# Chapter 3

## Michelle Imber

Chapter 3 of Beyond the Third Dimension was a lot of really amusing stuff about slicing things and the relation of slices to dimensionality. I've noticed that people tend to reiterate the contents of the chapter at the beginning of their html pages, such that it's almost formulaic; I'm going to try not to do that here. (It was starting to get this "book reportish"feel--I was having bad fifth grade flashbacks.) So...I'll try really hard not to do so (although I already started in my first sentence. Sorry.) Anyhow, I'll cut right to the interesting stuff in the chapter that seem worthy of mention:

First of all, on page 51 is a famous optical illusion involving two arrows that is known as the Muller-Lyer illusion (named after its discoverer). The illusion: two line segments of equal length are drawn, one directly above the other, but one is given inverted arrowheads at either end and one is given regular outward-pointing ones. The line segment with the outward-pointing arrowtips appears to be shorter than the longish one. I thought I would mention briefly a theory or two about why this illusion works (hypothetically, anyway). Illusions work because they take advantage of some pre-existing neural circuitry in our brains. Our brains like things to be nice and easy, and therefore they take shortcuts wherever they can by making generalizations about the world (especially in the world of vision). A scientist of yore named R.L. Gregory thought that the Muller-Lyer illusion is the result of a generalization based on size perspective. We know intuitively that things which are farther away look smaller, and things which are closer look larger. Therefore, if we look at two things which appear to be the same size, and we know (or assume) that one is farther away than the other one, we will automatically assume that the farther one looks bigger. Now, here's the neat trick: if you set the Muller-Lyer arrows on end so that they are vertical, they simulate the corners of 1) the inside of a room or 2) the exterior of a building, kind of like this:

\ / /|\

| |

| |

/ \ \|/

My, my. That just looks terrible on the screen. This is where paper comes in handy. Anyhow, the corners of a room meet and produce something like the first example, where the vertical line represents a far corner to the viewer and therefore looks larger than the vertical line in example 2, which might be the exterior corner of a building coming out (convex) towards the viewer. It's just a hypothesis, but it makes some sense.

Now, here's where my problem with the book comes in. The book mentions the tetrahedron problem (how to put it together from two of the fold-out model which equals half of one). It attributes the difficulty most people have with the tetrahedron to a "three-dimensional equivalent" of the Muller-Lyer illusion. Now, what is causing this illusion in the THIRD dimension? Of course, the illusion in the SECOND dimension is caused by the fact that our brain must find shortcuts to help us extrapolate from a two-dimensional retina to a three-dimensional world. If we have shortcuts in the THIRD dimension, what would they be for? Now, granted, there are different types of three-dimensional illusions that would cause one to see one three-dimensional object in a different way. But what does our brain THINK it's seeing that tricks it in the case of a tetrahedron? I built a three-dimensional foldup model to see if it would help me (too pathetic-looking to bring into class for show-and-tell, I fear). It didn't. I mean, I see why it's unintuitive to put the triangle on top of the trapezoid. But it seems more like ít's the funny angle you have to tilt it at. This also brings me to a related topic that I don't have time to explore now: higher-dimensional illusions. Hard to find, to be sure--since our brain tends not to have assumptions about the fourth (or higher) dimensions. I can imagine seeing things in higher dimensions and perceiving them as illusions in lower dimensions, like A Square had the illusion of a circle being born and growing up and then shrinking. Can anyone else think of examples of dimensionally-induced illusions, or illusions that would be possible in higher dimensions, to us or to higher-dimensional beings? Weird stuff.

All I have time for now--maybe I'll add to this later.