“Dark matter” is stuff that we can’t see or touch, but we know it exists because it affects the motion of regular matter, which we can see. Although there were vague hints that dark matter might be a thing going back about a century, it wasn’t until the 1970s that direct observations of the movements of stars in galaxies revealed the presence of too much matter. *Way* too much.

There’s so much more dark than regular matter in the universe, that its gravitational influence more or less dictates where the biggest concentrations of matter (galaxies and even galaxy clusters) end up.

One (possible) prediction of string theory is that our universe is a “brane“, a “sheet” of spacetime floating about in a higher dimensional space called the “bulk”. Other branes (other universes) *might* be floating around nearby, almost overlapping our universe, some small distance away. All of the particles native to a brane are stuck to it, but the gravity they produce *might* not be. So *perhaps* what we perceive as dark matter is really just the gravitation from matter in another nearby universe. That would explain why we don’t see it and why it doesn’t bump into things.

The danger of this line of thinking is that it requires taking several steps beyond the shelter of direct experimental evidence. The bulk, other branes, and even string theory itself are just a bunch of interesting ideas, since none of the theory’s novel predictions have been supported by experiment (so far).

Before delving into brane stuff, it’s worth considering what we know about dark matter. By carefully tallying up the number of stars, the distribution of their masses, the amount of gas and dust (by looking at how the local starlight filters through it), and then figuring out the amount of gravity due to all of it, we can figure out how fast each star in a galaxy *should* be orbiting said galaxy. We found two things: first, that there’s many times more matter in play than we can see and second, that this unseen matter is distributed around every large galaxy as though it had never been subject to accretion. That last bit is important.

Accretion is why stuff in space tends to clump together or flatten out into disks (or both!). When stuff has the option of bumping into itself, collections of matter go from incomprehensibly big, fluffy clouds, to concentrated pin-points (like stars and planets).

By looking at the orbital speeds of stars vs. the distance to their galaxy’s cores we find that they are orbiting a disk of stuff as well as a diffuse cloud with around ten times the mass (the exact ratio varies from galaxy to galaxy). So whatever dark matter is, there’s lots of it and it doesn’t bump into anything.

The big difference between the 1970s and now is computer power, space telescopes, and the gradual muting of hairstyles. And nothing else. As our telescopes and computers improved slightly over the last few decades, we’ve been able to determine not just *that* dark matter exists, but *where* it exists and how much is around. Gravitational lensing bends light as it passes really heavy things. Determining the amount of mass in something based on how much it distorts the image of things behind it is something that’s very difficult to do with a slide ruler, but is pretty easy with a post-Pong-era computer.

If dark matter and regular matter show up in more or less the same place (which they usually do), then it’s difficult to tell the difference between the existence of dark matter and a misunderstanding of how gravity works. Luckily, dark matter is not only heavy but weakly interacting; a property that reveals itself when huge blobs of ordinary matter collide and their associated blobs of dark matter notably don’t.

The Bullet Cluster is actually a pair of colliding galaxy clusters. When galaxies or galaxy clusters collide their stars miss each other (because stars are stupid small compared to the distances between them), but their massive gas clouds find each other just fine. When they do, they heat up (not just from the physical impact but from being tossed around by their mutual magnetic fields) until they glow in the x-ray spectrum. Shining with x-rays makes these impacts stand out pretty clearly. So you’d expect that when galaxy clusters run into each other there should be a decent smearing of mass with a big chunk in the middle where the gas clouds pancake into each other. Instead, we see some hot gasses in the middle of the impact with the vast majority of the mass taking the form of two big chunks that cruised right through.

The point here is that dark matter is literal, actual stuff, not just a massively boneheaded mistake or miscalculation. You can point at where it is, say how much is there, and even talk about what it’s doing. We just can’t directly detect it because it doesn’t touch anything. As it turns out, that isn’t *completely* strange.

Different particles interact through different forces. For example, protons interact through all four fundamental forces: the strong nuclear force, the weak nuclear force, the electromagnetic force, and gravity. The strong force, keeps protons and neutrons packed tightly together in atomic nuclei. Electrons on the other hand interact through all but the strong force, which is why they buzz *around* the nucleus instead of being inside it.

Neutrinos only interact through the weak and gravitational forces. When you try and fail to walk through a wall, you’re interacting through the electromagnetic force. Without the electromagnetic force, neutrinos are free to pass right through damn near anything. On the order of a trillion pass through your body every second, and yet our best, biggest neutrino detectors only pick up a couple dozen a day (mostly from the Sun, and the detectors work both day and night since neutrinos don’t really notice the Earth). Since they interact through the weak force (and not very well) neutrinos are “weakly interacting”.

Whatever dark matter is, it seems to take things one step further: interacting through gravity alone. Without the strong force it wouldn’t form elements. Without the electromagnetic force it passes through ordinary matter and itself. And without the weak force we’re left with no way to directly detect it. But if there’s enough of it (and there seems to be), then we can detect its gravity.

Right now the search for dark matter is a little like setting out traps in your house because your cheese keeps going missing and you can hear skittering in the walls. There’s definitely *something* there and you know more or less where it is and what it’s like, but it would be nice to know exactly what you’re dealing with.

So to actually answer that question from way back when: could dark matter be the gravity of stuff in another universe (or brane or whatever) bleeding into ours? Probably not. Dark matter doesn’t interact with ordinary matter, which *could* be because it’s on a different brane. But even so, based on how it gets distributed we know that it doesn’t interact with itself either. So bringing extra universes into play doesn’t answer any questions, it just muddies up an existing question with untested ideas. It’s like using honey to get gum out of your hair; the situation is different, maybe even thought provoking, but not better.

The mathematical model of the universe picture is from here.

The “brane” picture is from here.

The space stuff is all from NASA.

The chewed up wire picture is from here.

]]>π is the ratio of a circle’s circumference to its diameter. Everything about π boils down to that definition.

That the distance across a fixed shape is proportional to the distance around it is nothing special. The same is true for absolutely every shape. If you double the size of any shape, both the distance across and around with double as well, so their ratio stays the same.

Knowledge of the fact that the ratio of the circumference to the diameter of circles is *some* particular fixed number is older than recorded history. The fact that the ratio isn’t exactly three takes a little care and precision. Figuring out that the ratio has an infinite and non-repeating decimal expansion (that starts with 3.14159265358979323846264…) takes a little math and time.

If you can nail one end of a piece of string down and tie a quill or piece of charcoal to the other, then you can draw a damn-near-perfect circle. With a longer piece of string (at least 2π times as long, in fact) and a ruler, you can measure the circumference and as long as you’re careful, it’s pretty obvious that π≠3. As long as the error in your measurements is below 4%, you can tell the difference. The Babylonians created the Code of Hammurabi and some pretty impressive buildings, so presumably they could create a meter stick with centimeter marks (or rather, a kuš stick with šu-si marks). It turns out the tablet in the picture above is really a “cheat sheet” for *roughly* approximating the area of circles. They were known to have used , an approximation within half a percent of the correct value. Which, for the bronze age, is fine.

Since π shows up practically anytime you’re doing math involving circles, there were a heck of a lot of opportunities for people in the ancient world to notice it and we can’t be sure which first caught their eye (this is the drawback of discoveries that are older than history). For example, the volume, v, of a barrel with height h and diameter d is . If you want to predict how much water a cylindrical barrel can hold to within a couple percent, then you need to know π to at least a couple digits.

All of this is to say that π, while full of mathematical mysteries or whatever, is not some abstract idea. It’s something you can physically measure. Well… you can tell it’s not three anyway. The larger the circle and the better your measurements, the more digits of π you can discover, but the law of diminishing returns kicks in faster than an all-you-can-eat-week-old-sushi buffet.

By the time you know π out to N digits, you know the ratio of the circumference to the diameter of a circle to on the order of 1 part in 10^{N}. For example, if you know that π≈3.14, then you can fit a bike tire onto a rim to within a cm. If you know that π≈3.1415, you can build a fence around a circular acre to within a cm. And if you know that π≈3.1415926535, you can wrap a cable around the Earth with less than a cm of wasted cable. Arguably, knowing π out to ten or more digits is aggressively pointless, but that has never stopped mathematicians from practicing their cruel craft. Not once.

The definition of π not only gives us a recipe for *physically* measuring it, but also hundreds of ways to *mathematically* derive it, and that’s where the real precision comes from. Mathematicians like Archimedes and Liu Hui, and some nameless Egyptian a couple millennia before them, were able to approximate π using polygons. Liu Hui calculated π accurately out to three digits, a record slightly better than Archimedes’, that held for about a thousand years. Which is strange.

Either Archimedes and/or historians really dropped the ball, or people in ancient Greece were more level-headed about knowing lots of digits of π than we are today. For a given circle, the points on an “inscribed polygon” touch the circle (so it’s inside) and the middle of the edges of a “circumscribed polygon” touch the circle as well (so it’s outside). Archie approximated π by inscribing and circumscribing 96-gons in and around the circle and calculating their perimeters. An inscribed polygon gives a lower bound, and a circumscribed polygon gives an upper bound, for the value of π. But here’s the thing: Archie didn’t just find the perimeter of 96-gons, he invented an iterative algorithm to calculate the perimeter of 2n-gons given n-gons. That is, he started with hexagons (6-gons) and bootstrapped up to 12-gons, 24-gons, 48-gons, and then inexplicably *stopped* at 96-gons. Evidently he had better things to do than calculate more digits of π. Which, to be fair, is not a high bar. He may have just declared the problem solved, since anyone following his procedure could find as many digits as they’d want, and moved on to heat rays or something (seriously, dude tried to build a solar heat ray to defend Syracuse).

Every time Archimedes’ iterative algorithm is used the estimate for π gets about 4 times better (its rate of convergence is 1/4). Which is less impressive than it sounds; that’s about 3 decimal digits every 5 iterations. Archie used it 4 times to go from hexagons to enneacontahexagons for an accuracy of about 3 decimal places. If he had bothered to repeat the process, say, ten more times, he would have known the first 9 decimal digits. Definitely *not* useful, but potentially brag-worthy.

Modern algorithms put those old approximations to shame instantly. Archimedes’ technique converges linearly to the true value of π (you gain about the same number of digits every time you use the algorithm). Things didn’t really start to pick up until we invented quadratically converging algorithms, which *double* the number of known digits with each iteration. That is: if you know ten, then after the next iteration you’ll know twenty. The fastest algorithms today converge *nonically* (the accurate decimal expansion gets nine times longer with each step).

The definition of π, the ratio of the circumference to the diameter of a circle, allows us to measure it directly, but inaccurately, or calculate it precisely, but pointlessly. The more abstract properties of π, like going on forever without repeating (which it does) or containing every possible pattern (which it might), require more than brute force discovery of digits. These more abstract properties are based on the definition of π, not its value, and can typically be proven or disproven without knowing even a single digit. Math is useful in the physical world, but it doesn’t “live here”. π has physical significance, but we use its mathematical properties to learn about it.

**Answer Gravy**: Ancient people were very clever and when they lived long enough they even got to prove it. This gravy is the math behind Archimedes’ algorithm. This isn’t *exactly* the way he wrote it. Ancient Greek mathematicians suffered from the demonstrably false belief that a minimum of words is the secret to a maximum of understanding. So, even translated, their proofs still read like Greek.

Mr. Medes’ method was this. If I_{n} is the perimeter of the inscribed polygon and C_{n} is the perimeter of the outer n-gon, then:

You *could* rigorously prove that as you increase the number of sides the perimeters will get closer to π (the circumference of the circle) using lots of equations or something, *or* you can just draw a picture and say “look… they clearly do”.

If you stick six triangles together, you get a hexagon and with a little trigonometry you find that if your circle has a diameter of 1, then the inscribed hexagon’s perimeter is I_{6} = 3 and the circumscribed hexagon’s perimeter is C_{6} = 2√3 ≈ 3.46.

To find the perimeters for dodecagons, plug C_{6} and I_{6} into the iterative equations:

These perimeters are both closer to π than those before the iteration and since for all n, this gives us an ever-diminishing range where π can be found. Here’s why it works:

In/circum-scribing n-gons on a circle produces some useful symmetries. In particular, it allows us to draw some triangles and rapidly figure out their angles.

A full circle is 360 degrees, so each side of an n-gon spans 360/n degrees. In this case, a is half of one of those angles, so a=180/n.

Both b’s are complimentary to a (they sum to 90), since the sum of angles in a triangle is 180 and the third angle is 90 (a symmetry we get from the fact that radial lines always hit a circle at a right angle). So, b = 90-180/n.

c and b are complimentary, so c = a = 180/n.

c+d = 180, so d = 180-c = 180-180/n.

The sum of angles in a triangle is 180, so d+e+e = 180 and e = 90-d/2 = 90/n.

Finally, b+e+f = 90, so f = 90-b-e = 90/n.

Since f=e, the two red triangles have the same angles: they’re “similar triangles“. Similarly, since c=a the two blue triangles have the same angles and are also similar. When two triangles are similar, the ratios of their sides are the same.

Using the similarities on the blue triangles we can say:

and of course, using the red triangles:

So, starting with a little geometry and the definition of π we can construct a bootstrapping method for approximating it better the longer we bother to work at it.

Multiplication and long division can be done by hand pretty easily. The hardest part of this iterative algorithm facing any ancient person is the square root. Also making the scratch paper. Fortunately, there are tricks for that too. For example, if you want to take the square root of S, just make a guess, x, then calculate and what you get will be a number closer to than your original guess, x. This method was known to the Babylonians and Archimedes (it is, in fact, “the Babylonian method”) and it converges quadratically, so it achieves whatever (reasonable) accuracy you’re hoping for almost instantly.

The point here is that you can, through the injudicious use of reason, time, and scratch paper alone, find π to as many digits as you would ever need.

You can read a little more about the Babylonian tablet here.

]]>This is a seriously old problem that needed to be solved before we became a routinely globe-trotting species. If you have the latitude and longitude of your location and a destination, you *could* just move east/west until you match the destination’s longitude then move north/south until you match the latitude. But that’s a big waste of time, as the so-intuitively-obvious-it-barely-deserves-a-name “triangle inequality” will tell you.

As you may have heard, the shortest distance between two points is a straight line. For curved surfaces the closest you can get to a straight line is a “geodesic”, which is a straight line on small scales, but it you step back may be going all over the place. If you wrap a ribbon around a gift (or any manner of festive package) and manage to keep it flat, you’ve found a geodesic, because if the path of the ribbon changes direction, the ribbon kinks. On a sphere like Earth, where you’re stuck moving along the surface (instead of drilling straight through like some kind of compulsive, fire-proof mole), the geodesics are “great circles”. If you walk in a straight line, you’re walking on a great circle.

Doing geometry on a sphere is more difficult than geometry on a flat plane, but not by a hell of a lot. Geometry, despite what some pudunk Greek philosophers may say, is basically just playing with triangles. A nice, universal rule for any triangle is the Law of Cosines which relates the lengths of the three sides, a, b, and c, with the angle opposite one of those sides, C: .

Slap that triangle on the side of a sphere and that law becomes the Spherical Law of Cosines:

Here the meaning of a, b, and c has changed a little. Lengths on a sphere can be described by angles (drawn from the center to the surface). If you’re using radians (which you always should), those angles are . This is generally easier to use than the actual distance. For example, the angle from the north pole to the equator is 90° (obviously) while the physical distance is about 6,200 miles (different on every planet).

Lucky for us, positions on Earth are usually described in terms of angles (latitude and longitude) and not distances, so the spherical law of cosines is ready to go. Just put the corners at the north pole, where you’re at, and where you’re planning to go.

The law of cosines (either of them) relates four things: all three sides and one of the angles. If you have any three of those pieces of information you can solve for the forth. With the latitudes we have two sides and with the difference of the longitudes we got an angle.

First (because you always should) convert your angles from degrees to radians by multiplying by : , , and .

and therefore

Boom! There’s the distance. And with three sides in hand (the two given by the latitudes and the distance from, like, two seconds ago) you can find your “bearing”, the angle between north and your destination.

and so

For example, if you happen to be traveling from Nueva Guinea, Nicaragua (11.6932° N, 84.4540° W) to Nesflaten, Norway (59.6466° N, 6.7997° E), the angles involved are , , and . So the distance is

and the bearing is

If you want to walk in a straight line from downtown Nueva Guinea to uptown Nesflaten, you’d face due north, turn about 30 degrees to the right, and walk in a straight line for about two and a half months, while ignoring the Atlantic Ocean to the best of your ability.

So how do you calculate distance and bearing? A small bucket of math. Today if you want to calculate sines and cosines you break out a computer. So you may be wondering how they did this math back in the days of sailing ships (without GPS). They didn’t. Calculating was done once, then written down, then looked up many times. Once upon a time books were useful!

The people who actually did the calculating were “computers”; pitiable folk who sat in dark rooms and slowly went mad and rarely got to sail around the world.

]]>Clearly, the issue is that a spoon and a bowl aren’t osculating curves. The radius of a spoon’s curvature is typically less than an inch while a bowl’s is several inches. With different curvatures, when you drag a spoon across the bottom of an ice-cream-laden bowl, they only contact each other at a tiny point and you can only remove ice cream along a thin strip (Matching the curvatures is why you instinctively start using the spoon sideways when the ice cream is low). So if you wanna get your bowl perfectly clean, you gotta figure out how wide that strip is.

Around the point of contact there’s a gap between the spoon and bowl that gets bigger the farther away you get. But so long as the ice cream molecules are bigger than that gap, they’ll be caught. Water, at about a quarter of a nanometer across, is the smallest molecule in ice cream’s molecular menagerie.

The bottom of a circle with a radius R can be closely approximated by a parabola of the form (There’s nothing too special about parabolas; every curve that doesn’t have sharp corners can be closely approximated by circles and vice versa, it’s just that parabola math is easy). Spoons have a radius of around 1 cm. Bowls have a radius of around 7 cm. The distance from the center (the point of contact) to the edge of the strip, x, such that water molecules won’t be able to slip through the gap between the bowl and the passing spoon is given by . Solve for x and you get , so the strip should be about 5 micrometers across; on the order of a tenth of a hair’s width.

The wide ice-cream-free swaths you see in practice aren’t regions where all of the ice cream has been removed, but regions where it’s been left thin enough to see though. So, if you *really* wanted to scrape a bowl clean of every molecule of ice cream, you’d need to carefully “scan” one strip after another, about four million and change times. Just to make sure you don’t miss any (else you and your IC OCD would be forced to start over), it couldn’t hurt for those strips to overlap. Call it an even ten million passes. At, say, two passes per second (the approximate speed of a kid charging headlong into a brain freeze), it would take you almost two months to scour your bowl clean. About a week into that, you’ll get the overwhelming urge to ask for seconds. Or at least do something else.

So far, this is all a bit idealized. The Platonic perfection of parabolas doesn’t apply to actual spoons and bowls. If you zoom in close enough, ceramics are like moonscapes with plenty of room for ice cream to hide and the tip of a spoon looks more like a stainless steel mountain range. Truly, a rocky road.

As undeniably clever as the approach above is, that “10,000,000 passes with a spoon” estimate is built on unrealistic premises. Sadly, you can’t eat all the ice cream in a conventional bowl using compulsive spooning alone.

But that doesn’t mean it’s impossible eat all of the ice cream, you just have to stretch the rules a little. For example, you could eat as much as you would like, then put the bowl into a kiln. By and large, organic molecules (sugar, fat, cellulose, etc.) burn at temperatures below 500°F and turn into new compounds, and a good kiln heats things up to 1700°F. A ceramic bowl would survive the heat intact, but the chemicals that make ice cream ice cream (as opposed to ash) wouldn’t.

So if you define “eating all the ice cream in the bowl” as “there was ice cream, I ate ice cream, now it’s all gone”, a kiln is a good way to do it. But if you think destroying the evidence is somehow cheating, there is one more, perhaps not as delicious, alternative guaranteed to work.

Eat the bowl.

The empty ice cream bowl picture is from here.

The extreme close-up of a ceramic is from here.

The kiln picture is from here.

The Willy Wonka picture is from Willy Freaking Wonka.

]]>Fans of Star Trek, on the other hand, think they just read “photon torpedoes” twice.

According to the *perfectly* named Wookieepedia, proton torpedoes release clouds of high energy protons on impact. Now technically, that’s what all explosives do. By mass, atoms are almost entirely made of protons and neutrons, each of which are around 2000 times more massive than electrons, and (for most elements) show up in roughly equal numbers. So about half of the mass of practically *anything* is protons (the glaring exception is hydrogen, which is 1 proton and 1 electron).

Incidentally releasing clouds of protons, because that’s what’s in clouds of *anything*, is clearly not the idea behind proton torpedoes nor the spirit of this question. Neutron bombs (which are real things) release a heck of a lot of neutrons and not *too* much else. Proton torpedoes should be like neutron bombs, just with protons.

Once again physics, the queen of sciences and mother of buzzkills, has things to say. The thing about protons is that long before you notice them doing damage by physically impacting things, you notice their electrical charge. A proton is like a mosquito when you’re trying to sleep; the mass is not what’s important.

There are incomprehensible electrical charges buried inside of everything, we just don’t notice them because they’re normally almost perfectly balanced. One gram of protons, without accompanying electrons to bring their net charge to zero, would create an electric field strong enough strip electrons from their host atoms and ionize air more than 15 km away. It wouldn’t be too healthy for any other materials either; a field this large and intense could pull lightning bolts through granite at a range of 10 km.

So a proton torpedo with a one gram payload does a heck of a lot of damage just by existing. Luke’s aim didn’t really need to be that good.

Ironically, the big effect of detonating the torpedo and releasing the protons is that the electric field actually drops. As the last post talked about too much, you can understand a lot about electric and gravitational fields by “drawing bubbles” around them. The total field flowing out of a bubble is proportional to the amount of charge inside of that bubble. So if you’re standing far enough away, the field you feel stays about the same whether or not the torpedo has exploded. As the shell of protons expands and repels itself, the field outside of it stays about the same while the field inside of it disappears. If you’re up close, when the expanding shell of protons passes you (assuming you’re alive) you’d find yourself in a region with little or no electric field. The most destructive thing you can do with a pile of protons is not blow it up, but keep it together in as small a place as possible.

The only way to block the electric field created by a bucket of charge is to surround it with the same amount of the opposite charge. Both positive and negative charges create electric fields, but those fields counteract each other. That’s why, despite containing many kg of protons, you doesn’t destroy everything around you.

Unfortunately for BlasTech Industries (famed maker of clumsy, random blasters), counteracting electric fields is exactly what charges always do, given the chance.

So even if you intentionally store protons in you proton torpedo with the hope of weaponizing its electric field, the first thing that will happen is any nearby matter will ionize. The loosed electrons will fly toward and coat the torpedo while the stripped nuclei (which are full of protons) beat a hasty retreat. Assuming the casing is a perfect insulator, in the end you’d have a torpedo with a gram of extra protons inside, about half a milligram of electrons outside (because electrons are about 2000 times less massive than protons), and no devastating electric field.

You could keep the protons and electrons together until the torpedo hits its target, then release the protons but keep the electrons in place (I mean… somehow). Then you could transport your munitions without preemptively killing everything around them. But that would be like firing bullets tied to (very strong) rubber bands; the electrons and protons would rather fly back together than do anything else.

So “proton torpedo” may just be a code name like “tomahawk missile” (noteworthy for its lack of tomahawks). That, or George Lucas is playing a little fast and loose with physics.

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