Chapter 4 of Rudy Rucker's The Fourth Dimension: Toward a Geometry of Higher Reality is called "Through the Looking Glass." In this chapter, Rucker focuses on that important Common Core transformation -- the reflection.
As we already know, reflections reverse orientation. Indeed, reflections are unlike translations and rotations in that it's easy to take a real object and translate it or rotate it. But we can't very easily reflect a real object -- transform a right shoe into a left show, for example.
The basic idea of this chapter is that a reflection in n-dimensional space is equivalent to a 180-degree rotation in (n+1)-dimensional space. So we convert a transformation that's physically impossible, a reflection, into one that's physically possible, a rotation.
Rucker describes a new dream of A Square in which our quadrilateral hero visits the king of Lineland again -- only this time he rotates the monarch 180 degrees, centered at his midpoint. This is equivalent to a 1-D reflection.
Then A Cube visits Flatland and rotates A Square 180 degrees, which A Square's midsegment as the axis of rotation. This is equivalent to a 2-D reflection.
And so Rucker explains that it's possible to reflect a 3-D object like A Cube by rotating him 180 degrees in the fourth dimension. The center of a 4-D rotation is a plane -- the same plane as the mirror of the desired 3-D reflection. Rotating a right shoe 180-degrees in 4-D results in a left shoe.
Rucker ends the chapter by describing a "Necker cube." A cube is drawn just as it is in Lesson 1-5 of the U of Chicago text, in non-perspective -- except that none of the sides are dotted. Now it is ambiguous which side of the cube is the front. Rucker points out that there are two ways of viewing the Necker cube as an actual cube -- with one of the cubes the reflection image of the other.
Here are the puzzles in this chapter:
A cube that intersects a plane at right angles makes a square cross section. Would it be possible to place a cube so that it intersects a plane in a triangular cross section? How? What other cross-sectional shapes could a cube show?
In cross section, a cube can appear to be a square, a triangle, a rectangle, or a hexagon. This is illustrated in a drawing by Claude Bragdon.
Commentary: Cross-sections of 3-D figures appear in Lesson 9-4 of the U of Chicago text (where they are called "plane sections"). In particular, Exploration Question 28 in Lesson 9-4 is identical to Rucker's Puzzle 4.1. Of course, we skipped Chapter 9 altogether here on the blog.
Here is a very remarkable Necker-type illustration. Can the little man see the beetle or not?
This illustration was sent to me [Rucker -- dw] by Orville L. Parrinello of Brazoria, Texas.
Commentary: I'll throw the rest of this puzzle out -- it doesn't really work without the visual.
Looking at a Necker cube can help us think about the fourth dimension. Can you think of a "Flatland Necker Illusion" that A Square could use to help think of the third dimension?
If A Square were transparent, then he could seem to reverse when seen edge on, just as the transparent A Cube reverses. The reversal would seem sort of as if the Square were "pulled through himself," just as a left glove can be turned into a right glove by turning it inside out.
Commentary: Here is the best link I could find about the Necker cube. Rucker's Necker Cube has a face -- to represent A Cube's face -- rather than a red dot as in the link. (Just pretend that the red dot is A Cube's nose!)
Today is another activity day -- because I have so many links from back in February about the volumes and surface areas of prisms and cylinders.
Joel Speranza is an Australian secondary math teacher. He wrote two versions of this lesson -- the original lesson five years ago where the students are assigned a block letter and must calculate its surface area and volume, and a new version which incorporates the block letters into a sign.
This is what I wrote to Speranza in the comments: