Math Goes to the Movies

Chapter 15 Survival in the Fourth Dimension

Chapter 15 Survival in the Fourth Dimension In The Simpsons episode “Homer3 ,” Homer is seeking to hide from Marge’s sisters and discovers a portal to the third dimension. Rather than face the sisters, he takes his chances in the mystery world, and gets trapped. LISA: So where’s my dad? PROFESSOR FRINK: Well, it should be obvious to the most dimwitted individual who holds an advanced degree in hyperbolic topology, m’hey, that Homer Simpson has stumbled into [lights go off] the third dimension! LISA [switches the light back on]: Sorry. FRINK: Here is an ordinary square. Fig. 15.1 Explaining the Frinkahedron. CHIEF WIGGUM: Whoa, whoa. Slow down, egghead. FRINK: But suppose we extend the square beyond the two dimensions of our universe along the hypothetical z-axis, there. EVERYONE: Gasp! FRINK: This forms a three-dimensional object known as a “cube,” or a “Frinkahedron,” named in honor of its discoverer. M’hey, m’hey. HOMER: Help me! Are you helping me, or are you just going on and on? FRINK: Oh, right. And, of course, within we find the doomed individual. Looking for a footnote? There isn’t one. “Homer3” is actually the title of the episode. 161 162 15 Survival in the Fourth Dimension Of course, the two-dimensional Homer is totally unprepared to meet the challenges of a three-dimensional world.1 Similarly, we suspect that most of our three-dimensional readers would be very lost if suddenly thrown into a four-dimensional world. With this in mind, and to alleviate the great concern created by this ever-present danger, we offer here our survival guide to the fourth dimension. 15.1 Time, Space, Both, or What? In the Devil Girl From Mars (1954), the devil girl Nyah is describing the ultimate weapon: 0:28 NYAH: As fast as matter was created, it was changed by its molecular structure into the next dimension, and so destroyed itself. PROFESSOR HENNESSEY: So there is a fourth dimension! The Professor seemed very convinced by Nyah, though we’re not sure why. Indeed, ever since Einstein, “the fourth dimension” has been a hugely popular topic of discussion, but seldom accompanied by much understanding. As part of Einstein’s theory of relativity, the three spatial dimensions are woven together with time to create a four-dimensional space-time universe, a beautiful and mysterious concept. It’s also not at all what we wish to discuss. For mathematicians, the fourth dimension is (usually) different, and it was a mathematical reality at least fifty years prior to Einstein’s space-time ingenuity. To describe it, we first note that the position of a point in everyday three-dimensional space (our three-dimensional Euclidean space) is determined by its three coordinates (x, y, z). Analogously, a point in fourdimensional space is pinned down by four coordinates (x, y, z, w). And if you want more dimensions, it’s easy: for example, a point in seven-dimensional space is simply pinned down by seven coordinates (x, y, z, w, p, q, r). Now, what all these dimensions mean is a matter of context and of debate. However, the mathematics of higher dimensions is quite easy. These mathematical worlds are not fundamentally different from our three-dimensional world, and they can be played with and analyzed in very similar and familiar ways. Our goal is to describe how, using movie math as our platform. What then has all this to do with time being the fourth dimension? Not a whole lot. In relativity, there are also four dimensions, pinned down by the four coordinates (x, y, z, t). However, whereas in our four-dimensional world all the dimensions are alike, this is definitely not the case for space-time: the time dimension is fundamentally different from the space dimensions. For example, we can easily move back and forth in space, whereas moving back in time requires the purchase of a time machine. In the following, we’ll 1 All the scenes in the three-dimensional world can be experienced in full 3D, as part of the IMAX movie Cyberworld (2000). 15.2 The Hypercube Via Analogy 163 focus on a fourth spatial dimension, steering clear of the special and trickier concept of time as the fourth dimension.2 15.2 The Hypercube Via Analogy The movie Cube 2: Hypercube (2002) is the sequel to Cube (1997); see chapter...