Rucker includes six puzzles in this chapter. I can summarize this chapter easily simply by posting the six puzzles and their answers -- with commentary as usual.
In four dimensions it is possible to have two 3-D spaces "perpendicular" to each other. Two such spaces would have only a plane in common. Suppose now that there is a 3-D space perpendicular to ours, a space with people moving around in it. Use a Flatland analogy to figure out how these people would appear to us.
If there were a 3-D space intersecting ours only in a plane, then we would see something like this: a plane of light slanting up from the ground into the sky, with odd-shaped globs drifting around in the plane, sliding up and down like holy Frisbees. The globs would be very thin and would feel solid to the touch.
If we assume that A Square's eye stays the same when he is lifted into 3-D space, he will not really be able to see whole 2-D objects as we do. What will he see? How might he build up a mental image of the full 2-D Flatland?
A Square's retina is a line segment designed to take in light from the plane of A Square's body. Now it seems that, looking down at Flatland from the third dimension, A Square would really see just those objects of Flatland that intersect the plane of his vision. The situation would be exactly like that of a Flatlander in a perpendicular world, as described in puzzle 3.1.
See if you can complete this table:
Corners Edges Faces Solids
Point 1 0 0 0
Segment 2 1 0 0
Square 4 4 1 0
Commentary: As you can probably figure out, a hypercube is the 4-D analogue of a cube -- Rucker also calls it a "tesseract." Therefore a hyperhypercube is the 5-D analogue of a hypercube.
Cube 8 12 6 1
Hypercube 16 32 24 8
Hyperhypercube 32 80 80 40
It is pretty easy to see that the number of corners will double each time we go up a dimension. But what about the other entries? How, without actually counting the lines, do we know that the hypercube has 32 edges? The idea is that a hypercube is gotten by starting with a cube in inital position, moving the cube one unit ana, and then having the cube in final position. The initial and final cube each contribute 12 eges, and the cubes eight corners each trace out an edge during the ana motion. 12 + 8 + 12 = 32. Similar reasoning will justify the remaining entries.
Commentary: It's also possible to figure this out using polynomials. We must find (x + 2)^d, where d is the dimension. Since (x + 2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16, this tells us that a hypercube has 16 vertices (0-D objects), 8 edges (1-D objects), 24 faces (2-D objects), 8 solids (3-D objects), and one hypersolid (4-D object). Note that Rucker uses the words "ana" and "kata" to refer to travel in the fourth dimension -- just as "forward" and "backward" imply the first dimension, "left" and "right" the second, and "up" and "down" the third.
This figure shows an unfolded hypercube. Try to figure out which sides should get glued to which if the thing is to be folded back up into a four-dimensional hypercube. Specifically, which faces get glued to the faces of the bottom cube?
The very bottom face must join up with the very top face, and the side faces of the bottom cube connect to the four open faces of the main part. This is easy to see if we consider the analogous process of folding up a crucifix up into a cube.
The volume of a cube S feet of a side is given by the formula S^3. What do you think would be the formula for the hypervolume of a hypercube S feet on a side? Specifically, what would be the hypervolume of a 2-by-2-by-2-by-2 hypercube?
The formula is S^4, and the specific hypercube mentioned would have a hypervolume of 16 hypercubic feet.
Commentary: Recall that I chose to discuss the fourth dimension now because our Geometry students are now learning about the third dimension. We can therefore imagine a Hypervolume Postulate similar to yesterday's Volume Postulate:
a. Uniqueness Property: Given a unit hypercube, every polyhedroidal solid has a unique hypervolume.
b. Hyperbox Volume Formula: The hypervolume of a hyperbox with dimensions l, w, h, and a is lwha.
c. Congruence Property: Congruent figures have the same hypervolume.
d. Additive Property: The hypervolume of the union of two nonoverlapping hypersolids is the sum of the hypervolumes of the hypersolids.
The two points at the end of a line segment have the pleasant property of being equally far from each other. If we move into 2-D space, we can find a third point so that now all three points are equally far from each other. The three points, of course, are the vertices of an equilateral triangle. In 3-D space we can go out of the triangle's plane and get a fourth point so that now all four points are the same distance from each other. These four points make up the corners of a triangular pyramid, also known as a tetrahedron. What kind of 4-D figure do you get if you continue this procedure one more step?
If we go into 4-D space, then it is possible to find a fifth point (by moving ana from the tetrahedron's center), so that now all five points are the same distance from each other. These five points are the corners of the so-called pentahedroid.
This is what I wrote last year about today's lesson:
As I mentioned yesterday, we are moving on to Lesson 10-5 of the U of Chicago text, which is on volumes of prisms and cylinders. As we proceed with volume, let's look at what some of our other sources say about the teaching of volume. Dr. David Joyce writes about Chapter 6 of the Prentice-Hall text -- the counterpart of U of Chicago's Chapter 10:
Chapter 6 is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.
There are 11 theorems, the only ones that can be proved without advanced mathematics are the ones on the surface area of a right prism (box) and a regular pyramid. Most of the theorems are given with little or no justification. The area of a cylinder is justified by unrolling it; the area of a cone is unjustified; Cavalieri's principle is stated as a theorem but not proved (it can't be proved without advanced mathematics, better to make it a postulate); the volumes of prisms and cylinders are found using Cavalieri's principle; and the volumes of pyramids and cones are stated without justification. The only argument for the surface area of a sphere involves wrapping yarn around a ball, and that's unlikely to get within 10% of the formula. Finally, a limiting argument is given for the volume of a sphere, which is the best that can be done at this level.
In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered. Alternatively, surface areas and volumes may be left as an application of calculus.
There are a few things going on here. First of all, Joyce writes that surface areas and volumes should be treated after the "basics of solid geometry," but he doesn't explain what these "basics" are. It's possible that the U of Chicago's Chapter 9 is actually a great introduction to the basics of solid geometry -- after all, Chapter 9 begins by extending the Point-Line-Plane postulate to 3-D geometry, then all the terms like prism and pyramid are defined, and so on. Only afterward, in Chapter 10, does the U of Chicago find surface areas and volumes.
So I bet that Joyce would appreciate Chapter 9 of the U of Chicago text. Yet most texts, like the Prentice-Hall text, don't include anything like the U of Chicago's Chapter 9, and I myself basically skipped over most of it and went straight to Chapter 10. Why did I do this?
It's because most sets of standards -- including the Common Core Standards, and even most state standards pre-Common Core -- expect students to learn surface area and volume and essentially nothing else about solid geometry. Every single problem on the PARCC and SBAC exams that mentions 3-D figures involves either their surface area or volume. And so my duty is to focus on what the students are expected to learn and will be tested on, and that's surface area and volume.
Joyce writes that something called "Cavalieri's principle" is stated as a theorem but not proved, and it would be better if it were a postulate instead. Indeed, the U of Chicago text does exactly that:
e. (Cavalieri's Principle) Let I and II be two solids included between parallel planes. If every plane P parallel to the given plane intersects I and II in sections with the same area, then
Volume(I) = Volume (II).
According to the text, Francesco Bonaventura Cavalieri was the 17th-century Italian mathematician who first realized the importance of this theorem. Cavalieri's Principle is mainly used in proofs -- as Joyce points out above, the volumes of prisms and cylinders is derived from the volume of a box using Cavalieri's Principle, and the U of Chicago also derives the volumes of oblique prisms and cylinders using Cavalieri.
The text likens Cavalieri's Principle to a stack of papers. If we have a stack of papers that fit in a box, then we can use the formula lwh to find its volume. But if we shift the stack of papers so that it forms an oblique prism, the volume doesn't change. This is Cavalieri's Principle.
Notice that we don't need Cavalieri's Principle if one is simply going to be handed the volume formulas without proving that they work. But of course, doing so doesn't satisfy Joyce. Indeed, the U of Chicago goes further than Prentice-Hall in using Cavalieri to derive volume formulas -- as Joyce points out, we can find the volume of some pyramids without advanced mathematics (Calculus). But then the U of Chicago uses Cavalieri to extend this to all pyramids as well as cones. Finally -- and this is the grand achievement -- we can even use Cavalieri's Principle to derive the volume of a sphere, using the volumes of a cylinder and a cone! As we'll soon see, Joyce is wrong when he says that a limiting argument is the best that we can do to find the volume of a sphere. Cavalieri's Principle will provide us with an elegant derivation of the sphere volume formula.
Here are the Common Core Standards where Cavalieri's Principle is specifically mentioned
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
But not everyone is excited about the inclusion of Cavalieri's Principle in the standards. The following link is to the author Dr. Katharine Beals, a traditionalist opponent of Common Core:
Beals begins by showing a question from New York State -- recall that NY state has one of the most developed Common Core curricula in the country. This question involves Cavalieri's principle. Then after stating the same three Common Core Standards that I did above, Beals asks her readers the following six "extra credit questions":
1. Will a student who has never heard the phrase "Cavalieri's principle" know how to proceed on this problem?
2. Should a student who has never heard the phrase "Cavalieri's principle" end up with fewer points on this problem than one who has?
3. Should a student who explains without reference to "Cavalieri's principle" why the two volumes are equal get full credit for this problem?
4. Is it acceptable to argue that the volumes are equal because they contain the same number of equal-volume disks?
5. To what extent does knowledge of "object permanence," typically attained in infancy, suffice for grasping why the two stacks built from the same number of equal-sized building blocks have equal volume?
6. To what degree does this problem test knowledge of labels rather than mastery of concepts?
The question involves two stacks of 23 quarters -- one arranged as a cylinder, the other arranged not so neatly -- and asks the students to use Cavalieri to explain why they have the same volume.
Notice that in four of the "extra credit questions," Beals criticizes the use of the phrase "Cavalieri's principle," arguing that the label makes a simple problem needlessly complicated. But Cavalieri's Principle is no less a rigorous theorem than the Pythagorean Theorem. Imagine a test question which asks a student to use the Pythagorean Theorem to find the hypotenuse of a right triangle, and someone responding with these "extra credit questions":
1. Will a student who has never heard the phrase "Pythagorean Theorem" know how to proceed on this problem?
2. Should a student who has never heard the phrase "Pythagorean Theorem" end up with fewer points on this problem than one who has?
3. Should a student who explains without reference to "Pythagorean Theorem" what the hypotenuse is get full credit for this problem?
6. To what degree does this problem test knowledge of labels rather than mastery of concepts?
As for the other two "extra credit questions," yes, the volumes are equal because they contain the same number of equal-volume disks. We don't need Cavalieri's Principle to prove this, since the Additive Property -- part (d) of the Volume Postulate from yesterday's lesson -- tells us so.
The true power of Cavalieri's Principle is not when we divide a solid into finitely (in this case 23) many pieces, but only when we divide it into infinitely many pieces. In higher mathematics, we can't simply extract from finite cases to an infinite case without a rigorous theorem or postulate telling us that doing so is allowed. I doubt that the infant mentioned in "extra credit question 5" above is intuitive enough to apply object permanence to infinitely divided objects.
In fact, in the year since I wrote this, I've discussed the Banach-Tarski Paradox. That paradox tells us that we can divide a sphere into finitely many pieces and reassemble them to form two balls. I'd like to see someone try to apply an infant's intuition of object permanence to Banach-Tarski.
The Volume Postulate fails for Banach-Tarski because even though there are finitely many pieces, the pieces are non-measurable (i.e., they don't have a volume). The Volume Postulate fails for the oblique cylinder because we're dividing it into uncountably many flat pieces. In both cases we need something else to help us find the volume -- and in the latter case, that something is Cavalieri.
I wonder how Beals would have responded had the question been, "These two cylinders have the same radius and height, but one is oblique, the other right. The right cylinder has volume pi r^2 h. Use Cavalieri's Principle to explain why the oblique cylinder also has volume pi r^2 h." Nothing less rigorous than Cavalieri gives us a full proof of the oblique cylinder volume formula.
Furthermore, in another post a few weeks before this one, Beals tells the story of a math teacher -- the traditionalist Barry Garelick -- who would only allow those who successfully derive the Quadratic Formula to date his daughter:
I claim that deriving the sphere volume formula from Cavalieri's Principle in Geometry is just as elegant as deriving the Quadratic Formula from completing the square in Algebra I.
So this has obviously turned into our traditionalist-labeled post for the week. But I said that I would discuss the Andrew Hacker article in my next traditionalist post -- oops!
Well, after having a super-long post earlier this week, we don't need another long post. The Hacker article can wait until next week.