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Mathematics Goes to the Movies
by Burkard Polster and Marty Ross
This is part of a regular Simpsons episode. What qualifies it for inclusion
in this collection is the fact that it is also part of an IMAX 3D movie.
To escape Marge’s sisters who are about to visit Homer hides behind a cupboard. There, in the wall, he finds a gateway into the third dimension. Homer takes his chances and steps through.
Homer becomes 3-dimensional. The strange mathematical world he finds himself in consists of a square grid as a floor and is populated by geometric shapes such as spheres, cubes, cylinders and cones, a Greek temple, a fish pond, a “coordinate street sign,” as well as some numbers and mathematical formulae.
1 + 1 = 2
46 72 69 6E 6B 20 72 75 6C 65 73 21
(translated into ASCII this reads “Frink Rules!”
This refers to Professor Frink (below)
Expression on the right is the “critical density of the universe”
critical density of the Universe.
If the average mass density in the Universe is equal to or less than this, then the Universe behaves as if it is unbound and it will expand forever. If the average mass density of the Universe is greater than the critical density, then it will expand to a maximum scale length, then recollapse, behaving like it is a bound system.
The case with a density greater than critical gives as a closed universe with positive curvature, finite volume, which will expand for some time, then begin to recollapse. The case with a sub-critical density corresponds to an open universe with negative curvature which will expand forever, and is infinite in volume. A universe with the critical density is flat, infinite, and will expand forever though slowing down toward zero at infinite time in the future.
The question of whether we live in an open, flat, or closed universe is a matter of what the mass density of the Universe is relative to the critical density 3H^2 /8Pi G. For a Hubble constant of H = 82 km/s per Mpc as measured by HST, the critical density is 1.26 x 10^-26 kg/m^3. This seems really tiny, but space is really big. (Q: What is this critical density in units of solar masses per cubic Megaparsec?) In fact, it appears that our Universe may have only about 30% of the critical density, and we might be living in an open universe. On the other hand, this is a hard measurement to make, and there are some indications (as well as some theoretical prejudices) that we live in a flat (or very nearly flat) Universe.
After it pokes him Homer tosses one of the cones in the air. As it hits the ground it punches a hole. The hole starts expanding and develops into a black hole the swallows everything up.
Back in the Simpson’s house
LISA: Where is my dad?
FRINK: Well, it should be obvious to the most dimwitted individual who holds an advanced degree in hyperbolic topology, n’gee, that Homer Simpson has stumbled into (lights go off)
the 3rd dimension.
LISA: switches the light back on Sorry.
FRINK: Here is an ordinary square.
POLICE CHIEF: 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.
FRINK: This forms a 3-dimensional object known as a “cube,” or a “Frinkahedron”, named in honour of its discoverer who n’hey, n’hey.
HOMER: Help me. Are you helping me, or are you going on and on?
PROFESSOR: Oh, right. And of course within we find the doomed individual.