math, science

Life in Hyperbolic Space 1: Primer

It’s rare that weird, brain-bending ideas and video games meet, but in my experience, when they do, it’s pretty glorious. Portal, Prey, Antichamber, The Stanley Parable, and SUPERHOT are examples I’ve played personally. Also Miegakure (if it ever comes out, grumble grumble) will probably land instantly in that category, being a 4D puzzle game. But my most recent weird-game obsession has been HyperRogue. HyperRogue is awesome. Like Dwarf Fortress, it’s got a bit of a retro look with clean, minimalist graphics. And like Dwarf Fortress, it’s pretty obvious that a lot of love has gone into it. Here’s what the game looks like:

HyperRogue.png

Like many roguelikes (modeled after the ancient ASCII game Rogue), it’s played on a sort of chessboard. It looks at first glance like one of those board wargame or D&D-type hexagonal chessboards, but instead of being just hexagons, it’s got heptagons (7-sided polygons) too. You know what? That reminds me of the description of another geometric object:

Truncated Icosahedron

(Screenshot from the awesome polyhedron program polyHédronisme)

That is a truncated icosahedron, but I’d wager that most people know it better as either A) a soccer ball/football, or B) a molecule of C60: a fullerene, a buckyball.

Don’t worry. It’ll be clear in a moment what the hell I’m going on about. You see, HyperRogue takes place in the hyperbolic plane. A flat piece of paper is a Euclidean plane. The surface of a globe (or the Earth, or a buckyball) is a sphere. The hyperbolic plane is the third brother in the trio, so to speak, and it’s the weird brother. Their relationship makes more sense to me if I think in terms of polygons. Here’s another picture:

1000px-hexagonal_tiling-svg

(From the Wikimedia commons.)

The Euclidean plane (the hexagonal tiling above) consists of hexagons, each of which is bordered by six hexagons. The buckyball (football/soccer ball) farther above, representing spherical geometry, consists of pentagons, each of which is bordered by five hexagons. And at the very top of the page, the world of HyperRogue, representing the hyperbolic plane, consists of heptagons (7-sided polygons), each of which is bordered by seven hexagons. The hyperbolic plane is what would happen if you tried to sew up a soccer ball using heptagonal and hexagonal pieces of leather, rather than the usual pentagonal and hexagonal ones. Here’s what that looks like:

convoluted

(From the page of Frank Sotille, who has awesome templates so you can make your own hyperbolic football. I would’ve done it myself, but all of my Scotch tape has vanished.)

The only buckyball-type tiling that lays flat is the one with hexagons surrounded by hexagons. Pentagons surrounded by hexagons curves into a sphere, and heptagons surrounded by hexagons curls up into what HyperRogue’s creator calls a hypersian rug.

But here’s a more intuitive (though less precise) way to understand the hyperbolic plane. Consider the flat Euclidean plane. Pick a point. Draw a circle centered on that point. Measure the circumference of that circle. Draw another circle with the same center, but twice the radius. Measure the circumference of that circle. The farther you get from your starting point, the larger the circumference, but the increase is very predictable and linear. As a matter of fact, the circumference (the “amount of space”) you cover as you increase the radius of your circle increases exactly like this:

save (1)(1)

(Graphed with FooPlot.com)

Now do the same thing on a sphere: centered on the north pole, draw a very small circle with a given radius (with the radius measured across the curved surface of the sphere, not straight from point to point). Draw another circle with twice the radius. Up until you hit the equator, the circumference will increase, but eventually it maxes out and goes back down:

save (3)

(Circumference in the Euclidean plane in black, circumference on the sphere in red)

There’s another important effect to consider here: the sphere’s radius matters. That plot assumed a sphere of radius 1. Here’s what it would look like with a sphere of radius 2:

save (2)

It doesn’t make much sense to ask for the “scale” or “radius” of the Euclidean plane, because the answer is “infinity.” Any Euclidean plane is indistinguishable from any other, no matter how you swell or shrink it. Spheres, though, are distinguished by their radii: each has an inherent positive curvature.

Of course, on a sphere, the largest circle you can draw has a radius (measured across the surface of the sphere) of ½πR. the circumference of that circle lies right on the sphere’s equator. After that, the circumference decreases as the radius increases, because your circle’s shrinking as it approaches the opposite pole. It reaches zero when the radius is πR, and your radius-line stretches from one pole to the other.

The Euclidean plane has zero curvature. The sphere has positive curvature. The hyperbolic plane has negative curvature. The “radius” of a hyperbolic plane is defined as 1 / sqrt(-K), where K is a measurement called “Gaussian curvature.” (For comparison, the Gaussian curvature of a sphere is 1/(R²) ). K is the thing that’s zero for the plane, positive for the sphere, and negative for the hyperbolic plane. For a hyperbolic plane of K = -1, with a “radius” of 1, the circumference increases like this:

save (1)

(Hyperbolic circumference is the green line. The red line for a sphere with R = 1 (K=1) is included for comparison.)

That’s the weird thing about hyperbolic geometry: a sphere of infinite radius behaves exactly like the Euclidean plane (it is the Euclidean plane). But as the radius shrinks, the sphere contains less and less space (so to speak). To put it another way: if you were knitting a Euclidean plane (which people do, because mathematical knitting is a thing, which is awesome), then you’d need to knit in twice as much thread at radius 2 than you’d needed at radius 1. To knit a sphere, you’d need to knit less than twice as much thread in at twice the radius. And to knit a hyperbolic plane, you’d need to knit in more than twice as much yarn at twice the radius (according to the formula 2 * pi * R * sinh(r/R), where R = 1/sqrt(-K), and r is the radius measured through the plane; the equivalent relation for the sphere is 2 * pi * R * sin(r/R)). Where the Euclidean knit would give you a flat circular rug, and the spherical knit would give you a hacky-sack ball, the hyperbolic knit would give you something like this:

crochet_02

(Knitted by Daina Taimina, exhibited on the website of the Institute for Figuring.)

But the weird thing is that you can still do almost exactly the same geometry on the hyperbolic plane that you can do in the Euclidean plane. In fact, that’s why hyperbolic geometry is interesting: apart from a couple of weird quirks, it behaves just like a plane. That means you can make rigorous, valid, geometric proofs in the hyperbolic plane.

The Euclidean plane gets its name from Euclid of Alexandria, who is responsible for the infinite misery of people (like me) who just couldn’t get along with high-school geometry. But he condensed many centuries of Greek (and other) geometry into a set of postulates (axioms) from which you can prove pretty much anything that’s true in geometry. Here they are:

  1. You can draw a straight line between any two points.
  2. You can extend an existing straight line as far as you like.
  3. Using a compass, you can draw a circle with any center and any radius you want.
  4. All right angles are the same.
  5. If you’ve got two lines running alongside each other and another line running through both of those, and the angles on the insides of the intersections add up to less than a right angle, then the lines will intersect if you extend them far enough, and they’ll intersect on the side where the angle-sum is less than two right angles.

That last one, called the parallel postulate was a thorn in geometry’s side for a long time, because it seems a lot less elegant than the others, and it seems like the kind of thing you might be able to prove from the other axioms, which would mean it’s not an axiom, since axioms are your starting rules and theorems are what you prove using them. A less messy way to write the parallel postulate is:

5. Given any straight line, and given a point which doesn’t lie on that line, there is exactly one straight line through that point which never intersects the first line (the second line being the parallel).

Spherical geometry and hyperbolic geometry are both based on changing the parallel postulate. In spherical geometry, it becomes:

5. Given any straight line, and given a point which doesn’t lie on that line, there are zero straight lines through that point which never intersect the first line.

Or, to put it in simpler terms: because you’re on a sphere, lines that should be parallel (that is, lines which form a total of two right angles when intersected by a third line), inevitably intersect. Think of the north and south poles as two points. Draw the prime meridian from North to South. Now draw another line passing through the equator at 1° East longitude, 0 ° North latitude. At that point, the two lines are as parallel as lines get: they form ninety-degree intersections with the equator, and therefore, the interior angles on either side form exactly two right angles. But keep drawing those lines, and they’re going to intersect at the south pole, even though they should, according to Euclidean intuition, have stayed parallel.

In hyperbolic geometry, the parallel postulate is modified to the other extreme:

5. Given any straight line L, and given a point P which doesn’t lie on that line, there are an infinite number of straight lines through P which do not intersect L.

Poincare Parallels

That’s the Poincaré disk model, which fits the infinite hyperbolic plane into a finite circle. It’s elegant and simple, and it’s the model HyperRogue uses, so I’m going to stick with it. The red lines are the boundaries of the infinite set of lines which pass through P but never intersect L. It looks like the red lines intersect L at one end each, but the Poincaré disk model distorts distances more and more the closer you get to the edge. A picture’s worth a thousand words, so here’s a series of circles of equal radius starting from the center of the disk and moving outward:

CircleChain

(Both images rendered with the awesome (and free) geometry software GeoGebra)

I know they don’t look like circles of equal radius, but that’s just an artifact of the projection. The same way a Mercator map distorts Greenland so that it looks like it’s bigger than North America (when in reality it’s not much bigger than Quebec or Mexico), the Poincaré disk model distorts distances the closer you get to the edge of the circle. As a matter of fact, that bounding circle, as measured within the hyperbolic plane, is infinitely far from the center. You can’t ever reach it: it’s infinitely far from everywhere. So those intersections that seem to exist in the picture with the red and blue lines, they don’t actually exist, because you’d have to go infinitely far to get there. And keep in mind that, from any given point, it feels like you’re at the center of a Poincaré disk, so those lines don’t actually get closer to the line L the way the projection makes it seem. The projection is a necessary evil. You can do spherical geometry on an ordinary globe and remove all distortion, but you saw how messy and rumpled the hyperbolic version of a globe became: just look at the red crochet thing above. That’s not gonna happen. You have to live with the distortion.

The cool thing about the Poincaré disk model, though, is that it preserves angles, which makes it easy to do the kind of straightedge-and-compass geometry that’s so handy for geometric proofs.

And guess what? Just like we experience the world an almost-flat Euclidean space (relativity says it’s not perfectly flat, but it’s very close in our neighborhood, which I have to say to stop the nitpickers from yelling at me), there are three-dimensional spaces with spherical curvature, and there are three-dimensional spaces with hyperbolic curvature. In the next part, I’m going to talk about life in a highly-curved hyperbolic space. But before I go, let me leave you with a picture of something. In the Euclidean plane, you can only pack six equilateral triangles (all angles and edge-lengths the same) around a single point. The result looks like this:

1-uniform_n11.svg.png

(Source.)

In the hyperbolic plane, though, you can fit seven equilateral triangles around a vertex, and it looks like this (and don’t forget, those triangle are just as perfect and equilateral as the ones above; it just doesn’t look like it, because of the projection):

1024px-H2_tiling_237-4.png

(Source.)

You can actually fit eight triangles around a vertex too, although the triangles have to be larger (largeness being a slightly complicated concept in hyperbolic geometry, but we’ll get to that next time):

1024px-H2_tiling_238-4.png

(Source.)

And actually, it’s perfectly allowable to fit an infinite number of equilateral triangles around every vertex. That looks like this:

H2_tiling_23i-4.png

(Source.)

And remember, those triangles are still perfect and equilateral and regular. Hyperbolic space is weird. Remember that thing Christopher Lloyd said in Back to the Future? Get ready: we’re gonna see some serious shit.

Standard
Cars

Weird Engines

You’ve probably noticed I’ve been on an automotive kick lately. Don’t worry, I don’t think I’m in any danger of becoming a babbling gear-head. Firstly, I don’t have the patience or mechanical skill to actually put an engine together without blowing myself up and setting the neighborhood on fire. Secondly, where I live, working on cars isn’t a cheap hobby, and the fact that I’m considering going to grad school means all my pennies are spoken for.

I’m interested in engines the same way a little boy would be. They’re big, loud, powerful, complicated, and mechanical. I could probably come up with fancy reasons for being curious about engines, but the fact is, I just like them. But I like them in kind of a shallow way. The more intricate details of engines still confuse me. There’s a reason I call myself Hobo Sullivan: in pretty much all the areas I talk about, I’m like a hobo being dragged into an art gallery. I can say “That painting looks like shit” or “That’s a really pretty painting,” but if you start trying to teach me about composition or Postmodernism, my eyes’ll glaze over and I’ll start asking when I get my bowl of soup.

Which is a really long-winded way of saying that this post isn’t intended for engine experts. If you’re into engines, you probably know every single thing on this list. This is a post for people who are casually interested like me. This is me emerging from the Google Caves with a handful of funny-shaped crystals and saying “Look at this cool stuff I found!” This list has nothing to do with with deciding on the best engine or anything fancy like that. This is a list of the engines I’ve found in my bizarre curiosity that made me say “That’s kinda cool…”

The Boxer-6

20150320_143648-5515cf5d4d285

(Image from CarThrottle.com)

The boxer-6 isn’t all that weird, since it’s still in common use. That’s partly because you’ll find it in sports cars like the Porsche 911, and in some of Subaru’s current cars and SUVs. When I first started learning about weird engines, I thought the boxer-6 was just a V6 where the V got all flattened out. Engines like that do exist, but they’re not called boxer-6s. The difference is, in a boxer-6, pairs of pistons opposite each other move in and out simultaneously. Unlike in a lot of V-engines, each piston’s connecting rod has its own bearing on the crankshaft. What does any of this matter? Well, it’s unusual, for one. For two, the peculiarities of engine dynamics (which I don’t pretend to understand) make the boxer-6 a very smooth-running, well-balanced, low-vibration engine. Also, the fact that its cylinders aren’t crammed cheek-to-cheek in a V means that even fairly high-power boxer-6s can be air-cooled (though some are still water-cooled).

But you guys know me by now. You know I always go to extremes if given the chance. I’m not going to be satisfied with a six-cylinder engine, no matter how nice.

The H-16

brm_h16_engine

(From Wikipedia.)

We’ve made quite a leap. While you could go out right now and buy a car with a boxer-6 in it, the only way you could have an H-16 engine is time travel, or by being an insane millionaire and having one built for you.

The engine above is British Racing Motors’ H16 engine. There are plenty of weird, esoteric terms used in automotive circles, but the weird letters that show up in engine names are perfectly sensible (mostly). My car is powered by an I4 engine, which means it’s an inline-four: four cylinders in a straight line. A V6 has six cylinders in two banks of three, with the pistons angled so that their connecting rods form a V-shape, with the crankshaft at the tip of the V. You’d refer to the boxer-6 above as an F6, for flat-6.

What, then, is an H16 engine? Well, it’s horrifying, is what it is:

h-engine

(Again, from Wikipedia.)

An H-16 is two flat-8s stacked on top of each other, each with its own crankshaft. The crankshafts are connected at the end by gears. Like I said, I don’t know engines, but that seems like a bad idea.

For some applications, it’s actually not. H-engines are mechanically pretty well-balanced, for one. For two, they’re a bit more compact than, say, a V-engine with the same number of cylinders, which made them popular for high-power airplanes.

But car enthusiasts will know that the H16 I showed at the top of the section didn’t come from an airplane. It came from British Racing Motors’ ill-fated P83 Formula 1 car, which has a peculiar, and if you ask me, slightly unpleasant sound.

BRM’s H16 was plagued with problems. For one thing, having two sets of cylinder heads on opposite sides meant it needed two radiators and a split fuel system. It also needed dual camshafts (which are the mechanical clockwork-type devices that tell a cylinder’s valves when to open the intake valves and let in fuel, and when to open the exhaust valves to let out burned fuel). It needed dual camshafts for each cylinder head. That’s bad enough in a V-engine, where you have two cylinder heads, but the H16 had four. For another, it was more complicated and fuel-hungry than even its competitor, the mighty V16. Unsurprisingly, the engine wasn’t a success, partly because a lot of them just blew up during races. Apparently, British Racing Motors’ troubles at the time, and their obsession with complicated engines, earned them the nickname “British Racing Misery.”

Still, it’s a weird, interesting engine, and somebody had some serious gonads to say “You know what? A V16 is just too damn simple.”

The Napier Deltic

napier_deltic_animation

(From Wikipedia, again.)

You remember that scene in the first Back to the Future where Dr. Brown is talking about inventing the Flux Capacitor? “I slipped, hit my head on the edge of the sink, and when I woke up, I drew this.” I’ve gotta figure a bonk on the head inspired the Napier Deltic. For those who don’t know much about engines, let me explain why the picture above shows one of the weirder engines ever invented.

  1. Opposed pistons. In a common piston engine, the air-fuel mixture burns, and the pressure from the hot gas pushes a piston down, which applies torque to the crankshaft, which drives whatever machine the engine is running. There’s just a cylinder head holding the hot gas in. In an opposed-piston engine, though, there isn’t a cylinder head, but rather, two pistons in a headbutt configuration in every cylinder. The fuel-air mixture enters the space between the two heads and burns there, pushing the two pistons apart, driving two crankshafts.
  2. The delta design. As you can see, the Deltic isn’t even as simple as a regular opposed-piston engine. That picture shows three cylinders, containing six pistons, driving three crankshafts. It’s weird, but it’s got a certain geometric appeal to it. It was also, apparently, a lot more compact and powerful than similar engines of the time, which made it attractive for British torpedo boats and locomotives.
  3. It was a two-stroke. If you don’t know, two-stroke engines are usually what you find powering small power tools like weed-whackers and chainsaws.The engines of most modern cars are four-stroke. The differences are subtle. In a four-stroke engine (which most gasoline engines are), the piston has to go up or down a total of four times to complete a full cycle: It moves down to suck in in air and fuel, moves up to compress the air and fuel before ignition, is pushed down by the burning air and fuel, and finally rises up to push out the exhaust. In a two-stroke engine, the intake and exhaust steps happen at the same time: often, the piston pushes down on a fuel-air mixture in the crankcase, and the pressure pushes the mixture into the engine, which pushes the exhaust out. The major advantage of a two-stroke engine is that it doesn’t need mechanically-operated intake and exhaust valves, since the piston controls intake and exhaust. This saves weight and complexity, which is vital when your engine is powering, say, an airplane, or a boat, or a chainsaw that needs to be light enough to hold up for long periods, without getting tired and dropping it on their foot.

The Deltic was fairly successful, since it packaged eighteen or more cylinders (two pistons each) into a smaller package than was possible with other designs of the same cylinder number. It was a lot lighter than comparable engines, too.

Why aren’t there more Deltic-type engines around? Well, a little research suggests that’s mostly because Napier was bought out after World War 2, and switched to making things like turbine engines and turbochargers, the former of which were rapidly filling a lot of the niches the Deltic once occupied. I suspect the peculiarity of the delta design also played a part, since my first reaction to it was “there’s no way that was a success.” I was wrong about that, but I bet the Deltic scared off more than a few mechanics.

The W12

napier_lion_ii

(Still Wikipedia.)

We’re not done with Napier yet. Above, you see the Napier Lion.

Although it’s not entirely accurate, you wouldn’t be far off if you described a V8 engine as two I4 engines driving the same crankshaft. The Napier Lion is a broad-arrow W12, which is what you’d get if you took a V8 and jammed another I4 down the V (sounds like a weird fetish fantasy…)

You’ve probably heard of W engines before, even if you’re not into cars. A W16 is the powerplant behind the Bugatti Veyron, which is famous for two things: holding the Guinness record for fastest street-legal car, and being a rare example of a supercar that isn’t absolutely horrible to look at. (I mean, it’s still not great, if you ask me. It looks like ergonomic furniture. But look at a Lamborghini Aventador and tell me the Veyron isn’t better-looking, if only because it’s not as…pointy.) But the W16 in the Veyron is a different type: it has two banks with eight cylinders each, and those cylinders are jammed just about as close as you can get them. Each of the banks is essentially a VR8: a V8 with a very narrow V. Since it only has one crankshaft, W isn’t exactly a good letter to use, but as a double-vee, I guess I get it. (I’m tempted to call the Napier Lion a Ш12, but Sh-12 isn’t exactly catchy.)

What the hell are you getting at, I hear you ask.  I get that question a lot. The reason I bring up the Napier Lion is that it demonstrates, better than the Veyron’s W16, that you can take a pretty ordinary set of cylinders (banks of I4s) and turn them into bizarre, high-power variants. It was weird aircraft engines like the Lion that got me interested in weird engines to begin with. Unsurprisingly, someone took a pair of W12 Lions and put them in a car, breaking a land-speed record and becoming the first to break the 350 mph barrier. And trust me, it’s not the last time we’re going to see weird engines dropped in cars.

The Radial

radial_engine_timing-small

(Wikipedia.)

For some reason, I think that looking at that animation while you had a really bad fever (or after eating funny mushrooms) would be really scary. But there’s not that much scary about it. It’s just a radial-5 engine, which proves that, there are more interesting ways to arrange pistons than lines and Vs.

I really like radial engines. There’s something pleasing about that five-fold symmetry. It’s like a starfish made of explosions. It also offers some serious advantages if you’re looking for an engine to stick in a propeller-driven airplane: compact and powerful, but with pistons still far enough apart that you can air-cool them, which eliminates the radiator, which is a notoriously brittle piece of hardware. And you don’t want brittle stuff on, say, a fighter plane.

For symmetry and smoothness, nearly all four-stroke radial engines have odd numbers of cylinders. A two-cylinder radial is just a two-cylinder boxer. I’ve seen videos of three-cylinder radials, but I like pentagons better than triangles, so I won’t bother with those. Plus, talking about five-cylinder radials gives me an excuse to show you a video of a Toyota being powered by an airplane engine. And this whole section has pretty much been an elaborate segue to talk about the Plymouth Air Radial Truck:

016-1939-plymouth-radial-airplane-truck-gary-corns

(From Motor Trend.)

Because, sometimes, you wake up in the morning and think “You don’t see enough nine-cylinder hot rods out there.” That is a 300 horsepower Jacobs 9-cylinder radial airplane motor from the 50s. There are plenty of things about hot rods that I don’t like (they’re always alarmingly low to the ground, for one, which bothers me for some reason), I will never deny that hot rodders are insanely creative in all the right ways. My proof? Look at that picture again.

Naturally, the radial engine lost a lot of its popularity when turboprop and turbojet engines became reliable and affordable for airplanes. It didn’t help that, eventually, engines like the I4 and F4 became refined enough and powerful enough to do the same job as a radial without needing the specialized radial construction.

But there was a period when the radial was king of the air. Within that period was a really wild period when planes needed more power than an ordinary radial could give them. Trouble is, if you want to make a radial engine with more than 11 cylinders, you have to make it really wide so that those cylinders can stick out to be air-cooled. And if you do that, then you’ve just stuck what amounts to a dinner plate on the front of your airplane, which is no good for aerodynamics. Engine manufacturers solved this with multi-bank radial engines.

The Wasp Major

biggest_rotary_cutaway

(Bet you can’t guess the source.)

That is a cutaway of the Pratt and Whitney Wasp Major, which is a beast of a machine. It has 28 cylinders (four banks of seven cylinders) and a built-in supercharger. It has a displacement of 71 liters (compare that to the Bugatti Veyron or Dodge Viper, two high-power cars, both of which displace all of 8 liters.) It produced up to 3,500 horsepower in its original form, and over 4,000 once they added turbochargers. It also sounds pretty fucking awesome. If I was a millionaire and wanted to build a hot rod, I think this is the engine I’d put in it. Either this, or our next weird engine…

The Zvezda M503

3630728158_62398cce2a

(From Flickr, this time.)

I live in North Carolina. We’re one of the home states of NASCAR, which might be the ultimate in redneck racing. I went to a monster truck show last weekend, at which one of the events was lawnmower racing. That’s kinda what we do down here. We like big trucks, noisy cars, and bizarre racing. Tractor pulling is popular down here, too.

If you don’t know, tractor-pulling is an odd sport where you hook a tractor to a weighted sled and try to pull it a specified distance (usually 300 feet or 100 meters). On top of the sled is a sliding weight which is geared to the wheels so that it moves forward in proportion to the distance pulled, making the front of the sled dig into the dirt, making it harder to pull the farther you pull. Unsurprisingly, that takes a lot of power. I’ve seen pictures of tractors with as many as five or six supercharged V8s, tractors with 18-cylinder radial engines, tractors with two diesel V12s, and tractors with helicopter turboshaft engines.

I always assumed tractor pulling was exclusive to American rednecks. Clearly, I was ignorant, because a lot of the really good tractor-pulling videos are from Germany, the Netherlands, and Australia, where it’s apparently a really popular sport. And my favorite tractor by far is the German “Dragon Fire.” I don’t care if you don’t like big engines or tractor pulls. I insist you watch this video of Dragon Fire in action.

As much as I like big engines and weird engines, I’ll freely admit that tractor pulling seems a bit excessive. Five supercharged V8’s? Just looks silly to me. Like a truckload of engines collided with a truckload of bad Ikea chandeliers.

Dragon Fire, though, has a more elegant solution. Well, kind of. It depends how you define “elegant,” I guess. Dragon Fire isn’t powered by a bunch of V8s. It’s powered by a single 42-cylinder radial engine: the Zvezda M503, built for use in Soviet missile boats. The 503 has 42 cylinders (6 rows of 7). It weighs five times (almost five and a half times) as much as my car. It displaces 143 liters (which is about the volume of a bathtub, according to Wolfram Alpha). In its stock form, it produced nearly 4,000 horsepower. And it’s a diesel, too.

I should mention that the M503 powering Dragon Fire is running on methanol, not diesel fuel. And it, apparently, produces closer to 8,000 horsepower, which is absolutely ridiculous, and which makes me happy.

The Swashplate

swash20plate20motor3

(From the website of Douglas Self, who’s compiled a whole bunch of awesomely weird engines.)

I wouldn’t want to be the guy who proposed that design. I’m sure some stuffy executive looked at that, took out his monocle, took his cigar out of his mouth and said “Stop bringing me nonsense and bring me a real engine!” But the swashplate engine is real, and it’s pretty damn cool. It operates just about the same as any other piston engine, but instead of turning a crankshaft, the swashplate engine pushes on an eccentrically-mounted disk. Not only does this squeeze four (or more) pistons into a compact package, but it also eliminates the heavy crankshaft and provides a natural gear reduction. Apparently, while it has been used on a few cars, and was considered for use in airplanes, its main use is as the powerplant in torpedoes, where its small frontal area and corresponding tiny drag is vital.

For some reason, I really like the swashplate engine. It’s about as far as you can take a traditional fixed-cylinder design. It’s just delightfully weird, and it’s weird that it actually works. (Make it any weirder, and you’ll get awesome stuff like the Duke engine, which didn’t make the list because its cylinders aren’t fixed, and that was one of the arbitrary qualifications I came up with.)

I also like that the swashplate engine has been built in Lego form, by the awesome YouTuber DrDudeNL.

The Chrysler A57 Multibank

18ncol1fdw8s5jpg

(Found via Jalopnik.)

When the United States entered World War 2 in 1941, apparently, Chrysler was tasked with producing a high-power engine for the M4 Sherman tank as quickly as possible. They delivered a 30-cylinder engine. 30 cylinders isn’t as many as 42, so the A57 shouldn’t be as impressive as the Zvezda M503. But it is, for my money. Because Chrysler did a clever thing to meet the deadline. To allow them to use existing engineering, they didn’t exactly build a 30-cylinder radial. They built a 30-cylinder engine made of five Chrysler I6 engines, attached to the main output shaft by gears. Here’s what that looked like:

chrysler-a57-multibank-gears

(From Old Machine Press.)

Using five existing I6s saved Chrysler design time, and they already had the machinery set up to make I6s, so they didn’t have to retool any factories, presumably. The A57 really is just five straight-6 engines stuck together. That’s the kind of Mad Max/McGyver creativity I like. Sure, the engine only produced 450 horsepower, but it was surprisingly compact, and I love this kind of simple innovation.

Honorable Mentions

I wanted to include the Junkers Jumo 205, but I’m running long as it is, and I already talked about opposed-piston two-stroke engines when talking about the Napier Deltic. I wanted to include the Duke Engine, but it’s got rotating cylinders, which disqualified it. For the same reason, I didn’t talk about the weird rotary radials developed early in piston-engined flight. I didn’t include the Wankel rotary engine, because it has no pistons. I didn’t mention awesomely weird concepts like the nutating engine or the gerotor combustion engine for the same reason. Plus, there’s only so much time in the day, and I’ve taken up enough of yours. But if you’ve got any weird engines you think I should’ve included, leave them in the comments. I might stick then in an addendum later on.

 

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A great big pile of money.

When I was little, there was always that one kid on the playground who thought he was clever. We’d be drawing horrifying killer monsters (we were a weird bunch). I would say “My monster is a thousand feet high!” Then Chad would say “My monster is a mile high!” Then I would say “Nuh-uh, my monster is a thousand miles high!” Then Taylor would break in, filling us with dread, because we knew what he was going to say: “My monster is infinity miles high!” There would then follow the inevitable numeric arms race. “My monster is infinity plus one miles high!” “My monster is infinity plus infinity miles high!” “My monster is infinity times infinity miles high!” Our shortsighted teachers hadn’t taught us about Georg Cantor, or else we would have known that, once you hit infinity, pretty much all the math you do just gives you infinity right back.

But that’s not what I’m getting at here. As we got older and started (unfortunately) to care about money, the concept of “infinite money” inevitably started coming up. As I got older still and descended fully into madness, I realized that having an infinite amount of printed money was a really bad idea, since an infinite amount of mass would cause the entire universe to collapse into a singularity, which would limit the number of places I could spend all that money. Eventually, my thoughts of infinite wealth matured, and I realized that what you really want is a machine that can generate however much money you want in an instant. With nanomachines, you could conceivably assemble dollar bills (or coins) with relative ease. As long as you didn’t create so much money that you got caught or crashed the economy, you could live really well for the rest of your life.

But that’s not what I got hung up on. I got hung up on the part where I collapsed the universe into a Planck-scale singularity. And that got me thinking about one of my favorite subjects: weird objects in space. I’ve been mildly obsessed with creating larger and larger piles of objects ever since. Yes, I do know that I’m weird. Thanks for pointing that out.

Anyway, I thought it might be nice to combine these two things, and try to figure out the largest pile of money I could reasonably accumulate. My initial thought was to make the pile from American Gold Eagle coins, but I like to think of myself as a man of the world, and besides, those Gold Eagles are annoyingly alloyed with shit like copper and silver, and I like it when things are pure. So, instead, I’m going to invent my own currency: the Hobo Sullivan Dragon’s Egg Gold Piece. It’s a sphere of 24-karat gold with a diameter of 50 millimeters, a mass of 1,260 grams, and a value (as of June 29, 2014) of $53,280. The Dragon’s Egg bears no markings or portraits, because when your smallest unit of currency is worth $53,280, you can do whatever the fuck you want. And you know what? I’m going to act like a dragon and pile my gold up in a gigantic hoard. But I don’t want any arm-removing Anglo Saxon kings or tricksy hobbitses or anything coming and taking any of it, so instead of putting it in a cave under a famous mountain, I’m going to send it into space.

Now, a single Dragon’s Egg is already valuable enough for a family to live comfortably on for a year, or for a single person to live really comfortably. But I’m apparently some kind of ridiculous royalty now, so I want to live better than comfortably. As Dr. Evil once said, I want one billion dollars. That means assembling 18,769 Dragon’s Eggs in my outer-space hoard. Actually, now that I think about it, I’m less royalty and more some kind of psychotic space-dragon, which I think you’ll agree is infinitely cooler. 18,769 Dragon’s Eggs would weigh in at 23,649 kilograms. It would form a sphere with a diameter of about 1.46 meters, which is about the size of a person. Keen-eyed (or obsessive) readers will notice that this sphere’s density is significantly less than that of gold. That’s because, so far, the spheres are still spheres, and the closest possible packing, courtesy of Carl Friedrich Gauss, is only 74% sphere and 26% empty space.

You know what? Since I’m being a psychotic space-dragon anyway, I think I want a whole golden planet. Something I can walk around while I cackle. A nice place to take stolen damsels and awe them with shining gold landscapes.

Well, a billion dollars’ worth of Dragon’s Eggs isn’t going to cut it. The sphere’s surface gravity is a pathetic 2.96 microns per second squared. I’m fascinated by gravity, and so I often find myself working out the surface gravity of objects like asteroids of different compositions. Asteroids have low masses and densities and therefore have very weak gravity. The asteroid 433 Eros, one of only a few asteroids to be visited and mapped in detail by a space probe (NEAR-Shoemaker), has a surface gravity of about 6 millimeters per second squared (This varies wildly because Eros is far from symmetrical. Like so many asteroids, it’s stubbornly and inconveniently peanut-shaped. There are places where the surface is very close to the center of mass, and other places where it sticks way up away from it.) The usual analogies don’t really help you get a grasp of how feeble Erotian gravity is. The blue whale, the heaviest organism (living or extinct, as far as we know) masses around 100 metric tons. On Earth, it weighs 981,000 Newtons. You can also say that it weighs 100 metric tons, because there’s a direct and simple equivalence between mass and weight on Earth’s surface. Just be careful: physics dorks like me might try to make a fool out of you. Anyway, on Eros, a blue whale would weigh 600 Newtons, which, on Earth, would be equivalent to a mass of 61 kilograms, which is about the mass of a slender adult human.

But that’s really not all that intuitive. It’s been quite a while since I tried to lift 61 kilograms of anything. When I’m trying to get a feel for low gravities, I prefer to use the 10-second fall distance. That (not surprisingly) is the distance a dropped object would fall in 10 seconds under the object’s surface gravity. You can calculate this easily: (0.5) * (surface gravity) * (10 seconds)^2. I want you to participate in this thought experiment with me. Take a moment and either stare at a clock or count “One one thousand two one thousand three one thousand…” until you’ve counted off ten seconds. Do it. I’ll see you in the next paragraph.

In those ten seconds, a dropped object on Eros would fall 30 centimeters, or about a foot. For comparison, on Earth, that dropped object would have fallen 490 meters. If you neglect air resistance (let’s say you’re dropping an especially streamlined dart), it would have hit the ground after 10 seconds if you were standing at the top of the Eiffel Tower. You’d have to drop it from a very tall skyscraper (at least as tall as the Shanghai World Financial Center) for it to still be in the air after ten seconds.

But my shiny golden sphere pales in comparison even to Eros. Its 10-second fall distance is 148 microns. That’s the diameter of a human hair (not that I’d allow feeble humans on my golden dragon-planet). That’s ridiculous. Clearly, we need more gold.

Well, like Dr. Evil, we can increase our demand: 1 trillion US dollars. That comes out to 18,768,769 of my golden spheres. That’s 23,649,000 kilograms of gold. My hoard would have a diameter of 14.6 meters and a surface gravity of 29.6 microns per second squared (a 10-second fall distance of 1.345 millimeters, which would just barely be visible, if you were paying close attention.) I am not impressed. And you know what happens when a dragon is not impressed? He goes out and steals shit. So I’m going to go out and steal the entire world’s economy and convert it into gold. I’m pretty sure that will cause Superman and/or Captain Planet to declare me their nemesis, but what psychotic villain is complete without a nemesis?

It’s pretty much impossible to be certain how much money is in the world economy, but estimates seem to be on the order of US$50 trillion (in 2014 dollars). That works out to 938,438,439 gold balls (you don’t know how hard I had to fight to resist calling my currency the Hobo Sullivan Golden Testicle). That’s a total mass of 1.182e9 kilograms (1.182 billion kilograms) and a diameter of 54 meters (the balls still aren’t being crushed out of shape, so the packing efficiency is still stuck at 74%). 54 meters is pretty big in human terms. A 54-meter gold ball would make a pretty impressive decoration outside some sultan’s palace. If it hit the Earth as an asteroid, it would deposit more energy than the Chelyabinsk meteor, which, even though it exploded at an altitude of 30 kilometers, still managed to break windows and make scary sounds like this:

This golden asteroid would have a surface gravity of 0.108 millimeters per second squared, and a 10-second fall distance of 0.54 centimeters. Visible to the eye if you like sitting very still to watch small objects fall in weak gravitational fields (and they say I have weird hobbies), but still fairly close to the kind of micro-gravity you get on space stations. I can walk across the room, get my coffee cup, walk back, and sit down in 10 seconds (I timed it), and my falling object would still be almost exactly where I left it.

Obviously, we need to go bigger. Most small asteorids do not even approach hydrostatic equilibrium: they don’t have enough mass for their gravity to crush their constituent materials into spheres. For the majority of asteroids, the strength of their materials is greater than gravitational forces. But the largest asteroids do start to approach hydrostatic equilibrium. Here’s a picture of 4 Vesta, one of the other asteroids that’s been visited by a spacecraft (the awesome ion-engine-powered Dawn, in this case.)

(Image courtesy of NASA via Wikipedia.)

You’re probably saying “Hobo, that’s not very fucking spherical.” Well first of all, that’s a pretty damn rude way to discuss asteroids. Second of all, you’re right. That’s partly because of its gravity (still weak), partly because its fast rotation (once every 5 hours) deforms it into an oblate spheroid, and partly because of the massive Rheasilvia crater on one of its poles (which also hosts the solar system’s tallest known mountain, rising 22 kilometers above the surrounding terrain). But it’s pretty damn spherical when you compare it to ordinary asteroids, like 951 Gaspra, which is the shape of a chicken’s beak. It’s also large enough that its interior is probably more similar to a planet’s interior than an asteroid’s. Small asteroids are pretty much homogenous rock. Large asteroids contain enough rock, and therefore enough radioactive minerals and enough leftover heat from accretion, to heat their interiors to the melting point, at least briefly. Their gravity is also strong enough to cause the denser elements like iron and nickel to sink to the center and form something approximating a core, with the aluminosilicate minerals (the stuff Earth rocks are mostly made of) forming a mantle. Therefore, we’ll say that once my golden asteroid reaches the same mass as 4 Vesta, the gold in the center will finally be crushed sufficiently to squeeze out the empty space.

It would be convenient for my calculations if the whole asteroid melted so that there were no empty spaces anywhere. Would that happen, though? That’s actually not so hard to calculate. What we need is the golden asteroid’s gravitational binding energy, which is the amount of energy you’d need to peel the asteroid apart layer by layer and carry the layers away to infinity. This is the same amount of energy you’d deposit in the asteroid by assembling it one piece at a time by dropping golden balls on it. A solid gold (I’m cheating there) asteroid with Vesta’s mass (2.59e20 kg) would have a radius of 147 kilometers and a gravitational binding energy of about 1.827e25 Joules, or about the energy of 37 dinosaur-killing Chicxulub impacts. That’s enough energy to heat the gold up to 546 Kelvin, which is less than halfway to gold’s melting point.

But, you know what? Since I don’t have access to a supercomputer to model the compressional deformation of a hundred million trillion kilograms of close-packed gold spheres, I’m going to streamline things by melting the whole asteroid with a giant draconic space-laser. I’ll dispense with the gold spheres, too, and just pour molten gold directly on the surface.

You know where this is going: I want a whole planet made of gold. But if I’m going to build a planet, it’s going to need a name. Let’s call it Dragon’s Hoard. Sounds like a name Robert Forward would give a planet in a sci-fi novel, so I’m pleased. Let’s pump Dragon’s Hoard up to the mass of the Earth.

Dragon’s Hoard is a weird planet. It has the same mass as Earth, but its radius is only 66% of Earth’s. Its surface gravity is 22.64 meters per second squared, or 2.3 earth gees. Let’s turn off the spigot of high-temperature liquid gold (of course I have one of those) for a while and see what we get.

According to me, we get something like this:

Gold Tectonics

The heat content of a uniform-temperature sphere of liquid gold depends on its volume, but since it’s floating in space, its rate of heat loss depends on surface area (by the Stefan-Boltzmann law). The heat can move around inside it, but ultimately, it can only leave by radiating off the surface. Therefore, not only will the sphere take a long time to cool, but its upper layers will cool much faster than its lower layers. Gold has a high coefficient of thermal expansion: it expands more than iron when you heat it up. Therefore, as the liquid gold at the surface cools, it will contract, lose density, and sink beneath the hotter gold on the surface. It will sink and heat up to its original temperature, and will eventually be displaced by the descent of cooler gold and will rise back to the surface. When the surface cools enough, it will solidify into a solid-gold crust, which is awesome. Apparently, my fantasies are written by Terry Pratchett, which is the best thing ever. I’ve got Counterweight Continents all over the place!

Gold is ductile: it’s a soft metal, easy to bend out of shape. Therefore, the crust would deform pretty easily, and there wouldn’t be too many earthquakes. There might, however, be volcanoes, where upwellings of liquid gold strike the middle of a plate and erupt as long chains of liquid-gold fountains. It would behave a bit like the lava lake at Kilauea volcano, in Hawai’i. See below:

What a landscape this would be! Imagine standing (in a spacesuit) on a rumpled plain of warm gold. To your right, a range of gold mountains glitter in the sun, broken here and there by gurgling volcanoes of shiny red-hot liquid. Flat, frozen puddles of gold fill the low spots, concave from the contraction they experienced as they cooled. To your right , the land undulates along until it reaches another mountain range. In a valley at the foot of this range is an incandescent river of molten gold, fed by the huge shield volcano just beyond the mountains. Then a psychotic space-dragon swoops down, flying through the vacuum (and also in the face of physics), picks you up in his talons, carries you over the landscape, and drops you into one of those volcanoes.

Yeah. It would be something like that.

As fun as my golden planet is, I think we could go bigger. Unfortunately, the bigger it gets, the more unpredictable its properties become. As we keep pouring molten gold on it, its convection currents will become more and more vigorous: it will have more trapped heat, a larger volume-to-surface area ratio, and stronger gravity, which will increase the buoyant force on the hot, low-density spots. Eventually, we’ll end up with convection cells, much like you see in a pot of boiling water. They might look like this:

Benard Cells

Those are Rayleigh-Bénard cells, which you often get in convective fluids. I used that same picture in my Endless Sky article. But there, I was talking about supercritical oxygen and nitrogen. Here, it’s all gold, baby.

Eventually, the convection’s going to get intense enough and the heat’s going to get high enough that the planet will have a thin atmosphere of gold vapor. If it rotates, the planet will also develop a powerful magnetic field: swirling conductive liquid is believed to be the thing that creates the magnetic fields of Earth, the Sun, and Jupiter (and all the other planets). This planet is going to have some weird electrical properties. Gold is one of the best conductors there is, second only to copper, silver, graphene, and superconductors. Therefore, expect some terrifying lightning on Dragon’s Hoard: charged particles and ultraviolet radiation from the Sun will ionize the surface, the gold atmosphere (if there is one) or both, and Dragon’s Hoard will become the spherical terminal in a gigantic Van de Graaff generator. As it becomes more and more charged, Dragon’s Hoard will start deflecting solar-wind electrons more easily than solar-wind protons (since the protons are more massive), and will soak up protons, acquiring a net positive charge. It’ll keep accumulating charge until the potential difference explosively equalizes. Imagine a massive jet or bolt of lightning blasting up into space, carrying off a cloud of gold vapor, glowing with pink hydrogen plasma. Yikes.

After Dragon’s Hoard surpasses Jupiter’s mass, weird things will begin happening. Gold atoms do not like to fuse. Even the largest stars can’t fuse them. Therefore, the only things keeping Dragon’s Hoard from collapsing altogether are the electrostatic repulsion between its atoms and the thermal pressure from all that heat. Sooner or later, neither of these things will be enough, and we’ll be in big trouble: the core, compressed to a higher density than the outer portions by all that gold, will become degenerate: its electrons will break loose of their nuclei, and the matter will contract until the electrons are squeezed so close together that quantum physics prevents them from getting any closer. This is electron degeneracy pressure, and it’s the reason white dwarf stars can squeeze the mass of a star into a sphere the size of a planet without either imploding or exploding.

The equations involved here are complicated, and were designed for bodies made of hydrogen, carbon, and oxygen instead of gold (Those short-sighted physicists never consider weird thought experiments when they’re unraveling the secrets of the universe. The selfish bastards.) The result is that I’m not entirely sure how large Dragon’s Hoard will be when this happens. It’s a good bet, though, that it’ll be somewhere around Jupiter’s mass. This collapse won’t be explosive: at first, only a fraction of the matter in the core will be degenerate.As we add mass, the degenerate core will grow larger and larger, and more and more of it will become degenerate. It will, however, start to get violent after a while. Electron-degenerate matter is an excellent conductor of heat, and its temperature will equalize pretty quickly. That means that we’ll have a hot ball of degenerate gold (Degenerate Gold. Add that to the list of possible band names.) surrounded by a thin layer of hot liquid gold. Because of the efficient heat transfer within the degenerate core, it will partly be able to overcome the surface-area-versus-volume problem and radiate heat at a tremendous rate.  The liquid gold on top, though, will have trouble carrying that heat away fast enough, and will get hotter and hotter, and thanks to its small volume, will eventually get hot enough to boil. Imagine a planet a little larger than Earth, its surface white-hot, crushed under a gravity of 500,000 gees, bubbles exploding and flinging evaporating droplets of gold a few kilometers as gaseous gold and gold plasma jet up from beneath. Yeah. Something like that.

But in a chemical sense, my huge pile of gold is still gold. The nuclei may be uncomfortably close together and stripped of all of their electrons, but the nuclei are still gold nuclei. For now. Because you know I’m going to keep pumping gold into this ball to see what happens (That’s also a line from a really weird porno movie.)

White dwarfs have a peculiar property: the more massive they are, the smaller they get. That’s because, the heavier they get, the more they have to contract before electron degeneracy pressure balances gravity. Sirius B, one of the nearest white dwarfs to Earth, has a mass of about 1 solar mass, but a radius similar to that of Earth. When Dragon’s Hoard reached 1.38 solar masses, it would be even smaller, having a radius of around 3000 kilometers. The stream of liquid gold would fall towards a blinding white sphere, striking the surface at 3% of the speed of light. The surface gravity would be in the neighborhood of 2 million gees. If the gravity were constant (which it most certainly would not be), the 10-second fall distance would be 2.7 times the distance between Earth and moon. Now we’re getting into some serious shit.

Notice that I specified three significant figures when I gave that mass: 1.38 solar masses. That is not by accident. As some of you may know, that’s dangerously close to 1.39 solar masses, the Chandrashekar limit, named after the brilliant and surprisingly handsome Indian physicist Subrahmanyan Chandrasekhar. (Side note: Chandrashekhar unfortunately died in 1995, but his wife lived until 2013. Last year. She was 102 years old. There’s something cool about that, but I don’t know what it is.) The Chandrasekhar limit is the maximum mass a star can have and still be supported by electron degeneracy pressure. When you go above that, you’ve got big trouble.

When ordinary white dwarfs (made mostly of carbon, oxygen, hydrogen, and helium) surpass the Chandrasekhar limit by vacuuming mass from a binary companion, they are unable to resist gravitational contraction. They contract until the carbon and oxygen nuclei in their cores get hot enough and close enough to fuse and make iron. This results in a Type Ia supernova, which shines as bright as 10 billion suns. It’s only recently that our supercomputers have been able to simulate this phenomenon. The simulations are surprisingly beautiful.

I could watch that over and over again and never get tired of it.

Unfortunately, even though it’s made of gold (which, as I said, doesn’t like to fuse), the same sort of thing will happen to Dragon’s Hoard. When it passes the Chandrasekhar limit, it will rapidly contract until the nuclei are touching. This will trigger a bizarre form of runaway fusion. The pressure will force electrons to combine with protons, releasing neutrinos and radiation. Dragon’s Hoard will be heated to ludicrous temperatures, and a supernova will blow off its outer layers. What remains will be a neutron star, which, as I talked about in The Weather in Hell, is mostly neutrons, with a thin crust of iron atoms and an even thinner atmosphere of iron, hydrogen, helium, or maybe carbon. Most or all of the gold nuclei will be destroyed. The only thing that will stop the sphere from turning into a black hole is that, like electrons, neutrons resist being squeezed too close together, at least up to a limit.

But you know what? That tells us exactly how much gold you can hoard in one place: about 1.38 solar masses. So fuck you, Taylor from kindergarten! You can’t have infinity dollars! You can only have 0.116 trillion trillion trillion dollars (US, and according to June 2014 gold prices) before your gold implodes and transmutes itself into other elements! So there!

But while I’m randomly adding mass to massive astronomical objects (that’s what space dragons do instead of breathing fire), let’s see how much farther we can go.

The answer is: Nobody’s exactly certain. The Chandrasekhar limit is based on pretty well-understood physics, but the physics of neutron-degenerate matter at neutron star pressures and temperatures (and in highly curved space-time) is not nearly so well understood. The Tolman-Oppenheimer-Volkoff limit (Yes, the same Oppenheimer you’re thinking of.) is essentially a neutron-degenerate version of the Chandrasekhar limit, but we only have the TOV limit narrowed down to somewhere between 1.5 solar masses and 3 solar masses. We’re even less certain about what happens above that limit. Quarks might start leaking out of neutrons, the way neutrons leak out of nuclei in a neutron star, and we might get an even smaller, denser kind of star (a quark star). At this point, the matter would stop being matter as we know it. It wouldn’t even be made of neutrons anymore. But to be honest, we simply don’t know yet.

Sooner or later, though, Karl Schwarzschild is going to come and kick our asses. He solved the Einstein field equations of general relativity (which are frightening but elegant, like a hyena in a cocktail dress) and discovered that, if an object is made smaller than its a certain radius (the Schwarzschild radius), it will become a black hole. The Schwarzschild radius depends only on the object’s mass, charge, and angular momentum. Dragon’s Hoard, or rather what’s left of it, doesn’t have a significant charge or angular momentum (because I said so), so its Schwarzschild radius depends only on mass. At 3 solar masses, the Schwarzschild radius is 8.859 kilometers, which is only just barely larger than a neutron star. Whether quark stars can actually form or not, you can bet your ass they’re going to be denser than neutron stars. Therefore, I’d expect Dragon’s Hoard to fall within its own Schwarzschild radius somewhere between 3 and 5 solar masses. Let’s say 5, just to be safe. There are suspected black holes with masses near 5.

That’s the end of Dragon’s Hoard. The physics in the center gets unspeakably weird, but the gold-spitting space dragon doesn’t get to see it. He’s outside the event horizon, which means the collapse of his hoard is hidden to him. He just sees a black sphere with a circumference of 92.77 kilometers, warping the images of the stars behind it. It doesn’t matter how much more gold we pour into it now: it’s all going to end up inside the event horizon, and the only noticeable effect will be that the event horizon’s circumference will grow larger and larger. But fuck that. If I wanted to throw money down a black hole, I’d just go to Vegas. (Heyo!) Dragon’s Hoard isn’t getting any more of my draconic space-gold.

But one last thing before I go. Notice how I suddenly went from saying Schwarzschild radius to talking about the event horizon’s circumference. That’s significant. Here’s a terrible picture illustrating why I did that:

CurvedSpacetime

Massive objects create curvature in space-time. Imagine standing at the dot on circle B, in the top picture. If you walk to the center-point along line A, you’ll measure a length a. If you then walk around circle B, you’ll get the circle’s circumference. You’ll find that that circumference is 2 * pi * a. The radius is therefore (circumference) / (2 * pi) But that only holds in flat space. When space is positively curved (like it is in the vicinity of massive objects), the radius of a circle will always be larger than (circumference) / (2 * pi). That is to say, radius C in the bottom picture is significantly longer than radius A in the top one, and longer than you would expect from the circumference of circle D.

In other words, the radius of a massive object like a star, a neutron star, or a black hole, differs from what you would expect based on its circumference. The existence of black holes and neutron stars has not actually been directly confirmed (because they’re so small and so far away). It is merely strongly suspected based on our understanding of physics. The existence of spacetime curvature, though, has been confirmed in many experiments.

Imagine you’re standing in a field that looks flat. There’s a weird sort of bluish haze in the center, but apart from that, it looks normal. You walk in a circle around the haze to get a better look at it. It only takes you fifteen minutes to walk all the way around and get back to your starting point. The haze makes you nervous, so you don’t walk straight into it. Instead, you walk on a line crossing the circle so that it passes halfway between the haze and the circle’s edge at its closest approach. Somehow, walking that distance takes you twenty minutes, which is not what you’d expect. When you walk past the haze at a quarter-radius, it takes you an hour. When you walk within one-eighth of a radius, it takes you so long you have to turn back and go get some water. Each time, you’re getting closer and closer to walking along the circle’s radius towards its center, but if you actually tried to walk directly into its center, where the haze is (the haze is because there’s so much air between you and the stuff beyond the haze, which is the same reason distant mountains look blue), you would find that the distance is infinite.

That’s how black holes are. They’re so strongly-curved that there’s way more space inside than there should be. The radius is effectively infinite, which is why it’s better to talk about circumference. As long as the black hole is spherically symmetric, circumferences are still well-behaved.

But the radius isn’t actually infinite. When you consider distance scales close to the Planck length, Einstein’s equations butt heads with quantum mechanics, and physicists don’t really know what the fuck’s going on. We still don’t know what happens near a black hole’s central singularity.

Incidentally, the Planck length compared to the diameter of an atom is about the same as the diameter of an atom compared to the diameter of a galaxy. The Universe is a weird place, isn’t it?

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