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By Root 1933 0

the left edge, that’s tantamount to identifying the entire right edge with the left edge. Were the screen flexible (made of thin plastic, say) this identification could be made explicit by rolling the screen into a cylindrical shape and taping the right and left edges together. When we declare that crossing the upper edge brings you to the lower edge, that too is tantamount to identifying those edges. We can make this explicit by a second manipulation in which we bend the cylinder and tape the upper and lower circular edges together. The resulting shape has the usual doughnutlike appearance. A misleading aspect of these manipulations is that the surface of the doughnut looks curved; were it coated with reflective paint, your reflection would be distorted. This is an artifact of representing the torus as an object sitting within an ambient three-dimensional environment. Intrinsically, as a two-dimensional surface, the torus is not curved. It is flat, as is clear when it’s represented as a flat video-game screen. That’s why, in the text, I focus on the more fundamental description as a shape whose edges are identified in pairs.

10. The mathematically inclined reader will note that by “judicious slicing and paring” I am referring to taking quotients of simply connected covering spaces by various discrete isometry groups.

11. The quoted amount is for the current era. In the early universe, the critical density was higher.

12. If the universe were static, light that had been traveling for the last 13.7 billion years and has only just reached us would indeed have been emitted from a distance of 13.7 billion light-years. In an expanding universe, the object that emitted the light has continued to recede during the billions of years the light was in transit. When we receive the light, the object is thus farther away—much farther—than 13.7 billion light-years. A straightforward calculation using general relativity shows that the object (assuming it still exists and has been continually riding the swell of space) would now be about 41 billion light-years away. This means that when we look out into space we can, in principle, see light from sources that are now as far as roughly 41 billion light-years. In this sense, the observable universe has a diameter of about 82 billion light-years. The light from objects farther than this distance would not yet have had enough time to reach us and so are beyond our cosmic horizon.

13. In loose language, you can envision that because of quantum mechanics, particles always experience what I like to call “quantum jitter”: a kind of inescapable random quantum vibration that renders the very notion of the particle having a definite position and speed (momentum) approximate. In this sense, changes to position/speed that are so small that they’re on par with the quantum jitters are within the “noise” of quantum mechanics and hence are not meaningful.

In more precise language, if you multiply the imprecision in the measurement of position by the imprecision in the measurement of momentum, the result—the uncertainty—is always larger than a number called Planck’s constant, named after Max Planck, one of the pioneers of quantum physics. In particular, this implies that fine resolutions in measuring the position of a particle (small imprecision in position measurement) necessarily entail large uncertainty in the measurement of its momentum and, by association, its energy. Since energy is always limited, the resolution in position measurements is thus limited too.

Also note that we will always apply these concepts in a finite spatial domain—generally in regions the size of today’s cosmic horizon (as in the next section). A finite-sized region, however large, implies a maximum uncertainty in position measurements. If a particle is assumed to be in a given region, the uncertainty of its position is surely no larger than the size of the region. Such a maximum uncertainty in position then entails, from the uncertainty principle, a minimum amount of uncertainty in momentum measurements—that is, limited resolution