Tonight, Slava Turyshev sent me a link to an article that was actually published three months ago on medium.com but until now, escaped our attention.

It is a very nice summary of the work that we have been doing on the Solar Gravitational Lens to date.

It really captures the essence of our work and the challenges that we have been looking at.

And there is so much more to do! Countless more things to tackle: image reconstruction of a moving target, imperfections of the solar gravitational field, precision of navigation… not to mention the simple, basic challenge of attempting a deep space mission to a distance four times greater than anything to date, lasting several decades.

Yes, it can be done. No it’s not easy. But it’s a worthy challenge.

One thing follows another…

I’m listening to old MP3 files on my computer, one of which contains this once popular song by Paper Lace, The Night Chicago Died.

The page mentions, among other things, how then Chicago mayor Daley hated the song. Daley? The same Daley who demolished Meigs Field airport, the island airport serving downtown Chicago that was the starting location of Microsoft Flight Simulator for many years?

Indeed. (Well, almost. There were two Daleys, father and son, Richard J and Richard M.) And Wikipedia tells me that the island has indeed since been turned into a park and nature preserve. But there are few pictures, so I figured I’d check it out using Google Maps.

So I typed Chicago into Google Maps and was greeted with this message in response:

I don’t know but this seems… a tad embarrassing isn’t it. Unless of course Chicago actually did die last night, and was promptly removed from Google Maps in response.

But no, Chicago is still there. The Google Maps thing was just a glitch. As is Northerly Island, which once hosted that ill-fated airport, its future as uncertain as it has always been in the past century or so.

A popular Internet meme these days is to present an arithmetic expression like, say, 6/3(4−2) and ask the poor souls who follow you to decide the right answer. Soon there will be two camps, each convinced that they know the truth and that the others are illiterate fools: According to one camp, the answer is 4, whereas the other camp will swear that it has to be 1.

In reality it is neither. Or both. Flip a coin, take your pick. There is no fundamental mathematical truth hidden here. It all boils down to human conventions. The standard convention is that multiplication and division have the same precedence and are evaluated from left to right: So 6/3×(4−2) is pretty unambiguous. But there is another, unwritten convention that when the multiplication sign is omitted, the implied multiplication is assumed to have a higher precedence.

Precisely because of these ambiguities, when you see actual professionals, mathematicians or physicists, write down an expression like this, they opt for clarity: they write, say, (6/3)(4−2) or 6/[3(4−2)] precisely so as to avoid any misunderstanding. Or better yet, they use proper math typesetting software such as LaTeX and write 2D formulas.

A tragedy took place in Beirut yesterday.

The actual power of the massive explosion is yet to be estimated accurately (probably not quite as large as the largest non-nuclear, accidental explosion that took place in Halifax, Nova Scotia 103 years ago though it comes close), but the images and videos are horrifying.

Reportedly, windows were shattered as far as 25 kilometers away from the epicenter.

The audio on one of the many videos showing the moment of the explosion accurately captures the event: “What the actual fuck?” asks a woman’s voice incredulously.

In light of the scope of the disaster, I expect that the final death toll will far exceed the 78 deceased that we know about for now.

It now appears that it was an industrial accident: welding work setting off a fire that in turn spread to a warehouse where thousands of tons of ammonium nitrate was stored.

Naturally, it didn’t prevent America’s “stable genius” from talking about an “attack”. When asked, he even referred to his “generals” who, according to him, told him that it was likely an attack.

I have no doubt that he made it all up on the spot. But his pronouncement had predictable consequences. It was like pouring oil on the fire, as it gave an excuse for every closet antisemite to come out and spread the conspiracy theory that it was an attack by Israel. Twitter accounts spreading this inflammatory nonsense include a Robert de Niro parody account; for a brief moment, I thought it was the real Robert de Niro, which would have been terribly disappointing.

Wildfires in Australia, locusts in Africa, a global pandemic, widespread racial riots in the United States, “murder hornets” spreading in North America… The Internet was already full of joke calendars for this year with disaster memes, as well as speculation that perhaps that infamous Mayan calendar was misinterpreted, as it referred to 2020, not 2012. In light of this catastrophe in Beirut, I am inclined to ask, what next? Alien invasion? The Yellowstone caldera? Global thermonuclear war? Giant asteroid impact? I won’t even try to guess, just note that we still have nearly five months left of this year.

A few weeks ago, Christian Ready published a beautiful video on his YouTube channel, Launch Pad Astronomy. In this episode, he described in detail how the Solar Gravitational Lens (SGL) works, and also our efforts so far.

I like this video very much. Especially the part that begins at 10:28, where Christian describes how the SGL can be used for image acquisition. The entire video is well worth seeing, but this segment in particular does a better job than we were ever able to do with words alone, explaining how the Sun projects an image of a distant planet to a square kilometer sized area, and how this image is scanned, one imaginary pixel at a time, by measuring the brightness of the Einstein-ring around the Sun as seen from each pixel location.

We now understand this process well, but many more challenges remain. These include, in no particular order, deviations of the Sun from spherical symmetry, minor variations in the brightness of the solar corona, the relative motion of the observing probe, Sun, exosolar system and target planet therein, changing illumination of the target, rotation of the target, changing surface features (weather, perhaps vegetation) of the target, and the devil knows what else.

Even so, lately I have become reasonably confident, based on my own simulation work and our signal-to-noise estimates, as well as a deconvolution approach under development that takes some of the aforementioned issues into consideration, that a high-resolution image of a distant planet is, in fact, obtainable using the SGL.

A lot more work remains. The fun only just began. But I am immensely proud to be able to contribute to of this effort.

I met Gabor David back in 1982 when I became a member of the team we informally named F451 (inspired by Ray Bradbury of course.) Gabor was a close friend of Ferenc Szatmari. Together, they played an instrumental role in establishing a business relationship between the Hungarian firm Novotrade and its British partner, Andromeda, developing game programs for the Commodore 64.

In the months and years that followed, we spent a lot of time working together. I was proud to enjoy Gabor’s friendship. He was very knowledgeable, and also very committed to our success. We had some stressful times, to be sure, but also a lot of fun, frantic days (and many nights!) spent working together.

I remember Gabor’s deep, loud voice, with a slight speech impediment, a mild case of rhotacism. His face, too, I can recall with almost movie like quality.

He loved coffee more than I thought possible. He once dropped by at my place, not long after I managed to destroy my coffee maker, a stovetop espresso that I accidentally left on the stove for a good half hour. Gabor entered with the words, “Kids, do you have any coffee?” I tried to explain to him that the devil’s brew in that carafe was a bitter, undrinkable (and likely unhealthy) blend of burnt coffee and burnt rubber, but to no avail: he gulped it down like it was nectar.

After I left Hungary in 1986, we remained in sporadic contact. In fact, Gabor helped me with a small loan during my initial few weeks on Austria; for this, I was very grateful.

When I first visited Hungary as a newly minted Canadian citizen, after the collapse of communism there, Gabor was one of the few close friends that I sought out. I was hugely impressed. Gabor was now heading a company called Banknet, an international joint venture bringing business grade satellite-based Internet service to the country.

When our friend Ferenc was diagnosed with lung cancer, Gabor was distraught. He tried to help Feri with financing an unconventional treatment not covered by insurance. I pitched in, too. It was not enough to save Feri’s life: he passed away shortly thereafter, a loss I still feel more than two decades later.

My last conversation with Gabor was distressing. I don’t really remember the details, but I did learn that he suffered a stroke, and that he was worried that he would be placed under some form of guardianship. Soon thereafter, I lost touch; his phone number, as I recall, was disconnected and Gabor vanished.

Every so often, I looked for him on the Internet, on social media, but to no avail. His name is not uncommon, and moreover, as his last name also doubles as a first name for many, searches bring up far too many false positives. But last night, it occurred to me to search for his name and his original profession: “Dávid Gábor” “matematikus” (mathematician).

Jackpot, if it can be called that. One of the first hits that came up was a page from Hungary’s John von Neumann Computer Society, their information technology history forum, to be specific: a short biography of Gabor, together with his picture.

And from this page I learned that Gabor passed away almost six years ago, on November 10, 2014, at the age of 72.

Well… at least I now know. It has been a privilege knowing you, Gabor, and being able to count you among my friends. I learned a lot from you, and I cherish all those times that we spent working together.

Seventy-five years ago this morning, a false dawn greeted the New Mexico desert near Alamagordo.

At 5:29 AM in the morning, the device informally known as “the gadget” exploded.

“The gadget” was a plutonium bomb with the explosive power of about 22 kilotons of TNT. It was the first nuclear explosion on planet Earth. It marked the beginning of the nuclear era.

I can only imagine what it must have been like, being part of that effort, being present in the pre-dawn hours, back in 1945. The war in Europe just ended. The war in the Pacific was still raging. This was the world’s first high technology war, fought over the horizon, fought with radio waves, and soon, to be fought with nuclear power. Yet there were so many unknowns! The Trinity test was the culmination of years of frantic effort. The outcome was by no means assured, yet the consequences were clear to all: a successful test would mean that war would never be the same. The world would never be the same.

And then, the most surreal of things happens: minutes before the planned detonation, in the pre-dawn darkness, the intercom system picks up a faint signal from a local radio station, and music starts playing. It’s almost as if reality was mimicking the atmosphere of yet-to-be-invented computer games.

When the explosion happened, the only major surprise was that the detonation was much brighter than anyone had expected. Otherwise, things unfolded pretty much as anticipated. “The gadget” worked. Success cleared the way to the deployment of the (as yet untested) simpler uranium bomb to be dropped on Hiroshima three weeks later, followed by the twin of the Trinity gadget, which ended up destroying much of Nagasaki. The human cost was staggering, yet we must not forget that it would have been dwarfed by the costs of a ground invasion of the Japanese home islands. It was a means to shorten the war, a war not started by the United States. No responsible commander-in-chief could have made a decision other than the one Truman made when he approved the use of the weapons against Imperial Japan.

And perhaps the horrors seen in those two cities played a role in creating a world in which the last use of a nuclear weapon in anger occurred nearly 75 years ago, on August 9, 1945. No one would have predicted back then that there will be no nuclear weapons deployed in war in the coming three quarters of a century. Yet here we are, in 2020, struggling with a pandemic, struggling with populism and other forces undermining our world order, yet still largely peaceful, living in a golden age unprecedented in human history.

Perhaps Trinity should serve as a reminder that peace and prosperity can be fragile.

This may not be an all-time record-breaking day according to Environment Canada (supposedly, the peak temperature today at Ottawa Airport was 34.8 C at 2 PM) but it sure is hot.

You could be forgiven if you thought that this measurement is of the body temperature of a COVID-19 patient with mild symptoms, not the outdoor temperature on our balcony, measured in the shade:

As I said… really hot. Praise be to air conditioning.

Long overdue, but I just finished preparing the latest Maxima release, version 5.44.

I am always nervous when I do this. It is one thing to mess with my own projects, it is another thing to mess with a project that is the work of many people and contains code all the way back from the 1960s.

In case anyone doubted that modern birds are descendants of dinosaurs, here is a reminder: the shoebill.

These amazing creatures are apparently quite docile with humans, but eat baby crocodiles for lunch, which they kill by decapitating them.

They really look like survivors of the K-T asteroid impact. They are… I think they are beautiful.

I don’t always like commercial publishers. Some of their textbooks are prohibitively expensive, yet often lacking in quality. (One persistent exception is Dover Publications, who published some of the best textbooks I own, as low-cost paperbacks.)

Last night, however, I was very pleasantly surprised by Springer, who made several hundred textbooks across a range of disciplines available for free, on account of COVID-19.

I did not get greedy. I didn’t download titles indiscriminately. But I did find several titles that are of interest to me, and I gladly took advantage of this opportunity.

Thank you, Springer.

I am one of the maintainers of the Maxima computer algebra system. Maxima’s origins date back to the 1960s, when I was still in kindergarten. I feel very privileged that I can participate in the continuing development of one of the oldest continuously maintained software system in wide use.

It has been a while since I last dug deep into the core of the Maxima system. My LISP skills are admittedly a bit rusty. But a recent change to a core Maxima capability, its ability to create Taylor-series expansions of expressions, broke an important feature of Maxima’s tensor algebra packages, so it needed fixing.

The fix doesn’t amount to much, just a few lines of code:

It did take more than a few minutes though to find the right (I hope) way to implement this fix.

Even so, I had fun. This is the kind of programming that I really, really enjoy doing. Sadly, it’s not the kind of programming for which people usually pay you Big Bucks… Oh well. The fun alone was worth it.

One of the most fortunate moments in my life occurred in the fall of 2005, when I first bumped into John Moffat, a physicist from The Perimeter Institute in Waterloo, Ontario, Canada, when we both attended the first Pioneer Anomaly conference hosted by the International Space Science Institute in Bern, Switzerland.

This chance encounter turned into a 15-year collaboration and friendship. It was, to me, immensely beneficial: I learned a lot from John who, in his long professional career, has met nearly every one of the giants of 20th century physics, even as he made his own considerable contributions to diverse areas ranging from particle physics to gravitation.

In the past decade, John also wrote a few books for a general audience. His latest, The Shadow of the Black Hole, is about to be published; it can already be preordered on Amazon. In their reviews, Greg Landsberg (CERN), Michael Landry (LIGO Hanford) and Neil Cornish (eXtreme Gravity Institute) praise the book. As I was one of John’s early proofreaders, I figured I’ll add my own.

John began working on this manuscript shortly after the announcement by the LIGO project of the first unambiguous direct detection of gravitational waves from a distant cosmic event. This was a momentous discovery, opening a new chapter in the history of astronomy, while at the same time confirming a fundamental prediction of Einstein’s general relativity. Meanwhile, the physics world was waiting with bated breath for another result: the Event Horizon Telescope collaboration’s attempt to image, using a worldwide network of radio telescopes, either the supermassive black hole near the center of our own Milky Way, or the much larger supermassive black hole near the center of the nearby galaxy M87.

Bookended by these two historic discoveries, John’s narrative invites the reader on a journey to understand the nature of black holes, these most enigmatic objects in our universe. The adventure begins in 1784, when the Reverend John Michell, a Cambridge professor, speculated about stars so massive and compact that even light would not be able to escape from its surface. The story progresses to the 20th century, the prediction of black holes by general relativity, and the strange, often counterintuitive results that arise when our knowledge of thermodynamics and quantum physics is applied to these objects. After a brief detour into the realm of science-fiction, John’s account returns to the hard reality of observational science, as he explains how gravitational waves can be detected and how they fit into both the standard theory of gravitation and its proposed extensions or modifications. Finally, John moves on to discuss how the Event Horizon Telescope works and how it was able to create, for the very first time, an actual image of the black hole’s shadow, cast against the “light” (radio waves) from its accretion disk.

John’s writing is entertaining, informative, and a delight to follow as he accompanies the reader on this fantastic journey. True, I am not an unbiased critic. But don’t just take my word for it; read those reviews I mentioned at the beginning of this post, by preeminent physicists. In any case, I wholeheartedly recommend The Shadow of the Black Hole, along with John’s earlier books, to anyone with an interest in physics, especially the physics of black holes.

Heaven knows why I sometimes get confused by the simplest things.

In this case, the conversion between two commonly used cosmological coordinate systems: Comoving coordinates vs. coordinates that are, well, not comoving, in which cosmic expansion is ascribed to time dilation effects instead.

In the standard coordinates that are used to describe the homogeneous, isotropic universe of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, the metric is given by

$$ds^2=dt^2-a^2dR^2,$$

where $$a=a(t)$$ is a function of the time coordinate, and $$R$$ represents the triplet of spatial coordinates: e.g., $$dR^2=dx^2+dy^2+dz^2.$$

I want to transform this using $$R’=aR,$$ i.e., transform away the time-dependent coefficient in front of the spatial term in the metric. The confusion comes because for some reason, I always manage to convince myself that I also have to make the simultaneous replacement $$t’=a^{-1}dt.$$

I do not. This is nonsense. I just need to introduce $$dR’$$. The rest then presents itself automatically:

\begin{align*} R’&=aR,\\ dR&=d(a^{-1}R’)=-a^{-2}\dot{a}R’dt+a^{-1}dR’,\\ ds^2&=dt^2-a^2[-a^{-2}\dot{a}R’dt+a^{-1}dR’]^2\\ &=(1-a^{-2}\dot{a}^2{R’}^2)dt^2+2a^{-1}\dot{a}R’dtdR’-d{R’}^2\\ &=(1-H^2{R’}^2)dt^2+2HR’dtdR’-d{R’}^2, \end{align*}

where $$H=\dot{a}/a$$ as usual.

OK, now that I recorded this here in my blog for posterity, perhaps the next time I need it, I’ll remember where to find it. For instance, the next time I manage to stumble upon one of my old Quora answers that, for five and a half years, advertised my stupidity to the world by presenting an incorrect answer on this topic.

This, incidentally, would serve as a suitable coordinate system representing the reference frame of an observer at the origin. It also demonstrates that such an observer sees an apparent horizon, the cosmological horizon, given by $$1-H^2{R’}^2=0,$$, i.e., $$R’=H^{-1},$$ the distance characterized by the inverse of the Hubble parameter.

So here I am, reading about some trivial yet not-so-trivial probability distributions.

Let’s start with the uniform distribution. Easy-peasy, isn’t it: a random number, between 0 and 1, with an equal probability assigned to any value within this range.

So… what happens if I take two such random numbers and add them? Why, I get a random number between 0 and 2 of course. But the probability distribution will no longer be uniform. There are more ways to get a value in the vicinity of 1 than near 0 or 2.

And what happens if I add three such random numbers? Or four? And so on?

The statistics of this result are captured by the Irwin-Hall distribution, defined as

$$f_{\rm IH}(x,n)=\dfrac{1}{2(n-1)!}\sum\limits_{k=1}^n(-1)^k\begin{pmatrix}n\\k\end{pmatrix}(x-k)^{n-1}{\rm sgn}(x-k).$$

OK, so that’s what happens when we add these uniformly generated random values. What happens when we average them? This, in turn, is captured by the Bates distribution, which, unsurprisingly, is just the Irwin-Hall distribution, scaled by the factor $$n$$:

$$f_{\rm B}(x,n)=\dfrac{n}{2(n-1)!}\sum\limits_{k=1}^n(-1)^k\begin{pmatrix}n\\k\end{pmatrix}(nx-k)^{n-1}{\rm sgn}(nx-k).$$

For what it’s worth, here is the Maxima script to generate the Irwin-Hall plot:

fI(x,n):=1/2/(n-1)!*sum((-1)^k*n!/k!/(n-k)!*(x-k)^(n-1)*signum(x-k),k,0,n);
plot2d([fI(x,1),fI(x,2),fI(x,4),fI(x,8),fI(x,16)],[x,-2,18],[box,false],
[legend,"n=1","n=2","n=4","n=8","n=16"],[y,-0.1,1.1]);

And this one for the Bates plot:

fB(x,n):=n/2/(n-1)!*sum((-1)^k*n!/k!/(n-k)!*(n*x-k)^(n-1)*signum(n*x-k),k,0,n);
plot2d([fB(x,1),fB(x,2),fB(x,4),fB(x,8),fB(x,16)],[x,-0.1,1.1],[box,false],
[legend,"n=1","n=2","n=4","n=8","n=16"],[y,-0.1,5.9]);

Yes, I am still a little bit of a math geek at heart.

My lovely wife, Ildiko, woke up from a dream and asked: If you have a flower with 7 petals and two colors, how many ways can you color the petals of that flower?

Intriguing, isn’t it.

Such a flower shape obviously has rotational symmetry. Just because the flower is rotated by several times a seventh of a revolution, the resulting pattern should not be counted as distinct. So it is not simply calculating what number theorists call the $$n$$-tuple. It is something more subtle.

We can, of course, start counting the possibilities the brute force way. It’s not that difficult for a smaller number of petals, but it does get a little confusing at 6. At 7 petals, it is still something that can be done, but the use of paper-and-pencil is strongly recommended.

So what about the more general case? What if I have $$n$$ petals and $$k$$ colors?

Neither of us could easily deduce an answer, so I went to search the available online literature. For a while, other than finding some interesting posts about cyclic, or circular permutations, I was mostly unsuccessful. In fact, I began to wonder if this one was perhaps one of those embarrassing little problems in combinatorial mathematics that has no known solution and about which the literature remains strangely quiet.

But then I had another idea: By this time, we both calculated the sequence, 2, 3, 4, 6, 8, 14, 20, which is the number of ways flowers with 1, 2, …, 7 petals can be colored using only two colors. Surely, this sequence is known to Google?

Indeed it is. It turns out to be a well-known sequence in the online encyclopedia of integer sequences, A000031. Now I was getting somewhere! What was especially helpful is that the encyclopedia mentioned necklaces. So that’s what this problem set is called! Finding the Mathworld page on necklaces was now easy, along with the corresponding Wikipedia page. I also found an attempt, valiant though only half-successful if anyone is interested in my opinion, to explain the intuition behind this known result:

$$N_k(n)=\frac{1}{n}\sum_{d|n}\phi(d)k^{n/d},$$

where the summation is over all the divisors of $$n$$, and $$\phi(d)$$ is Euler’s totient function, the number of integers between $$1$$ and $$d$$ that are relative prime to $$d$$.

Evil stuff if you asked me. Much as I always liked mathematics, number theory was not my favorite.

In the case of odd primes, such as the number 7 that occurred in Ildiko’s dream, and only two colors, there is, however, a simplified form:

$$N_2(n)=\frac{2^{n-1}-1}{n}+2^{(n-1)/2}+1.$$

Substituting $$n=7$$, we indeed get 20.

Finally, a closely related sequence, A000029, characterizes necklaces that can be turned over, that is to say, the case where we do not count mirror images separately.

Oh, this was fun. It’s not like I didn’t have anything useful to do with my time, but it was nonetheless a delightful distraction. And a good thing to chat about while we were eating a wonderful lunch that Ildiko prepared today.

Our most comprehensive paper yet on the Solar Gravitational Lens is now online.

This was a difficult paper to write, but I think that, in the end, it was well worth the effort.

We are still investigating the spherical Sun (the gravitational field of the real Sun deviates ever so slightly from spherical symmetry, and that can, or rather it will, have measurable effects) and we are still considering a stationary target (as opposed to a planet with changing illumination and surface features) but in this paper, we now cover the entire image formation process, including models of what a telescope sees in the SGL’s focal region, how such observations can be stitched together to form an image, and how that image compares against the inevitable noise due to the low photon count and the bright solar corona.

I just saw the news: Alexei Leonov died.

Leonov was a Soviet cosmonaut. The first man to ever take a spacewalk (which, incidentally, nearly killed him, as did his atmospheric re-entry, which didn’t exactly go as planned either.)

Leonov was also an accomplished artist. Many of his paintings featured space travel. Here is a beautiful picture, from a blog entry by Larry McGlynn, showing Leonov with one of his paintings, in 2004 in Los Angeles.

So Leonov now joins that ever growing list of brave souls from the dawn of the space age who are no longer with us. Rest in peace, Major General Leonov.

I just came across this XKCD comic.

Though I can happily report that so far, I managed to avoid getting hit by a truck, it is a life situation in which I found myself quite a number of times in my life.

In fact, ever since I’ve seen this comic an hour or so ago, I’ve been wondering about the resistor network. Thankfully, in the era of the Internet and Google, puzzles like this won’t keep you awake at night; well-reasoned solutions are readily available.

Anyhow, just in case anyone wonders, the answer is 4/π − 1/2 ohms.

Yesterday, we posted our latest paper on arXiv. Again, it is a paper about the solar gravitational lens.

This time around, our focus was on imaging an extended object, which of course can be trivially modeled as a multitude of point sources.

However, it is a multitude of point sources at a finite distance from the Sun.

This adds a twist. Previously, we modeled light from sources located at infinity: Incident light was in the form of plane waves.

But when the point source is at a finite distance, light from it comes in the form of spherical waves.

Now it is true that at a very large distance from the source, considering only a narrow beam of light, we can approximate those spherical waves as plane waves (paraxial approximation). But it still leaves us with the altered geometry.

But this is where a second observation becomes significant: As we can intuit, and as it is made evident through the use of the eikonal approximation, most of the time we can restrict our focus onto a single ray of light. A ray that, when deflected by the Sun, defines a plane. And the investigation can proceed in this plane.

The image above depicts two such planes, corresponding to the red and the green ray of light.

These rays do meet, however, at the axis of symmetry of the problem, which we call the optical axis. However, in the vicinity of this axis the symmetry of the problem is recovered, and the result no longer depends on the azimuthal angle that defines the plane in question.

To make a long story short, this allows us to reuse our previous results, by introducing the additional angle β, which determines, among other things, the additional distance (compared to parallel rays of light coming from infinity) that these light rays travel before meeting at the optical axis.

This is what our latest paper describes, in full detail.