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