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:
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?
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.
Suppose we were to discover a big bright star with nothing inside it but light and empty space. What might we conclude?
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.
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?
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.
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?
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.
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?
He turns into his own mirror image!
Commentary: Yes, David Kung does exactly the same thing in one of his lectures!
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?
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.
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?
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.