There’s no better way to celebrate 3/14/16 than chatting about how π is a mess but it’s connected to the shape of the Universe, all while enjoying a nice piece of pie. I’ll have a slice of that Neil Gaiman Country Apple, please.
The ancient Greeks didn’t quite know what to do about π. For the Pythagoreans it transgressed a basic tenet of their religious faith — all numbers are supposed to be integers or at least ratios of integers. Alas for the faithful, π misbehaves. The ratio of the circumference of a circle to its diameter just refuses to match the ratio of any pair of integers.
The best Archimedes could do about 250 BCE was determine that π is somewhere between 22/7 (0.04% too high) and 223/71 (0.024% too low). These days we know of many different ways to calculate π exactly. It’s just that each of them would take an infinite number of steps to come to a final result. Nobody’s willing to wait that long, much less ante up the funding for that much computer time. After all, most engineers are happy with 3.1416.
Nonetheless, mathematicians and cryptographers have forged ahead, calculating π to more than a trillion digits. Here for your enjoyment are the 99 digits that come after digit million….
Why cryptographers ? No-one has yet been able to prove it, but mathematicians are pretty sure that π’s digits are perfectly random. If you’re given a starting sequence of decimal digits in π, you’ll be completely unable to predict which of the ten possible digits will be the next one. Cryptographers love random numbers and they’re in π for the picking.
Another π-problem the Greeks gave us was in Euclid’s Geometry. Euclid did a great job of demonstrating Geometry as an axiomatic system. He built his system so well that everyone used it for millennia. The problem was in his Fifth Postulate. It claimed that parallel lines never meet, or equivalently, that the angles in every triangle add up to 180o.
Neither “fact” is necessarily true and Euclid knew that — he’d even written a treatise (Phaenomena) that used spherical geometry for astronomical calculations. On our sweetly spherical Earth, a narwhale can swim a mile straight south from the North Pole, turn left and swim straight east for a mile, then turn left again and swim north a mile to get back to the Pole. That’s a 90o+90o+90o=270o triangle no problem. Euclid’s 180o rule works only on a flat plane.
Back to π. The Greeks knew that the circumference of a circle (c) divided by its diameter (d) is π. Furthermore they knew that a circle’s area divided by the square of its radius (r) is also π. Euclid was too smart to try calculating the area of the visible sky in his astronomical work. He had two reasons — he didn’t know the radius of the horizon, and he didn’t know the height of the sky. Later geometers worked out the area of such a spherical cap. I was pleased to learn that π is the ratio of the cap’s area to the square of its chord, s2=r2+h2.
The Greeks never had to worry about that formula while figuring our how many tiles to buy for a circular temple floor. The Earth’s curvature is so small that h is negligible relative to r. Plain old πr2 works just fine.
Astrophysicists and cosmologists look at much bigger figures, ones so large that curvature has to be figured in. There are three possibilities
- Positive curvature, which you get when there’s more growth at the center than at the edges (balloons and waistlines)
- Zero curvature, flatness, where things expand at the same rate everywhere
- Negative curvature, which you get when most of the growth is at the edges (curly-leaf lettuce or a pleated skirt)
Near as the astronomers can measure, the overall curvature of the Universe is at most 10-120. That positive but miniscule value surprised everyone because on theoretical grounds they’d expected a large positive value. In 1980 Alan Guth explained the flatness by proposing his Inflationary Universe theory. Dark energy may well figure into what’s happening, but that’s another story.
Oh, that was tasty pie.
~~ Rich Olcott
We can investigate things that take longer than an instrument’s characteristic time by making repeated measurements, but we can’t use the instrument to resolve successive events that happen more quickly than that. We also can’t resolve events that take place much closer together than the instrument’s characteristic length.
A wave happens in a system when a driving force and a restoring force take turns overshooting an equilibrium point AND the away-from-equilibrium-ness gets communicated around the system. The system could be a bunch of springs tied together in a squeaky old bedframe, or labor and capital in an economic system, or the network of water molecules forming the ocean surface, or the fibers in the fabric of space (whatever those turn out to be).
An isolated black hole is surrounded by an intense gravitational field and a corresponding compression of spacetime. A pair of black holes orbiting each other sends out an alternating series of tensions, first high, then extremely high, then high…
Almost a century later, James Clerk Maxwell (the bearded fellow at left) wrote down his electromagnetism equations that explain how light works. Half a century later, Einstein did the same for gravity.
Gravitodynamics is completely unlike electrodynamics. Gravity’s transverse “force” doesn’t act to move a whole mass up and down like Maxwell’s picture at left. Instead, as shown by Einstein’s picture, gravitational waves stretch and compress while leaving the center of mass in place. I put “force” in quotes because what’s being stretched and compressed is space itself. See 
The experiment consists of shooting laser beams out along both arms, then comparing the returned beams.
Grammie always grimaced when Grampie lit up one of his cigars inside the house. We kids grinned though because he’d soon be blowing smoke rings for us. Great fun to try poking a finger into the center, but we quickly learned that the ring itself vanished if we touched it.
For instance, suppose Fred and Ethel collaborate on a narwhale research project. Fred is based in San Diego CA and Ethel works out of Norfolk VA. They fly to meet their research vessel at the North Pole. Fred’s plane follows the green track, Ethel’s plane follows the yellow one. At the start of the trip, they’re on parallel paths going straight north (the dotted lines). After a few hours, though, Ethel notices the two planes pulling closer together.
The line rotates as a unit — every skater completes a 360o rotation in the same time. Similarly, everywhere on Earth a day lasts for exactly 24 hours.
Now suppose our speedy skater hits a slushy patch of ice. Her end of the line is slowed down, so what happens to the rest of the line? It deforms — there’s a new center of rotation that forces the entire line to curl around towards the slow spot. Similarly, that blob near the Equator in the split-Earth diagram curls in the direction of the slower-moving air to its north, which is counter-clockwise.
See that little guy on the bridge, suspended halfway between all the way down and all the way up? That’s us on the cosmic size scale.
So that’s the size range of the Universe, from 1.6×10-35 up to 2.6×1026 meters. What’s a reasonable way to fix a half-way mark between them?
Newton definitely didn’t see that one coming. He has an excuse, though. No-one in in the 17th Century even realized that electricity is a thing, much less that the electrostatic force follows the same inverse-square law that gravity does. So there’s no way poor Isaac would have come up with quantum mechanics.
If Newton loved anything (and that question has been discussed at length), he loved an argument. His battle with Leibniz is legendary. He even fought with Descartes, who was a decade dead when Newton entered Cambridge.
Hard Science Ain't Hard 