In my previous post, I showed that an elevator falling from the surface through the center of the earth due to gravity alone would obey the dynamics of a simple harmonic oscillator. I did not know what would happen if the shaft went through some arbitrary chord through the earth. Rick Gerkin believed that it would take the same amount of time for all chords and it turns out that he is correct. The proof is very simple. Consider any chord (straight path) through the earth. Now take a plane and slice the earth through that chord and the center of the earth. This is always possible because it takes three points to specify a plane. Now looking perpendicular to the plane, you can always rotate the earth such that you see
Let the blue dot represent the elevator on this chord. It will fall towards the midpoint. The total force on the elevator is towards the center of the earth along the vector . From the previous post, we know that the gravitational acceleration is . The force driving the elevator is along the chord and will have a magnitude that is given by times the cosine of the angle between and . But this has magnitude exactly equal to ! Thus, the acceleration of the elevator along the chord is and thus the equation of motion for the elevator is , which will be true for all chords and is the same as what we derived before. Hence, it will take the same amount of time to transit the earth. This is a perfect example of how most problems are solved by conceptualizing them in the right way.
The 2012 remake of the classic film Total Recall features a giant elevator that plunges through the earth from Australia to England. This trip is called the “fall”, which I presume to mean it is propelled by gravity alone in an evacuated tube. The film states that the trip takes 17 minutes (I don’t remember if this is to get to the center of the earth or the other side). It also made some goofy point that the seats flip around in the vehicle when you cross the center because gravity reverses. This makes no sense because when you fall you are weightless and if you are strapped in, what difference does it make what direction you are in. In any case, I was still curious to know if 17 minutes was remotely accurate and the privilege of a physics education is that one is given the tools to calculate the transit time through the earth due to gravity.
The first thing to do is to make an order of magnitude estimate to see if the time is even in the ballpark. For this you only need middle school physics. The gravitational acceleration for a mass at the surface of the earth is . The radius of the earth is 6.371 million metres. Using the formula that distance (which you get by integrating twice over time), you get . Plugging in the numbers gives 1140 seconds or 19 minutes. So it would take 19 minutes to get to the center of the earth if you constantly accelerated at . It would take the same amount of time to get back to the surface. Given that the gravitational acceleration at the surface should be an upper bound, the real transit time should be slower. I don’t know who they consulted but 17 minutes is not too far off.
We can calculate a more accurate time by including the effect of the gravitational force changing as you transit through the earth but this will require calculus. It’s a beautiful calculation so I’ll show it here. Newton’s law for the gravitational force between a point mass m and a point mass M separated by a distance r is
where is the gravitational constant. If we assume that mass M (i.e. earth) is fixed then Newton’s 2nd law of motion for the mass m is given by . The equivalence of inertial mass and gravitational mass means you can divide m from both sides. So, if a mass m were outside of the earth, then it would accelerate towards the earth as
and this number is g when r is the radius of the earth. This is the reason that all objects fall with the same acceleration, apocryphally validated by Galileo on the Tower of Pisa. (It is also the basis of the theory of general relativity).
However, the earth is not a point but an extended ball where each point of the ball exerts a gravitational force on any mass inside or outside of the ball. (Nothing can shield gravity). Thus to compute the force acting on a particle we need to integrate over the contributions of each point inside the earth. Assume that the density of the earth is constant so the mass where is the radius of the earth. The force between two point particles acts in a straight line between the two points. Thus for an extended object like the earth, each point in it will exert a force on a given particle in a different direction. So the calculation involves integrating over vectors. This is greatly simplified because a ball is highly symmetric. Consider the figure below, which is a cross section of the earth sliced down the middle.
The particle/elevator is the red dot, which is located a distance r from the center. (It can be inside or outside of the earth). We will assume that it travels on an axis through the center of the earth. We want to compute the net gravitational force on it from each point in the earth along this central axis. All distances (positive and negative) are measured with respect to the center of the earth. The blue dot is a point inside the earth with coordinates (x,y). There is also the third coordinate coming out of the page but we will not need it. For each blue point on one side of the earth there is another point diametrically opposed to it. The forces exerted by the two blue points on the red point are symmetrical. Their contributions in the y direction are exactly opposite and cancel leaving the net force only along the x axis. In fact there is an entire circle of points with radius y (orthogonal to the page) around the central axis where each point on the circle combines with a partner point on the opposite side to yield a force only along the x axis. Thus to compute the net force on the elevator we just need to integrate the contribution from concentric rings over the volume earth. This reduces an integral over three dimensions to just two.
The magnitude of the force (density) between the blue and red dot is given by
To get the component of the force along the x direction we need to multiple by the cosine of the angle between the central axis and the blue dot, which is
(i.e. ratio of the adjacent to the hypotenuse of the relevant triangle). Now, to capture the contributions for all the pairs on the circle we multiple by the circumference which is . Putting this together gives
The y integral extends from zero to the edge of the earth, which is . (This is R at (the center) and zero at (the poles) as expected). The x integral extends from one pole to the other, hence -R to R. Completing the y integral gives
The second term comes from the 0 limit of the integral, which is . The square root of a number has positive and negative roots but the denominator here is a distance and thus is always a positive quantity and thus must include the absolute value. The first term of the above integral can be completed straightforwardly (I’ll leave it as an exercise) but the second term must be handled with care because r-x can change sign depending on whether r is greater or less than x. For a particle outside of the earth r-x is always positive and we get
Inside the earth, we must break the integral up into two parts
The first term of (*) integrates to
Using the fact that , we get
(We again need the absolute value sign). For , the particle is outside of the earth) and . Putting everything together gives
Thus, we have explicitly shown that the gravitational force exerted by a uniform ball is equivalent to concentrating all the mass in the center. This formula is true for too.
For we have
Remarkably, the gravitational force on a particle inside the earth is just the force on the surface scaled by the ratio r/R. The equation of motion of the elevator is thus
(Recall that the gravitational acceleration at the surface is ). This is the classic equation for a harmonic oscillator with solutions of the form . Thus, a period (i.e. round trip) is given by . Plugging in the numbers gives 5062 seconds or 84 minutes. A transit through the earth once would be half that at 42 minutes and the time to fall to the center of the earth would be 21 minutes, which I find surprisingly close to the back of the envelope estimate.
Now Australia is not exactly antipodal to England so the tube in the movie did not go directly through the center, which would make the calculation much harder. This would be a shorter distance but the gravitational force would be at an angle to the tube so there would be less acceleration and something would need to keep the elevator from rubbing against the walls (wheels or magnetic levitation). I actually don’t know if it would take a shorter or longer time than going through the center. If you calculate it, please let me know.
Inflation, the steady increase of prices and wages, is a nice example of what is called a marginal mode, line attractor, or invariant manifold in dynamical systems. What this means is that the dynamical system governing wages and prices has an equilibrium that is not a single point but rather a line or curve in price and wage space. This is easy to see because if we suddenly one day decided that all prices and wages were to be denominated in some other currency, say Scooby Snacks, nothing should change in the economy. Instead of making 15 dollars an hour, you now make 100 Scooby Snacks an hour and a Starbucks Frappuccino will now cost 25 Scooby Snacks, etc. As long as wages and prices are in balance, it does not matter what they are denominated in. That is why the negative effects of inflation are more subtle than simply having everything cost more. In a true inflationary state your inputs should always balance your outputs but at an ever increasing price. Inflation is bad because it changes how you think about the future and that adjustments to the economy always take time and have costs.
This is why our current situation of price increases does not yet constitute inflation. We are currently experiencing a supply shock that has made goods scarce and thus prices have increased to compensate. Inflation will only take place when businesses start to increase prices and wages in anticipation of future increases. We can show this in a very simple mathematical model. Let P represent some average of all prices and W represent average wages (actually they will represent the logarithm of both quantities but that will not matter for the argument). So in equilibrium P = W. Now suppose there is some supply shock and prices now increase. In order to get back into equilibrium wages should increase so we can write this as
where the dot indicates the first derivative (i.e. rate of change of W is positive if P is greater than W). Similarly, if wages are higher than prices, prices should increase and we have
Now notice that the equilibrium (where there is no change in W or P) is given by W=P but given that there is only one equation and two unknowns, there is no unique solution. W and P can have any value as long as they are the same. W – P = 0 describes a line in W-P space and thus it is called a line attractor. (Mathematicians would call this an invariant manifold because a manifold is a smooth surface and the rate of change does not change (i.e. is invariant) on this surface. Physicists would call this a marginal mode because if you were to solve the eigenvalue equation governing this system, it would have a zero eigenvalue, which means that its eigenvector (called a mode) is on the margin between stable and unstable.) Now if you add the two equations together you get
which implies that the rate of change of the sum of P and W, which I call S, is zero. i.e. there is no inflation. Thus if prices and wages respond immediately to changes then there can be no inflation (in this simple model). Now suppose we have instead
The second derivative of W and P respond to differences. This is like having a delay or some momentum. Instead of the rate of S responding to price wage differences, the rate of the momentum of S reacts. Now when we add the two equations together we get
If we integrate this we now get
where C is some nonnegative constant. So in this situation, the rate of change of S is positive and thus S will just keep on increasing forever. Now what is C? Well it is the anticipatory increases in S. If you were lucky enough that C was zero (i.e. no anticipation) then there would be no inflation. Remember that W and P are logarithms so C is the rate of inflation. Interestingly, the way to combat inflation in this simple toy model is to add a first derivative term. This changes the equation to
which is analogous to adding friction to a mechanical system (used differently to what an economist would call friction). The first derivative counters the anticipatory effect of the second derivative. The solution to this equation will return to a state zero inflation (exercise to the reader).
Now of course this model is too simple to actually describe the real economy but I think it gives an intuition to what inflation is and is not.
The novel coronavirus SARS CoV-2 responsible for the COVID-19 pandemic and SARS CoV-1 responsible for the SARS epidemic of 2002-2003 share an ancestor yet evolved to have much different transmissibility and global impact 1. A previously developed thermodynamic model of protein conformations hypothesized that SARS CoV-2 is very close to a new thermodynamic critical point, which makes it highly infectious but also easily displaced by a spike-based vaccine because there is a tradeoff between transmissibility and robustness 2. The model identified a small cluster of four key mutations of SARS CoV-2 that predicts much stronger viral attachment and viral spreading compared to SARS CoV-1. Here we apply the model to the SARS-CoV-2 variants Alpha (B.1.1.7), Beta (B.1.351), Gamma (P.1) and Delta (B.1.617.2)3 and predict, using no free parameters, how the new mutations will not diminish the effectiveness of current spike based vaccines and may even further enhance infectiousness by augmenting the binding ability of the virus.
This paper is based on the ideas of physicist Jim Phillips, (formerly of Bell Labs, a National Academy member, and a developer of the theory behind Gorilla Glass used in iPhones). It was only due to Jim’s dogged persistence and zeal that I’m even on this paper although the persistence and zeal that ensnared me is the very thing that alienates most everyone else he tries to recruit to his cause.
Jim’s goal is to understand and characterize how a protein will fold and behave dynamically by utilizing an amino acid hydrophobicity (hydropathy) scale developed by Moret and Zebende. People have been developing hydropathy scores for many decades as a way to understand proteins with the idea that hydrophobic amino acids (residues) will tend to be on the inside of proteins while hydrophillic residues will be on the outside where the water is. There are several existing scores but Moret and Zebende, who are physicists and not chemists, took a different tack and found how the solvent-accessible surface area (ASA) scales with the size of a protein fragment with a specific residue in the center. The idea being that the smaller the ASA the more hydrophobic the residue. As protein fragments get larger they will tend to fold back on themselves and thus reduce the ASA. They looked at several thousand protein fragments and computed the average ASA with a given amino acid in the center. When they plotted the ASA vs length of fragment they found a power law and each amino acid had its own exponent. The more negative the exponent the smaller the ASA and thus the more hydrophobic the residue. The (negative) exponent could then be used as a hydropathy score. It differs from other scores in that it is not calculated in isolation based on chemical properties but accounts for the background of the amino acid.
M and Z’s score blew Jim’s mind because power laws are indicative of critical phenomena and phase transitions. Jim computed the coarse-grained hydropathy score (over a window of 35 residues) at each residue of a protein for a number of protein families. When COVID came along he naturally applied it to coronaviruses. He found that the coarse-grained hydropathy score profile of the spike protein of SARS-CoV-1 and SARS-CoV-2 had several deep hydrophobic wells. The well depths were nearly equal with SARS-CoV-2 being more equal than SARS-CoV-1. He then hypothesized that there was a selection advantage for well-depth symmetry and evolutionary pressure had pushed the SARS-CoV-2 spike to be near optimal. He argues that the symmetry allows the protein to coordinate activity better much like the way oscillators synchronize easier if their frequencies are more uniform. He predicted that given this optimality the spike was fragile and thus spike vaccines would be highly effective and that spike mutations could not change the spike much without diminishing function.
My contribution was to write some Julia code to automate this computation and apply it to some SARS-CoV-2 variants. I also scanned window sizes and found that the well depths are most equal close to Jim’s original value of 35. Below is Figure 3 from the paper.
What you see is the coarse-grained hydropathy score of the spike protein which is a little under 1300 residues long. Between residue 400 and 1200 there are 6 hydropathic wells. The well depths are more similar for SARS-CoV-2 and variants than SARS-CoV-1. Omicron does not look much different from the wild type, which makes me think that Omicron’s increased infectiousness is probably due to mutations that affect viral growth and transmission rather than spike binding to ACE2 receptors.
Jim is encouraging (strong arming) me into pushing this further, which I probably will given that there are still so many unanswered questions as to how and why it works, if at all. If anyone is interested in this, please let me know.
Here are the slides for my talk at the “Howard University Math-Bio Virtual Workshop on Mitigation of Future Pandemics” last Saturday. One surprising thing (to me) the model predicted, shown on slide 40, is that the median fraction of those previously infected or vaccinated (or both) was 40% or higher during the omicron wave. I was pleased and relieved to then find that recent CDC serology results validate this prediction.
I’ve been thinking a lot about what we should do for the next (and current) disaster. The first thing to say is that I am absolutely positively sure that I could not have done any better than what had been done for Covid-19. I probably would have done things differently but I doubt it would have led to a better (and probably a worse) outcome. I still think in aggregate, we are doing about as well as we could have. The one thing I do think we need to do is to figure out a way to partition risk. The biggest problem of the current pandemic is that people do not realize or care that their own risky behavior puts other people at risk. I do not care if a person wants to jump off of a cliff in a bat suit because they are mostly taking the risk upon themselves (although they do take up a bed in an ER ward if they get injured). However, not wearing a mask or getting vaccinated puts other people, including strangers, at risk. If you knowingly attend a wedding with a respiratory illness then you have the potential to infect tens if not hundreds of people and killing a fraction of them.
I do think people should be allowed to take risks as long as there are limited consequences to others. Thus, in a pandemic I think we should figure out a way for people to not get vaccinated or wear masks without affecting others. Currently, the main bottleneck is the health care system. If we allow people to wantonly get infected then there is a risk that they overwhelm hospitals. This affects all people who may need healthcare. Now is not a good time to try to repair your roof because if you fall you may not be able to get a bed in an ER ward. Thus, we really do need to think about stratifying health care according to risk acceptance. People who choose to lead risky lives should get to the back of the line when it comes to treatment. These policies should be made clear. Those who refuse to be vaccinated should just sign a form that they could be delayed in receiving health care. If you want to attend a large gathering then you should sign the same waiver.
I think that people should be allowed to opt out of the Nanny State but they need to absorb the consequences. I personally like to live in a highly regulated state but I think people should have a choice to opt out. They can live in a flood zone if they wish but they should not be bailed out after the flood. If banks want to participate in risky activities then fine but we should not bail them out. We should have let every bank fail after the 2008 crisis. We could have just let them all go under and saved homeowners instead (who should have been made better aware of the risks they were taking). Bailing out banks was a choice not a necessity.
In light of the new omicron variant and breakthrough infections in people who have been vaccinated or previously infected, I was asked to discuss what a model would predict. The simplest model that includes reinfection is an SIRS model, where R, which stands for recovered, can become susceptible again. The equations have the form
I have ignored death due to infection for now. So like the standard SIR model, susceptible, S, have a chance of being infected, I, if they contact I. I then recovers to R but then has a chance to become S again. Starting from an initial condition of S = N and I very small, then S will decrease as I grows.
The first thing to note that the number of people N is conserved in this model (as it should be). You can see this by noting that the sum of the right hand sides of all the equations is zero. Thus and thus the integral is a constant and given that we started with N people then there will remain N people. This will change if we include births and deaths. Given this conservation law, then the dynamics have three possibilities. The first is that it goes to a fixed point meaning that in the long run the numbers of S, I and R will stabilize to some fixed number and remain there forever. The second is that it oscillates so S, I, and R will go up and down. The final one is that the orbit is chaotic meaning that S, I and R will change unpredictably. For these equations, the answer is the first option. Everything will settle to a fixed point.
To show this, you first must find an equilibrium or fixed point. You do this by setting all the derivatives to zero and solving the remaining equations. I have always found the fixed point to be the most miraculous state of any dynamical system. In a churning sea where variables move in all directions, there is one place that is perfectly still. The fixed point equations satisfy
There is a trivial fixed point given by S = N and I = R = 0. This is the case of no infection. However, if is large enough then this fixed point is unstable and any amount of I will grow. Assuming I is not zero, we can find another fixed point. Divide I out of the second equation and get
Solving the third equation gives us
which we can substitute into the first equation to get back the second equation. So to find I, we need to use the conservation condition S + I + R = N which after substituting for S and R gives
which we then back substitute to get
The fact that and must be positive implies is necessary.
The next question is whether this fixed point is stable. Just because a fixed point exists doesn’t mean it is stable. The classic example is a pencil balancing on its tip. Any small perturbation will knock it over. There are many mathematical definitions of stability but they essentially boil down to – does the system return to the equilibrium if you move away from it. The most straightforward way to assess stability is to linearize the system around the fixed point and then see if the linearized system grows or decays (or stays still). We linearize because linear systems are the only types of dynamical systems that can always be solved systematically. Generalizable methods to solve nonlinear systems do not exist. That is why people such as myself can devote a career to studying them. Each system is its own thing. There are standard methods you can try to use but there is no recipe that will always work.
To linearize around a fixed point we first transform to a coordinate system around that fixed point by defining , , , to get
So now s = h = r = 0 is the fixed point. I used lower case h because lower case i is usually . The only nonlinear term is , which we ignore when we linearize. Also by the definition of the fixed point the system then simplifies to
which we can write as a matrix equation
, where and . The trace of the matrix is so the sum of the eigenvalues is negative but the determinant is zero (since the rows sum to zero), and thus the product of the eigenvalues is zero. With a little calculation you can show that this system has two eigenvalues with negative real part and one zero eigenvalue. Thus, the fixed point is not linearly stable but could still be nonlinearly stable, which it probably is since the nonlinear terms are attracting.
That was a lot of tedious math to say that with reinfection, the simplest dynamics will lead to a stable equilibrium where a fixed fraction of the population is infected. The fraction increases with increasing or and decreases with . Thus, as long as the reinfection rate is much smaller than the initial infection rate (which it seems to be), we are headed for a situation where Covid-19 is endemic and will just keep circulating around forever. It may have a seasonal variation like the flu, which is still not well understood and is beyond the simple SIRS equation. If we include death in the equations then there is no longer a nonzero fixed point and the dynamics will just leak slowly towards everyone dying. However, if the death rate is slow enough this will be balanced by births and deaths due to other causes.
Like many youth of my generation, I watched the original Star Trek in reruns and Next Generation and Deep Space Nine in real time. I enjoyed the shows but can’t really claim to be a Trekkie. I was already in graduate school when Next Generation began so I could not help but to scrutinize the shows for scientific accuracy. I was impressed that the way they discovered life in a baby universe created in one episode was by detecting localized entropy reduction, which is quite sophisticated scientifically. I bristled each time the star ship was on the brink of total failure and about to explode but the artificial gravity system still didn’t fail. I celebrated the one episode that actually had an artificial gravity failure and people actually floated in space! I thought it was ridiculous that almost every single planet they visited was always at room temperature with a breathable atmosphere. That doesn’t even describe many parts of earth. I mostly let these inaccuracies slide in the interest of story but I could never let go of one thing that always left me feeling somewhat despondent about the human condition, which was that even in a supposed super advanced egalitarian democratic society where material shortages no longer existed, Star Fleet was still an absolute autocracy. Many of the episodes dealt with strictly obeying the chain of command and never disobeying direct orders. A world with a democratic federation of planets, transporters and faster than light travel still believed that autocracy was the most efficient way to run an organization.
For most people throughout history and including today, the difference between autocracy and democracy is mostly abstract. People go to jobs where a boss tells them what to do. Virtually no one questions that corporations should be run autocratically. Authoritarian CEO’s are celebrated. Religion is generally autocratic. It only makes sense that the military backs autocrats given that autocracy is already the governing principle of their enterprise. Julius Caesar crossed the Rubicon and became the Dictator of Rome (he was never actually made Emperor) because he had the biggest army and it was loyal to him, not the Roman Republic. The only real question is how democracies even persist. People may care about freedom but do they really care all that much about democracy?
One outcome of the pandemic is that I have not had any illness (knock on wood), nary a cold nor sniffle, in a year and a half. On the other hand, my skin has fallen apart. I am constantly inflamed and itchy. I have no proof that the two are connected but my working hypothesis is that my immune system is hypersensitive right now because it has had little to do since the spring of 2020. It now overreacts to every mold spore, pollen grain, and speck of dust it runs into. The immune system is extremely complex, perhaps as complex as the brain. Its job is extremely difficult. It needs to recognize threats and eliminate them while not attacking itself. The brain and the immune system are intricately linked. How many people have gotten ill immediately after a final exam or deadline? The immune system was informed by the brain to delay action until the task was completed. The brain probably takes cues form the immune system too. One hypothesis for why asthma and allergies have been on the rise recently is that modern living has eliminated much contact with parasites and infectious agents, making the immune system hypersensitive. I for one, always welcome vaccinations because it gives my immune system something to do. In fact, I think it would be a good idea to get inoculations of all types regularly. I would take a vaccine for tape worm in a heartbeat. We are now slowly exiting from a global experiment in depriving the immune system of stimulation. We have no idea what the consequences will be. That is not to say that quarantine and isolation was not a good idea. Being itchy is clearly better than being infected by a novel virus (or being dead). There can be long term effects of infection too. Long covid is likely to be due to a miscalibrated immune system induced by the infection. Unfortunately, we shall likely never disentangle all the effects of COVID-19. We will not ever truly know what the long term consequences of infection, isolation, and vaccination will be. Most people will come out of this fine but a small fraction will not and we will not know why.
I read an article recently about an anti-vaccination advocate exclaiming at a press conference with the governor of Florida that vaccines against SARS-CoV-2 “change your RNA!” This made me think that most people probably do not know much about RNA and that a little knowledge is a dangerous thing. Now ironically, contrary to what the newspapers say, this statement is kind of true although in a trivial way. The Moderna and Pfizer vaccines insert a little piece of RNA into your cells (or rather cells ingest them) and that RNA gets translated into a SARS-CoV-2 spike protein that gets expressed on the surface of the cells and thereby presented to the immune system. So, yes these particular vaccines (although not all) have changed your RNA by adding new RNA to your cells. However, I don’t think this is what the alarmist was worried about. To claim that something that changes is a bad thing implies that the something is fixed and stable to start with, which is profoundly untrue about RNA.
The central dogma of molecular biology is that genetic information flows from DNA to RNA to proteins. All of your genetic material starts as DNA organized in 23 pairs of chromosomes. Your cells will under various conditions transcribe this DNA into RNA, which is then translated into proteins. The biological machinery that does all of this is extremely complex and not fully understood and part of my research is trying to understand this better. What we do know is that transcription is an extremely noisy and imprecise process at all levels. The molecular steps that transcribe DNA to RNA are stochastic. High resolution images of genes in the process of transcription show that transcription occurs in random bursts. RNA is very short-lived, lasting between minutes to at most a few days. There is machinery in the cell dedicated to degrading RNA. RNA is spliced; it is cut up into pieces and reassembled all the time and this splicing happens more or less randomly. Less than 2% of your DNA codes for proteins but virtually all of the DNA including noncoding parts are continuously being transcribed into small RNA fragments. Your cell is constantly littered with random stray pieces of RNA, and only a small fraction of it gets translated into proteins. Your RNA changes. All. The. Time.
Now, a more plausible alarmist statement (although still untrue) would be to say that vaccines change your DNA, which could be a bad thing. Cancer after all involves DNA mutations. There are viruses (retroviruses) that insert a copy of its RNA code into the host’s DNA. HIV does this for example. In fact, a substantial fraction of the human genome is comprised of viral genetic material. Changing proteins can also be very bad. Prion diseases are basically due to misfolded proteins. So DNA changing is not good, protein changing is not good, but RNA changing? Nothing to see here.
Even though Covid- is going to be with us forever, I actually think on the whole the pandemic turned out better than expected, and I mean that in the technical sense. If we were to rerun the pandemic over and over again, I think our universe will end up with fewer deaths than average. That is not to say we haven’t done anything wrong. We’ve botched up many things of course but given that the human default state is incompetence, we botched less than we could have.
The main mistake in my opinion was the rhetoric on masks in March of 2020. Most of the major Western health agencies recommended against wearing masks at that time because they 1) there was already a shortage of N95 masks for health care workers and 2) they thought that cloth and surgical masks were not effective in keeping one from being infected. Right there is a perfect example of Western solipsism; masks were only thought of as tools for self-protection, rather than as barriers for transmission. If only it was made clear early on that the reason we wear masks is not to protect me from you but to protect you from me. (Although there is evidence that masks do protect the wearer too, see here). This would have changed the rhetoric over masks we are having right now. The anti-maskers would be defending their right to harm others rather than the right to not protect themselves from harm.
The thing we got right was in producing effective vaccines. That was simply astonishing. There had never been a successful mRNA-based drug of any type until the BioNTech and Moderna vaccines. Many things had to go right for the vaccines to work. We needed a genetic sequence (Chinese scientists made it public in January), from that sequence we needed a target (the coronavirus spike protein), we needed to be able to stabilize the spike (research that came out of the NIH vaccine center), we needed to make mRNA less inflammatory (years of work especially at Penn), we needed a way to package that mRNA (work out of MIT), and we needed a sense of urgency to get it done (Western governments). Vaccines don’t always work but we managed to get one in less than a year. So many things had to go right for that to happen. The previous US administration should be taking a victory lap because it was developed under their watch, instead of bashing it.
As I’ve said before, I am skeptical we can predict what will happen next but I am going to predict now that there will not be a variant in the next year that will escape from our current vaccines. We may need booster shots and minor tweaks but the vaccines will continue to work. Part of my belief stems from the work of JC Phillips who argues that the SARS-CoV-2 spike protein is already highly optimized and thus there is not much room for it to change and to become infectious. The virus may mutate to replicate faster within the body but the spike will be relatively stable and thus remain a target for the vaccines. The delta variant wave we’re seeing now is a pandemic of the unvaccinated. I have no idea if those against vaccinations will have a change of heart but at some point everyone will be infected and have some immune protection. (I just hope they approve the vaccine for children before winter). SARS-CoV-2 will continue to circulate just like the way the flu strain from the 1918 pandemic still circulates but it won’t be the danger and menace it is now.
Although cryptocurrencies have been mainstream for quite a while, I still think the popular press has not done a great job explaining the details of how they work. There are several ideas behind a cryptocurrency like Bitcoin but the main one is the concept of a cryptographic hash. In simple terms, a hash is a way to transform an input sequence of characters (i.e. a string) into an output string such that it is hard to recreate the input string from the output string. A transformation with this property is called a one-way function. It is a machine where you get an output from an input but you can’t get the input from the output and there does not exist any other machine that can get the input from the output. A hash is a one-way function where the output has a standard form, e.g. 64 characters long. So if each character is a bit, e.g. 0 or 1, then there are different possible hashes. What makes hashes useful are two properties. The first, as mentioned, is that it is a one-way function and the second is that two different inputs do not give the same hash, called collision avoidance. There have been decades of work on figuring out how to do this and institutions like the US National Institutes of Standards and Technology (NIST) actually publish hash standards like SHA-2.
Hashes are an important part of your life. If a company is responsible, then only the hash of your password is stored on their servers, and when you type your password into a website, it goes through the hashing function and the hash is checked against the stored version. That way, if there is security breach, only the hash list is stolen. If the company is very responsible, then your name and all of your information is also only stored in hash form. Part of the problem with past security breaches is that the companies stored actual information instead of hashed information. However, if you use the same password on different websites then the hash would be same if the same standard was used. Some really careful companies will “salt” your password by adding a random string to it (that is hopefullly stored separately) before hashing. Or they will rehash your hash with salt. If you had a perfect hash, then the only way to break it would be to guess different inputs and see if it matches the desired output. The so-called complex math problem that Bitcoin solves before validating a transaction (and getting a reward) is finding a hash with a certain property but more on this later.
Now, one of the problems with hashing is that you need to deal with inputs of various sizes but you want the output to have a single uniform size. So even though a hash could have enough information capacity (i.e. entropy) to encode all of the world’s information ten times over, it is computationally inconvenient to just feed the complete text of Hamlet directly into a single one-way function. This is where the concept of a Merkle tree becomes important. You start with some one-way function that takes inputs of some fixed length and it scrambles the characters in some way that is not easily reversible. If the input string is too short then you just add extra characters (called padding) but if it is too long you need to do something else. The way a Merkle tree works is to break the text into chunks of uniform size. It then hashes the first chunk, adds that to the next chunk, hash the result and repeat until you have included all the chunks. This repeated recursive hashing is the secret sauce of crypto-currencies.
Bitcoin tried to create a completely decentralized digital currency that could be secure and trusted. For a regular currency like the US dollar, the thing you are most concerned about is that the dollar you receive is not counterfeit. The way that problem is solved is to make the manufacturing process of dollar bills very difficult to reproduce. So the dollar uses special paper with special marks and threads and special fonts and special ink. There are laws against making photocopiers with a higher resolution than the smallest print on a US bill to safeguard against counterfeiting. A problem with digital currencies is that you need to prevent double spending. The way this is historically solved is to have all transactions validated by a central authority.
Bitcoin solves these problems in a decentralized system by using a public ledger, called a blockchain that is time stamped, immutable and verifiable. The block chain keeps track of every Bitcoin transaction. So if you wanted to transfer one Bitcoin to someone else then the blockchain would show that your private Bitcoin ID has one less Bitcoin and the ID of the person you transferred to would have one extra Bitcoin. It is called a blockchain because each transaction (or set of transactions) is recorded into a block, the blocks are sequential, and each block contains a hash of the previous block. To validate a transaction you would need to validate each transaction leading up to the last block to validate that the hash on each block is correct. Thus the blockchain is a Merkle tree ledger where each entry is time stamped, immutable, and verifiable. If you want to change a block you need to change all the blocks before it.
However, the blockchain is not decentralized on its own. How do you prevent two blocks with two different hashes? The way to achieve that goal is to make the hash used in each block have a special form that is hard to find. This underlies the concept of “proof of work”. Bitcoin uses a hash called SHA-256 which consists of a hexadecimal string of 64 characters (i.e. a base 16 number, usually with characters consisting of the digits 0-9 plus letters a-f). Before each block gets added to the chain, it must have a hash that has a set number of zeros at the front. In order to do this, you need to add some random numbers to the block or rearrange it so that the hash changes. This is what Bitcoin miners do. They try different variations of the block until they get a hash that has a certain number of zeros in front and then they race to see who gets it first. The more zeros you need the more guesses you need and thus the harder the computation. If it’s just one zero then one in 16 hashes will have that property and thus on average 16 tries will get you the hash and the right to add to the blockchain. Each time you require an additional zero, the number of possibilities decreases by a factor of 16 so it is 16 times harder to find one. Bitcoin wants to keep the computation time around 15 minutes so as computation speed increases it just adds another zero. The result is an endless arms race. The faster the computers get the harder the hash is to find. The incentive for miners to be fast is that they get some Bitcoins if they are successful in being the first to find a desired hash and earning the right to add a block to the chain.
The actual details for how this works is pretty complicated. All the miners (or Bitcoin nodes) must validate that the proposed block is correct and then they all must agree to add that to the chain. The way it works in a decentralized way is that the code is written so that a node will follow the longest chain. In principle, this is secure because a dishonest miner who wants to change a previous block must change all blocks following it and thus as long as there are more honest miners than dishonest ones, the dishonest ones can never catch up. However, there are issues when two miners simultaneously come up with a hash and they can’t agree on which to follow. This is called a fork and has happened at least once I believe. This gets fixed eventually because honest miners will adopt the longest chain and the chain with the most adherents will grow the fastest. However, in reality there are only a small number of miners that regularly add to the chain so we’re at a point now where a dishonest actor could possibly dominate the honest ones and change the blockchain. Proof of work is also not the only way to add to a blockchain. There are several creative ideas to make it less wasteful or even make all that computation useful and I may write about them in the future. I’m somewhat skeptical about the long term viability of Bitcoin per se but I think the concepts of the blockchain are revolutionary and here to stay.
2021-06-21: some typos fixed and clarifying text added.
Interesting article in the New York Times today about how people to this day still do not know how magician David Berglass did his “Any Card At Any Number” trick. In this trick, a magician asks a person or two to name a card (e.g. Queen of Hearts) and a number (e.g. 37) and then in a variety of ways produce a deck where that card appears at that order in the deck. The supposed standard way to do the trick is for the magician to manipulate the cards in some way but Berglass does his trick by never touching the cards. He can even do the trick impromptu when you visit him by leading you to a deck of cards somewhere in the room or from his pocket that has the card in the correct place. Now, I have no idea how he or anyone does this trick but one way to do the trick is to use “full enumeration”, i.e. hide decks where every possibility is accounted for and then the trick is to remember which deck has that choice. So then the question is how many decks would you need? Well the minimal number of decks is 52 because a particular card could be in one of 52 positions. But how many more decks would you need? The answer is zero. 52 is all you need because for any particular arrangement of cards, each card is in one position. Then all you do is rotate all the cards by one, so the card in the first position is now in the second position for deck 2 and 52 moves to 1 and so on. What the magician can do is to then hide 52 decks and remember the order of each deck. In the article he picked the reporter’s card to within 1 but claimed he may only be able to do it to within 2. That means he’s hiding the decks in groups of 3 and 4 say and then points you to that location and lets you choose which deck.
I sometimes listen to the podcast “How I built this“, where host Guy Raz interviews successful entrepreneurs like Herb Kelleher, who founded Southwest Airlines, Reid Hoffman of Linkedin, Stacy Madison of Stacy’s Pita Chips, and so on. The story arc of each interview is similar – some scrappy undervalued person comes up with a novel idea and then against all odds succeeds by hard work, unrelenting drive, and taking risks. The podcast fully embraces the American myth of the hero entrepreneur although Guy tries to do his best to extend it beyond the stereotypical Silicon Valley one typified by Steve Jobs or Elon Musk. At the end of each interview Guy will ask the subject how much of their success was due to luck and how much due to their ingenuity and diligence. Most are humble or savvy enough to say that some large fraction of the success was luck. While I have no doubt that each successful entrepreneur is bright, hard working, and possesses unique skills, there are countless others who are equally talented and yet did not succeed. Each success story is an example of survivor bias. We sometimes hear about spectacular failures, like the Edsel , but rarely do we hear about the story of “How I almost built this”.
There is a stock market scam where you email blocks of 1024 prospective marks a prediction of what a stock will do that week. For one half, you say the stock will go up and for the other half you say it will go down. Then for the half for which you were correct, you do the same thing and half of them (one quarter of the original) will receive a correct prediction. Finally after ten weeks, one of the original 1024 will have received 10 correct predictions in a row and think that you are either a genius or have inside information and will be primed to sign up for whatever scam you are selling. The lucky (or unlucky) person is fooled because they lack the information that 1023 others did not receive perfect predictions. Obviously, this also works for sports predictions.
While, I think most success is luck there do seem to be outliers. Elon Musk seems to be one. He manages to invent new industries and succeed with regularity. Warren Buffet does seem to be able to beat the market. However, it is for us as a society to decide how winners should be rewarded. In many industries there is a winner-take-all dynamic, where the larger you get the easier it is to crush the competition. Mark Zuckerberg is clearly skilled but Facebook is dominant right now because it is a monopolist; it simply buys up as many competitors as it can. The same goes for Google, Amazon, and AT&T until the government broke it up. Finance works that way too. The bigger a bank or hedge fund gets, the easier it is to succeed. A small fluctuation that propels one firm a little ahead of the rest at the right time will be exponentially amplified. While, I do think it is a positive thing to reward success I don’t think the reward needs to be so disparate. Right now, a very small difference in ability (or none at all) and a lot of luck can be the difference between flying to your house in the Hamptons in a helicopter or selling hotdogs from a cart on Fifth Avenue.
The end of the Covid-19 pandemic is within reach. The vaccines have been a roaring success and former Bell Labs physicist J.C. Phillips predicted it (see here). He argued that the spike protein, which is the business end of the SARS-CoV-2 virus, has been optimized to such a degree in SARS-CoV-2 that even a small perturbation from a vaccine can disrupt it. While the new variants perturb the spike slightly and seem to spread faster, they will not significantly evade the vaccine. However, just because the end is within sight doesn’t mean we should not still be vigilante and not mess this up. Europe has basically scored multiple own goals these past few months with their vaccine rollout (or lack thereof) that is a combination of both gross incompetence and excessive conservatism. The Astra-Zeneca vaccine fiasco was a self-inflicted wound by all parties involved. The vaccine is perfectly fine and any side effects are either not related to the vaccine or of such low probability that it should not be a consideration for halting its use. By artificially slowing vaccine distribution, there is a chance that some new mutation could arise that will evade the vaccine. Europe needs to get its act in gear. The US has steadily ramped up vaccinations and is on course to have all willing adults vaccinated by start of summer. Although there has been a plateauing and even slight rise recently because of relaxation from social distancing in some areas, cases and deaths will drop for good by June everywhere in the US. North America will largely be back to normal by mid-summer. However, it is imperative that we press forward and vaccinate the entire world. We will also all need to get booster shots next fall when we get our flu shots.
Since, the discovery of exoplanets nearly 3 decades ago most astronomers, at least the public facing ones, seem to agree that it is just a matter of time before they find signs of life such as the presence of volatile gases in the atmosphere associated with life like methane or oxygen. I’m an agnostic on the existence of life outside of earth because we don’t have any clue as to how easy or hard it is for life to form. To me, it is equally possible that the visible universe is teeming with life or that we are alone. We simply do not know.
But what would happen if we find life on another planet. How would that change our expected probability for life in the universe? MIT astronomer Sara Seager once made an offhand remark in a podcast that finding another planet with life would make it very likely there were many more. But is this true? Does the existence of another planet with life mean a dramatic increase in the probability of life in the universe. We can find out by doing the calculation.
Suppose you believe that the probability of life on a planet is (i.e. fraction of planets with life) and this probability is uniform across the universe. Then if you search planets, the probability for the number of planets with life you will find is given by a Binomial distribution. The probability that there are planets is given by the expression , where is a factor (the binomial coefficient) such that the sum of from one to is 1. By Bayes Theorem, the posterior probability for (yes, that would be the probability of a probability) is given by
where . As expected, the posterior depends strongly on the prior. A convenient way to express the prior probability is to use a Beta distribution
where is again a normalization constant (the Beta function). The mean of a beta distribution is given by and the variance, which is a measure of uncertainty, is given by . The posterior distribution for after observing planets with life out of will be
where is a normalization factor. This is again a Beta distribution. The Beta distribution is called the conjugate prior for the Binomial because it’s form is preserved in the posterior.
Applying Bayes theorem in equation (*), we see that the mean and variance of the posterior become and , respectively. Now let’s consider how our priors have updated. Suppose our prior was , which gives a uniform distribution for on the range 0 to 1. It has a mean of 1/2 and a variance of 1/12. If we find one planet with life after checking 10,000 planets then our expected becomes 2/10002 with variance . The observation of a single planet has greatly reduced our uncertainty and we now expect about 1 in 5000 planets to have life. Now what happens if we find no planets. Then, our expected only drops to 1 in 10000 and the variance is about the same. So, the difference between finding a planet versus not finding a planet only halves our posterior if we had no prior bias. But suppose we are really skeptical and have a prior with and so our expected probability is zero with zero variance. The observation of a single planet increases our posterior to 1 in 10001 with about the same small variance. However, if we find a single planet out of much fewer observations like 100, then our expected probability for life would be even higher but with more uncertainty. In any case, Sara Seager’s intuition is correct – finding a planet would be a game breaker and not finding one shouldn’t really discourage us that much.
Liberalism, as a philosophy, arose during the European Enlightenment of the 17th century. It’s basic premise is that people should be free to choose how they live, have a government that is accountable to them, and be treated equally under the law. It was the founding principle of the American and French revolutions and the basic premise of western liberal democracies. However, liberalism is inherently conflicted because when I exercise my freedom to do something (e.g. not wear a mask), I infringe on your freedom from the consequence of that thing (e.g. not be infected) and there is no rational resolution to this conflict. This conflict led to the split of liberalism into left and right branches. In the United States, the term liberal is exclusively applied to the left branch, which mostly focuses on the ‘freedom from’ part of liberalism. Those in the right branch, who mostly emphasize the ‘freedom to’ part, refer to themselves as libertarian, classical liberal, or (sometimes and confusingly to me) conservative. (I put neo-liberalism, which is a fundamentalist belief in free markets, into the right camp although it has adherents on both the left and right.) Both of these viewpoints are offspring of the same liberal tradition and here I will use the term liberal in the general sense.
Liberalism has never operated in a vacuum. The conflicts between “freedom to” and “freedom from” have always been settled by prevailing social norms, which in the Western world was traditionally dominated by Christian values. However, neither liberalism nor social norms have ever been sufficient to prevent bad outcomes. Slavery existed and was promoted by liberal Christian states. Genocide of all types and scales have been perpetrated by liberal Christian states. The battle to overcome slavery and to give equal rights to all peoples was a long and hard fought battle over slowly changing social norms rather than laws per se. Thus, while liberalism is the underlying principle behind Western governments, it is only part of the fabric that holds society together. Even though we have just emerged from the Dark Years, Western Liberalism is on its shakiest footing since the Second World War. The end of the Cold War did not bring on a permanent era of liberal democracy but may have spelled it’s eventual demise. What will supplant liberalism is up to us.
It is often perceived that the American Democratic party is a disorganized mess of competing interests under a big tent while the Republicans are much more cohesive but in fact the opposite is true. While the Democrats are often in conflict they are in fact a fairly unified center-left liberal party that strives to advocate for the marginalized. Their conflicts are mostly to do with which groups should be considered marginalized and prioritized. The Republicans on the other hand are a coalition of libertarians and non-liberal conservatives united only by their desire to minimize the influence of the federal government. The libertarians long for unfettered individualism and unregulated capitalism while the conservatives, who do not subscribe to all the tenets of liberalism, wish to halt encroaching secularism and a government that no longer serves their interests.
The unlikely Republican coalition that has held together for four decades is now falling apart. It came together because the more natural association between religious conservatism and a large federal bureaucracy fractured after the Civil Rights movements in the 1960’s when the Democrats no longer prioritized the concerns of the (white) Christian Right. (I will discuss the racial aspects in a future post). The elite pro-business neo-liberal libertarians could coexist with the religious conservatives as long as their concerns did not directly conflict but this is no longer true. The conservative wing of the Republican party have discovered their new found power and that there is an untapped population of disaffected individuals who are inclined to be conservative and also want a larger and more intrusive government that favors them. Prominent conservatives like Adrian Vermeule of Harvard and Senator Josh Hawley are unabashedly anti-liberal.
This puts the neo-liberal elites in a real bind. The Democratic party since Bill Clinton had been moving right with a model of pro-market neo-liberalism but with a safety net. However they were punished time and time again by the neo-liberal right. Instead of partnering with Obama, who was highly favorable towards neoliberalism, they pursued a scorched earth policy against him. Hilary Clinton ran on a pretty moderate safety-net-neo-liberal platform and got vilified as an un-American socialist. Now, both the Republicans and Democrats are trending away from neo-liberalism. The neo-liberals made a strategic blunder. They could have hedged their bets but now have lost influence in both parties.
While the threat of authoritarianism looms large, this is also an opportunity to accept the limits of liberalism and begin to think about what will take its place – something that still respects the basic freedoms afforded by liberalism but acknowledges that it is not sufficient. Conservative intellectuals like Leo Strauss have valid points. There is indeed a danger of liberalism lapsing into total moral relativism or nihilism. Guardrails against such outcomes must be explicitly installed. There is value in preserving (some) traditions, especially ancient ones that are the result of generations of human engagement. There will be no simple solution. No single rule or algorithm. We will need to explicitly delineate what we will accept and what we will not on a case by case basis.
For the past four years, I have been unable to post with any regularity. I have dozens of unfinished posts sitting in my drafts folder. I would start with a thought but then get stuck, which had previously been somewhat unusual for me. Now on this first hopeful day I have had for the past four trying years, I am hoping I will be able to post more regularly again.
Prior to what I will now call the Dark Years, I viewed all of history through an economic lens. I bought into the standard twentieth century leftist academic notion that wars, conflicts, social movements, and cultural changes all have economic underpinnings. But I now realize that this is incorrect or at least incomplete. Economics surely plays a role in history but what really motivates people are stories and stories are what led us to the Dark Years and perhaps to get us out.
Trump became president because he had a story. The insurrectionists who stormed the Capitol had a story. It was a bat shit crazy lunatic story but it was still a story. However, the tragic thing about the Trump story (or rather my story of the Trump story) is that it is an unintentional algorithmically generated story. Trump is the first (and probably not last) purely machine learning president (although he may not consciously know that). Everything he did was based on the feedback he got from his Twitter Tweets and Fox News. His objective function was attention and he would do anything to get more attention. Of the many lessons we will take from the Dark Years, one should be how machine learning and artificial intelligence can go so very wrong. Trump’s candidacy and presidency was based on a simple stochastic greedy algorithm for attention. He would Tweet randomly and follow up on the Tweets that got the most attention. However, the problem with a greedy algorithm (and yes that is a technical term that just happens to coincidentally be apropos) is that once you follow a path it is hard to make a correction. I actually believe that if some of Trump’s earliest Tweets from say 2009-2014 had gone another way, he could have been a different president. Unfortunately, one of his early Tweet themes that garnered a lot of attention was on the Obama birther conspiracy. This lit up both racist Twitter and a counter reaction from liberal Twitter, which led him further to the right and ultimately to the presidency. His innate prejudices biased him towards a darker path and he did go completely unhinged after he lost the election but he is unprincipled and immature enough to change course if he had enough incentive to do so.
Unlike standard machine learning for categorizing images or translating languages, the Trump machine learning algorithm changes the data. Every Tweet alters the audience and the reinforcing feedback between Trump’s Tweets and its reaction can manufacture discontent out of nothing. A person could just happen to follow Trump because they like The Apprentice reality show Trump starred in and be having a bad day because they missed the bus or didn’t get a promotion. Then they see a Trump Tweet, follow the link in it and suddenly they find a conspiracy theory that “explains” why they feel disenchanted. They retweet and this repeats. Trump sees what goes viral and Tweets more on the same topic. This positive feedback loop just generated something out of random noise. The conspiracy theorizing then starts it’s own reinforcing feedback loop and before you know it we have a crazed mob bashing down the Capitol doors with impunity.
Ironically Trump, who craved and idolized power, failed to understand the power he actually had and if he had a better algorithm (or just any strategy at all), he would have been reelected in a landslide. Even before he was elected, Trump had already won over the far right and he could have started moving in any direction he wished. He could have moderated on many issues. Even maintaining his absolute ignorance of how govening actually works, he could have had his wall by having it be part of actual infrastructure and immigration bills. He could have directly addressed the COVID-19 pandemic. He would not have lost much of his base and would have easily gained an extra 10 million votes. Maybe, just maybe if liberal Twitter simply ignored the early incendiary Tweets and only responded to the more productive ones, they could have moved him a bit too. Positive reinforcement is how they train animals after all.
Now that Trump has shown how machine learning can win a presidency, it is only a matter of time before someone harnesses it again and more effectively. I just hope that person is not another narcissistic sociopath.
On some rare days when the sun is shining and I’m enjoying a well made kouign-amann (my favourite comes from b.patisserie in San Francisco but Patisserie Poupon in Baltimore will do the trick), I find a brief respite from my usual depressed state and take delight, if only for a brief moment, in the fact that mathematics completely resolved Zeno’s paradox. To me, it is the quintessential example of how mathematics can fully solve a philosophical problem and it is a shame that most people still don’t seem to know or understand this monumental fact. Although there are probably thousands of articles on Zeno’s paradox on the internet (I haven’t bothered to actually check), I feel like visiting it again today even without a kouign-amann in hand.
I don’t know what the original statement of the paradox is but they all involve motion from one location to another like walking towards a wall or throwing a javelin at a target. When you walk towards a wall, you must first cross half the distance, then half the remaining distance, and so on forever. The paradox is thus: How then can you ever reach the wall, or a javelin reach its target, if it must traverse an infinite number of intervals? This paradox is completely resolved by the concept of the mathematical limit, which Newton used to invent calculus in the seventeenth century. I think understanding the limit is the greatest leap a mathematics student must take in all of mathematics. It took mathematicians two centuries to fully formalize it although we don’t need most of that machinery to resolve Zeno’s paradox. In fact, you need no more than middle school math to solve one of history’s most famous problems.
The solution to Zeno’s paradox stems from the fact that if you move at constant velocity then it takes half the time to cross half the distance and the sum of an infinite number of intervals that are half as long as the previous interval adds up to a finite number. That’s it! It doesn’t take forever to get anywhere because you are adding an infinite number of things that get infinitesimally smaller. The sum of a bunch of terms is called a series and the sum of an infinite number of terms is called an infinite series. The beautiful thing is that we can compute this particular infinite series exactly, which is not true of all series.
Expressed mathematically, the total time it takes for an object traveling at constant velocity to reach its target is
which can be rewritten as
where is the distance and is the velocity. This infinite series is technically called a geometric series because the ratio of two subsequent terms in the series is always the same. The terms are related geometrically like the volumes of n-dimensional cubes when you have halve the length of the sides (e.g. 1-cube (line and volume is length), 2-cube (square and volume is area), 3-cube (good old cube and volume), 4-cube ( hypercube and hypervolume), etc) .
For simplicity we can take . So to compute the time it takes to travel the distance, we must compute:
To solve this sum, the first thing is to notice that we can factor out and obtain
The quantity inside the bracket is just the original series plus 1, i.e.
and thus we can substitute this back into the original expression for and obtain
Now, we simply solve for and I’ll actually go over all the algebraic steps. First multiply both sides by 2 and get
Now, subtract from both sides and you get the beautiful answer that . We then have the amazing fact that
I never get tired of this. In fact this generalizes to any geometric series
for any that is less than 1. The more compact way to express this is
Now, notice that in this formula if you set , you get , which is infinity. Since for any , this tells you that if you try to add up an infinite number of ones, you’ll get infinity. Now if you set you’ll get a negative number. Does this mean that the sum of an infinite number of positive numbers greater than 1 is a negative number? Well no because the series is only defined for , which is called the domain of convergence. If you go outside of the domain, you can still get an answer but it won’t be the answer to your question. You always need to be careful when you add and subtract infinite quantities. Depending on the circumstance it may or may not give you sensible answers. Getting that right is what math is all about.