflection! The main character, Maisie Gleaves, is a young woman who misses high school so much that she wants to go back in time to her school days. So she undergoes a time-reflection. Like all reflections, a time-reflection reverses orientation -- except that it's the time dimension that is reversed. Maisie sees everyone seeming to walk and talk backwards -- and to everyone else, Maisie is the one walking and talking backwards.

Before I conclude this post, let me announce that tonight on PBS, the TV show NOVA is airing a repeat of last year's episode, "The Great Math Mystery." I described this episode in detail on the blog almost exactly a year ago (back in April). I highly recommend watching it if you missed it last year!

Thus concludes my spring break post. I'll post once more during the vacation period.

Lesson 10-9 Activity: Spring Spheres and Easter Eggs (Day 131)

Chapter 9 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality is called "Spacetime Diary." It's the start of Part III, "How to Get There" (to the fourth dimension, of course).

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:

Puzzle 9.1:
If we say that the fourth dimension is time, then it is possible to construct a hypersphere in space and time. How?

Answer 9.1:
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.

Puzzle 9.2:
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?

Answer 9.2:
"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:

Puzzle 9.3:
"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!)

Answer 9.3:
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:

Puzzle 9.4:
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?

Answer 9.4:
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.

http://oilf.blogspot.com/2016/03/barry-garelicks-latest-book-is-out.html

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?

http://oilf.blogspot.com/2016/03/math-problems-of-week-reducing-words.html

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?
[60 words]

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:

http://www.timeanddate.com/calendar/determining-easter-date.html

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 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:

http://www.webexhibits.org/calendars/calendar-christian-easter.html

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."

http://mynorthwest.com/11/459668/Seattle-school-renames-Easter-eggs-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.

http://quasar.as.utexas.edu/BillInfo/ReligiousCalendars.html

(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.

Chapter 10 Test (Day 130)

Chapter 8 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality is called "Magic Doors to Other Worlds." In this chapter, Rucker proposes that there are parallel universes -- a common science fiction trope -- and that one must travel through the fourth dimension to reach them.

He writes:

People have always enjoyed thinking about such magical doors. The perfect symbol of the mind's freedom from the body's spatial limitations, magic doors occur throughout fantastic literature, from Lewis Carroll to C. S. Lewis to Robert Heinlein. As a rule, writers of fiction have been very vague about how magic doors might actually be "tunnels through hyperspace." But as it turns out, modern cosmologists have developed some good ways of thinking about magic doors (also known as Einstein-Rosen bridges or Schwarzschild wormholes).

To get the picture, we turn as usual to A Square. Suppose that Flatland is a plane, and parallel to it is another plane called Globland. Ordinarily, there is no way that an inhabitant of one of these two-dimensional universes can get to the other universe. But suppose that somehow a flap of space from each world has been snipped out, and say that the two flaps are sewn together. Now the Globbers can visit Flatland, and the Flatlanders can visit Globland.

So we see that Rucker considers a new 2-D universe, Globland, that's parallel to Flatland. There is an Einstein-Rosen bridge -- which he defines as a "magic door" -- leading from Flatland to Globland.

This leads to the first of three puzzles of this chapter:

Puzzle 8.1:
What would happen if the Globber were to choke the throat of the space tunnel down to point size?

Answer 8.1:
The two sheets of space might well snap apart. If the snap were too abrupt, it might do some damage to the Globber in question.

Commentary: This puzzle is highly dependent on the picture. It suffices to say that the Einstein-Rosen between Flatland and Globland is cylindrical, so that the door leading to Globland -- and from A Square's perspective, Globland itself -- appears to be a circle.

Puzzle 8.2:
An Einstein Rosen bridge would look something like a spherical mirror, with the odd property that the world in the mirror was actually different from the world outside the mirror. Now imagine an ordinary flat mirror with the property that the world seen in the mirror is not the same as the world on our side of the mirror. What kind of connection between two spaces is being described here?

Answer 8.2:
This is the type of connection represented [by a simple rectangular slide leading from one world to the other -- dw] although the holes in the spaces could be eliminated. The image of a strip of space joining two distinct spaces together. The strip could, of course, be made very short by bulging the spaces down to meet each other at the surface of the mirror. (Keep in mind here that just as a mirror in our space is a piece of a plane, a mirror in Flatland is a piece of a line.) This kind of link between spaces is exactly what Lewis Carroll deals with in Through the Looking-Glass. Marcel Duchamp was also obsessed with the notion of mirrors as doors to alternative universes. He was struck by the fact that a point approaching a mirror has the choice, in principle, of either breaking through the mirror and continuing in normal space, or of moving out of our space and into the alternative space we see inside the mirror. Thus, for Duchamp, a mirror represented a sort of railroad switch where one chooses between two spaces: real space and mirror space. See Linda Dalrymple Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art, 1983.

Puzzle 8.3:
In the last chapter [7 -- dw], I [Rucker -- dw] said there were three ways to fir Flatland into a basement, but I only described two ways: bend Flatland into a sphere, or infinitely shrink it to it in a square. What is the third way?

Answer 8.3:
Install an Einstein-Rosen bridge in the basement, and have this hyperspace tunnel lead to an infinite space empty except for the endless plane of Flatland.

This is what I wrote last year about today's test:

Here is the Chapter 10 Test. Let me include the answers as well as the rationale for including some of the questions that I did.

1. 4.
2. 24 square units.
3. 9pi cubic units.
4. sqrt(82) * pi square units, 3pi cubic units.
5. 48,000 cubic units.
6. 98 square units.
7. 30,576pi square units, 691,488pi cubic units. (Yes, I had to change this question from last year's sphere to this year's cylinder.)
8. 5.5. Section 10-3 of the U of Chicago text asks the students to estimate cube roots. If one prefers to make it a volume question, simply change it to: The volume of a cube is 165 cubic units. What is the length of its sides to the nearest tenth?
9. Its volume is multiplied by 343. This is a big PARCC question!
10. Its volume is multiplied by 25 -- not 125 because only two of the dimensions are being multiplied by 5, not the thickness.

I had to cut off the test at ten questions since the next ten questions originally deal with problems from our upcoming circle and spheres unit.

Review for Chapter 10 Test (Day 129)

Chapter 7 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality is called "The Shape of Space." This chapter adds a whole new level to some concepts that I've talked about before on the blog.

In this chapter, Rucker proposes the idea that the universe is the surface of a 4-D hypersphere. That is, one can travel on the hypersphere in three dimensions, but if one travels far enough, it's possible to travel all the way around the universe and return to the place of departure. As usual, Rucker provides a Flatland analogy -- A Square's world could actually be the surface of a sphere, and it's possible for A Square to move all the way around the sphere and return to his starting point.

This chapter has seven puzzles, so let's get to those:

Puzzle 7.1:
Although no line on a curved surface is really straight, some lines are straighter than others: straighter in the sense of being shortest paths. Such straightest possible lines are called the geodesics of the surface. What kinds of lines do you think are geodesics on a sphere?

Answer 7.1:
A geodesic on a sphere is a so-called great circle, that is, a circle, such as the equator, which is as big as possible.. Relative to the earth's surface, the equator is "straight" because it bends neither north nor south. A smaller circle, such as the Arctic Circle, can be seen to bend on the surface of a sphere and is not regarded as a geodesic.

Commentary: Hey, this sounds familiar! This is because we spent last summer discussing spherical geometry right here on the blog. And yes, we learned that the great circles are the geodesics, or the lines, of spherical geometry. In this chapter, Rucker is describing a sort of 3-D spherical geometry, as opposed to the 2-D spherical geometry of Legendre.

Puzzle 7.2:
Suppose we were to discover a big bright star with nothing inside it but light and empty space. What might we conclude?

Answer 7.2:
One natural explanation of a massless star would be to say that space is hyperspherical, and that the massless star is the virtual image of a real star at the opposite end of the universe. Unfortunately, even if space really is a hypersphere, we are not likely to actually observe any such "fake stars." The problem is that space is marred by medium-scale irregularities that will prevent the perfect focusing of the stars light rays at the point furthest away from the star. Another difficulty is that space contains clouds of dust here and there, dust which will absorb most of a star's light long before the light makes it halfway round the universe. Were it not for these two problems, we would in general expect to find a virtual image of any observed star at a diametrically opposite spot in the sky -- provided, of course, that space really is hyperspherical.

Commentary: Rucker explains in his text book that in a hypersphere, we would expect the light waves sent out from any object to travel in all directions throughout the hypersphere -- and then they would rejoin on the other side of the hypersphere to form a "ghost image" of the object. The ghost image would appear at the point directly opposite the original object -- the antipodal point.

By the way, since this chapter is so visual, let me provide a link to another website. Here Scott Burns, a former Illinois professor, describes a hyperspherical universe similar to Rucker's.

Puzzle 7.3:
Most cosmologists assume that any one region of our universe is more or less like any other region. This assumption is known as the cosmological principle. There is no overwhelming body of evidence for the cosmological principle. People just like it because it makes things simpler. But now suppose that the cosmological principle is wrong. Suppose that there is a single most important object in our universe -- a unique mammoth object that is very much more massive than anything else. If you combine this supposition with the assumption that space curves back on itself like a hypersphere, what kind of universe do you get? Can you draw a Flatland/Sphereland-style picture of such a space?

Answer 7.3:
It would look like a light bulb, or an ice cream cone: a ball with a big bulge on it. It is conceivable that our space actually has such a lopsided structure. If we were just about on the opposite side from the superstar that makes the bulge, then we wouldn't necessarily have to be able to see it directly. Space dust and intermediate space bumps could dissipate the monster's image. Such a model for our space is discussed in Paul Davies, The Edge of Infinity, 1983.

Puzzle 7.4:
Here is a two-dimensional pattern of lines. [Note: This image appears to be the closest to what Rucker draws -- dw] Suppose that in 3-D space you were to stretch this surface so that the distances between each neighboring pair of lines became the same. What shape would the surface take?

Answer 7.4:
We would get something like this: a square surface with a peak in the middle. The first and second pictures are two different ways of representing the same fact: there is more space toward the middle of this surface than one would ordinarily expect to find.

Puzzle 7.5:
A Mobius strip is formed by taking a strip of paper, giving it a half-twist, and taping the two ends together. Think of A Square as an ink pattern that soaks right through the paper of a Mobius strip. If he slides around the strip, what happens to him?

Answer 7.5:
He turns into his own mirror image!

Commentary: Yes, David Kung does exactly the same thing in one of his lectures!

Puzzle 7.6:
Suppose that there is a hole in Flatland, but that no one can fall into it because the closer they get to it, the more they shrink. Can you draw a curved-space picture of this situation?

Answer 7.6:
The space near the hole is stretched up into an endless "chimney." No Flatlander ever gets to the end of the chimney; no Flatlander ever slides into the hole.

Puzzle 7.7:
The Gabriel's Horn surface shown in this figure has a very strange property: although its length is infinite, its surface area is finite. Can you think of any way to cut a unit square into infinitely many pieces so as to make up an infinitely long surface with an area of one?

Answer 7.7:
The idea is to use an endless halving procedure to cut the square into infinitely many pieces: a piece of height 1/2, one of height 1/4, 1/8, 1/16, and so on. Zeno strikes again! If we line up the regions obtained, we get something that is infinitely long, but of unit area.

Commentary: This last puzzle brings together several ideas from previous posts. First of all, David Kung brought up Zeno's Paradoxes, so we know about the 1/2 + 1/4 + 1/8 + 1/16 + ... trick. As for Gabriel's Horn, Rucker was giving this as an example of a surface that satisfies hyperbolic -- rather than Euclidean or spherical -- geometry. The Flatland version of hyperbolic geometry is often known as the Poincare disc -- Rucker draws several Poincare discs but does not refer to them by that name.

You may notice that today's blog entry is called "Review for Chapter 10 Test." At this point you're probably wondering -- how can there be a Chapter 10 Test already? After all, we haven't covered the surface area or volume of a sphere yet (to say nothing of the hypervolume of Rucker's hypersphere)!

The problem, of course, is that next week is spring break. Chapter 10 is long, and the Easter holiday ends up splitting the chapter. This is actually a domino effect caused by Pi Day falling on Monday -- I wanted to cover pi -- part of Lessons 8-8 and 8-9 -- on Pi Day Monday, so we didn't start Chapter 10 until Tuesday. So that ended up pushing back Lessons 10-8 and 10-9 on the sphere.

We know that the formulas in Chapter 10 are hard for students to remember -- that is, after all, why the U of Chicago text devotes a full lesson, 10-6, just for remembering formulas! So imagine how much harder the formulas will be to remember when we have a week of spring break separating the start of Chapter 10 from its end!

And so I decided to declare this week to be the end of the unit and give a test this week. This is the same rationale for the Early Start Calendar -- we want to test the students before they have a chance to forget the material over the vacation weeks.

But in some ways, this is a blessing in disguise. Let's think about it:

-- Last week, the week of March 14th was devoted to prisms and cylinders. We began the week with pi and the circumference and area of a circle, and then we immediately applied both formulas to the surface area of a cylinder.

-- This week, the week of March 21st, is devoted to pyramids and cones. The formulas for both are taught, and then a unit test covering last week's and this week's material is given.

-- Next week, the week of March 28th, will be spring break.

-- The week of April 4th then will be devoted to spheres. Indeed, it will mark the start of a new unit -- a unit on spheres and circles. This includes some of the circle lessons that are spread out among Chapters 11, 13, and 15 in the U of Chicago text. Last year, I squeezed some of these circle lessons in at the end of Chapter 10, but this year I'll consider them to be a separate unit.

Think about it -- Lesson 13-5, one of the lessons we still have yet to cover, is titled "Tangents to Circles and Spheres," so there is continuity from the surface area and volume of spheres to tangents to the same spheres. And, as I promised a few months ago, we will end the unit the same way the U of Chicago ends the book -- with the Isoperimetric Inequalities of Lessons 15-8 and 15-9. We'll see that of all 2-D figures with the same perimeter, the circle has the most area, and of all 3-D solids with the same surface area, the sphere has the most volume.

(Suppose we were to ask, of all 4-D hypersolids with the same surface volume, which one has the greatest hypervolume? A good guess would be the hypersphere -- and as it turns out, that guess is exactly right.)

Here is the review sheet from last year. Notice that last year the worksheet was two-sided and consisted of 20 questions, but this year I include only the front side, since the back side contained material that we're saving for the circle and sphere unit. Even then, there are still questions on the front side that mention spheres (Questions 5, 11, and 12). I will fix these problems when we reach the actual test tomorrow.

Lesson 10-7: Volumes of Pyramids and Cones (Day 128)

Chapter 6 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality is called "What We're Made Of." It's the first chapter of Part II, which he simply titles "Space."

He begins:

We are accustomed to thinking of the world as made up of lumps of matter floating in empty space. Matter is something, and space is nothing. But is this really correct? In the past, many powerful thinkers have held that the space between visible objects is filled with a subtler material, a smooth and continuous substance, a "universal plenum," or aether.

As we can see, this chapter leans more towards physics than mathematics. We usually think of aether as a pre-Einstein concept that the theory of relativity has ruled out. But according to Rucker, Einstein still incorporated aether in his theory to explain how gravity can act at a distance.

Of course, there has been much discussion about Einstein's theory in the news lately, first with the 100th anniversary of the theory and then the discovery of gravitational waves. The newly discovered waves explain how gravity acts a distance without any need for an ether. Still, we can't fault Rucker for writing about aether 30 years before the gravitational waves were discovered.

This chapter, like Chapter 3, contains six puzzles, so I can summarize this chapter much more efficiently by posting the puzzles:

Puzzle 6.1:
In Astria, Charles H. Hinton's version of Flatland, it is explicitly assumed that the two-dimensional creatures have a slight thickness, and that they are sliding around on a hard surface. They are like cold cuts lying on a flat tray of space, a space that serves as an elastic medium for transmitting all kinds of vibrations. Deep within the tissue of his or her body, each Astrian has a sharp little 3-D bump, a sort of "astral vibrator." This vibrator, about the size of a phonograph needle, oscillates with the rhythm of each Astrian's thoughts so as to set up sympathetic vibrations in the underlying space. Any other Astrian who is nearby can, without really knowing how, pick up the flavor of these thoughts from the vibrations of his own bump. What kind of impression would the Astrians get of one of their fellows who had been turned over in the third dimension?

Answer 6.1:
The turned-over Astrian would have his or her vibrator uselessly sticking up away from space. Such a person might seem to be without atmosphere or sensitivity, a zombie, a man without qualities, a burnt-out case. Is it interesting here to recall that some early thinkers, including Rene Descartes, thought that the pineal gland, located in the brain's center, might serve as a sort of third eye that perceives other people's auras or astral vibrations.

Commentary: This gives a new brand-new meaning to "I think, therefore I am"!

Puzzle 6.2:
As in puzzle 6.1, we will think about Astrians as being two-dimensional creatures who slide about on top of their space-aether. Each Astrian has a kind of higher-dimensional tooth that digs down into the firm, underlying space. How might the Astrians use meditation techniques to levitate?

Answer 6.2:
If an Astrian could dig his little 3-D thorn into underlying space hard enough, then he could oppose a force of gravity pulling him across space. In Charles H. Hinton's 1907 novel, An Episode of Flatland, this is exactly what the Astrians do....

Commentary: At this point Rucker quotes the novel written by Hinton, a contemporary of Edwin Abbott Abbott (the author of the original Flatland). I find no need to repeat the quote.

Puzzle 6.3:
We assume that Flatlanders [Abbott's original version, not Hinton's -- dw] are actually in their space, like inkblots in thin paper, or like color swirls in a soap film. A Flatlander cannot exist out of his or her space. But how, then, was A Cube able to lift A Square out of Flatland's space and turn him over [the reflections from Chapter 4 -- dw]?

Answer 6.3:
A Cube snipped the piece of space containing A Square and turned the piece over. Now you might worry about the space of Flatland "popping" from having a hole in it. Well, perhaps A Cube took care of that by using something like embroidery rings to clamp down around the edges of the hole and of the cut-out piece. Or maybe there was just a hole in space for a while. What would it be like to have a hole in space? We'll get back to the question in Chapter 7.

Puzzle 6.4:
Suppose we believe, along with Einstein, that it is possible [I believe Rucker errs here -- he means to say "impossible," so that the rest of the question makes sense -- dw] to permanently mark or label a given point in space. But now suppose that we find a hole in space. Doesn't the hole serve to single out a definite space location?

Answer 6.4:
No, for we cannot tell whether the hole is moving toward space. This is analogous to the fact that a particular matter bump does not serve to mark out some particular region of the fabric of space: as a wave moves across the water, its component water bits are constantly changing. We might compare a moving space hole to a bubble floating up through a liquid. Although the bubble's shape and size stay the same, the bits of liquid that lie at the bubble's boundary are changing as the bubble moves. An offbeat idea suggested by these considerations is that perhaps the smallest components of matter are not vortices or space bumps but actual holes in space.

Puzzle 6.5:
Quasars are very bright, very distant objects. Recently astronomers found two quasars that seem to be unusually close to each other. Further analysis of these two spots of light revealed that what we are seeing is actually two images of the same quasar. Can you find a "space bump" diagram that show how one quasar's image can be split into two?

Answer 6.5:
The idea is that the galaxy's bump is on a line directly between us and the quasar. Light from the quasar to us can travel along two alternative shortest paths: one around either side of the galaxy's big space bump. The splitting of a quasar's image was definitely observed in 1979. (See Frederic Chaffee, "The Discovery of a Gravitational Lens," Scientific American, November 1980.) The phrase "gravitational lens" is an exciting way of expressing the fact that space curvature can bend light. It is amusing to think of a vast supertelescope based on gravitational lenses millions of miles across.

Puzzle 6.6:
Generally we have represented the space curvature caused by a bit of mass as a rounded-off hump. Suppose that bits of mass were actual points. What kind of space shape would best represent such a mass point?

Answer 6.6:
A point-sized mass can be most simply represented as a sharp cusp in space. Alternatively, one might imagine this cusp to be pulled all the way out to infinity.

Commentary: Ah yes, this is the famous "delta function," which isn't really a function at all.

This is what I wrote last year about today's lesson:

Lesson 10-7 of the U of Chicago text is on the volumes of pyramids and cones. And of course, the question on everyone's mind during this section is, where does the factor of 1/3 come from?

The U of Chicago text provides two ways to determine the factor of 1/3, and these appear in Exploration Questions 22 and 23. Notice that without the 1/3 factor, the volume formulas for pyramid and cone reduce to those of prism and cylinder, respectively -- so what we're actually saying is that the volume of a conic surface is one-third that of the corresponding cylindric surface. So Question 22 directs the students to create a cone and its corresponding cylinder and see how many conefuls of sand fill the cylinder. The hope, of course, is that the students obtain 3 as an answer. This is the technique used in Section 10.6 of the MacDougal Littell Grade 7 text that I mentioned in yesterday's post as well.

(By the way, I just realized that I mentioned four different math texts in yesterday's post! Now Rucker's book is not a math text, but it is a book -- which means that I referred to five different books in that post. And I did mention the Pappas calendar also. Even though a calendar isn't quite a book, it does mean that I mentioned six items with pages full of math in a single post! And so once again, I end up with another bloated post.)

But of course, here in High School Geometry, we expect a more rigorous derivation. In Question 23, students actually create three triangular pyramids of the same base area and height and join them to form the corresponding prism, thereby showing that each pyramid has 1/3 the prism's volume. But this only proves the volume formula for a specific case. We then use Cavalieri's Principle to show that therefore, any pyramid or cone must have volume one-third the base area times height -- just as we used Cavalieri a few weeks ago to show that the volume of any prism, not just a box, must be the base area times height.

I decided not to include either of the activities from Questions 22 or 23. After all, there was just an activity yesterday and I wish to avoid posting activities on back-to-back days unless there is a specific reason to.

In Question 11, we learn that the largest monument ever built is the Quetzalcoatl at Cholula de Rivadabia, a pyramid about 60 miles southeast of Mexico City. So in particular, the Quetzalcoatl is larger than any of the more famous Egyptian pyramids mentioned in yesterday's lesson. (But apparently, the Quetzalcoatl is well-known enough to avoid being flagged as a spelling error here on the Blogger editor.) Naturally, the students are asked to determine the volume of this large pyramid.

Question 15 invokes one of the simplest solids of revolution -- a BC Calculus staple. Of course, we only need to use the cone formula to find the volume and not anything from Calculus.

Finally, in the lesson I incorporated some of Lesson 10-6, "Remembering Formulas." I like the idea of having a section devoted solely to remembering these formulas -- since these are notoriously difficult for students to remember. The problem is that the way my lessons are set up, there is precious little time to devote an entire day's lesson just to remembering formulas. But I squeeze it in here by showing the same hierarchy of three-dimensional figures that appear in Lesson 10-6.