fomosapiens

Carl Sagan filmed a free seven-minute explanation of the fourth dimension that has now been watched more than five million times. Universities charge tens of thousands of dollars a year for cosmology and geometry courses that never reach this material with the same clarity. He posted the entire sequence for nothing. Almost no one who casually debates higher dimensions has watched it to the end. The clip comes from Episode 10 of Cosmos, “The Edge of Forever,” recorded in 1980. Sagan begins by placing the viewer inside Flatland, a two-dimensional world inhabited by squares and circles who can move left-right and forward-back but have never heard of up or down. Into that plane he lowers a three-dimensional apple. The Flatlanders experience only a series of expanding and shrinking circles as the apple passes through their world. They never see the whole fruit, only successive slices, and they invent elaborate theories to explain the changing shapes that appear and vanish without cause. Sagan then constructs a cube by moving a square through a third dimension at right angles to itself, and asks what happens if the same operation is performed once more: take the cube and move it at right angles to all three existing directions. The result is a four-dimensional hypercube, a tesseract. He cannot show a true tesseract because both he and the viewer remain trapped in three dimensions, but he can display its three-dimensional shadow: two nested cubes with every vertex connected by lines. In the real four-dimensional object every edge would be equal and every angle a right angle; the distorted projection is simply the cost of losing a dimension. Every modern discussion of curved spacetime, finite but unbounded universes, and higher-dimensional geometry rests on the same progression he lays out with an apple and a cube. The sequence is free on YouTube. The willingness to sit through those seven minutes before speaking about the fourth dimension is a much rarer commodity than the confidence to argue without having watched it.