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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.

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Could you kill the human race with a BB gun?

As you might have guessed, this post is heavily influenced by xkcd’s brilliant weekly What-If blog. 

While pondering meteorites striking the earth with absurd velocities, I got to wondering whether or not you could actually kill the human race with a single BB. Because physics is a frightening place, the answer to questions like this is usually yes.

To simplify the first stage of calculations, we need to know how much energy is required to kill the human race. I will call this constant the “ohgod,” and I will set it equal to the kinetic energy of a 15-kilometer-wide stony asteroid traveling at 22 kilometers per second, which would be more than sufficient to cause a mass extinction which would almost certainly wipe out the human race. One ohgod is approximately 1.283 x 10^24 joules, or about 2.6 Chicxulubs (or, as people who fear the awesomeness of Mesoamerican words put it, 2.6 dinosaur-killers). 

The mass of a BB is surprisingly hard to find, although there is a very handy chart listing the masses of high-end BBs in grains, which can easily be converted to grams. By my reckoning, a standard 4.5-millimeter (0.177 caliber) BB should weigh about 0.4 grams. In order to figure out how fast a 0.4-gram BB would have to be moving to have 1 ohgod of kinetic energy, we must solve the relativistic kinetic energy equation for velocity. The relativistic kinetic energy equation is a little unwieldy:

E = [(1/sqrt(1-v^2/c^2)) + 1] * m * c^2

I actually had to get out pen and paper to solve this equation for v. Here’s my math, to prove that I’m not a lazy cretin:

Image

When I calculated v and plugged my numbers back into the relativistic kinetic energy formula on WolframAlpha, I was greeted with one of the most satisfying things a nerd can ever see: I got back exactly 1.283 x 10^24 joules, which means I didn’t have to do all that algebra again.

As it turns out, in order to have a kinetic energy of a species-killing asteroid, a BB would have to be traveling quite fast. I would have to be traveling at 0.9999999999999999999996074284163612948528545037647345 c, in fact. That speed is slower than the speed of light by only a few parts in 10^22, which is to say by only a few parts in 10 billion trillion. A few parts in 10 billion trillion equates to a bacterium-sized drop of water added to an Olympic-sized swimming pool. Feel free to insert your own joke about homeopathy here.

Our lethal BB would be traveling almost as fast as the fastest-moving particles ever detected. I’m thinking here about the Oh-my-God particle, which was (probably) a proton that hit Earth’s atmosphere at 0.9999999999999999999999951 c. The Oh-my-God particle is faster by a hair. But our BB is still traveling a ludicrous speeds. Light is fast. A beam of light could circle the Earth in 133 milliseconds, which, if you look at reaction-time data, is about half the time it takes a human being to detect a stimulus and for the nerve impulse to travel down their arm and make a muscle contract. Very few physical objects can hold their own against light. But our BB could. If you raced our BB and a beam of light head-to-head, after a thousand years, the BB would only be lagging 3 millimeters behind, which is about the diameter of a peppercorn, which is ridiculous.

But the question here is not “How fast would a BB have to be traveling to have the same kinetic energy as a humanity-ending asteroid.” The question is “Could you actually use a hyper-velocity BB to kill the human race,” which is much more interesting and complicated. 

First of all, the BB would have an insane amount of kinetic energy. E = m * c^2, of course, and from that, we know that our BB’s kinetic energy (its energy alone) would have a mass of 1.428 * 10^7 kilograms, which is about the mass of five Boeing 747 jet airliners. I would have to be a very gifted physicist to tell you what happens when you’ve got atoms with that kind of energy, but I suspect that there would be very weird quantum effects (aren’t there always?) which would conspire to slow the BB down. Because of quantum randomness, I imagine the BB would constantly be emitting high-energy gamma rays, which would decay into electron-positron and proton-antiproton pairs. Which is to say that our BB would be moving so fast that, rather than leaving behind a wake of Cherenkov radiation, it would leave behind a wake of actual physical matter, conjured seemingly from the ether by the conversion of its kinetic energy. 

As for what would happen when the BB actually hit the Earth, that’s beyond my power to calculate, on account of I don’t have access to a fucking supercomputer. But we can assume that the BB would pass straight through the Earth with no physical impact: all of its interactions with our planet would probably be on the level of ultra-high-energy particle physics. And from that, we can estimate its effects.

The BB would cut a cylindrical path 4.5 millimeters across and 1 earth diameter long. If it deposited all of its kinetic energy along this track, it would raise the temperature of the rock by 10^15 Kelvin, which would make it one million times hotter than a supernova, which would most certainly be more than enough to kill all of us and vaporize a significant fraction of the Earth.

But the BB would only spend 44 milliseconds passing through the Earth, and somehow I doubt that regular matter would stop it entirely. Let’s assume instead that only one tenth of its energy got deposited in its track. We’re still talking a temperature a hundred thousand times that in the center of a supernova, which is ridiculous and would, once again, kill us all and peel the skin off the planet.

What if the BB only loses one one hundredth of its energy as it passes through the Earth? Same result: the Earth is replaced by a ball of radioactive lava.

But if, because of its ridiculous speed, it only loses one one millionth of its kinetic energy interacting with Earth, it still heats its needle-thin track to 3 billion kelvin, which is hot enough to fuse Earth’s silicon into iron and produce a violent explosion that would spawn earthquakes and firestorms and might, in spite of the energy losses, kill us all anyway.

But when you consider how much energy even the mighty Oh-my-God particle (which was, let me remind you, moving so fast that light was having trouble staying ahead of it) deposited just by hitting the atmosphere, I’d say the BB would lose quite a bit more than one one-millionth of its kinetic energy on impact. And I’d say that that kinetic energy would be spread over a fairly wide area. I’m thinking it would leave behind a column of hydrogen-bomb-temperature fusion plasma in the atmosphere, then hit the crust and fan out within an ice-cream-cone-shaped volume of the mantle. The nastiness of the results depend entirely on how big an ice-cream cone we’re talking, but it’s likely to be fairly narrow and fairly long, so we’re probably looking at a near-supernova-temperature column of fusing rock plasma with a length measured in kilometers. The explosion would be worse than anything the Earth has ever seen and would, yes, almost certainly kill all of us. If the immediate radiation didn’t get us, then the explosion would expose the mantle and lift enough dust to darken the sky for years.

You know, at the start of this, I thought I had an idea I could pitch to Daisy Outdoor Products. Now that I think about it, I think I’ll put the proposal in a drawer and forget about it.

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The curse of dimensionality.

(The coolest part of this article will be that “The Curse of Dimensionality” is an actual legitimate term from computer science, even though it sounds like something from H.P. Lovecraft’s nightmares. But it’s a real scientific thing. Take a look.)

I like cellular automata. A cellular automaton is a bunch of cells arranged on some kind of grid. Each cell has a “state” variable. Each time-step, it decides which state to go into next according to a rule which looks at its current state and the state of all the cells around it. It’s about as simple as a computer simulation gets, but even in one dimension, it can produce amazing psychedelic patterns like this one:

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(The picture is two-dimensional, of course. Each timestep, the one-dimensional line of cells is rendered as a line of colored pixels, with its distance from the top increasing as time goes on. Click on the picture to get a version that isn’t all compressed and horrible.)

It doesn’t take much processing power to run a program like this. For instance, a 1-D cellular automaton with 1,000 cells and 8 possible states would only require 1,000 bytes of memory.

2-D automata take up a little more space, but they can still run smooth as butter on most computers, especially with neat, streamlined code. In two dimensions, an 8-state cellular automaton on a grid measuring 1,000 by 1,000 (1,000,000 cells) would only require a megabyte.

But you know me. I don’t like it when things are easy. I like it when things are weird. What if, for instance, we imagined a 3-dimensional cellular automaton? Here, each cell has 27 neighbors (itself and the cells adjacent to it and touching its corners and edges). If this cube of cells measured 1,000 cells on an edge (and each cell could still be in one of 8 states), then the cellular automaton would need 1 gigabyte of memory. I’m old enough to remember when that was an impressive amount, and indeed, processing a gigabyte of at 30 frames per second takes a bit of power. More power than a Pentium III chip had in 1999, at least.

But couldn’t we go bigger? OF COURSE! In a 4-dimensional cellular automaton, every cell has 81 neighbors, we’ve got a trillion cells, a terabyte of data (you’re probably going to need an extra hard drive), and our throughput would strain even a modern graphics card.

But let’s go BIGGER! I like the number 6, so let’s imagine a cellular automaton in 6 dimensions. Every cell has 729 neighbors adjacent to it. (Remember the good old days, back in 2 dimensions, when every cell had a measly 9 neighbors?). We’re going to need 1 million hard drives to store all those 8-bit numbers (there are 1,000,000,000,000,000,000 of them, after all), and we’re going to need a supercomputer more powerful than any currently in existence to handle the 60 exaFLOPS needed to maintain a smooth 60-FPS framerate. We will also need an air conditioning unit the size of a house, but don’t we always when I’m around?

But this is baby stuff. When scientists talk about high-dimensional mathematics, they’re not talking about hypercubes in 4 or 6 dimensions. They’re talking about crazy curves and datasets in, say, 256 dimensions! 256-D sounds like fun, so let’s go there!

…

Okay. I think I went too far. Again. For a start, every cell in our 256-D automaton will have 139008452377144732764939786789661303114218850808529137991604824430036072629766435941001769154109609521811665540548899435521 neighbors, which is more than a googol, which is a LOT. Second, the full 1,000 by 1,000 by 1,000 (by 1,000 until you hit 256) automaton would require

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

bytes to store. That’s 768 zeroes. We’re talking numbers without names here.

What the hell happened? I mean, 256 isn’t exactly a small number. It’s certainly larger than the number of pounds of cilantro you’d want to eat, but it’s hardly huge.

Well, that’s the Curse of Dimensionality. A cube’s N-dimensional “content” (which is an analogue for 3-dimensional volume or 2-dimensional area) is equal to S^N, where S is the length of an edge. And as anybody who’s ever played with numbers knows, throwing exponents into things makes numbers get scary very fast.

And there’s another odd problem that you don’t notice much in two or three dimensions. A circle with a diameter of 2 has an area of (approximately) 3.142, while a square with an edge length of 2 has an area of 4. The ratio of square area to circle area is 1.273. A sphere with a diameter of 2 has a volume of 4.189, while a cube with an edge length of 2 has a volume of 8. The ratio is now 1.910. Well, the ratio keeps going up. And it goes up FAST. A 256-dimensional cube with edge length 2 has a hyper-volume (content) of 115792089237316195423570985008687907853269984665640564039457584007913129639936, which is a lot. Meanwhile, a 256-dimensional sphere with a diameter of 2 has a content of 1.1195×10^-152. That’s an absurdly small number. 1.1195×10^-152 universe diameters (and in case you forgot, the universe is so large that trying to fit its size into your head would probably physically kill you) is ten million trillion trillion trillion trillion trillion trillion trillion times smaller than the smallest distance which makes any physical sense (Side note: isn’t it weird that there’s a smallest physically-sensible distance? Reality has a finite resolution!)

What this all boils down to is that the vast majority of a hyper-cube’s content is in its corners. And, as anybody who’s ever tried to dust a room knows, corners are a pain in the ass. Which is why it’s so hard to work on data-sets in high-dimensional space.

Here’s another interesting fact. I was talking about Lovecraftian horrors at the start. And, although he gets binned as a horror writer, he was actually kind of a horror/sci-fi writer. He liked thinking about non-Euclidean geometry and extra dimensions and other such stuff that was wild and revolutionary at the time. Before I go get some much-needed sleep (I keep having weird dreams about having to take tests so I can work at a pizza parlor. Trust me, they’re exhausting.), let’s work out just how large a 256-dimensional creature would be.

A human being contains approximately 7×10^27 atoms. That’s a lot. If we take (7×10^27)^(1/3) (the cube root), we get how many atoms would lie along the edge of a cube with that many atoms. A cube about the density of a human containing about the same amount of stuff as a human would be 40 centimeters on an edge (Don’t try to imagine ways to make a human into a perfect cube. None of them are nice.) What about a 6-dimensional human? Well, because you can pack so much stuff into a 6-cube, a creature with as many atoms as a human and the same approximate atom-to-atom distance as a human would form a cube 8.7 microns on an edge, about as large as a medium-large bacterium or a human cell. In 256 dimensions, you’d actually run out of atoms before you ran out of places to put them. Imagine you put down one atom at one corner of a big 256-D grid. Then, you start putting atoms in the neighboring cells. Well, in 256-D, there are 115792089237316195423570985008687907853269984665640564039457584007913129639935 neighboring cells, which is more than the number of atoms in a person, so our 256-D human would be no more than two atoms across (no larger than a molecule, in other words) along any dimension.

I don’t know about you, but I’ve succeeded in tripping all the circuit breakers in my head. I’m going to lie down. But I’ll see you again soon, after I find something else headache-inducing to think about.

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