## Q: Is there anything unique about our solar system?

Physicist: Maybe.

As far as stars go, ours is a dull as dishwater, middle of the road, dime a dozen, main sequence star.  You can’t spit in space without hitting a Sun-like star.  So this question is really about the stuff around our Sun.  But, beyond their stars, comparing solar systems is a little tricky.

A solar system is basically a star (sometimes a couple of them) with a tiny bit of grit left over.  Not only does our Sun comprise 99.86% of the mass of our solar system, but it’s disproportionately “loud”.  While the Earth may have a smattering of radio antennae pumping signals into space, the Sun is a screaming ball of electromagnetic noise almost a million miles across.  And not for nothing: it’s seriously bright.  So finding stuff around other stars is difficult, not just because everything other than stars is tiny, but because of the stars themselves.  In fact, most of the techniques we have for detecting stuff around other stars involves looking at the effect of said stuff on their host stars.  Ultimately, if you really want to get a good look at other solar systems you’ve got to get off of your podunk planet and go there.

That said, we are now living in the century of planetary discovery.  Every civilization has known about 6 planets: Mercury, Venus, Earth, Mars, Jupiter, and Saturn.  Uranus, Neptune, and Ceres were discovered in the 19th century.  Pluto and the first few exoplanets were discovered in the 20th century.  But in the first decade and a half of this century we’ve discovered dozens of new dwarf planets in our solar system and thousands of exoplanets in other solar systems.  The rate of discovery is ramping up exponentially.

The planets that are easiest to detect are massive (vertical axis) and close to their stars (horizontal axis).  We would probably miss most of our planets (blue) if we were looking for them from another solar system.

The methods we use to detect planets around other stars are much better at detecting planets that are big (which isn’t surprising) and close to their stars.  So the easiest planets to find are “hot Jupiters“.  But based on what we’ve seen so far it looks like damn near every star in the sky, even binary and trinary stars, have planets in orbit around them.  Despite the difficulties in finding any planets, we’ve confirmed hundreds of solar systems with multiple planets (which strongly implies that practically all of them have multiple planets).  And that’s based on methods that look at only a tiny fraction of the sky and miss the majority of planets.

In this picture our solar system is in the center and the exoplanets we’ve found are red dots.  The majority of the exoplanets have been found using the Kepler space observatory (guess where it’s pointing).  The point is: we’ve discovered thousands of planets around other stars and we’ve barely scratched the surface.

Even better!  By far the most common element in the universe is hydrogen with oxygen a distant third.  So we can expect to find H2O (water) all over the universe, including on exoplanets, and when we look that’s exactly what we find.  That doesn’t necessarily mean that there’s life out there, but if it’s around it’ll definitely have lots of places to do its living.

So here’s the point.  It’s really hard to see planets around other stars, we miss most of them, and we’ve only looked at a tiny fraction of the nearby stars.  And yet!  We’re finding planets freaking everywhere.  Big, small, hot, cold, rocky, gaseous, short years, long years, dry, or wet.  The shocking variety and preponderance of planets has completely rewritten our ideas about planetary and solar system formation.  We once thought that solar systems would all more or less “follow the same script” during their formation and end up as variations on our own.  Instead we find that each is surprising and unique.  Our solar system is unique like a snowflake in a blizzard is unique.

Some podunk planet.

The top two pictures are from the (free as of writing) exoplanet app and that picture of Earth is from space.

Posted in -- By the Physicist, Astronomy, Philosophical | 5 Comments

## Q: What is dark energy?

Physicist: The universe is expanding and the rate of that expansion is accelerating.  If the universe were full of only matter (both regular and dark) and energy, then we’d expect that the expansion would be slowing.  Dark energy is the thing that is responsible for that acceleration.  Now as for what exactly that thing is: we have interesting and vaguely informed guesses.

Back in the day Einstein came to a few realizations after he spent some time thinking about falling elevators and rockets.  This lead to a way of describing the nature of gravity (falling and whatnot) in terms of the geometry of spacetime and a relation between that geometry and the amount of matter and energy that’s around.  Einstein’s field equations can be used to describe how time and space are influenced by the presence of big chunks of matter like planets and stars, manages to explain/predict Mercury’s weird orbit and a bucket of other stuff, and has passed every test it’s been put to with flying colors.  That last bit is important: it’s easy to come up with crazy new theories, but hard to come up with theories that precisely predict results that were not previously understood.

Enter Friedmann, who happened to be thinking about the entire universe one day.  Einstein’s field equations related mass/energy with the shape of spacetime.  So, pondered Friedmann, what happens when you apply those equations to the universe as a whole?  First he assumed that space can expand or contract over time (why not?), then he threw that assumption at Einstein’s field equations to see if they’d stick.  Somewhat surprisingly, this is easier than it sounds.

Mr. Алекса́ндр Алекса́ндрович Фри́дман (Alexander Friedmann), a man who looks exactly like you’d hope he would.

On a totally over-the-top-large scale (billions of lightyears) the matter and energy of the universe is distributed roughly uniformly.  On this scale we describe the matter and energy in the universe as the “cosmological fluid”, because on the largest scales even the largest clumps of matter are as indistinguishable as atoms in water.  These “largest chunks” (as far as we know) are galactic super clusters; ours is about half a billion lightyears across and there are possibly many millions of super clusters in the observable universe.  And that’s only the observable universe.  Guessing the size of the entire universe based only on the part of it that we can see is like trying to guess the size of the Earth by standing in a featureless open field.

So you’re somewhere.  How big is it?

Words utterly fail when trying to describe how much “out there” is out there.  Point is: all that’s important for figuring out the behavior of the universe as a whole is the average density of matter and energy, which is roughly the same everywhere.  You don’t need to stress about every star and galaxy in the universe, or even about the size of the universe as a whole, any more than you need to stress about every grain of sand in a pile of sand.

When Friedmann said space can expand or contract over time he said it this way: $ds^2 = \left[a(t)\right]^2dx^2-c^2dt^2$.  That’s the spacetime interval (which is how distance is measured in spacetime) with the addition of a scaling term, a(t).  If a(t) doubles, that means that the distance between any two points in space has doubled.  So, if you figure out how a(t) changes over time, you can describe how the universe is expanding or contracting over time.  Incidentally, a(t) also describes the cosmological redshift of light; if a(t) doubles between when some light is emitted and absorbed, then the wavelength of that light (like the space it’s traveling through) will double and longer wavelengths mean redder.  The oldest light in the universe was emitted when a(t) was about 1100 times smaller than it is now.

It turns out (this is not obvious) that Einstein’s equations dictate that $a(t) = kt^\frac{2}{3(w+1)}$, where w is the weirdly named “equation of state” and k is a constant number.  $w=\frac{1}{3}$ and $a(t)\propto t^\frac{1}{2}$ for a universe filled with radiation (light, neutrinos, and matter moving near light speed).  $w=0$ and $a(t)\propto t^\frac{1}{3}$ for a universe filled with regular matter (including you, your stuff, basically everything else, and dark matter too).  In both of these cases the universe expands, and the rate of that expansion slows down, forever.

As space expands the density of matter and radiation decreases, because there’s more space for it to be spread out in.  But radiation not only gets more spread out, but redshifted as well and that means that the energy density of radiation drops faster than the energy density of ordinary matter.  So, we can expect that w, which describes a combination of both matter and radiation, should be somewhere in the range $0\le w\le \frac{1}{3}$.  Since the expansion of space decreases the energy density of radiation more than matter, over time w should drift closer to 0 as matter becomes more dominant.

The expansion of space decreases the energy density of matter by spreading it out and decreases the energy density of radiation by spreading it out as well as redshifting it. But the expansion of space doesn’t decrease the energy density of dark energy; instead it just seems to create more

But here’s the thing!  Since the late 90’s we’ve been able to show that the expansion of the universe is speeding up, not slowing down.  In order for that to happen the equation of state of the universe must be $w\le-\frac{1}{3}$.  This came as a bit of a shock to cosmologists.  But being brow-beaten by experiment and observation is what good science is all about.  Onward.

Upon closer examination of the expanding universe we find that including “stuff” with the equation of state w=-1 is a good fit.  This corresponds to a uniform negative energy with a constant density.  There are a couple of things about that which are… a little strange.  A constant density means that as space expands there’s more of this stuff around.  Matter and radiation are indeed being thinned out more and more by the expansion of the universe, but this very bizarre new stuff doesn’t get thinned out.  The fact that it has negative energy is just icing on the weirdness cake.

Not knowing what else to call it, physicists have dubbed this uniform, undiluting, negative-energy stuff “dark energy”.  That’s not to say that we have any idea what it is, but that’s no reason to not give something a name.  Unfortunately, dark energy and dark matter are difficult to study (hence the names) so it’s tricky to figure out exactly how much of each is around, but of the energy and matter around today roughly 75% is dark energy, 20% is dark matter, and 5% or less is regular matter.  Over time the percentage of dark energy should approach 100% as the expansion of space waters down all of the matter and radiation.

We can infer the existence of dark matter in large part because models of the universe that take it into account do a good job of describing the observed evolution of the universe, while models that don’t take dark energy into account do a terrible job.  Because the amount of dark energy seems to be proportional to the amount of space around, it seems fairly reasonable to say that dark energy is the “energy of empty space”.  What in the hell that’s supposed to mean is now a lively topic of discussion and debate.  There are a bunch of theorists and experimentalists running around trying to directly detect and/or describe dark energy, and with any luck they just might do it.

Posted in -- By the Physicist, Astronomy, Physics, Relativity | 28 Comments

## Q: What are “actual pictures” of atoms actually pictures of?

Something IBM made with some very flat, very clean, very cold copper and a few hundred carbon monoxide molecules.

Physicist: Actual pictures of atoms aren’t actually pictures at all.

There are a few good rules of thumb in physics.  Among the best is: light acts like you’d expect on scales well above its wavelength and acts weird on scales below.  In order to take a picture of a thing you need light to bounce off of it in a reasonable way and travel in straight lines (basically: behave like you’d expect).  But the wavelength of visible light is about half a micrometer (a two-millionth of a meter) and atoms are around one ångström (a ten-billionth of a meter) across.  On the scale of atoms, visible light acts too wonky to be used for photographs.

Atoms are literally too small to see.

(Left) A photograph of a 3 ball.  (Right) What we have in lieu of a photograph of an atom.

You could try using light with a shorter wavelength, but there are issues with that as well.  When light has a wavelength much shorter than an atom is wide, it takes the form of gamma rays and each photon packs enough energy to send atoms flying and/or strip them of their electrons (it is this characteristic that makes gamma rays dangerous).  Using light to image atoms is like trying to get a good look at a bird’s nest by bouncing cannonballs off it.

There are “cheats” that allow us to use light to see the tiny.  When the scales are so small that light behaves more like a wave than a particle, then we just use its wave properties (what else can you do?).  If you get a heck of a lot of identical copies of a thing and arrange them into some kind of repeating structure, then the structure as a whole will have a very particular way of interacting with waves.  Carefully prepared light waves that pass through these regular structures create predictable interference patterns that can be projected onto a screen.  Using this technique we learned a lot about DNA and crystals and all kinds of stuff.  This is the closest thing to a photograph of an atom that is possible using light and, it’s fair to say, it’s not really what anyone means by “photograph”.  It’s less what-the-thing-looks-like and more blurry-rorschach-that-is-useful-to-scientists.  Even worse, it’s not really a picture of actual individual atoms, it’s information about a repeating structure of atoms that happens to take the form of an image.

By passing light (left) or even electron streams (right) through a regular, crystalline structure we create an interference pattern that gives us information about the structure of the crystal (but never pictures of individual atoms).  The picture on the left (left) is a pattern created by DNA (species unimportant).  Notice how not obvious the helical structure is.  The picture on the right is created by an electron beam passing through some simple mineral or salt.

These techniques are still in use today (are relatively cheap), but since 1981 we’ve also had access to the Scanning Tunneling Electron Microscope (STM).  However, despite the images it creates, the STM isn’t taking a photograph either.  The STM sees the world the way a blind person on the end of a tiny robotic arm sees the world.

The essential philosophy behind the Scanning Tunneling Electron Microscope is what allows this dude to know more about the bottom of this chili cauldron than you do.

The STM is basically a needle with a point that is a single atom (literally, it is the pointiest thing possible) which it uses to measure subtle electrical variations (such as a stray atom sitting on what was otherwise a very flat, clean surface).  The “Tunneling Electron” bit of the name refers to the nature of the electrical interaction being used to detect the presence of atoms; when the tip is brought close to an atom electrons will quantum tunnel between them and the exchange of electrons is a detectable as a current.  The “Scanning” bit of the name refers to how this is used to generate a picture: by scanning back and forth across a surface over and over until you’ve bumped every atom with your needle several times.  The pictures so generated aren’t photographs, they’re maps of what the STM’s needle experienced as it was moved over the surface.  The STM “sees” atoms using this needle in the same way you can “see” the bottom of a muddy river with a pokin’ stick.

An STM and some of the pictures it pokes into being.

This technology has been around for decades and, like the advent of the synth, has given rise to all manner of jackassery.

Posted in -- By the Physicist, Physics, Quantum Theory | 9 Comments

## Q: If you were shrunk to microscopic size would you be able to see normally? Would you be able to see microscopic things?

The original question was: In the ‘60s sci-fi classic “Fantastic Voyage,” the crew of a submarine crew are shrunk to microscopic size and injected into the body of an injured scientist.  I realize that this film is rather sloppy from a scientific point of view, but here’s my question:  The shrunken crew members are able to see microscopic objects, like white blood cells.  But, assuming you could scale a person down, does having smaller eyes necessarily mean you can see smaller – even microscopic — objects?  I can’t make out one-point type, even if it is printed clearly through some high-resolution photographic process.  Would I be able to if I were the size of an ant?  Would scaling down our eyes give them the same capabilities as a microscope?

The Fantastic Voyage, 1966.

Physicist: Yes and no, but mostly no.  This question basically boils down to: if you were scaled down so that tiny things were large compared to you, would they appear large to your now-tiny eyeballs, with all of their microscopic details made macro?

If you were shrunk down until a Lego brick appeared as large as this sound stage at Estudios Churubusco, then would it literally look like this to your tiny eyes?

The answer is yes: as lenses and eyeballs shrink, the world literally does look bigger.  But mostly no: the smaller you get, the darker the world will appear and if you’re shrunk to less than about one 10,000th of your size, the lenses in your eyes will cease to work on visible light.

A smaller lens made in the same shape and with the same material will focus light at a proportionately shorter distance.  This means that your eyes should continue to work normally; things that are now relatively larger, will appear larger.

The way light interacts with a lens is dictated by the material and the geometry of the lens.  Assuming your tiny eyes are the same shape and material as your present eyes (and they always appear to be in the movies), they should work normally.  If there’s a cell or Lego brick in front of you and about your size, it will appear to be about your size.  You should be able to see smaller details as though the tiny object were literally larger just by walking up to it.

There is an issue.  The amount of light bouncing off of tiny things, or flying into tiny eyes, is small.  So tiny eyes are always in the dark.  Overcoming this problem is why fancy microscopes have light bulbs.  Assuming that you, the things you’re interacting with, and the distance to the things you’re interacting with, are all scaled down by the same factor, x (e.g., you’re shrunk and injected into the bloodstream of Dr. Benes), then everything around you will appear to be darker by the inverse square of this factor, x-2.  Everything around you would appear to be x times bigger, but the lights would all be x-2 times dimmer.

Smaller eyes collect less light. By the time you’re the size of a cell, the amount of light needed for a human eye to see is more than enough to set you on fire.

So if you’re shrunk from being a couple meters tall to being a couple millimeters tall (shrunk by a factor of 103), then the tiny world around you would appear one millionth as bright (decreased by a factor of 10-6).  The noon-day Sun would appear about as bright as a full Moon to the milli-you.

In the Fantastic Voyage the ship is shrunk to one micrometer across; a factor of around ten million, 107.  The ambient light would need to be one hundred trillion times brighter in order for their environment to have appeared normally lit.  If you tried this, you’d see just fine for the fractions of a second before you were cooked.  1014 is a big factor.

Even worse, there’s a diffraction limit brought about by the wave nature of light.  Below scales about as large as its wavelength, light starts to act more wavy and less particly.  It oozes around corners and ripples around obstacles.  In a micro-meter eyeball visible light cannot be relied upon to propagate in a straight line; instead it would splash haphazardly onto your retina.  As you shrink through the diffraction limit (assuming there was still enough light to see) the world would get blurrier until it just “blurred out” entirely.  Visible light has a wavelength of about half a micrometer and our pupils are around 2 – 5 mm across; about ten thousand times bigger.  So, if you were shrunk by a factor of around ten thousand, then you’re eyes will no longer be able to focus incoming light and project it into useful images.  You’d basically be in a haze of the average light coming from every direction.

Left: Laser light through a wide aperture. Right: laser light through a very small aperture.  The scattering effect of diffraction prevents micrometer scale (or smaller) eyes from being able to form images.

White blood cells are about 10 μm across, or about 20 visible-light-wavelengths.  The diffraction limit would start to be an issue when they appear to be about the size of golfballs, and you’d be completely blind when the white blood cells appeared to be about the size of your head.

Despite the difficulties, micrometer sized eyes do exist.  But because of the difficulties, they’re crap.  Fortunately, the smallest eyes belong to (technically “are”) single celled critters, which are universally too dumb to notice the quality.  Synechocystis is cyanobacterium (uses photosynthesis for energy) that’s about 3 micrometers across and uses its entire body as an eyeball.  Light passing through the cell focuses a little more in the surface of the cell opposite from the light’s source.  This isn’t a trick that’s difficult to evolve; it’s something raindrops do just as well en route to the ground.  What makes Synechocystis an “eye” is the fact that it then reacts to that “image”.  By swimming away from the bright spot it swims toward the Sun (or any other bright source of light), which is a good move for a critter that eats light.  Because of the diffraction limit, this sloppy slightly-brighter-region is about the only image that the tiniest possible eyeball can create and we shouldn’t expect to find eyes much smaller.

Synechocystis: the smallest, or very nearly the smallest, eyes possible.

Posted in -- By the Physicist, Biology, Physics | 10 Comments

## Q: How does one attain an understanding of everything?

Metatem, High Chair of Truscience: The true path to enlightenment, born of the enlightenment, is obviously Truscience.  Truscience replaces the “old science”, which is full of complicated symbols that no one understands, crazy stories about ancient lizard monsters killed by a giant rock, and the laziest creation myth ever conceived: a “Big Bang”.  St. Bertrand forgive the fools!  It is common knowledge that if you can’t explain an idea to a 5-year-old, then you don’t understand it.  Clearly “scientists” don’t understand much of anything.  All of Truscience is, of course, completely backed up by modern scientific knowledge only without any of the wrong stuff (hence the name).

Truscience combines the true findings of the old science with modern quantum meditation.  Finally free of the shackles of “math” we can come to understand reality through the lens of the Truself.  It took trillions of years of old science for humanity to finally realize that it doesn’t take trillions of years of science to fully comprehend the manyverse.  To know the full scope of time and space we need only ask Laplace’s Demon.

The High Chancellor communing with followers of Truscience from other planets, post-singularity energy matrices, and the future.

I can’t underscore this enough: science is hard.  It takes decades of meditation, logic, and introspection to be granted total knowledge and at all times you’ll be assailed from all sides by the Unreasoning masses who have been fooled into believing that mathematics and expensive equipment are the path to knowledge.  Fortunately, you don’t have to understand Truscience to completely believe it.  In fact, striving to understand every little thing really gets in the way.  All you need is faith.  Faith and a nominal fee to participate in a few classes, workshops, and friendly, purifying get-togethers.

We (the followers of Truscience) have come a long way from our cargo-cult origins.  Whereas we once only went through the motions of science without a spiritual connection, now we have access to the manyversal truths.  For example, we now know that we will all live forever.  As St. Newton divined, “energy is neither created nor destroyed” and as St. Einstien proved “energy and matter are the same thing”.  Therefore, once our matter-bodies release our energy-selves each be sorted by Maxwell’s Demon; some will ascend into the Singularity and some will be left behind to be crushed under our robot feet.

We also now know that whatever we believe is true.  As St. Shrodinger showed when he killed a cat with his mind, quantum physics means that we can make of reality whatever we truly wish.  It wasn’t until after I earned all my degrees and found my true name, Metatem, that I came to understand Truscience.  Now that I’m 7th degree, it’s my turn be in the High Chair.

According to our most basic precepts, all people are followers of Truscience (zeroth degree) except of course for “the Unreasoning”; which includes such poisonous influences as concerned family members and old-science “scientists” who dispute the truth of Truscience.  The power of Truscience is that it’s true whether you believe it or not, because we believe it (and thus it is manifestly so).  And we should know.  The followers of Truscience are smarter than followers of every other religion and creed for a simple reason: the Aumann Agreement theorem, which says that all smart people will eventually agree.  Therefore all smart people either believe in Truscience, haven’t heard of it, or have been compromised by an Unreasoning.

Truly, how can something with so much science and smart, moral people not be true?

Posted in -- Guest Author, April Fools | 6 Comments

## Q: Can planes (sheets) be tied in knots in higher dimensions the way lines (strings) can be tied in knots in 3 dimensions?

Physicist: Yes!

Mathematicians are pretty good at talking about things in spaces with any number of dimensions.  Sometimes that math is fairly easy and even intuitive.  For example, a line has 2 sides (ends), a square has 4 sides, a cube has 6 sides, and a hypercube has __* sides.

Ordinary knots (that you can tie with string) can only exist in exactly 3 dimensions.  It’s impossible to create a knot in 2D since every knot involves some amount of “over-and-under-ing” and in 2D space there’s none of that.  Because it makes the math more robust, mathematicians always talk about knots being tied in closed loops rather than on a bight.  In part because once you’ve connected the ends of your string the knot you’ve got is the knot you’ve got, and that invariance is very attractive to math folk.

Left: In two dimensions, no matter how complicated and convoluted your string is it can never be tied in a knot.   Right: Even the simplest knot requires at least three over-under excursions into three dimensional space to get around self-intersections.

In 2D, if you have a dot inside of a circle, it’s stuck.  But if you have access to another dimension (“dimension” basically means “direction”), then you can get the dot out.  In exactly the same way, if you can “lift” part of a regular knot into a fourth dimension it’s like opening the loop and you’re free to untie your knot in the same way you’d untangle/untie anything.  Afterwards you just “lower” the segment of the string back so that it all sits in 3D and now you’ve just got a loop of unknotted string (very creatively, this is called an “unknot”).  So, you’ve managed to untied your knot without worrying about self-intersections and all it took was an extra dimension.

In 2D a dot can be stuck inside of a circle, but if we have the option to “lift” part of the circle in a new direction then the dot can get out.  From the perspective of the flat denizens of 2D space, this looks like part of the circle being removed.

All that was just to say: be excited, the way you tie your shoes is only possible in universes similar to (with the same number of dimensions as) our own.  You can’t tie a knot in a string in two dimensions and a knotted string in four (or more) isn’t really knotted at all.

The way we talk about ordinary knots is in the context of a loop (tie your knot and then splice the loose ends of the string).  The generalization of a loop (a 1-sphere) to higher dimensions is first the surface of a regular sphere (a 2-sphere), then the surface of a hyper-sphere (a 3-sphere) and so on.  An N-sphere can be tied in a knot in N+2 dimensional space.

An N-sphere can be tied in knots in N+2 dimensions. 1-spheres can be tied in knots in three dimensions (these are known colloquially as “knots”), which means that they can actually be created.  2-spheres (the surface of a ball) can be tied in knots in four dimensions.  The image here is only a cross-section of such a knot.

It turns out that if you have an ordinary knot, you can use it to create a higher dimensional knot.  There are a several ways to do this.  There’s “suspension“, which usually doesn’t work (the created knot is often not a “manifold“, which is kinda cheating), and there’s also “spinning” which always works.

The basic idea behind spun knots.

To create a “spun knot” you rotate it in a higher dimensional space and collect all of the points that it sweeps through.  The picture above is more symbolic than applicable.  In this picture a knot in 3D is spun to create a sphere that’s still in 3D space, but with a funky-shaped tube running around its equator.  That’s not a knot (knot at all).  This process needs to be done in four dimensions, where the added direction allows you to get around the self-intersection problem, but the basic idea is the same.  So for every knot that you can tie with a loop of rope in 3D, there’s a knot you can tie with a hollow sphere in 4D.

And yes: you can keep going into higher and higher dimensions using the same idea.

While you can’t directly picture a four dimensional knot, you can create cross-sections (the same way a 2-dimensional being might picture 3-dimensional objects using cross-sections).  This video (~0.4MB) shows 3D cross-sections of a rotating 4D knot.  But be warned: that video is, for lack of a better word, groovy.

Sometimes a group of scientists will get really involved with a particular subject and kinda disappear up their collective butts for a while (especially mathematicians).  Eventually one of them will emerge like a prairie dog and bark “fellow dudes and dudesses, we should really send a message to the world so they don’t worry about us” at which point a summarizing paper such as this or this is written (about higher dimensional knots in this case), in an attempt to convey to a slightly broader audience what they’ve been doing.

Mathematicians after a long think.

The tied sheets painting is by Teun Hocks and is from here.

The 4D knot picture and the video are from here.

The spun knot picture was lifted remorselessly from the second paper mentioned earlier.

*8

Posted in -- By the Physicist, Geometry, Math | 8 Comments