So at this point we have an idea of what entanglement is, and some reassurance that it doesn’t ruin all of physics with its existence! Now we turn to a very important question: how, in practice, do we produce entangled quantum particles?

In our discussions so far, we have imagined entanglement arising through the hypothetical decay of a neutral pion into an electron/positron pair, as illustrated below.

This is a fine idea in principle, but it is an utterly impractical method to reliably create entangled particles in a laboratory setting. First, let’s talk about the pion: a pion is a high-energy particle that we typically¹ only see as a product in high-energy particle physics experiments, such as those done at places like CERN. When produced, they are usually moving at relativistic speeds, and in a direction that is more or less random. They decay very quickly, on the order of 10 billionths of a billionth of a second when at rest, which means you can’t store them for future use. Even if you could store them, the direction of their decay products is random as well, which means we should revise our image above to appear as shown below.

In short: even if we managed to get a pion to sit still in one place, we wouldn’t know where to put our pair of detectors to spot the particles. We would miss the vast majority of entangled pairs.

So using pions as a source of entangled particles to do experiments is not practical. Fortunately, it turns out that we have a great source of particles available to us that are relatively easy to produce in an entangled state: photons, i.e. particles of light! By the use of a process known as spontaneous parameteric down conversion, we can reliably and easily produce photons with entangled polarization that appear in very predictable locations. Let’s see how!

We begin by noting that photons are fundamentally different than the particles we’ve been entangling so far, and the nature of their entanglement is somewhat different. Electrons and positrons are spin-1/2 particles, which we have seen can exist individually in a quantum superposition state of spin-up and spin-down.

Photons, however, are spin-1 particles, which can still be described by a superposition of two states, but the two states in question are horizontal and vertical polarization!

Here, I’ve used “V” and “H” for “vertical” and “horizontal.” In the classical wave description of light, the polarization of a light wave represents the way the electric field “waves”, or oscillates. It can oscillate along a line, vertically, horizontally, or along any angle, or it can oscillate in a circular path. The former case is known as linear polarization, the latter as circular polarization.

Just for some perspective, below is an image of what a vertically polarized electromagnetic wave would “look like” from the side: the electric field **E** wiggles up and down, the magnetic field **H** wiggles left to right, and the wave itself propagates in a direction perpendicular to both wiggling directions.

What we will soon is that we can entangle a pair of photons in such a way that if one is found to be horizontal, then the other is definitely vertical, with a quantum state that appears as follows.

To understand how one produces this state, however, we have to talk about *nonlinear optics*, so let us spend a little time explaining what this means.

One thing you may have noticed about beams of light under ordinary circumstances: they don’t interact with one another. If we send two beams of light on a collision course, they pass right through each other without effect. On a quantum level, we can say that the photons don’t interact with each other.²

This is only true, however, in vacuum. When light propagates in matter, the matter can “mediate” an interaction between two or more photons, causing photons to combine, split apart, or do even more unusual things. In order for these interactions to happen, however, photons have to be relatively densely packed or, equivalently, the beam of light itself has to be very intense. Such high-intensity beams only became possible with the invention of the laser in 1960, and the field of *nonlinear optics³*, describing the interactions of photons, was founded not long afterwards.

Two typical nonlinear processes are illustrated below. The first, called second-harmonic generation, involves the combination of two photons of frequency ω into a single photon of frequency 2ω. Because the energy of the photon is proportional to the frequency, this means that two photons combine into a single photon of twice the energy. If you’ve ever used a green laser pointer, you have probably seen second-harmonic generation in action: they typically achieve a green color by sending an infrared laser beam with wavelength 1064 nm through a nonlinear potassium titanyl phosphate (KTP) crystal , resulting in green output photons of wavelength 532 nm.

The second process pictured above is known as difference-frequency generation, and it involves a photon of frequency ω_{2} interacting with a photon of frequency ω_{1}. This causes the higher-frequency photon to split into a photon of frequency ω_{2}-ω_{1} and one of frequency ω_{1}, with the original photon of frequency ω_{1} remaining.

Both of these processes are known as *parametric* processes, which means that the net energy and momentum of the input and output photons are the same; no momentum or energy gets transferred to the nonlinear medium. This ends up being a bit of a problem, as in most materials the momentum of a photon of frequency 2ω is *larger* than the total momentum of two photons of frequency ω. This is solved by using crystal structures for which the *speed* of light depends on the *polarization* of light. Because there are two perpendicular polarization possible (think “horizontal” and “vertical” again), there are two speeds in such materials. This difference in speeds results in the phenomenon of double refraction, as seen in the photograph below. The two images result because the two different polarizations of light take slightly different paths through the crystal.

We don’t want to dive into all the complicated optics of crystals right now, but suffice to say that there are crystals for which the net momentum of two horizontal photons of frequency ω is equal to the momentum of one vertical photon of frequency 2ω, allowing second-harmonic generation to happen. Such a process is known as Type-I second-harmonic generation (there is also a Type-II).

But let’s think a bit more about the process of difference-frequency generation. It is what we refer to as a *stimulated* process, in that the additional photon of frequency ω_{1} is needed to “stimulate” the photon of frequency ω_{2} to split into two. A different type of stimulated process is in fact what makes a laser — “Light Amplification by *Stimulated* Emission of Radiation” — operate. In a laser, photons interact with excited atoms and cause those atoms to emit another photon, and this stimulated process results in the bright and directional beam of the laser.

However, we know that atoms can and will emit radiation without the need for being stimulated. Every ordinary light source — lightbulbs, lightning bugs, the sun — consists of a collection of atoms that *spontaneously* emit radiation. It is then natural to wonder: is it possible to have a photon spontaneously split into two in a nonlinear medium, as happens in difference-frequency generation but without the extra photon making it happen?

It can, in fact, in a process that is known as *spontaneous parametric down conversion (SPDC)*. “Spontaneous” refers to the fact that it can happen without a “helper” photon providing stimulation, while “parametric,” as mentioned earlier, indicates that it is a process that conserves energy and momentum among the photons. “Down conversion” is another way to refer to the specific case of difference-frequency generation in which one photon of frequency 2ω breaks into two photons of frequency ω: the frequency of the output photons goes down.

In Type-II SPDC, the two output photons always have perpendicular polarizations. The conservation of momentum results (in a way too difficult to explain here) in the output photons propagating along two intersecting conical surfaces. If one projects this output onto a screen, one will see two intersecting circles, as pictured below.^{4}

But let’s consider the positions of individual pairs of photons that are created. In the picture above, the original “pump” beam would be coming directly out of the page; because of momentum conservation, the pairs of photons must be on opposite sides of this central axis. This is illustrated below.

Part (a) of the figure gives a 3-dimensional perspective on the overlapping cones. In part (b), we visualize the positions of photon pairs as a series of discrete points: if one photon appears at position 5, then the other photon appears at position 5′. We imagine, for simplicity, that the photons on the upper circle are always vertically polarized, and the photons on the lower circle are always horizontally polarized.^{5 }It is impossible to predict where any individual pair will appear, thanks to the inherent randomness of quantum physics, but if one appears in an unprimed position, the other will appear in the corresponding primed position.

But now imagine what happens when the pair of photons appear in positions 1, 9′ and 1′, 9. It is possible for either a horizontal or vertical photon to appear in either place! We again have a situation where the law of physics tell us that the two photons must be perpendicular, but do not tell us whether the photon in 1, 9′, for instance, is horizontal or vertical. The combined quantum state of any pair of photons appearing in this pair of locations must therefore be the entangled state we wrote earlier!

If we stick a pair of photon detectors in the paths of the 1, 9′ and 1′, 9 positions, we will have a nearly perfect and reliable source of entangled photon pairs that we can measure. This is the most common way that optics researchers produce entangled particles in the laboratory today. It is still, in a sense, inefficient — for every pair going to 1, 9′ and 1′, 9, there is a pair going to 2, 2′ and 3, 3′ and… — but photons are cheap!

There is one curious complication to this strategy that is illuminating and, when you ponder it for a while, quite astonishing. We have mentioned that photons propagating in an anisotropic crystal — like the one used to create SPDC — travel at different speeds depending on their polarization. This means that the pair of photons created in SPDC end up traveling at slightly different speeds through the crystal, which means that they end up arriving at the detector at slightly different times.

This means, however, that the photons are again *distinguishable* — we could tell which photon came from which location simply by timing when they arrive. This distinguishability effectively kills the appearance^{6} of entanglement between the photons, which means we won’t be able to do interesting experiments with them, like study wave interference.

This problem can be fixed the same way it was created: through the use of more anisotropic crystals! Suppose that the horizontal photon would arrive before the vertical one, because horizontally-polarized photons move faster than vertical ones in the SPDC crystal. We can correct this by putting anisotropic crystals in from of the two positions 1, 9′ and 1′, 9 in which horizontal photons move slower and vertical photons move faster! With the right choice of crystal, we can completely eliminate the time delay between the photons and make them indistinguishable again. This is illustrated below.

This is a simple example of a device referred to as a “quantum eraser,” which removes, or “erases,” information about a photon or photons that gives explicit information about their behavior.

I will hopefully have more to say about quantum erasers in a future post, but for now I would like to point out that this idea introduces another aspect of quantum physics that I have not addressed directly before: *the wave properties of quantum particles are intimately connected to the possible behaviors of that particle being indistinguishable*. The classic example of this is Young’s double slit experiment, which we used to introduce the wave properties of electrons in the first place:

We noted in part 2 of this series that even a single electron evolves as a wave through the double slit experiment — though it hits the screen at a definite point, the probability of it arriving at that point is dictated by the wave properties. When enough electrons accumulate at the screen *M*, the interference pattern becomes visible.

A different way to look at this is that the interference pattern appears because we have no way of *distinguishing* which slit the electron actually goes through. The wave itself essentially goes through both slits, and this is what gives us interference.

Let us imagine that we were to add a sensor to the lower slit in this experiment that would detect the presence of electrons passing through the slit. The two possible outcomes of the experiment for each electron are now distinguishable — we will know which slit the electron actually passed through. We would find that the interference pattern completely disappears! The wave effects are directly connected to our inability to tell which slit the electrons pass through.

This principle, that interference effects require indistinguishability, is pretty much a universal principle in quantum physics. We will likely encounter it again in later parts of this series.

In the next post, however, I am going to try and explain the revolutionary discovery of physicist John Stewart Bell, one that fundamentally changed the way we think about the meaning of quantum physics!

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¹ Pions are actually produced by the collision of high energy particles with the atmosphere, but they only appear at high altitudes. The original pion discoveries were done on mountaintops.

² This is only approximately true, in fact. Photons can interact with each other even in vacuum, though the effect is very weak and requires high-energy physics experiments to detect.

³ Why is it called “nonlinear optics”? In ordinary optics, the relationship between the polarization of a medium and the electric field of the illuminating light is a linear one. In processes like second-harmonic generation, the polarization is proportional to the *square* of the electric field — it is nonlinear with respect to the electric field. For our purposes, we can say that nonlinear optics involves the interaction of two or more photons.

^{4} Image taken from the paper by Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko, and Shih, “New high-intensity source of polarization-entangled photon pairs,” Phys. Rev. Lett. 75 (1995), 4337.

^{5} In practice, the relative orientations of the polarizations of the two photons varies on each circle, but the two created photons are always at 90 degrees from each other. We pretend that one is always vertical and the other always horizontal for simplicity.

^{6} I chose the word “appearance” here carefully, because a proper interpretation of what is going on is a bit more subtle.

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In the last post, we finally introduced the concept of quantum entanglement. An example of an entangled state between two quantum particles is given by the decay of a spin-zero pion into a spin-1/2 positron and a spin-1/2 electron, as illustrated below.

This results in a combined quantum spin state for the electron and positron that may be written as:

We may read this as “the two spin-1/2 particles end up in a quantum state which is an equal superposition of the positron being spin-up and the electron being spin-down with the positron being spin-down and the electron being spin-up.”

This suggests that the electron and positron, when produced in the decay, might be considered to exist simultaneously in a state where the electron (-) is up and the positron (+) is down, and vice-versa — their fates are “entangled.” When we measure the state of one of the particles, say the electron, it is 50% likely to “choose” the spin-up state and 50% likely to choose the spin-down state. When it does, the positron, no matter how far away, must *instantly* take on the opposite spin state — at least according to the original Copenhagen interpretation of quantum physics. In short, after measurement, the combined state of the electron and positron is either:

But, note the use of the word “instantly.” Because angular momentum is conserved, if the electron is measured spin-down, the positron must be in a spin-up state. This collapse of the wavefunction must happen as soon as the electron is measured, otherwise there would be the possibility of measuring the positron also in a spin-down state, which would violate angular momentum conservation.

This would seem to suggest that the electron must send a “message” to the positron, and this message arrives instantaneously, regardless of the distance between them. However, according to Einstein’s special theory of relativity, nothing is supposed to be able to move faster than the vacuum speed of light.

This raises the question: does entanglement violate special relativity? And, if it does, can we use it to communicate over vast distances at superluminal speeds?

As it turns out, the correct answer is “neither.” Entanglement, when considered carefully in the context of the full quantum theory, turns out to be perfectly consistent with relativity. But, as we will see, it is quite a theoretical adventure to come to that conclusion!

We begin our adventure by pointing out that it is well-known that there are, in fact, all sorts of “things” that can move faster than the vacuum speed of light — for a sufficiently loose definition of “things.” A couple of classic examples will serve to illustrate the point. For the first example, imagine a tiny gnat flying in front of a laser beam that is being projected at the Moon. The gnat casts a shadow and, because the beam itself gets wider as it travels, so will the shadow. This means that the slow-moving gnat on the Earth creates a very fast-moving shadow on the Moon!

A beam of light keeps spreading the further it gets away from the source, which means that the motion of a shadow gets faster the further away it gets. In a previous post, I estimated that, on Pluto, the shadow would be moving at roughly 10 times the speed of light!

Another example is known as the “scissors paradox” in special relativity. You may not have noticed this before, but the intersection point between the scissors blades moves faster as the blades close. This increase in speed is why scissors make a “snip” sound when you close them — they make a low frequency “sss” when the intersection is moving slow, and a high frequency “nip” when it’s moving fast!

If we make a really, really long pair of scissors — say, on the order of several light years in length — we can easily make the intersection point move faster than the speed of light.

Now we ask: can we use these systems to send messages at a speed faster than light?

It is pretty clear that, in the shadow example, the answer is “no.” Imagine that we have two bases on Pluto, and want to use the shadow to send a message from Base A to Base B. In principle, it would seem that we could use the shadow as a message: Base B will definitely know when it is cloaked in shadow. However, because the shadow is controlled by the gnat on the Earth, Base A must first send a message to Earth to tell the gnat to move. This message must travel to Earth at the speed of light, and then the shadow itself will not appear on Pluto until it has propagated from Earth, also at the speed of light. Because the travel time for light from Earth to Pluto is on the order of 5 hours, it will take much, much longer to send a message with a shadow than just by using an ordinary beam of light. The shadow cannot be used to convey a message without the help of the intermediate “gnat signal,” which travels at the speed of light.

The scissors example is a little more subtle to explain, but the result is the same: it is still not possible to use them to communicate faster than light. Suppose a messenger A tries to use our giant scissors to cut a rope at a distant station B, making a bell ring. The messenger A starts to close the scissors, but the blades themselves will not move instantaneously: the force applied to the blades will travel along them at the speed of light, until the entire pair of scissors is in motion. Once the blades start moving, the intersection point will move faster than the speed of light, but this will always be lagging behind the motion of the blade tips, which only started moving once the force reached them.

For both our “superluminal” examples, we find that the actual transfer of a message using faster-than-light techniques still always lags behind signals that are sent at the speed of light.

So we need to refine our original statement that “nothing travels faster than the vacuum speed of light.” Instead, we should say that “nothing *useful* travels faster than the vacuum speed of light.” Any sort of trick that we manage to use to make something — shadows, blade crossings — move faster than light seems to end up not being able to carry useful information without first having something moving at the relativistic limit.

We now turn back to our original question: “does entanglement allow faster-than-light communication?” We may now restate this as: “can we transmit any useful information using entanglement?”

We consider a possible communication scenario of the simplest form to begin with. Let us imagine a single entangled electron-positron pair is sent to distant observers A and B, each of whom store their particle, unmeasured, until ready to “communicate.” This is sketched below.

When ready, observer A will open the box, and use the analyzer to determine whether her particle is spin-up or spin-down. This will cause the wavefunction to instantaneously collapse, and put the particle of observer B in the opposite state of that measured by A.

The first question we ask: can observer B tell from his measurement whether observer A actually performed a measurement on her particle? This would be a way to transmit a simple “yes” or “no” message: “yes (I’ve measured my particle)” or “no (I haven’t measured my particle).”

The answer (to our physics question, not the measurement) is “no.” When observer A opens her box and uses her analyzer, she sees a particle that is either definitely spin-up or definitely spin-down. The same is true for observer B. Regardless of who makes their measurement first, A or B, they each have a 50/50 chance of finding the particle spin-up or spin-down. The possible outcomes for observer B are the same regardless of whether observer A performs her measurement. And observer A has no control over the outcome of her measurement, so she has no control over the outcome of observer B’s measurement. The randomness inherent to quantum mechanics effectively ruins any hope of communication in this simple scheme.

But there is one limitation of our first naive attempt: observer A and observer B both attempt the same *type* of measurement of their particles. To be more specific, both of them measure the spin of their respective particles along the vertical axis, using their analyzers oriented vertically. If observer A measures her particle’s spin along a different axis, this will result in particle B ending up in a different quantum state. Perhaps this difference can be detected, and therefore used to send information? Observer A could send “yes” or “no” by measuring “vertical” or “horizontal,” if this scheme works.

To explain how this would work, we need to talk a little more about quantum spin and the idea of a *change of basis*. We have previously described quantum spin as the inherent angular momentum of a quantum particle, roughly analogous to the Earth spinning around its axis. However, when measured, the spin of an electron can only take on two possible values, +1/2 or -1/2, which we call spin-up and spin-down. Strangely, we have the same possibilities *regardless of how we choose to measure the particle*: if we measure the spin vertically, horizontally, or longitudinally, we will always get one of two results: +1/2 or -1/2. This is very different from the measurement of a classical spinning object. If we were to introduce a coordinate system to describe the rotation of a spinning globe, for instance, we would generally need 3 numbers to describe its rotation, corresponding to how much it is rotating around the *x*, *y* and *z*-axes. In the case of a spin-1/2 particle, once we’ve measured the spin along any axis, we know exactly what the particle’s spin quantum state is: in fact, thanks to wavefunction collapse, we have transformed the particle into that quantum state.

One implication of the simple nature of quantum spin is that the same quantum states that we use to describe a particle being spin-up or spin-down in a vertical measurement can also be used to describe a particle being spin-left or spin-right in a horizontal measurement. We can write the actual quantum state of the spin in terms of the up/down states or the left/right states.

In words, we may say: a particle that is in a definite spin-right state or spin-left state may be considered an equal mixture of spin-up and spin-down (with the sign of the sum determining which). This can be reversed, as well, to say that:

The choice of how we choose to describe the spin state — up/down or left/right — is called the *basis*. If we decide to write one set of states in terms of another — left/right in terms of up/down, for example — it is known as a *change of basis*.

Why does spin work this way, so that we can describe up/down spin in terms of left/right, and vice-versa? We would need to do some significant math to prove these relations, but they make a bit of intuitive sense. A particle that is definitely in a horizontal spin state must have equal mixtures of spin-up and spin-down that cancel out, which is exactly what we have above.

Now we come to the key question: what happens if observer A measures her particle with a horizontal analyzer? That is, our experiment pre-measurement now looks like this:

Let us suppose that observer A measures that the electron is spin-right. That means that the positron is in a definite spin-left state. But observer B is using a vertical analyzer, so they are measuring whether the particle is spin-up or spin-down. That means that the state of the positron sitting in observer B’s box is:

We can roughly illustrate what happens, prior to B’s measurement, by the following image.

However, this spin-left state is a state that, again, has a 50% chance of being measured in the spin-up position and 50% chance of being measured in the spin-down position, which is exactly the outcome we would expect if both analyzers were vertical! Apparently changing the *type* of measurement that observer A makes will also not effect the measured outcomes that observer B will make.

But wait — there’s one more possibility! Instead of looking at individual particles, let us imagine that we insert a “quantum duplicator” at the output of observer B’s box, that will create a large number of positrons all in the *exact same quantum state* as the original one. In short: when observer B decides to make a measurement, he opens his box, and the positron passes into the quantum duplicator, which makes a large number of exact copies of it. Then each of those copies goes through the analyzer.

We illustrate all four possible outcomes now — two possible measurements that observer A can make, and two possible outcomes for each of those measurements by observer A.

I’ve boxed the two distinct outcomes above. When observer A measures the electron with a horizontal analyzer, observer B will get a bunch of positrons in superposition states. When measured, the positrons will be 50% up, 50% down. However, when observer A measures the electron with a vertical analyzer, observer B will either get positrons 100% up, or 100% down!

This seems to suggest that observer A can signal, superluminally, to observer B by the choice of analyzer orientation. For instance, “horizontal” could mean “yes” and “vertical” could mean “no.” This scheme, however, *depends on the ability to make a quantum multiplier*, that can create multiple perfect copies of a given quantum particle. Is this possible?

In fact, it is not! There is a theorem in quantum physics known as the *no-cloning theorem* that explicitly states that it is not possible to copy the quantum state of a particle without wrecking the original quantum state. That is, we could make particle #2 have the same state as particle #1 originally had, but in the process we would have changed the state of particle #1. Making multiple perfect copies of a single quantum state is therefore not possible, and our superluminal communication scheme¹ using entangled photons simply won’t work.

Proving the no-cloning theorem is definitely outside the scope of this post and our abilities, but a little thought about measurement and wavefunction collapse shows why it is plausible. Wavefunction collapse is an indication that, whenever we make a measurement of a particle’s quantum state, we in general change that quantum state. Any attempt to “clone” a particle would, in essence, require that the quantum state of that particle be measured², changing it.

In the end, then, we conclude that entanglement does not allow us to communicate faster than light. Surprisingly, the inability to do so hinges upon the seemingly-unrelated inability to duplicate quantum states! The collapse of the entangled wavefunction falls into the category of “stuff that moves faster than light but isn’t useful” like shadows and giant scissors.

To conclude, I should note again that we have based our entire discussion of superluminal communication on the Copenhagen interpretation of quantum physics, which assumes that the wavefunction collapses when it is measured. There has not yet been any experiment that disproves the Copenhagen interpretation, but there are other interpretations of quantum physics that reach the same conclusions. We will discuss some of these interpretations in a future post.

However, in my next entanglement post, I want to talk about how physicists generate entangled quantum states in the laboratory! There are much, much better and more efficient ways to do it other than with decaying pions.

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¹ This argument was made in a classic paper by D. Diecks, “Communication by EPR devices,” Phys. Lett. 92A (1982), 271-272. Diecks shows that, if one assumes that a quantum multiplier exists, one runs into a contradiction with fundamental aspects of quantum physics, which implies that the multiplier cannot exist. Note that this paper appeared not too long ago — 1982! The arguments about entanglement, what it means, and what can be done with it are still ongoing.

² This isn’t really a great way of justifying the no-cloning theorem, because one could imagine the possibility of cloning a particle through quantum interactions without doing a classical measurement. That is, we could bring 2 or more quantum particles together and hypothetically make them interact in such a way that they come out in the same state. The no-cloning theorem states that such a cloning process is not possible in any way. Diecks’ paper, mentioned above, is one of the first mentions of “no-cloning.”

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Here, in part 3, we will at long last introduce entanglement! But, before we do, we need to be sure we really understand what the wave properties of a quantum particle imply about its behavior.

So, by the late 1920s, physicists knew that discrete bits of matter — electrons, for example — sometimes act like a wave and sometimes act like a particle. This seemingly contradictory nature is often referred to as *wave-particle duality*. It was not immediately obvious how to interpret the wave properties of matter, but physicists finally settled on what is referred to as the “Copenhagen interpretation,” after the city in which it was more or less developed. If we consider the motion of a single electron, the Copenhagen description of quantum behavior could be summarized as follows:

- While freely propagating through space, the electron and all of its properties evolve as a wave.
- The wave may be described as a “wave of possibilities”: the amplitude of the wave only describes the probability of the electron to be found in a certain position or configuration.
- While evolving as a wave, the electron in general has
*no definite position or configuration*— it is, roughly speaking, existing in all possible configurations simultaneously. In Young’s double slit experiment, for instance, it is often said that the electron goes through*both*slits. - When a property of the electron is measured in an experiment, where the property must take on a definite value, the electron “chooses” an outcome, based on the wave probabilities mentioned above. The wave “collapses” into that single outcome. For instance, if we have a detector to measure the position of the electron, we will see that electron at a single definite location on the detector.
- If the particle still is free to continue moving, the process repeats itself, the collapsed wave evolving again, usually resulting in a wave of many possible outcomes again.

You might have noticed that I have sneakily added the word “configuration” when describing the behavior of the electron. In previous posts, we have described the electron as being a wave that evolves in space and time, implying that the wave describes the probability of finding the particle in a particular place at a particular time. However, every aspect of a quantum object is similarly wrapped up in this “wave of possibilities,” and is uncertain until it is specifically measured. This includes the energy of the particle, the momentum of the particle and, what will be particularly important to us, the angular momentum of the particle. We usually refer to the entire collection of properties of the particle that can be known simultaneously as the *state* of the particle.

At this point, all of this talk of “measurement” and “probability” might seem a little abstract. Let us try and make it clear with a silly example: a quantum quarter!

Let us imagine that we are able to fabricate a really, really small quarter, of a size comparable to that of an atom. Then this quarter would be subject to the laws of quantum physics, and the Copenhagen interpretation applies. Let us imagine we flip this quantum coin — what happens?

- After the flip, but before looking at it, the quantum quarter is in a wave state that is simultaneously 50% heads and 50% tails.
- When we look at the coin, i.e. when we measure it, the quarter “chooses” one of the outcomes, and becomes either definitely heads or definitely tails.

The key difference between this quantum coin flip and an ordinary coin flip is in the first part of this description. When we flip an ordinary coin, we may not know the outcome — we covered the coin with our hand, for instance — but it is definitely either heads **or** tails under our hand. With the quantum quarter, it is heads **and** tails until we look at it and “collapse” the wave of the quarter.

At this point, it will be helpful to introduce a little mathematical notation. We aren’t going to do any complicated mathematics here, but the notation will make the discussion a little clearer. First of all, let me represent the state of the quarter, *after* being flipped but *before* being looked at, as follows.

This looks a little strange at first, but I have written the state of the quarter in exactly the form that a quantum physicist would. The combination |〉 symbolizes the quantum state of an object; by |quarter〉 we mean “the general quantum state of the quarter.” *a*_{heads} represents the amplitude of that part of the wave which represents the quarter landing on heads, and *a*_{tails} represents the amplitude of that part of the wave which represents the quarter landing on tails. The objects |heads〉 and |tails〉 represent the quarter being either heads up or tails up.

Our equation written above, then, states that “the quantum state of the quarter (before measurement) is a combination of the quarter being in the state heads up and the state tails up.”

If it is a fair quarter, we expect that the probabilities of heads or tails are equal, i.e. 1/2 for each. (Where we use 1 = 100% and 1/2 = 50%.) Our quantum state of the quarter may then be written as:

Whenever we describe the state of a particle as being some sum of distinct outcomes, we refer to it as a *superposition*.

Wait: why is it one over the square root of two, instead of 1/2? You may remember in a previous post that I mentioned the Born rule for relating the quantum wave of the system to the probability of a particular measurement; it turns out that the relation between the probability *p* of a particular outcome and the amplitude *a* of that outcome in the wave is:

We don’t need to worry about this too much in our discussion going forward, but I wanted to write the equation in its physically correct form.

So what happens to the quantum state of the quarter after measurement? We cannot predict with certainty what happens: the Copenhagen interpretation of quantum mechanics implies that the outcome is truly a random one. Let us suppose we find that the quarter is heads up; this means that the quantum state of the quarter has “collapsed” to the form

The part of the state that was |tails〉 has disappeared, leaving us with a quantum quarter that is definitely, 100% heads up.

If this idea of “wavefunction collapse” seems strange and somewhat unsatisfying to you, you’re not alone: this part of quantum physics troubles and confounds researchers to this day. Nowadays, the conundrum is known as the “measurement problem.” We will talk a lot about this in the near future, but let me just point out one issue with the idea of “measurement” right now: what counts as a measurement? The way we have described it, it sounds as if a human being observing a quantum particle is necessary for the wave to collapse. This is, however, clearly an unsatisfactory explanation, as there is no reason to think that humans, or any living creatures, have such a special influence on the state of reality. Such an interpretation of wavefunction collapse is more an indication of our human-centric bias in describing the universe than any profound physics.

A slightly better explanation is that a measurement occurs whenever a quantum particle like an electron interacts with a macroscopic system, i.e. any laboratory detectors. The thinking here is that, somehow, a large collection of atoms or other quantum particles act fundamentally different than a small number, and that the large collection forces the electron to “make up its mind,” so to speak. Call it a quantum-mechanical form of peer pressure, if you will! This is also not particularly satisfying, and we will have more to say about measurement going forward.

Now we are finally, finally ready to introduce entanglement! Let us do so by sticking with our quantum quarter example; however, we now consider the situation where we glue two quarters together, tail to tail. This means that the heads of the two coins are facing outwards. In what is a subtly important point, we scratch a number “1” on one of the outward faces of our double coin, and a number “2” on the other, so that the two faces are *distinguishable*. This is more or less what the system looks like:

Let’s think about the outcomes of a non-quantum version of this first. If we flip the coin and it lands with heads 1 up, that means that tails 2 is up; similarly, if heads 2 lands facing up, that means that tails 1 is up. These are the only two possibilities from the flip of this double coin: the two coins, by nature of their gluing, always land with opposite faces up.

Similar to our earlier argument, the flip of this non-quantum double coin lands with “heads 1, tails 2” **or** “tails 1, heads 2”. If we were to make a quantum version of this coin, then, before it is measured, it is in the state “heads 1, tails 2” **and** “tails 1, heads 2”. In terms of our quantum physics notation, we may write:

This is our first example of *quantum entanglement*. It should be noted that it is not possible to talk about the quantum state of quarter 1 without taking into account the behavior of quarter 2, because the behaviors of the two coins have a definite relationship to each other: their fates are “entangled,” somewhat like the fates of two prisoners chained together at the wrist are entangled!

A rough way of thinking about entanglement is that it arises when the physics of a problem forces a definite relationship between two quantum objects but does not force either object to take on a definite state. For our quantum double quarter, the two coins will always land with opposing faces pointing upward, but nothing predetermines whether it will be coin 1 or coin 2 that lands heads up.

Our idea of a “quantum coin” and “quantum double coin” may seem quite artificial, but it turns out that electrons and other elementary particles such as protons and neutrons possess a property somewhat analogous to it, known as *spin†*.

You are hopefully familiar with the concept of *angular momentum*, which may be roughly described as the “momentum of rotation.” Such angular momentum is usual broken into two types: orbital and spin. As the name implies, orbital angular momentum comes from the orbital motion of an object around some external axis of rotation, while spin angular momentum comes from the rotation of an object about its own axis. The motion of the Earth around the Sun has orbital and spin components, as roughly illustrated below.

The spin of quantum particles is a little different: the measured spin of an elementary particle is a fixed quantity that never changes, and is either an integer or half-integer multiple of Planck’s constant, “h-bar.” The spin of some elementary particles is tabulated below; we will often write “spin-1/2” as shorthand for .

Even stranger: the spin of a spin-1/2 particle like an electron will always be measured as either +1/2 or -1/2. That is, the spin of the particle is 1/2 but it can be measured as spinning in a clockwise sense or a counterclockwise sense, which we refer to as “spin-up” or “spin-down.”

You may see where we’re going with this now: a spin-1/2 particle acts very much like a quantum quarter! The mathematical formula for such a state can be written as:

This can be read as “the general quantum state of a spin-1/2 particle is a superposition of a spin-up state and a spin-down state.”

We may also create an entangled spin state between two quantum particles by using the conservation of angular momentum. Let us suppose that we have an unstable, zero electric charge, spin-0 particle such as a pion which decays into a negatively-charged electron and a postively-charged anti-electron (positron), as crudely illustrated below.

Because the total angular momentum of the system is conserved, the net spin momentum of the electron and positron together must equal zero, which was the spin of the pion. This means that if the electron is spin-up, the positron is spin-down, and vice-versa. However, there is nothing in the laws of physics that forces either the electron or the positron to be spin-up or spin-down, so the quantum state of the electron and positron together must be a superposition of both possibilities: we have an entangled state almost exactly like the double quarter from earlier!

We may interpret this equation¹ as saying that “the two spin-1/2 particles end up in a quantum state which is an equal superposition of the positron being spin-up and the electron being spin-down with the positron being spin-down and the electron being spin-up.”

As in the double quarter case, we expect that if we measure the electron as being in a spin-down state, then the wavefunction has collapsed so that the positron is definitely in the spin-up state. There is one *huge* difference between the electron-positron case and the double quarter case, however. The quarters were stuck together, and physically connected to each other; the electron and positron, however, can be allowed to propagate through space an arbitrary distance away from each other without measuring them. Then, in principle, the quantum state written above still holds even when the electron and positron are light-years away from each other; what happens when we measure the spin of, say, the electron?

Let us suppose that it is measured to be spin-up. According to the Copenhagen interpretation, the entire wavefunction immediately collapses into the spin-up state of the electron. However, because of angular momentum conservation, the positron must have instantly collapsed into the spin-down state. This collapse must, evidently, *happen instantaneously, that is, faster than the vacuum speed of light*; otherwise, a measurement of the positron in the meantime could also return a spin-up value, which would violate angular momentum conservation!² At first glance, then, it would seem like quantum entanglement allows violation of the universal speed limit that Einstein’s theory of relativity postulates.

This idea, which implies that entanglement allows the quantum state of separated particles to change instantaneously when one of them is measured, was referred to by Albert Einstein as “spooky action at a distance.” This phrase was not intended to be flattering. In a paper published in 1935, Einstein, Podolsky and Rosen³ (EPR) used “spooky action” to argue that the Copenhagen interpretation of quantum mechanics, and the idea of “wavefunction collapse,” must be incorrect.

In the Copenhagen interpretation, as we have seen, the behavior of a quantum particle or particles is truly random: that is, it exists in a superposition state, both spin-up and spin-down simultaneously, until it is measured, at which point it chooses an outcome. EPR instead concluded that the seeming randomness of a quantum system is simply a symptom of the fact that we do not know how to properly measure all the factors that determine whether a particular particle is spin-up or spin-down. The situation, they argued, is very much like the flipping of an ordinary coin. We treat the flipping of a coin as a random event, but in fact if we were able to properly measure all the variables of the flip — how high we tossed the coin, how much we made it spin, how much it will bounce on landing — we would be able to know with 100% certainty whether it will come up heads or tails. It only seems random because we don’t usually measure, or know how to measure, all the variables that decide whether a coin comes up heads or tails.

Similarly, EPR said that quantum particles like electrons must possess “hidden variables” that determine whether the particle is measured spin-up or spin-down. The process seems random to us, but that is only because we don’t know what those additional variables are. EPR argued that the state of the electron and positron are always perfectly well-defined, in contrast to “wave of possibilities” that the Copenhagen interpretation suggests.

Another objection to the Copenhagen interpretation of quantum physics was suggested in 1935 by Schrödinger^{4}, and it is also intimately connected to the measurement problem we discussed earlier. As he tells it:

One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The ψ-function [wavefunction] of the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.

This is, of course, the famous Schrödinger’s cat thought experiment which has inspired not only physics but science fiction and snarky internet memes. An illustration of the idea, from Wikipedia, is shown below.

The idea is simply this: by putting the cat in the box which is linked to the radioactive atoms, we are entangling the cat with the atoms. We may write the quantum state of the combined cat and box system as follows.

This goes back to the question: “what is a measurement, and who does the measuring?” If we consider the cat part of the quantum system, then it is simultaneously living and dead in the box until we open the box and see the result. Or does the cat count as a measuring device? Or is it the radioactive detector that detects the radioactive decay? These questions get right to the heart of the puzzle: how do we interpret quantum physics? This is a puzzle that we are still not sure how to solve.

We do, however, know and understand a lot more than I can cover in this post! In the next post in this series, we will take a look at the issue of entanglement and relativity discussed above, and see how to resolve the seeming conflict between them. In a future post, we will address the issue of “hidden variables” that EPR introduced, and the answer will surprise us! I also hope to discuss how physicists today generate entangled quantum particles in the laboratory, and also discuss some of the possible applications of quantum entanglement, such as quantum cryptography and quantum computing. I will also try and discuss going “beyond Copenhagen”: different interpretations of quantum physics that avoid the measurement problem and wavefunction collapse that we have discussed.

As you can see, we still have quite a lot of physics to untangle!

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† If you want to learn a little more about quantum spin, please see my article on experiments with neutron spin.

¹ You may have noticed that there is a minus sign in the equation for the electron and positron together, where we had a plus sign for our double quarter. When one rigorously calculates the quantum state of the electron positron combination, that minus sign is necessary for the total spin to be zero. It doesn’t affect our argument here; I only include it so that physicists don’t yell at me.

² I borrowed some wording here from my recently-published *Singular Optics* textbook, as I didn’t think I could improve upon it.

³ Einstein, Podolsky and Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47 (1935), 777–780.

^{4} Schrödinger, “Die gegenwärtige Situation in der Quantenmechanik (The present situation in quantum mechanics)”. Naturwissenschaften. 23 (1935), 807–812.

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This book will combine physics and history to tell the surprisingly long story of scientists and engineers studying the remarkable ability of a cat to (almost) always land on its feet when it falls from a height. In other words, I will talk about images such as the famous one below, which was the first series of high-speed photographs taken of a falling cat, back in the 1890s.

Scientists and engineers have been fascinated by falling cats for a remarkably long time, stretching back at least as far as the 1850s and continuing to some extent even today. The puzzle of “cat-turning,” as it was known in Victorian times, has played a noteworthy role in the history of photography, geophysics, robotics, and even space exploration. Furthermore, the basic mechanism by which a cat rights itself while falling is connected to very profound mathematics related to the propagation of light, quantum physics, the motion of pendulums, and even parallel parking.

In telling the story, we will come across a number of famous physicists, such as James Clerk Maxwell, the father of electromagnetism, and see that cats — with their cat-turning — have caused all sorts of mischief over the years. The book will contain many illustrations of cats in free-fall, as many photos have been taken over the years…

… and it will also include some images of my own cats, to help illustrate the physics without complicated mathematics!

My book, *Falling Felines and Fundamental Physics*, will take a light-hearted look at the history of the falling cat problem, and will at the same time use it to introduce fascinating and fun concepts in physics. The book is intended for anyone interested in physics, or cats, or both. And it will include cat pictures!

I’m really excited to tell you this story! I’ve still got a lot of work to do, both writing and research, but the book is due to be finished in mid-2018, which means you will hopefully be able to read it in the second half of that year!

I will keep everyone updated on my progress!

*(PS: haven’t forgotten the entanglement series of posts: was out of town for work this past week. Will return to it asap.)*

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So, by the mid 1920s, physicists had made significant progress in developing the new quantum theory. It had been shown that light and matter each possess a dual nature as waves and particles, and Schrödinger had derived a mathematical equation that accurately described how the wave part of matter evolves in space and time.

But it was not clear what, exactly, was doing the “waving” in a matter wave. Water waves arise from the oscillation (waving) of water, sound waves arise from the oscillation of molecules in the air, but what is oscillating in a matter wave? Or, to put it another way, what does such a wave represent?

We will try and answer this question by looking at how a matter wave manifests in an actual experiment. It turns out that the best example for demonstrating the wave properties of matter is also the best example for demonstrating the wave properties of light: Young’s classic double slit experiment! However, the double slit experiment was not done with electrons until decades after the foundation of quantum mechanics, so we must briefly step away from our historical discussion to investigate it.

For light, the double slit experiment was conceived by Thomas Young in 1804, and it was the first definitive proof that light behaves as a wave. His experimental setup is roughly illustrated below.

Light from a source is used to illuminate a pair of closely-spaced slits in an opaque screen *S*. The light that emerges from the pair of slits spreads out in all directions, like ripples spreading from a rock thrown in a pond. The waves from the two slits interfere with one another, forming regions of light and darkness; these regions are projected on a measurement screen *M*. Thomas Young’s beautiful illustration of this effect is shown below.

Where the ripples from the two slits overlap, the waves add together, producing a bright region. Where the ripples do not intersect, the waves cancel out, producing a dark region. Points C, D, E and F in Young’s figure represent dark regions.

If this is not clear, the result of the experiment is that the light tends to form bright bands on the measurement screen, separated by regions of darkness; a crude simulation of how this would appear on the screen is shown below, assuming that the slits are horizontally side-by-side.

As we have said, Young’s experiment can also be done with electrons! The wavelength of the electrons is much, much smaller than the wavelength of light, which means that the slits must be much closer together to get an observable interference pattern, but fundamentally the experiment is the same. It was first performed in 1961 by Claus Jönsson¹, and his image of the detected electrons is shown below.

So far, there is nothing noticeably different from Young’s experiment with light — we have the same interference pattern, but the bright lines are caused by electrons instead of photons.

But Jönsson’s experiment used a *constant* stream of electrons; what happens if electrons are sent one at a time towards the two slits? It is important to note that Schrödinger’s equation, which describes the wave properties of matter, applies to a single electron; we should therefore somehow see wave effects in such a case, but how do they manifest?

Such a modified experiment has been done a number of times, but one of the most elegant versions was done by A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa² in 1989. They sent the electrons through the system one by one, and tallied the locations where they hit the screen. Images of the measurement screen are shown below.

Left to right, top to bottom, shows the accumulation of electrons as time passes. We make a number of observations from these images:

- Individual electrons appear as points on the screen, not as waves. Evidently the electrons appear as localized particles when we
*measure*their position. - Individual electrons arrive somewhat
*randomly*on the screen. It is not possible to predict exactly where any particular electron will arrive. - When we look at the arrival of many, many electrons on the screen, we find that they reproduce the familiar wave interference pattern of Young’s experiment!

This is a bewildering set of observations. Let us focus on the second and third points first. Apparently, it is not possible to say with certainty where any particular electron will arrive, but electrons are more likely to arrive at places where the wave is strong, and unlikely to arrive at places where the wave is weak. Apparently, the matter wave of a quantum-mechanical particle such as an electron should be interpreted as relating to the *probability* of finding the particle at that point in space.

So, the answer to “what does a matter wave represent?” is, apparently: “a wave of possibilities.”

This idea, that the amplitude of a matter wave is directly related to the probability of finding the particle in that location, was originally stated by German physicist Max Born in 1926, and is now known as the “Born rule“. This was a profoundly new way of thinking about nature, and it led to Born winning 1/2 of the 1954 Nobel Prize in Physics.

Before we continue, we should note that there is a similar relationship between a light wave and the behavior of the photons in that wave. This was demonstrated elegantly in 2007 by T.L. Dimitrova and A. Weis³; they performed Young’s double slit experiment with a severely attenuated light beam, such that photons passed through the system one at a time. Their integrated result appears below, and is more or less indistinguishable from the electron case!

This behavior, exhibited by both matter and light, suggests that there is an inherent randomness to the behavior of elementary particles, and this idea was radical when Born suggested it. Before Born’s time, physics was assumed to be a deterministic science. That is: it was assumed that we live in a “clockwork universe,” whose pieces were set into motion at the beginning of time and have been moving predictably in accordance with the laws of physics ever since.

Born’s argument changed that. It suggests that the laws of physics themselves have randomness built into them. Even if we send two electrons through the same double slit experiment in exactly the same way, we cannot say with certainty where each of them will end up on the measurement screen. The best we can do is say where any individual electron is *likely* or *not likely* to appear.

Such a huge change to the philosophy of physics raised additional questions, and problems. Among them:

- Where is the electron during the time it is emitted from the source until it arrives as the measurement screen?
- What makes the electron “choose” where to hit the measurement screen?

These questions were pondered by Niels Bohr and his assistant Werner Heisenberg at Bohr’s institute in Copenhagen over the years 1925-1927. The outcome of their pondering became known as the Copenhagen interpretation of quantum mechanics and, in the context of Young’s experiment, we can describe it as follows:

- When the electron is emitted and traveling through the double slit experiment, it evolves as a wave in accordance with Schrödinger’s equation.
- While traveling as a wave,
*the electron has no definite position*: it is, in a sense, taking every possible path through the system simultaneously. The electron goes through both slits! - When the electron arrives at the measurement screen, i.e. it is measured, it “chooses” where to appear, with probabilities as given by the Born rule. The wave “collapses” and the electron is observed in a definite location.

This interpretation of quantum physics is consistent with experimental observations. Philosophically, however, it is rather a bit of a mess. The biggest problem lies with this idea of “measurement” and the “collapse of the wavefunction.” Bohr and Heisenberg roughly defined a “measurement” as an interaction of the quantum particle with a researcher’s detection apparatus. This is vague, however, and seems to imply that human beings play a unique role in quantum physics: that quantum particles exist in this uncertain wave state until we, as humans, observe them. Physicists had spent centuries demonstrating that human beings are bound by the same laws of physics as every other object in the universe, so this was an unwelcome step backwards.

The collapse of the wave upon measurement poses its own problem. Schrödinger’s equation, which describes how a matter wave evolves, does not include any mechanism for this collapse. Wavefunction collapse is, in essence, rather crudely duct-taped onto the quantum theory, with no rigorous theoretical model for it.

The problems with measurement and wavefunction collapse have led to a mantra and joke among physicists working in quantum theory: “shut up and calculate.” In other words: don’t ask whether the theory actually makes sense, just use it to predict and understand experimental results.

So, by the late 1920s, the model of quantum physics could be summarized as follows. Quantum particles travel throughout space as a probability wave, with no definite position, until they are measured by an experimental apparatus. When measured, the particle “chooses” a definite position based on the probability wave amplitudes and the wave “collapses” into that definite position.

This view of how quantum physics works, however, leads to very strange predictions when two or more quantum particles have a relationship with each other. This is what we will know as *quantum entanglement*, which we will at last introduce in Part 3 of this series of posts!

POSTSCRIPT: The Copenhagen interpretation is not the only way to interpret experimental results in quantum physics. Other models include the “many worlds” interpretation, and the “pilot wave” interpretation. For our purposes, however, Copenhagen is the easiest way to think about things for the moment.

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¹ C. Jönsson, Zeitschrift für Physik 161 (1961), 454. Reprinted in English as “Electron diffraction at multiple slits,” Am. J. Phys. 42 (1974), 4- 11.

² A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa, “Demonstration of single-electron buildup of an interference pattern,” Am. J. Phys. 57 (1989), 117-120.

³ T.L. Dimitrova and A. Weis, “The wave-particle duality of light: A demonstration experiment,” Am. J. Phys. 76 (2008), 137-141.

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But it is difficult for a non-physicist to learn more about quantum entanglement than this, because even understanding it in a non-technical sense requires a reasonably thorough knowledge of how quantum mechanics works.

In writing my recently-published textbook on *Singular Optics*, however, I had to write a summary of the relevant physics for a chapter on the quantum aspects of optical vortices. I realized that, with some modification, this summary could serve as an outline for a series of non-technical blog posts on the subject; so here we are!

It will take a bit of work to really get at the heart of the problem; in this first post, I attempt to outline the early history of quantum physics, which will be necessary to understand what quantum entanglement is, why it is important, and why it has caused so much mischief for nearly 100 years!

*Small disclaimer: though I am a physicist, I am not an expert on the weirder aspects of quantum physics, which have many pitfalls in understanding for the unwary! There is the possibility that I may flub some of the subtle parts of the explanation. This post is, in fact, an exercise for me to test my understanding and ability to explain things. I will revise anything that I find is horribly wrong.*

Near the end of the 19th century, there was a somewhat broad perception that the science of physics was complete; that is, there were no more important discoveries to be made. This is encapsulated perfectly in an 1894 statement by Albert Michelson, “… it seems probable that most of the grand underlying principles have been firmly established … An eminent physicist remarked that the future truths of physical science are to be looked for in the sixth place of decimals.”

By 1900, the universe seemed to be well-described as a duality. Matter consisted of discrete particles (atoms), whose motion could be described by Newton’s laws of motion and law of gravitation, and light consisted of waves, whose evolution could be described by Maxwell’s equations for electromagnetism. In short: matter was made of particles, light was made of waves, and that covered everything that we observed. We will, in shorthand, call this “classical physics” going forward.

But there were still a number of mysteries that were perplexing and unsolved at the time. One mystery was the nature of atoms: atoms clearly had some sort of structure, because they absorbed and emitted light at isolated frequencies (colors), but what was that structure? There was much speculation in the early years of the 20th century related to this.

Another unsolved mystery was the origin of the phenomenon known as the photoelectric effect. In short: when light shines onto a metal surface under the right conditions, it can kick off electrons, as illustrated crudely below.

However, the photoelectric effect didn’t seem to work as classical physics predicted it would. The energy of electrons being kicked off of the metal didn’t increase with the *brightness* of the light beam, as one would expect from the classical theory; it increased with the *frequency* of light. If the light was below a certain frequency, no electrons at all would be kicked off. The brightness of the light beam only increased the *number* of electrons ejected.

The puzzle was solved by none other than Albert Einstein. In a 1905 paper, he argued that the photoelectric effect could be explained if light not only behaved as a wave but also as a stream of particles, later dubbed photons, each of which has an energy proportional to frequency. Higher frequency photons therefore transfer more energy to the ejected electrons. Also, a brighter light beam has more photons in it, resulting in more electrons getting ejected.

This was the first illustration of the concept of *wave-particle duality*: the idea that light has a dual nature as a wave and a stream of particles. Depending on the circumstances, sometimes the wave properties are dominant, sometimes the particle properties are; sometimes, both must be taken into account.

Einstein’s argument was a profound one, and answered other questions that had been troubling physicists for a number of years. For instance, the shape of the upper curve in Fraunhofer’s spectrum above, which shows the relative brightness of the different colors of sunlight, is known as a blackbody spectrum. It can be shown that the shape of the curve arises from the particle nature of light. Einstein won the 1921 Nobel Prize in Physics for his work on the photoelectric effect, which provided clear evidence that there was still more to understand about the fundamentals of physics.

So light, which was long thought to only be a wave, turns out to also be a particle! One might naturally wonder if the reverse is true: that matter, long thought to consist of particles, might also have wave properties? This was the idea that occurred to French physicist and PhD candidate Louis de Broglie in the early 1920s. As he would later say in his 1929 Nobel Lecture,

I thus arrived at the following overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases.

Louis de Broglie put forth this hypothesis in his 1924 PhD dissertation, and though his work was considered radical at the time, the wave nature of electrons was demonstrated in 1927 in what is now known as the Davisson-Germer experiment.

The idea that electrons have wave properties resolved other physics mysteries. Remember the question about the structure of the atom? The first major piece of the puzzle to be found was the experimental discovery of the atomic nucleus in 1910 by Ernest Rutherford and his colleagues. It naturally followed that electrons must orbit the atomic nucleus, much like planets orbit the sun, but this still did not explain why atoms would only absorb and emit light at distinct frequencies.

In 1914, Danish physicist Niels Bohr solved the problem by introducing new physics. In the Bohr model of the atom, electrons are only allowed to orbit the nucleus with discrete values of orbital angular momentum, and could only release or absorb light by “jumping” between these discrete orbits. The orbits are labeled by an integer index *n*, as illustrated below. Bohr’s model reproduced exactly the emission and absorption spectrum of hydrogen, and was viewed as a major step in understanding atomic structure.

But why would electrons only orbit with those discrete values of angular momentum? This was a question that the physics of the time could not answer, and was in essence an unexplained assumption in Bohr’s model.

It so happened that de Broglie’s hypothesis, that electrons have wave properties, provided the explanation! de Broglie realized that, if the electron acted like a wave, then those waves could only “fit” around the nucleus when an integer number of wavelengths fit in an orbit. A rough illustration of this is below.

Louis de Broglie was actually inspired by a very mundane example: a vibrating string! As he recalled in his Nobel lecture,

On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too.

This is an experiment you can try at home with a string or a phone cord! Though you can shake a string at any frequency you want, there are only certain special isolated frequencies that will feel natural, known as resonance frequencies.

So, by 1924, physicists were aware that both matter and light possess a dual nature as waves and particles. However, the situation between matter and light was not entirely equal. Since James Clerk Maxwell’s work in the 1860s, physicists had a set of equations, known as Maxwell’s equations, that could be used to describe how a light wave evolves in space and time. But nobody had yet derived an equation or set of equations that could describe how the wave associated with matter evolves.

This was a challenge undertaken by Austrian physicist Erwin Schrödinger in 1925, soon after de Broglie had suggested matter has wave properties. Within a year, using very clever arguments and intuition, Schrödinger derived an equation that accurately modeled the wave properties of electrons, now known as the Schrödinger equation. With the Schrödinger equation, it became possible to quantitatively model the behavior of electrons in an atom and accurately predict how they would absorb and emit light.

So, by 1927, the new *quantum theory* of light and matter was in reasonably good shape. There was experimental evidence of both the particle and wave nature of both light and matter, and the particle nature of light and the wave nature of matter had been experimentally confirmed. This is summarized in the table below, for convenience.

But there was one big puzzle on the matter side, as illustrated in the lower right corner: what, exactly, is “waving”? In other words, what does the wave part of a matter wave represent? In water waves, it is the water “waving” up and down, and conveying energy through this motion. In sound waves, it is the molecules of the air “waving” forward and back. In light waves, it is the electric and magnetic fields that are “waving.” But there was no obvious way to interpret what was doing the waving in a matter wave.

It turns out that the initial answer to the question, which would be formulated right around the years 1926-1927, would lead to some very strange philosophical implications of the new quantum theory. This will be discussed in part 2 of this series of posts!

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It is hard to find the right words to describe these novels. Bewildering, intricate, confusing, surreal, thoughtful, haunting, poetic, horrific, terrifying, beautiful? It is all these things, and more.

The trilogy is centered upon an area of an evidently uninhabited and wild coastal area known only as Area X, which became isolated from the rest of the world some 30 years earlier due to an “event” whose nature is not understood. Since that time, a government organization known as the Southern Reach has been studying Area X, trying to understand its nature, its purpose, and whether it is a threat. They have been sending expeditions into the region for years, and those expeditions have more often than not come to unexplained or horrifying fates.

*Annihilation*, the first of the trilogy, is a first-person narrative of the twelfth expedition into Area X, told from the point of view of an expedition member initially known only as “the biologist.” The very use of personal names in Area X is apparently dangerous, so the team members are known only through their professions: biologist, anthropologist, psychologist, and surveyor. The team must be hypnotized in order to pass through the only portal through the mysterious barrier that surrounds the area. Once inside, things seem at first to be quite ordinary. However, it does not take long for the team to experience unsettling and inexplicable things: living writing on the walls of a structure that should not be, animals that do not seem to be quite animals, the sound of something unknown wailing in the swamp. The expedition members hold their own secrets, as well, which have a corrosive influence on their cooperation and mission. As the novel unfolds, the health and sanity of the team collapses as they make their way to a fate that seems inevitable.

*Authority* is set in the aftermath of the twelfth expedition, and is told from the point of view of the new head of the Southern Reach, a man who goes by the name Control. Through Control’s eyes, new to the enigma of Area X, we learn much more about its history and mystery, while at the same time watching Control attempt to survive the politics of the Reach and solve the “puzzle” of Area X himself, something his predecessor failed to do.

By the end of *Authority*, however, things have gone catastrophically — perhaps apocalyptically — wrong, and in *Acceptance* we see events unfold in the past and the present through a number of characters: the biologist and Control, the psychologist of the twelfth expedition, and a man who lived in Area X before it underwent its transformation. Can humanity stop the force or forces behind Area X from accomplishing their goal? And what is their goal? And should it be stopped at all?

I tend to view the books as progressing, in a sense, as the evolution of a very complex jigsaw puzzle. The first book, which plays out very much as a horror novel, gives us a small handful of pieces that seem to be utterly unrelated to each other. The second book, conversely, seems to throw an overwhelming number of pieces at us at once, giving us *too much* information to process. The third book then puts some of the puzzle together for us, at the same time revealing that there is certainly more of it than we can see.

One should not expect easy answers from the Southern Reach trilogy or, in many cases, answers at all. It appears that a large theme of the entire series is the idea that there are limits to what we can understand, and that some things may be so alien to us as to be unknowable. I got a strong sense of postmodernism in the story, in that our ideas of reality are unsuited to solving the problem of Area X.

This is not to say that there are no answers to be had, and that the story is incomprehensible. There is a clearly a method to Jeff VanderMeer’s madness: the books are *saturated* with ideas, and hints of the story behind Area X. A reader should pay attention and think about the implications of every bizarre occurrence, every strange theory that a character voices; not all of them are equally significant, but they all provide interesting, often profound, things to ponder. This is a book series that rewards careful, and multiple, readings.

If you don’t pay close enough attention, be prepared: the final book ends abruptly, and will likely leave you baffled and haunted. I am reminded somewhat, in a good way, of the ending of the classic 1967 television series *The Prisoner*. That 17-episode series ended almost completely in allegory, leaving viewers outraged that they were given no definite answers to questions that had plagued them through the show. Southern Reach doesn’t end only in symbolism — or does it? I’m still not sure — but it does leave you with a bewildered feeling.

If you’re like me, and you needed a little help understanding some of what was going on throughout the trilogy, this Goodreads discussion page (spoilers, obviously) can fill in some of the blanks wonderfully. I was impressed not only with the insightful comments of some of the readers, but also with the cleverness of VanderMeer; on reflection, you come to realize how almost every sentence is fraught with meaning.

I was reminded of a few other pieces of literature while reading the Southern Reach. One is the novella *Needing Ghosts*, by my favorite horror author Ramsey Campbell. That story slowly chips away at your sense of confidence in reality and your perception of it; it ranks as one of the only stories that left me feeling almost hysterical when I finished it. (Now I can add *Acceptance* to that very short list.) The similarities are not simply superficial: both authors are masters as instilling a sense of dread in their readers, and both appear to take almost obsessive care in the crafting of their stories and the words that they use.

The other novel that the Southern Reach reminded me of is the classic *Roadside Picnic* by the Strugatskys. *Roadside Picnic* also features a forbidden region of the Earth, created by an “event,” a region which is filled with dangers inconceivable and powers unknowable. In *Roadside Picnic*, the “Zones” are created by aliens making a pit-stop; in the Southern Reach, I wasn’t quite sure throughout the trilogy whether I was even reading a science fiction story for most of its length! (This is, again, to VanderMeer’s credit, and goes back again to the idea of unknowable and incomprehensible forces.)

I could perhaps go on speculating about the ideas and meanings of the Southern Reach trilogy for another 1000 words, but I am not sure if I understand it well enough to do it proper justice! It is a haunting, often horrifying, story filled with vivid images and thought-provoking ideas, and it is almost certainly unlike anything you’ve read before. Highly recommended.

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As is made clear from the cover image and the title, *Chills* centers on a danger that originates in the coldest weather. A team of detectives and investigators race, in the midst of a blizzard, to solve a series of grisly murders and prevent future ones from occurring. However, they soon learn that even worse things are happening, and that there are unnatural things in the whiteout…

I’ve blogged about a number of SanGiovanni’s works before, her chapbook *No Songs for the Stars* and her novels *Chaos* and *Thrall*. In my experience, her stories tend to be fast-paced and fall squarely into cosmic horror, and *Chills* follows in that vein.

As the story begins, a freak snowstorm has hit a small town in Connecticut in May. In the dangerous cold and whiteout conditions, a body is found hanging from a tree, branded with occult symbols and apparently mauled by some unknown sort of animal. Detective Jack Glazier suspects that this is only the first victim of a cult that is using the weather to mask its activities, and he calls in occult crime specialist Kathy Ryan for help. Kathy quickly realizes that the sacrifice is part of a larger ritual, one that could tear open the gates of our reality and let something else — something horrible — through. Her fears begin to be confirmed as more victims turn up all over down, and she and the officers realize that some *things* have already entered our world. What starts as a murder investigation becomes a fight for survival as well as a race against time to stop the Hand of the Black Stars cult from achieving its monstrous goals.

As I said, the novel is fast-paced. The first corpse has barely begun to warm up¹ when things escalate and the heroes must fight for their lives. I might characterize *Chills* as an action-horror novel, which mixes cosmic horror with well-thought-out fight scenes. Both the horror and the action work well, in my opinion.

The only thing that suffers a bit from the fast start is the character development, as we have just begun to know a bit about Jack and Kathy and others before they are reacting to supernatural threats. This isn’t a detriment to the overall story, but felt worth mentioning.

In *Chills*, SanGiovanni does an excellent job capturing the menace, danger, and even horror of extreme cold. Extreme winter weather is a natural subject for horror, in its relentlessness, its unforgiving nature, its ability to isolate its victims and its ability to horrifically preserve those that fall to it. Stephen King’s *The Shining* is, of course, another famous example of a tale that uses the cold as a setting and subject of horror. Other examples include Ramsey Campbell’s 1992 novel *Midnight Sun* and Dan Simmons’ 2007 novel *The Terror*, which uses the doomed 1845 Franklin expedition to find the Northwest Passage as a setting for supernatural horror.

Incidentally, the Franklin expedition was also used as the inspiration for a truly wonderful horror story in the Marvel Comics series Alpha Flight, centered on a villain named Pestilence. I quote the Wikipedia description of Pestilence’s origin below, which I cannot better:

In 1845, F.R. Crozier became the doctor and chief science officer for an Arctic expedition led by famed explorer Sir John Franklin, who sought the Northwest Passage. The expedition consisted of two ships, the Terror and the Erebus. During the intervening three years, Crozier became captain of one of those vessels.

Six months after the departure of the expedition, the ships became trapped in the Arctic ice. In November 1847, with supplies running low, Sir John led part of the crew in search of help and was never seen again. In April 1848, Crozier took the remainder of the crew and set out over the ice. Many of the crew died of exposure during the march and were left unburied, and a number of advance scouts were apparently flash-frozen where they stood. With almost no one left alive, Crozier mixed together unknown chemicals and ingested them, inducing a state of suspended animation that the surviving crew mistook for death. His plan was to remain where he fell, allowing the ice to preserve him until the weather warmed enough to revive him. What he did not anticipate was that out of respect for him and his position, his remaining crew decided to bury him. Interred in permafrost, the sun never reached Crozier and he spent the next 140 years buried alive and going insane.

The entombed Crozier is later revived by a hapless Alpha Flight, and his long entombment, combined with the chemicals he ingested and the magic of the area, gives him incredible supernatural powers. But he maintained his 1840s fashion, which made him particularly creepy:

But I’m getting away from the point of this post, as I am wont to do! I simply wanted to point out that winter and cold are a really good inspiration and setting for horror stories. Mary SanGiovanni’s *Chills* is an excellent, fast-paced example of this sort of icy horror.

By the way, I’m acquainted with Mary SanGiovanni on Twitter, and from that know that she has recently started up a Patreon to support her work. If you enjoy her horror, please consider becoming a patron!

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¹ See, ’cause the corpse was frozen from being out in the snow. Get it? GET IT?

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But these days, there are even more imaginative varieties! I’ve starting collecting dice of every shape and size, and my current collection is shown below, in order¹:

This is a really amazing variety of dice! This is my “special dice” collection in its entirety, and includes some duplicates (don’t ask how I got 4 identical d60s), but in some cases, such as the d7, d12 and d24, there are varieties in shapes even with the same number of faces!

This variety got me wondering: how does one design dice with a weird number of faces? What mathematical strategies does one use to make them? What other types of dice are possible? And, perhaps most important: are these dice “fair”?

I thought it would be fun to answer these questions with a blog post, in which we discuss the geometry of dice!

We will start with the most familiar types of dice, and work ourselves gradually into strange and unfamiliar territory. Before we begin, we should note that a major consideration for any type of die is that it be “fair”: that is, every number on the die should be equally likely to be rolled. The most obvious way to do that is to make the die have a lot of symmetry in its shape, which brings us to our first category…

**The Platonic solids.** The most symmetric shapes, as their name implies, formally date back to the Greek philosopher Plato (428-348 BCE) , though most of them were recognized, or at least crafted, even earlier. The Platonic solids include the d4, d6, d8, d12, and d20, as shown below.

The Platonic solids are the only polyhedra (multi-sided objects) which are convex (have no concavities) and *regular*. A “regular” polyhedron is one for which not only are all faces equivalent to one another, but so are all edges and all vertices (points). To put it another way: a Platonic solid can be rotated to make any edge, vertex or face look exactly like any other one. Obviously, this is ideal for making a fair die — there is no preferred edge, vertex or face on the solid.

The Platonic solids are somewhat profound, in that there are a small number of them, and no (obvious) reason why they have the number of faces that they do. This profundity captured the imagination of Plato, who gave the solids a fundamental role in his dialogue *Timaeus* (c. 360 BCE), assigning each of them to one of the elements. From *Timaeus*,

To earth, then, let us assign the cubical form; for earth is the most immoveable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature. Now, of the triangles which we assumed at first, that which has two equal sides is by nature more firmly based than that which has unequal sides; and of the compound figures which are formed out of either, the plane equilateral quadrangle has necessarily, a more stable basis than the equilateral triangle, both in the whole and in the parts. Wherefore, in assigning this figure to earth, we adhere to probability; and to water we assign that one of the remaining forms which is the least moveable; and the most moveable of them to fire; and to air that which is intermediate. Also we assign the smallest body to fire, and the greatest to water, and the intermediate in size to air; and, again, the acutest body to fire, and the next in acuteness to, air, and the third to water. Of all these elements, that which has the fewest bases must necessarily be the most moveable, for it must be the acutest and most penetrating in every way, and also the lightest as being composed of the smallest number of similar particles : and the second body has similar properties in a second degree, and the third body in the third degree. Let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire; and let us assign the element which was next in the order of generation to air, and the third to water. We must imagine all these to be so small that no single particle of any of the four kinds is seen by us on account of their smallness: but when many of them are collected together their aggregates are seen. And the ratios of their numbers, motions, and other properties, everywhere God, as far as necessity allowed or gave consent, has exactly perfected, and harmonised in due proportion.

So fire is a d4, earth is a d6, air is a d8, and water is a d20. Of the 5th Platonic solid, the d12 dodecahedron, Plato vaguely says, “There was yet a fifth combination which God used in the delineation of the universe. ”

Later thinkers also thought the Platonic solids played a fundamental role in the cosmos. In 1596, the German astronomer Johannes Kepler published the book *Mysterium Cosmographicum*, in which he speculated that the positions of the known planets corresponded to the Platonic solids inscribed within one another, as illustrated below.

Obviously, this model did not work out, but it illustrates how the set of 5 solids captured the imagination of philosophers trying to make sense of the universe.

There is a big problem with the d4 when being used as a die, though: it doesn’t roll very well! One solution in recent years has been to truncate the tips of the tetrahedron and place the actual numbers on the tiny flat surfaces. Such dice are easier to roll and to read the result. A traditional d4 and a newer one are shown below.

There is a curious illusion associated with the d8 that is worth noting before moving on. You may note that the d8 can be created by taking two square-base pyramids made of equilateral triangles and fixing their bases together. Looking at such a die, it seems like it is mirror-symmetric across the plane of the bases, but not so symmetric that every edge and vertex is the same!

I interpret this as an illusion due to the way the numbers are put on the die. They are all placed with their bottoms towards a single base plane, which makes it look like this is the only symmetry. However, if you look at the die from other edges, you can see that it looks the same! There are 3 different base planes.

If this isn’t obvious from looking at actual dice, it is much clearer looking at a transparent model. See if you can trace the 3 squares formed by the edges in the image below.

**Mirror-symmetric dice.** But the idea of mirror symmetry provides a way to make dice of any large even number of sides! We can make a “pyramid” of sorts with any number of faces and a base and glue two of them base to base; as noted, a d8 can be thought of as gluing two 4-faced pyramids together. In this way, we can construct the final member of the original Dungeons & Dragons dice, the d10. We can also, however, make a different form of d12, as well as a d14, d16, or higher! Such a structure is formally known as a trapezohedron. My d10 and d16 are shown below.

You may notice a slight difference between the d10 and d16: the faces of the d10 are shifted off-center across the mirror plane, while the faces of the d16 are aligned across this plane. This shift of the d10 is necessary because there are 5 faces on either side of the mirror plane. If the faces across the plane were lined up, then the die would come to rest on one face, but an edge would be facing upward! The shift means that a face is upward when the die comes to rest. This is necessary for any mirror-symmetric die with an odd number of faces on either side of the mirror plane.

One can take the idea of mirror symmetry to a ridiculous extreme. Below is pictured a d50, which uses the same principle. Note that it also has its faces shifted, because there are an odd number of faces on either side of the edge!

**Face doubling.** So far, all the dice we’ve considered have had an even number of faces. The trapezohedrons, however, suggest a very straightforward, if unimaginative, way to make dice with an odd number of faces: simply double up on the lower numbers! In this way, a d6 becomes a d3, a d10 becomes a d5, and a d14 becomes a d7, as shown below.

D&D players were doing this for years before actual dice were made: players simply said, for instance “a roll of 1-2 on a d6 is 1, 3-4 is 2, and 5-6 is 3.” We were certainly able to handle that, but it is just nice to have dice explicitly marked for this strategy.

My favorite of this type is the d2: the 2-sided die! As can be seen below, it is in essence a die for which 3 sides in a “U”-shape represent 1 and the other 3 sides in a “U” represent 2. Each trio of sides has been rounded together to make the die have, effectively, 2 rounded sides!

Lots of people will say, “Why do you need a 2-sided die? Can’t you just flip a coin?” You could, but gamers such as myself *love* to roll dice. Flipping a coin just doesn’t have the same feeling to it!

**Catalan solids.** We will have more to say about odd-numbered dice in a few moments, but first let’s return back to dice with an even number of faces. Is there any way, other than trapezohedron, to create more varieties?

In fact, we can, by removing some of the symmetries of the Platonic solids. The Platonics, if you recall, were symmetric in faces, edges, and vertices: that is, every face, edge and vertex was equivalent to every other one. But, for a fair die, it would seem that we really only need every *face* of the die to be equivalent: the Platonic solids have, in a sense, more symmetry than we need. If we throw out edge and vertex symmetry, this leads us to the d24, d48, d30, d60 and d120!

These objects are members of a group of solids known as Catalan solids, after the Belgian mathematician Eugène Charles Catalan who described them in 1865. The complicated names of those pictures above are, respectively, the tetrakis hexahedron, the disdyakis dodecahedron, the rhombic triacontahedron, the deltoidal hexecontahedron, and the disdyakis triacontahedron.

That’s a lot of wordage, but we can understand most of these Catalan solids as extensions of the Platonic solids. For example: suppose we take each face of a cube, and “pull” it outward to make it a narrow pyramid, as shown below.

We have now created a tetrakis hexahedron, as there are 4 sides per pyramid and six faces to the cube! 4 times 6 = 24, obviously! It should be noted that the vertices are no longer all the same: the peak of each pyramid is not equivalent to one of the corners of the former cube.

We can similarly think of the d48, the disdyakis dodecahedron, as making an 8-sided pyramid out of each face of a cube. Alternatively, we can think of it as making a 6-sided pyramid out of each face of a d8 (octahedron).

There is another Catalan solid that is a d24, as well; it is known as the deltoidal icositetrahedron, and is shown below.

This 24-sided die can be viewed as making a triangular pyramid out of each face of an octahedron.

The d60 and d120, the deltoidal hexecontahedron, and the disdyakis triacontahedron, may similarly be thought of as modifications of a d20. For the d60, we make a 3-sided pyramid out of each face of the d20, and for the d120, we make a 6-sided pyramid. I highlight these pyramids in the figure below, if it isn’t clear.

Some Catalan solids cannot be simply fashioned from the Platonic solids, however. The d30 is one of those, as is a different version of a d12, a shape known as a rhombic dodecahedron. These two are shown below.

The rhombic dodecahedron has the curious property that its shape can be used to perfectly fill a volume in three-dimensional space. That is: if you had enough of these d12s of the same size, you could stack them together to fill a region of space without any gaps, just as you could with a bunch of d6s.

There are a few other tricks we can do with these two. If we take the rhombus that forms the faces of the d30 and pyramid it into four equal sections, we end up with a d120! If we take the rhombic d12 and pyramid up each of the faces right in the middle, we end up with the d24. These divisions are illustrated below.

It should be noted that the d60 and d120 end up pushing the limits of what might be considered a useful shape for a die. They are both quite spherical (the d120 is larger than a golf ball), and will roll for a long, long time before coming to rest on a number. Furthermore, the numbers are so closely packed together that it can be somewhat tricky, at a glance, to figure out what the actual rolled result is! It may be possible to make dice which have an even larger range of numbers, but they wouldn’t be terribly practical.

**Crystal-shaped dice.** Let’s now return to dice with an odd number of sides. Dice makers are not, of course, limited to making dice that conform to some standard of geometric perfection! Inspired by the mirror symmetry dice described earlier, one can simply remove the edge across the mirror plane, effectively reducing the number of sides in half. If the faces are to remain flat, this means that the edges have to be rounded, resulting in something like the d3 pictured below.

If the image doesn’t make it perfectly clear, picture it as a filled tube with a triangular cross-section, and the edges are rounded. Provided the edges are rounded in the same way, it is still a fair die, because each of the 3 flat sides is the same.

This same idea has been used to design what are often referred to as “crystal dice,” which look like amethyst crystals or something similar. For instance, here’s a lovely set of crystal d4s, via Gamemaster Dice.

Here the edges aren’t rounded, but beveled to a peak, so that the die cannot rest stably on those flat edges. Some manufacturers have been particularly imaginative with the tube shape of these dice, making them look, for instance, like a rocket ship.

**The cylindrical d5.** As we carry on to even more varieties of dice, we begin to enter truly strange territory. For instance, consider the d5 pictured below.

We can view this as being similar to the d3 discussed above, except that the ends of the triangular cylinder have been made flat and turned into additional faces instead of being rounded. But now we have a problem: the faces of the sides of the cylinder are of a different size and shape than the faces of the ends of the cylinder: how can we possibly know if this die is fair or not?

Arguably one can do some calculations to prove it, or do brute force testing of dice of various lengths, but we will simply argue that there must be a size of die which is fair; presumably the dice-maker figured out what that size is! Imagine first a die which is made of a very, very long cylinder, for instance with the same length to thickness proportions as a pencil. Such a die will be almost certain to land on its length, on one of the three sides around it. If we now imagine a die which is made of a very, very short cylinder, like a coin, it is almost certain to land on one of its ends.

Somewhere between the long die which will land on its length and the short die which will land on its end, there must be a die of a length which will land 2/5ths of the time on an end, and 3/5ths of the time on a side. Because each of the ends are the same, and each of the sides are the same, this die must be fair! This idea is illustrated below.

In this case, we have again taken advantage of the symmetry of the triangular cylinder: because each of the ends is the same and each of the sides along the cylinder length is the same, there is only one free parameter to tune — the cylinder length — in order to get a fair die.

A 7-sided die designed in a similar manner is available, as well!

**Here be dragons.** Even after all of this, we still have a huge number of unusual dice to consider! They are shown below.

These dice have no particular symmetry at all — the different faces are different sizes and shapes. This raises two questions: how are they designed, and (of course) are they fair?

To answer the first question, we turn to the descriptions given at The Dice Shop at mathartfun.com:

This design is based on spacing points as equally as possible on a sphere and then cutting planar slices perpendicular to those directions.

In short: the plan is just to make the die as symmetric as possible, slicing faces at points that are roughly equally spaced on a sphere. It turns out that it is not even possible, except in the Platonic cases we’ve already considered, to space points equally on the sphere, and there are a number of different ways to define and calculate equal spacings; a bit of mathematical description is given here.

Are these dice fair? Odds are against it, but there’s one way to find out: roll it a lot of times and see what happens! On one lonely night in a hotel room during a recent work trip, I did just that with a d7. (I only tested the d7 because bigger dice would require many more rolls to get good statistics.)

The tabulated results are as follows:

- 1: 63
- 2: 57
- 3: 72
- 4: 113
- 5: 75
- 6: 81
- 7: 39

There were 500 total rolls; if the die was fair, I would expect about 71 rolls of each number. Clearly there is a bias towards rolling a 4 and a bias against rolling a 7! The other numbers are relatively balanced, though evidently not perfectly.

So the die is not fair. But we can ask: does it matter? As noted at the beginning of this post, such dice are usually employed in role-playing games, where there is usually a large amount of flexibility in the rules anyway. “Mildly biased” dice aren’t really an issue where the rules are often made up on the fly!

**Other stuff.** In this post, I’ve focused on the geometry of individual dice, but there’s even more interesting stuff that can be said about the mathematics of combinations of dice! There are the so-called Sicherman dice, which are differently numbered pairs of d6s that produce the same sums as two ordinary d6s! There are also non-transitive dice, in which each die of a set of 4 can always be beaten, on average, by another member of the set! I blogged about these a few years back, if you’re interested in learning more.

If you’ve read this far, congratulations! You’ve read 3500 words on the geometry of dice! Hopefully I’ve gotten across the point that the design of dice is not child’s play.

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¹ I have a *lot* more dice, but this is my bag of “special” dice.

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I’ve covered most of the topics that I would truly call “basic” — hence the long time since the last post — but I realized that I missed one concept that is truly fundamental: the law of reflection!

It may not seem like there’s much to say about reflection, but we’ll see that isn’t the case. Many interesting things can happen when light reflects off of a surface — the challenge will be to include as many as possible while keeping this post short!

So what is the *law of reflection*? Well, the classic law of reflection states that the angle that a ray of light reflects off of a smooth, flat surface (the angle of reflection ) is equal to the angle at which the light is incident upon the surface (the angle of incidence ).

I note that this applies to a smooth surface. It applies to light reflecting off of a sheet of glass or a mirror, but not light reflecting off of many other surfaces, such as a painted wall. In the latter case, the roughness of the surface makes the light bounce off in many directions — this case is known as *diffuse reflection*. The more familiar case of light bouncing off of glass is known as *specular reflection*, and when people talk of the “law of reflection,” this is what they are typically referring to. It is specular reflection that this post will be mainly concerned with, though we will say a few words about diffuse reflection at the end.

The law of reflection was the first quantitative law of light propagation discovered, and stretches back to antiquity. Who, exactly, formulated the law in writing first is unknown, though authors who have made mention of it include Euclid (c. 300 BCE), Aristotle (384-322 BCE), and Hero of Alexandria (10 CE – 70 CE).

Even these early authors recognized the use of it in designing image-forming systems. We have noted that the law of reflection applies to smooth surfaces, and this includes curved surfaces, like a mirror with a spherical shape. Locally, such a spherical mirror is flat, just as the Earth appears flat when one is standing on it, so the law of reflection applies locally at every point of the curved surface.

This is illustrated below, to show some of the image-forming properties of a spherical mirror. Two rays are drawn from the object — represented by an upright arrow — to the image, which ends up being inverted.

This is a simple illustration of graphical ray tracing: in order to find the position and size of an image, one can draw two rays to find where they intersect at the image. The two easiest rays to draw are the one that runs horizontally, and reflects through the focal point *f*, and the the one that passes through the focal point and reflects horizontally. The point *C*, incidentally, is the center of curvature of the mirror, and the focal point is half of this distance.

Images can, of course, also be formed using lenses, which use refraction to form an image, but lenses have one notable disadvantage: the location of the image depends on the color of the light. The result is what is known as chromatic aberration — the image for red light is at a slightly different position than the image for blue light — and it limits the image-forming resolution of the lens. This chromatic aberration arises because the law of refraction, given by

,

depends on the refractive indices and , which are themselves dependent on the frequency (color) of the light. The law of reflection is the same for all frequencies, and so images from mirrors do not suffer from chromatic aberration.

Now that we’re talking a bit about refraction (and reminder that I have a basics post on that, too), there are a number of interesting aspects of reflection that are intimately associated with refraction. Reflection and refraction, in essence, go hand-in-hand: when light hits a transparent surface, some fraction of the incident light is reflected, some is refracted. What is interesting, however, is that the amount of light reflected depends on the* polarization* of light.

For those unfamiliar, light is a transverse wave: that is, the electric field **E** and magnetic field **H** that make up a light wave oscillate in a direction perpendicular to the direction the wave is going. This is illustrated below (and I’ve written a basics post about it, of course).

When we describe the reflection and refraction of light at a surface, we can consider two distinct cases: the case for which the electric field lies in the plane of reflection and the case for which the electric field lies outside this plane, as illustrated below.

It turns out, in general, that the in-plane polarized light gets reflected less, and refracted more, than the out-of-plane light. You can actually see this by using a polarizer to look at car windshields on a sunny day.

This effect is of practical importance for a number of reasons. First, if you want to design polarized sunglasses, you would like them to reduce the amount of glare that one sees from reflection off of, for example, car windshields. Since more light gets reflected from the out-of-plane polarization, one designs polarized sunglasses to block that horizontally polarized light!

At one special angle, known as the Brewster angle, the in-plane light is in fact not reflected at all: it is totally refracted into the second medium. This means that, if unpolarized light (with an equal mixture of in-plane and out-of-plane) is incident on a surface at the Brewster angle, the light that is reflected will be entirely out-of-plane polarized: the in plane polarized light will be completely transmitted. This is illustrated below.

In the 1800s, this technique was the way in which most optics researchers produced a polarized beam of light. For example, Michael Faraday used it to demonstrate a link between light and magnetism in 1845.

But why is no in-plane light reflected at the Brewster angle? The answer appears in the image above. We may imagine the reflected ray is created by the incident ray interacting with the medium, resulting in reflection and refraction. But the refracted ray is perpendicular to the reflected ray, which means that the electric field of the refracted ray is *parallel* to the reflected one! However, as we noted earlier, light is a transverse electromagnetic wave, which means it must have an electric field oscillating perpendicular to the direction of travel. This doesn’t exist at the Brewster angle, and there is therefore no reflection.

So far, we have been looking at reflection and refraction when light goes from a “rare” medium (like air) to a “dense” medium (like water, or glass). But, of course, light can travel in the other direction, as well. When it does, another unusual reflection effect can occur, known as total internal reflection.

I discussed this in my basics post on refraction, but it is worth reiterating in short here. The law of refraction is reversible — that is, it works whether you consider the light to be coming from medium 1 or medium 2. In going from a rare medium to a dense medium, the 90° worth of incident angles are represented by a smaller range of refracted angles, as shown below.

But what happens if we send a ray from medium 2 into medium 1, but from an angle outside of those shaded? For instance, consider the ray below: how does it refract?

The answer: it doesn’t! The angle falls outside of those for which we can use the law of refraction, and it turns out that the ray is completely reflected; we therefore call it total internal reflection.

This total internal reflection forms the basis of fiber optics, in which light is used to convey information through a long, thin glass fiber. In a simplistic way, we may imagine that the light rays travel from one end of the fiber to the other by multiple total internal reflection bounces. Of course, fiber optics is a major part of our internet infrastructure, and it is in a sense all due to a humble reflection property!

Total internal reflection can be used for quite spectacular demonstrations; as I’ve discussed previously, one can use it to guide light in a falling stream of water, in a demonstration that dates back to Jean-Daniel Colladon in 1842.

One other surprise arises from total internal reflection. It was noted in 1943 by Fritz Goos and Hilda Hänchen that a beam of light, totally internally reflected, actually shifts along the surface before being reflected! After multiple reflections, this Goos-Hänchen shift can accumulate and become quite large. The illustration below is from their original paper, showing how a beam of light multiply reflected within a piece of glass ends up with a very large shift.

Why does light, in essence, “creep” along the surface before reflecting? Well, you’ll have to read the blog post I wrote about the effect last year.

There’s one other curious effect worth mentioning before we conclude. We have, so far, considered only specular reflection; that is, reflection off of a smooth surface. Most surfaces, however, are not smooth, and a beam of light shining off of them ends up bouncing off of the surface irregularities and going in all directions, as shown in the image below, from Wikipedia.

This is why, for instance, the walls of your house do not look like mirrors: the roughness of the painted surface disrupts the reflection. Curiously, though, it has been known for well over a hundred years¹ that certain rough surfaces can also produce specular reflection, provided one looks at light incident from a very large angle!

It takes a surface with juuuuust the right roughness to make this effect work, but you can a actually see it in action pretty well with an ordinary sheet of paper! Below, I look at a piece of paper from an ordinary angle, and then tilt it and look at a lamp from a very grazing angle. In the latter case, you can (somewhat) see the reflected image of the lamp beyond.

Why would this work? It’s a bit tricky to explain, but we’ll give it a try! A lightwave, like the electromagnetic wave we drew earlier, has a characteristic wavelength associated with it. This wavelength is, literally, the spatial distance between two peaks of the wave. When light encounters structures smaller than the wavelength of light — for example a rough surface with surface variations smaller than the wavelength — it doesn’t “see” those variations.

This is why specular reflection exists at all, in fact. All surfaces have some degree of roughness, as they are all made out of atoms, but “smooth” surfaces like metals have roughness much smaller than the wavelength.

Now consider the illustration below. We first draw a normally incident wave, with the rays drawn a wavelength apart from each other for clarity. This wave encounters a surface where the variations also are about the size of the wavelength: the light wave will therefore “see” these variations and be scattered in all directions.

Now imagine that we shine the lightwave on the surface from an angle. The actual perpendicular distance between the rays is the same as before, but the horizontal distance between the rays is much, much larger. It turns out that this effective wavelength is now bigger than the size of the roughness — in this case, the wave no longer “sees” the roughness!

This explanation may seem a bit incomplete, and it is, but it is about as good as we can do without diving into the math in some detail.

At this point, this blog post is over 2000 words, which means it’s probably a good time to wrap it up. There is, as we have seen, a surprising amount that can be said about the simple process of reflection! Something to reflect upon…

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¹ For instance, there is the paper by T.K. Chinmayanandam, “On the specular reflection from rough surfaces,” Phys. Rev. 13 (1919), 96. He, however, talks about work by Lord Raleigh and others. It seems like an effect that was noticed by many people over the years and has no definite discoverer — somewhat like the law of reflection itself.

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