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
engineering, math, physics, thought experiment

The Treachery of Plumb-Lines

I’m pretty sure that’s my most pretentious article title to date, but really, the only pretentious thing about it is that it’s a Rene Magritte reference, because if you read it literally, that’s exactly what this article is about.

Imagine two skyscrapers. Both start from ordinary concrete foundations 100 meters by 100 meters, and each will be 1,000 meters high, when finished. We’ll call the first skyscraper Ruler, and the second skyscraper Plumb, for reasons I’ll explain.

Ruler is built exactly according to architectural specifications. Every corner is measured with a high-grade engineer’s square and built at precisely 90 degrees. Importantly, Ruler is constructed so that every floor is precisely 10 meters above the previous one, and every floor is 100 meters by 100 meters. This is done, of course, using a ruler. Because it’s kept so straight and square at every stage, Ruler is a very straight, square building.

Plumb, on the other hand, is kept straight and square using one of the oldest tricks in the architect’s book: the plumb-bob. True story: plumb-bobs are called that because, back in the day, they were almost always made of lead, and the Latin for lead is plumbus (or something like that; I took Latin in high school, but the teacher got deathly ill like two weeks in, so I never learned much). A well-made and well-applied plumb-bob is an excellent way to make sure something is absolutely vertical.

The builders of Plumb do use a ruler, but only to mark off the 10-meter intervals for the floors. They mark them off at the corners of the building, and they make sure the floors are perfectly horizontal using either a modified plumb-bob or a spirit level (which is largely the same instrument).

One might assume that Plumb and Ruler would turn out to be the exact same building. But anybody who’s read this blog knows that that’s the kind of sentence I use to set up a twist. Because Plumb was kept straight using plumb-bobs, and because plumb-bobs point towards the center of the Earth, and because the 100-meter difference between the east and west (or north and south walls) gives the bobs an angle difference of 0.009 degrees, Plumb is actually 11 millimeters wider at the top than at the bottom. Probably not enough to matter in architectural terms, but the difference is there.

Not only that, but Plumb’s floors aren’t flat, either, at least not geometrically flat. The Earth is a sphere, and because Plumb’s architects made its floors level with a spirit level or a plumb-bob, those floors aren’t geometrically flat: they follow the spherical gravitational equi-potential contours. Over a distance of 100 meters, the midpoint of a line across the Earth’s surface sits 0.2 millimeters above where it would were the line perfectly, geometrically straight. This difference decreases by the time you reach the 100th floor (the top floor) because the sphere in question is larger and therefore less strongly curved. But the difference only decreases by around a micron, which is going to get swamped out by even really small bumps in the concrete.

“Okay,” you might say, “so if you blindly trust a plumb-bob, your building will end up a centimeter out-of-true. What does that matter?” Well, first of all, if you came here looking for that kind of practicality, then this blog is just gonna drive you insane. Second, it doesn’t matter so much for ordinary buildings. But let’s say you’re building a 2,737-meter-long bridge (by total coincidence, the length of the Golden Gate Bridge). If you build with geometric flatness in mind, your middle pier is going to have to be 14.7 centimeters shorter than the ones at the ends. That’s almost the length of my foot, and I’ve got big feet. It’s not a big enough difference that you couldn’t, say, fill it in with concrete or something, but it’d certainly be enough that you’d have to adjust where your bolt-holes were drilled.

What’s the moral of this story? It’s an old moral that probably seems fairly ridiculous, but is nonetheless true: we live on the surface of a sphere. And, when it comes down to it, that’s just kinda fun to think about.

Standard
biology, science, silly, thought experiment

Life at 1:1000 Scale, Part 1

You can’t see it, but out in the real world, I look like a Scottish pub brawler. I’ve got the reddish beard and the roundish Scots-Irish face and the broad shoulders and the heavy build I inherited from my Scotch and Irish ancestors (the hairy arms come from my Italian ancestors).

What I’m saying is that I’m a bulky guy. I stand 6 feet, 3 inches tall. That’s 190.5 centimeters, or 1,905 millimeters. Keep that figure in mind.

When I was a kid, the motif of someone getting shrunk down to minuscule size was popular. It was the focus of a couple of books I read. There was that one episode of The Magic Schoolbus which was pretty much just The Fantastic Voyage in cartoon form. There was the insufferable cartoon of my late childhood, George Shrinks.

As a kid, I was very easily bored. When I got bored waiting in line for the bathroom, for instance, I would imagine what it would actually be like to be incredibly tiny. I imagined myself nestled among a forest of weird looping trees: the fibers in the weird multicolored-but-still-gray synthetic carpet my school had. I imagined what it would be like to stand right beneath my own shoe, shrunk down so small I could see atoms. I realized that the shoe would look nothing like a shoe. It would just be this vast plain of differently-colored spheres (that was how I envisioned atoms back then, because that’s how they looked in our science books).

Now, once again, I find myself wanting to re-do a childhood thought experiment. What if I were shrunk down to 1/1000th of my actual size? I’d be 1.905 millimeters tall (1,905 microns): about the size of those really tiny black ants with the big antennae that find their way into absolutely everything. About the size of a peppercorn.

Speaking of peppercorns, let’s start this bizarre odyssey in the kitchen. I measured the height of my kitchen counter as exactly three feet. But because I’m a thousand times smaller, the counter is a thousand times higher. In other words: two-thirds the height of the intimidating Mount Thor:

mount_thor

(Source.)

I remember this counter as being a lot smoother than it actually is. I mean, it always had that fine-textured grainy pattern, but now, those textural bumps, too small to measure when I was full-sized, are proper divots and hillocks.

I don’t care how small I am, though: I intend to have my coffee. Anybody who knows me personally will not be surprised by this. It’s going to be a bit trickier now, since the cup is effectively a mile away from the sugar and the jar of coffee crystals, but you’d better believe I’m determined when it comes to coffee.

Though, to be honest, I am a little worried about my safety during that crossing. There’s a lot more wildlife on this counter than I remember. There’s a sparse scattering of ordinary bacteria, but I don’t mind them: they’re no bigger than ants even at this scale, so I don’t have to confront their waxy, translucent grossness. There is what appears to be a piece of waxy brown drainage pipe lying in my path, though. It’s a nasty-looking thing with creepy lizard-skin scales up and down it. I think it’s one of my hairs.

I’m more concerned about the platter-sized waxy slab lying on the counter next to the hair. There are two reasons for this: First, I’m pretty sure the slab is a flake of sloughed human skin. Second, and most important, that slab is being gnawed on by a chihahua-sized, foot-long monstrosity:

8f87b071ca15a175804fa780020feade

I know it’s just a dust mite, but let me tell you, when you see those mandibles up close, and those mandibles are suddenly large enough to snip off a toe, they suddenly get a lot more intimidating. This one seems friendly enough, though. I petted it. I think I’m gonna call it Liam.

My odyssey to the coffee cup continues. It’s a mile away, at my current scale, but I know from experience I can walk that far in 20 minutes. But the coffee cup is sitting on a dishcloth, drying after I last rinsed it out, and that dishcloth is the unexpected hurdle that shows up in all the good adventure books.

The rumpled plateau that confronts me is 10 meters high (32 feet, as tall as a small house or a tree), and its surface looks like this:

cover-12-3_1

(Source.)

Those creepy frayed cables are woven from what looks like translucent silicone tubing. Each cable is about as wide as an adult man. If I’d known I was going to be exposed to this kind of weird-textured information overload, I never would’ve shrunk myself down. But I need my coffee, and I will have my coffee, so I’m pressing forward.

But, you know, now that I’m standing right next to the coffee cup, I’m starting to think I might have been a little over-ambitious. Because my coffee cup is a gigantic ceramic monolith. It’s just about a hundred meters high (333 feet): as tall as a football field (either kind) is long–as big as a 19-story office building. I know insects my size can lift some ridiculous fraction of their body weight, but I think this might be a bit beyond me.

All’s not lost, though! After another twenty-minute trek, I arrive back at the sugar bowl and the jar of coffee. Bit of a snag, though. It seems some idiot let a grain of sugar fall onto the counter (that grain is now the size of a nightstand, and is actually kinda pretty: like a huge crystal of brownish rock salt), which has attracted a small horde of HORRIFYING MONSTERS:

pharoh-ant4-x532-new

(Source.)

That is a pharaoh ant. Or, as we here in the Dirty South call them, “Oh goddammit! Not again!” In my ordinary life, I knew these as the tiny ants that managed to slip into containers I thought tightly closed, and which were just about impossible to get rid of, because it seemed like a small colony could thrive on a micron-thin skid of ketchup I’d missed when last Windexing the counter.

Trouble is that, now, they’re as long as I am tall, and they’re about half my height at the shoulder. And they’ve got mandibles that could clip right through my wrist…

Okay, once again, I shouldn’t have panicked. Turns out they’re actually not that hostile. Plus, if you climb on one’s back and tug at its antennae for steering, you can ride it like a horrifying (and very prickly-against-the-buttock-region) pony!

I’m naming my new steed Cactus, because those little hairs on her back are, at this scale, icepick-sized thorns of death. I’m glad Cactus is just a worker, because if she was a male or a queen, I’m pretty sure she would have tried to mate with me, and frankly, I don’t like my chances of coming out of that intact and sane. Workers, though, are sterile, and Cactus seems a lot more interested in cleaning herself than mounting me, for which my gratitude is boundless.

I’ve ridden her to my coffee spoon, because I’m thinking I can make myself a nice bowl of coffee in the spoon’s bowl.

I’ve clearly miscalculated, and quite horribly, too: the bowl of this spoon is the size of an Olympic swimming pool: 50 meters (160 feet) from end to end. Plus, now that I’m seeing it from this close, I’m realizing that I haven’t been doing a very good job of cleaning off my coffee spoon between uses. It’s crusted with a patchy skin of gunk, and that gunk is absolutely infested with little poppy-seed-sized spheres and sausages and furry sausages, all of which are squirming and writing a little too much like maggots for my taste. I’m pretty sure they’re just bacteria, but I’m not going to knowingly go out and touch germs. Especially not when they’re just about the right size to hitch a ride on my clothes and covertly crawl into an orifice when I’m sleeping.

You know what? If I can’t have my coffee, I think this whole adventure was probably a mistake. I think I’m going to return to my ordinary body. Conveniently (in more ways than one), I’ve left my real body comatose and staring mindlessly at the cabinets above the counter. He’s a big beast: a mile high, from my perspective. An actual man-mountain. I’ll spare you the details of climbing him, because he wears shorts and I spent far too long climbing through tree-trunk-sized leg hairs with creepy-crawly skin microflora dangerously close to my face.

Now, though, I’m back in my brain and back at my normal size. And now that my weird little dissociative fugue is over, I can tell you guys to look out for part two, when I’ll tell you all the reasons there’s no way to actually shrink yourself down like that and live to tell about it.

Standard
update

Stair Power!

Let’s say I, a human male massing 132 kg and carrying a 9-kilogram box of dusty old knick-knacks and other junk, ascend a 10-meter-high flight of stairs in 15 seconds. During that time, I’m expending 922 Watts, which is over nine times a human’s resting metabolism, and about one and a quarter horsepower.

This isn’t the start of a thought experiment. This is just my silly way of letting you guys know why I’ve been silent lately: helping people move is kind of a pain. Good exercise, though, if they, say, live on the third floor.

Standard
physics, science, silly, thought experiment

The Overkill Oven

As I was lying in bed last night, I started wondering: “What if I had an oven that could heat its contents up to nuclear-fusion temperatures?” This is why I have trouble sleeping: my brain is very badly-wired. But still, that’s a perfect question for this blog. But as I was preparing to write the article, I got to thinking: Why limit myself to nuclear-fusion temperatures? Nuclear fusion only requires a few billion Kelvin. There are processes (particle-accelerator impacts and cosmic-ray collisions) that reach a trillion trillion kelvin!

Overkill Oven 450 Kelvin

Here’s my new oven. You’ll notice it has quite a few temperature knobs. That’s because, if I tried to fit 10^24 Kelvin all on one knob, that knob would have to be the size of a galaxy before the 100-Kelvin interval marks were far enough apart to see with the naked eye. The cool thing about the decimal number system, though, is that I only (“only”) need 24 knobs, each only marked with 10 intervals, to set temperatures hot enough to melt protons.

This new oven has a couple of interesting features. The first is the patented ceramic bowl-schist lining. Bowl-schist is an exotic metamorphic rock I imported from a parallel universe. Its heat conductivity is so low you could put a block of it next to a supernova and it’d be just fine. The second important feature is the power supply. Naturally, I can’t plug a fancy oven like this into a standard 240-Volt U.S. oven socket. Instead, the cable passes through a very narrow wormhole into the Handwavium Universe, which is stuck mid-big-bang, and therefore is absolutely flooded with energy. With all that set up, let’s cook! To celebrate my new oven, I think I’ll make a big beef roast, with some potatoes, peas, carrots, onions, and herbs and spices.

0.001 Kelvin

The trouble with the Oven of Doom is that the controls are a little difficult to get used to. But hey, I play Dwarf Fortress, so I’m no stranger to shockingly opaque controls. Still, starting out, I accidentally set the oven almost to zero Kelvin. I didn’t realize this until I saw the fur of oxygen and nitrogen ice growing all over my roast. Luckily, the Death Oven is also completely hermetically sealed during operation, to prevent operator death, so I didn’t freeze out all the air in the house. And defrosting was easy.

450 Kelvin

Overkill Oven 450 Kelvin

After that initial hiccup, my roast is coming along nicely. I’m making a brisket roast, so I should probably cook it long and low and slow, so it gets nice and tender. I just hope I don’t run out of patience before

1,000 Kelvin

Overkill Oven 1000 Kelvin

Well that could have gone better. In my defense, this oven has a lot of knobs, and if there’s anything resembling a knob or switch, I am compelled to fiddle with it. The roast was on fire for a few minutes, but once most of the fat burned off, it settled down. Now I’m left with an oven full of glowing orange soot and carbonized meat and vegetables. I can probably find some creature willing to eat it…

5,000 Kelvin

Overkill Oven 5000 Kelvin

The trouble with having a fancy high-power oven is that it’s really tempting to turn it up unnecessarily high in the hopes of getting your food finished as quick as possible. I think there might be something to all this “slow food” stuff I keep hearing about. Trying to cook my roast at 5,000 Kelvin has reduced it to a cloud of white-hot soot with a pale yellow vapor of sodium, potassium, and iron simmering over it. Still, at least I can be sure it’s safe for the people who insist on having their beef well-done.

10,000 Kelvin

Overkill Oven 10000 Kelvin

You know, I should probably close the shutter over that porthole… It’s getting awfully bright in there. I’m pretty sure the roast hasn’t escaped, but truth be told, when I look in there, all I see is this screaming blue-white fog of ionized carbon. On the plus side, if I hurry up and buy a second roast, I can cook it with the light from the first one.

100,000 Kelvin

Overkill Oven 1e5 K.png

I think I’m starting to understand now why the oven’s window is more of a peephole. It’s only three inches across, but already I shouldn’t be able to stand in front of it without my legs evaporating. Actually, I shouldn’t be able to have the peephole open without my house exploding in a horrendous fireball. The oven’s emitting more power from radiant heat alone than the Three Gorges Dam. But I can hold my hand in front of the porthole, no problem. I think I’m starting to see why the department store I bought it from was called BS & Sons…

5,000,000 Kelvin

Overkill Oven 5e6 K.png

I don’t think I have the right to keep calling this thing a roast, do I? It’s really just a soup of highly-ionized carbon, oxygen, iron (from the myoglobin in the meat, and from what used to be my nice new roasting pan), nitrogen, sulfur, and trace metals. On the plus side, I’ve got my own pet solar flare now!

10,000,000 Kelvin

It’s not all bad news, though. The oven is now self-powering. All those hydrogen atoms that used to be part of things like fats, proteins and starches have long since evaporated into a searing plasma. Now, though, they’re colliding fast enough that they’re starting to fuse. Not only am I getting extra energy from this, but I’m making homemade helium, too! Cooking’s fun!

100,000,000 Kelvin

Well, I’ve gone and overdone it again. I burned up all the helium I just made! Now it’s gone and fused to make more carbon vapor. I should probably call somebody about this. Frankly, at this point, I’m afraid to turn the oven off. I mean, since the thermal conductivity is pretty much zero, it’s never going to cool down. And if I open the door, I’m going to release as much energy as detonating 30 tons of TNT. I think I’ll just wall off the kitchen and pretend none of this ever happened…

500,000,000 Kelvin

I am now essentially cooking my roast with a continuous nuclear explosion. Also, I’m pretty sure that, even if I managed to cool it down, not even a physicist with a mass-spectrometer would be able to identify what the roast used to be. That’s partly because, of course, it’s been thoroughly vaporized. But also, the carbon nuclei have started fusing to form weird stuff like neon. If you find an organism that likes to eat neon, send it my way. I’ve got a roast for it.

1,500,000,000 Kelvin

My oven now contains as much energy as a half-kiloton nuclear explosion. The oxygen nuclei are fusing to form things like phosphorus, magnesium, and silicon. If the peephole wasn’t made of pure handwavium crystal, it would be emitting more power (briefly) than the Sun.

3,000,000,000 Kelvin

The good news is that I got my roasting pan back, and then some! All the light atoms have pretty much fused into heavier elements, which have fused to form Nickel-56. If I opened the door, I would be violently vaporized, but after the fallout cooled, the Nickel-56 would decay into Cobalt-56 and then Iron-56, and I’d be able to re-cast my roasting pan!

12,000,000,000 Kelvin

All that brilliant blue-white death-light that filled the oven is finally starting to fade. The bad news is that that’s only fading because the thermal radiation is so intense that it’s actually spontaneously turning into matter and antimatter, forming electron-positron pairs. The other bad news is that I’ve lost my roasting pan again: the energy of the particles in the oven has exceeded the binding energy per nucleon of iron, which is the tightest-bound atomic nucleus. In other words, my stupid iron atoms are starting to melt and shed protons and neutrons. Oh well. Maybe I’ll make some really exotic elements and get them named after me. And if IUPAC won’t name them after me, I’ll threaten to open my death-oven, which has long since become a weapon of mass destruction.

5,900,000,000,000 Kelvin

By now, the iron nuclei should have melted. All I need to do is heat them a little more to get that nice gooey brown crust. Except, I just checked, and I’m pretty sure the protons and neutrons are also melting. It’s just a very thin soup of quarks and un-named nonsense particles in there. Just like the Standard Model, amirite? Sorry. I shouldn’t be joking about particle physicists. Actually, speaking of particle physicists, could somebody call one of them? Because I’ve got three kilograms of pure quark-gluon plasma that they’ll probably want to study. You know, if they’re obscenely brave and not concerned about the 1.9 megatons of thermal energy packed into my oven. To be fair, if the door was gonna fail, I’m pretty sure it would have done it by now.

I’m really glad I spent the extra money on the Handwavium Universe power connector. In the 15 minutes it took me to obliterate my roast and put the entire Earth in jeopardy, the oven was drawing 8,830 terawatts. I’ll have to check the electrical panel, but I’m pretty sure 37 billion amps is above the rating of the breaker for the kitchen. Now all I need to do is call BS & Sons customer service and see if there’s a way to dump what’s left of my roast back into the Handwavium Universe. I don’t think I’ll be hurting anything: the HU is way hotter than my oven can get. Actually, the HU is so hot that the laws of physics themselves are above their melting point.

Standard
astronomy, physics, short

Weight of the World

According to this report, the Earth’s mass (M⊕) is

5,972,190,000,000,000,000,000,000 kilograms

You might notice that there are an awful lot of zeros in that number. That’s because the report doesn’t actually directly specify the Earth’s mass. Like a lot of astronomical papers, it instead uses the Earth’s gravitational parameter, which is the Earth’s mass multiplied by the Newtonian gravitational constant. You see, when it comes to gravity, the force is ultimately determined by the gravitational parameter, rather than directly by the mass. As a result, the gravitational parameter is, as a rule, known to much higher accuracy than the mass. Newton’s gravitational constant is hard to measure, since it’s so tiny, so the report only gives it to six significant digits. So six significant digits is what I gave for the Earth’s mass.

I imagine you’re wondering why the hell I’m talking about all this. Well, I was thinking about planets, whose masses are very often measured in Earth masses. That made me wonder what the mass of say, a person, is, compared to the mass of the Earth. So, without further nonsense, here’s my big list of random objects measured in Earth masses. (I probably need to come up with a better name.)

2.78045 × 10-51 M⊕ : Hydrogen atom.

1.13926 × 10-24 M⊕ : a dumbbell

2.279 × 10-23 M⊕ : me

1.674 × 10-22 M⊕ : my car

7.023 × 10-20 M: the International Space Station

9.878 × 10-16 M⊕ : the Great Pyramid of Giza

1.671 × 10-12 M : Comet 67P/Churyumov-Gerasimenko

8.620 × 10-7 M⊕ (not quite a millionth): The Earth’s atmosphere

4.470 × 10-5 M : asteroid 4 Vesta.

1.590 × 10-4 M : asteroid 1 Ceres (the largest in the solar system)

2.344 × 10-4 M (two ten thousandths and change): the Earth’s oceans

 0.00219 M⊕ : Pluto

0.0123 M⊕ : the Moon

0.0552 M: Mercury

0.107 M: Mars (I always forget how small Mars actually is…)

0.815 M⊕ : Venus (Venus was my second-favorite planet as a kid, after Pluto, which was still a planet back then)

1.000 M⊕ : Earth (Might as well stick it in the list…)

10 M: Planet Nine (Lower bound. If it exists.)

14.536 M⊕ : the mass of Uranus (I still think it’s funny…)

17.148 M⊕ : Neptune

95.161 M⊕ : Saturn

317.828 M⊕ :  Jupiter

332,949 M⊕ : the Sun (1 solar mass, 1 M. Guess who finally learned how to do subscripts!)

26,600 M⊕ : the mass of TRAPPIST-1, which is significant for being one of the smallest stars ever observed, for having seven rocky planets, and for having three planets in its habitable zone. If there’s radio-communicating life on one of them, and we send a message right now, some of you might still be alive if we get the response. Not me. I’d be 98, and I suspect I’m gonna fall into a vat of curry or something stupid like that before then.

672,600 M⊕ : Sirius A, the brightest star in the sky (besides the Sun, obviously)

710,850 M⊕ : Vega, a fairly bright nearby star distorted into a lozenge shape by its rapid rotation.

1,270,000 M⊕ : Alcyone, the brightest star in the Pleiades

2,830,000 M⊕ : UY Scuti, a likely candidate for the largest known star as of March 2017. It’s around 1,700 times the diameter of the Sun, and if you placed it where the Sun is, it’d engulf Jupiter and come close to engulfing Saturn.

3,862,000 M⊕ : Betelgeuse, the bright reddish star on the shoulder of Orion (cue Rutger Hauer.) It’s also an enormous, lumpy star. If you put it where the Sun is, it’d reach at least as far as the orbit of Mars.

33,295,000 M⊕ : the larger component of Eta Carinae, an enormous, extremely bright, angry multiple star that’s so massive and so hot that it’s vomiting its own guts into space and making a pretty nebula in the process.

38,622,000 M: the poetically-named NGC 3603-A1. With 116 times the Sun’s mass, this is the largest star (as of March 2017, blah blah blah) whose mass is known with any certainty. There are other stars predicted to be more massive, but while their masses are estimated from models of stellar evolution, NGC 3603-A1’s mass is inferred from the orbital period of it and its binary companion, which is much more precise and less guess-y.

2.331 × 1015 M: the mass of the Small Magellanic Cloud, one of the Milky Way’s small galactic neighbors.

2.830 × 1017 M: the mass of our Milky Way galaxy (roughly).

4.994 × 1017 M: the mass of the Andromeda galaxy (roughly).

1.647 × 1028 M: mass of ordinary matter in the observable universe (atoms and other familiar stuff) (very roughly)

3.349 × 1029 M: mass of the observable universe, including weird stuff like dark matter and dark energy (very roughly)

Standard
astronomy, image, pixel art, science, short, Space, Uncategorized

Pixel Solar System

pixel-solar-system-grid

(Click for full view.)

(Don’t worry. I’ve got one more bit of pixel art on the back burner, and after that, I’ll give it a break for a while.)

This is our solar system. Each pixel represents one astronomical unit, which is the average distance between Earth and Sun: 1 AU, 150 million kilometers, 93.0 million miles, 8 light-minutes and 19 light-seconds, 35,661 United States diameters, 389 times the Earth-Moon distance, or a 326-year road trip, if you drive 12 hours a day every day at roughly highway speed. Each row is 1000 pixels (1000 AU) across, and the slices are stacked so they fit in a reasonably-shaped image.

At the top-left of the image is a yellow dot representing the Sun. Mercury and Venus aren’t visible in this image. The next major body is the blue dot representing the Earth. Next comes a red dot representing Mars. Then Jupiter (peachy orange), Saturn (a salmon-pink color, which is two pixels wide because the difference between Saturn’s closest and furthest distance from the Sun is just about 1 AU), Uranus (cyan, elongated for the same reason), Neptune (deep-blue), Pluto (brick-red, extending slightly within the orbit of Neptune and extending significantly farther out), Sedna (a slightly unpleasant brownish), the Voyager 2 probe (yellow, inside the stripe for Sedna), Planet Nine (purple, if it exists; the orbits are quite approximate and overlap a fair bit with Sedna’s orbit). Then comes the Oort Cloud (light-blue), which extends ridiculously far and may be where some of our comets come from. After a large gap comes Proxima Centauri, the nearest (known) star, in orange. Alpha Centauri (the nearest star system known to host a planet) comes surprisingly far down, in yellow. All told, the image covers just over 5 light-years.

Standard