Science – Page 7 – Spinor Info

Number theory, petals and necklaces

May 022020

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:

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

Unfortunately, this form is for “free necklaces”, which treats mirror images as equivalent. For $n<6$ it makes no difference, but substituting $n=7$, we get 18, not 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.

A difficult paper

Astronomy, Personal, Physics No Responses »

Feb 172020

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.

Good-bye, Alexei Leonov

History, Space No Responses »

Oct 112019

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.

Nerd sniping

Humor, Physics No Responses »

Oct 112019

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.

Imaging extended sources with the solar gravitational lens

Astronomy, Personal, Physics, Space No Responses »

Aug 072019

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.

Was Apollo 11 not PC enough?

History, Politics, Space No Responses »

Jul 192019

The world is celebrating the 50th anniversary of one of the most momentous events in human history: the first time a human being set foot on another celestial body.

It is also a triumph of American ingenuity. Just as Jules Verne predicted a century earlier, it was America’s can-do spirit that made the Moon landing, Armstrong’s “one small step” possible.

And today, just like 50 years ago, their success was celebrated around the world, by people of all nationality, religion, gender or ethnicity.

But that’s not good enough for some New York Times columnists.

Instead of celebrating the Moon landing, Mary Robinette Kowal complains about the gender bias that still exists in the space program. Because, as we learn from her article, this evil male chauvinistic space program was “designed by men, for men”. Because, you know, men sweat in different areas of their body and all. Even in the office, temperatures are set for men, which leaves women carrying sweaters.

Sophie Pinkham goes further. Instead of celebrating America’s success on July 20, 1969, Pinkham goes on to praise the Soviet space program in a tone that might have been rejected even by the editors of Pravda in 1969 as too over-the-top. Because unlike America, the Soviets put the first woman in space! Their commitment to equality did not stop there: They also sent the first Asian man and the first black man into orbit. Because, we are told, “under socialism, a person of even the humblest origins could make it all the way up.”

Just to be clear, I am not blind to gender bias. We may have come a long way since the 1960s, but full gender equality has not yet been achieved anywhere: not in the US, not in Canada, not even in places like Iceland. And racism in America remains a palpable, everyday reality. Back in 1969, things were a lot worse.

But to pick the 50th anniversary of an event that, even back in the turbulent 1960s, had the power to unify humanity, to launch such petty rants? That is simply disgraceful. Or, as the New York Post described it, obscene.

The New York Post also makes mention of one of the female pioneers of the US space program, Margaret Hamilton, whose work was instrumental in making the Apollo landings possible. Yet somehow, neither Pinkham nor Kowal found it in their hearts to mention her name.

I have to wonder: Are columnists like Pinkham or Kowal secretly rooting for Donald Trump? Because they certainly seem to be doing their darnedest best to alienate as many voters as possible, from what appears to be an increasingly bitter, intolerant, ideological agenda on the American political left.

Moon shot

History, Space No Responses »

Jul 162019

Fifty years ago today, fifty years ago this very hour in fact, at 9:32 AM EDT on July 16, 2019, Apollo 11 was launched.

Moonbound Apollo 11 clears the launch tower. NASA photo

And thus began a journey that, arguably, remains the greatest adventure in human history to date.

I was six years old in 1969, hooked on the novels of Jules Verne. With Apollo 11, Verne’s bold imagination became the reality of the day.

Galileo down

Space, Technology, Transportation No Responses »

Jul 152019

Galileo is the world’s third global satellite navigation system, built by the European Union, operating in parallel with the American GPS system and Russia’s GLONASS. It has been partially operational since 2016, with a full constellation if satellites expected to enter service this year.

But as of early Monday, July 15, Galileo has been down for nearly four days, completely inoperative in fact:

As of the time of this writing, no explanation is being offered, other than one article mentioning an unspecified issue with Galileo’s ground-based infrastructure.

It really is difficult to comprehend how such a failure can occur.

It is even more difficult to comprehend the silence, the lack of updates, explanations, or any information about the expected recovery.

Maxima 5.43

Computer Algebra, Personal, Programming No Responses »

Jun 052019

And before I forget: Last week, wearing my release manager hat I successfully created a new version of Maxima, the open-source computer algebra system. As a result, Maxima is again named one of SourceForge’s projects of the week, for the week of June 10.

The release turned out to be more of an uphill battle than I anticipated, but in the end, I think everything went glitch-free.

Others have since created installers for different platforms, including Windows.

And I keep promising myself that when I grow up, I will one day understand exactly what git does and how it works, instead of just blindly following arcane scripts…

Is everything in physics knowable?

Astronomy, Physics No Responses »

May 312019

Here is a thought that has been bothering me for some time.

We live in a universe that is subject to accelerating expansion. Galaxies that are not bound gravitationally to our Local Group will ultimately vanish from sight, accelerating away until the combination of distance and increasing redshift will make their light undetectable by any imaginable instrument.

Similarly, accelerating expansion means that there will be a time in the very distant future when the cosmic microwave background radiation itself will become completely undetectable by any conceivable technological means.

In this very distant future, the Local Group of galaxies will have merged already into a giant elliptical galaxy. Much of this future galaxy will be dark, as most stars would have run out of fuel already.

But there will still be light. Stars would still occasionally form. Some dwarf stars will continue to shine for trillions of years, using their available fuel at a very slow rate.

Which means that civilizations might still emerge, even in this unimaginably distant future.

And when they do, what will they see?

They will see themselves as living in an “island universe” in an otherwise empty, static cosmos. In short, precisely the kind of cosmos envisioned by many astronomers in the early 1920s, when it was still popular to think of the Milky Way as just such an island universe, not yet recognizing that many of the “spiral nebulae” seen through telescopes are in fact distant galaxies just as large, if not larger, than the Milky Way.

But these future civilizations will see no such nebulae. There will be no galaxies beyond their “island universe”. No microwave background either. In fact, no sign whatsoever that their universe is evolving, changing with time.

So what would a scientifically advanced future civilization conclude? Surely they would still discover general relativity. But would they believe its predictions of an expanding cosmos, despite the complete lack of evidence? Or would they see that prediction as a failure of the theory, which must be remedied?

In short, how would they ever come into possession of the knowledge that their universe was once young, dense, and full of galaxies, not to mention background radiation?

My guess is that they won’t. They will have no observational evidence, and their theories will reflect what they actually do see (a static, unchanging island universe floating in infinite, empty space).

Which raises the rather unnerving, unpleasant question: To what extent exist already features in our universe that are similarly unknowable, as they can no longer be detected by any conceivable instrumentation? Is it, in fact, possible to fully understand the physics of the universe, or are we already doomed to never being able to develop a full picture?

I find this question surprisingly unnerving and depressing.

Extinction club

Nature, Politics No Responses »

May 082019

The Globe and Mail managed to publish today one of the saddest editorial cartoons I ever saw:

There really is nothing that I can add. The cartoon speaks for itself.

Patreon

Internet, Personal, Physics No Responses »

Apr 092019

My research is unsupported. That is to say, with the exception of a few conference invitations when my travel costs were covered, I never received a penny for my research on the Pioneer Anomaly and my other research efforts.

Which is fine, I do it for fun after all. Still, in this day and age of crowdfunding, I couldn’t say no to the possibility that others, who find my efforts valuable, might choose to contribute.

Hence my launching of a Patreon page. I hope it is well-received. I have zero experience with crowdfunding, so this really is a first for me. Wish me luck.

It’s cold in Toronto, but…

Television, Weather and Climate No Responses »

Mar 242019

Yes, it can get cold in Toronto. Usually not as cold as Ottawa, but winters can still be pretty brutal.

But this brutal, this late in the season?

Yes, according to The Weather Network earlier this morning, the temperature overnight will plummet to -59 degrees Centigrade next weekend.

Yikes. Where is global warming when we need it?

Some common, naive misunderstandings in cosmology

Physics No Responses »

Jan 162019

I run across this often. Well-meaning folks who read introductory-level texts or saw a few educational videos about physical cosmology, suddenly discovering something seemingly profound.

And then, instead of asking themselves why, if it is so easy to stumble upon these results, they haven’t been published already by others, they go ahead and make outlandish claims. (Claims that sometimes land in my Inbox, unsolicited.)

Let me explain what I am talking about.

As it is well known, the rate of expansion of the cosmos is governed by the famous Hubble parameter: $H\sim 70~{\rm km}/{\rm s}/{\rm Mpc}$. That is to say, two galaxies that are 1 megaparsec (Mpc, about 3 million light years) apart will be flying away from each other at a rate of 70 kilometers a second.

It is possible to convert megaparsecs (a unit of length) into kilometers (another unit of length), so that the lengths cancel out in the definition of $H$, and we are left with $H\sim 2.2\times 10^{-18}~{\rm s}^{-1}$, which is one divided by about 14 billion years. In other words, the Hubble parameter is just the inverse of the age of the universe. (It would be exactly the inverse of the age of the universe if the rate of cosmic expansion was constant. It isn’t, but the fact that the expansion was slowing down for the first 9 billion years or so and has been accelerating since kind of averages things out.)

And this, then, leads to the following naive arithmetic. First, given the age of the universe and the speed of light, we can find out the “radius” of the observable universe:

$$a=\dfrac{c}{H},$$

or about 14 billion light years. Inverting this equation, we also get $H=c/a$.

But the expansion of the cosmos is governed by another equation, the first so-called Friedmann equation, which says that

$$H^2=\dfrac{8\pi G\rho}{3}.$$

Here, \rho is the density of the universe. The mass within the visible universe, then, is calculated as usual, just using the volume of a sphere of radius $a$:

$$M=\dfrac{4\pi a^3}{3}\rho.$$

Putting this expression and the expression for $H$ back into the Friedmann equation, we get the following:

$$a=\dfrac{2GM}{c^2}.$$

But this is just the Schwarzschild radius associated with the mass of the visible universe! Surely, we just discovered something profound here! Perhaps the universe is a black hole!

Well… not exactly. The fact that we got the Schwarzschild radius is no coincidence. The Friedmann equations are, after all, just Einstein’s field equations in disguise, i.e., the exact same equations that yield the formula for the Schwarzschild radius.

Still, the two solutions are qualitatively different. The universe cannot be the interior of a black hole’s event horizon. A black hole is characterized by an unavoidable future singularity, whereas our expanding universe is characterized by a past singularity. At best, the universe may be a time-reversed black hole, i.e., a “white hole”, but even that is dubious. The Schwarzschild solution, after all, is a vacuum solution of Einstein’s field equations, wereas the Friedmann equations describe a matter-filled universe. Nor is there a physical event horizon: the “visible universe” is an observer-dependent concept, and two observers in relative motion or even two observers some distance apart, will not see the same visible universe.

Nonetheless, these ideas, memes perhaps, show up regularly, in manuscripts submitted to journals of dubious quality, appearing in self-published books, or on the alternative manuscript archive viXra. And there are further variations on the theme. For instance, the so-called Planck power, divided by the Hubble parameter, yields $2Mc^2$, i.e., twice the mass-energy in the observable universe. This coincidence is especially puzzling to those who work it out numerically, and thus remain oblivious to the fact that the Planck power is one of those Planck units that does not actually contain the Planck constant in its definition, only $c$ and $G$. People have also been fooling around with various factors of $2$, $\tfrac{1}{2}$ or $\ln 2$, often based on dodgy information content arguments, coming up with numerical ratios that supposedly replicate the matter, dark matter, and dark energy content.

My thoughts on the Apollo 8 Genesis reading, published

History, Personal, Religion, Space No Responses »

Jan 062019

I almost forgot: a couple of months ago, I was interviewed over the telephone by a journalist who wanted to know my thoughts about one of my favorite moments in manned space exploration: The Apollo 8 “Genesis” moment, the reading of the opening verses of the Old Testament, on Christmas Day, 1968, by the astronauts of Apollo 8 as their spacecraft emerged from behind the Moon.

Today, something reminded me of this interview and I did a quick search. Lo and behold, there it is: My words, printed in The Boston Globe on December 23, 2018:

“It was a beautiful moment, and Genesis is part of our Western cultural heritage,” said Viktor Toth, an atheist and a senior research fellow at Carleton University in Ottawa, Canada, who played the lead role in the investigation of the Pioneer Anomaly, the mysterious acceleration of the Pioneer 10 and 11 spacecrafts in deep space. “This was an awe-inspiring thing: Human beings for the first time cut off from the Earth, and then they reemerged and saw the Earth again. The message was entirely appropriate.”

Though shortened, this pretty accurately reflects what I actually said during that roughly 10-minute conversation with the journalist.

Most distant visited object

Music, Space No Responses »

Jan 042019

Even as China was celebrating the first successful landing of a spacecraft on the far side of the Moon, NASA’s New Horizons continued to radio back data from its New Year’s Day encounter with Ultima Thule: a strange, “contact binary” asteroid in the Kuiper belt, far beyond Pluto.

Ultima Thule will remain, for the foreseeable future, the most distant celestial object visited by spacecraft. While there is the odd chance that New Horizons may find another target within range (as determined by the on-board fuel available, which limits trajectory corrections, and the aging of its nuclear power source that provides electricity on board), chances are it won’t happen, and it won’t be until another deep space probe is launched, quite possibly decades from now, before we get a chance to see a world as distant as Ultima Thule.

Another piece of news from the New Horizons project is that so far, the probe found no moon orbiting Ultima Thule. No Moon At All.

https://youtu.be/4GxvSALRLzM

The dark side of the Moon

Space No Responses »

Jan 032019

OK, the far side of the Moon is not really dark, but it is kind of hard to see. But now, from the department of unqualified good news: China successfully landed its spacecraft, Chang’e 4 (named after the Chinese Moon goddess), on the dark side of the Moon, and it has already sent us back some pictures.

This is big. Really big. To make it happen, China first had to launch a lunar orbiter, Queqiao (“Magpie bridge”), in order to maintain communication with the lander. And being on the far side of the Moon, the lander is completely shielded from radio signals from the Earth, which means an unprecedented opportunity to study radio signals of extrasolar origin.

Chang’e 4 also carried a rover, Yutu-2, which has since been deployed.

By any reasonable measure, this is a huge success for China’s space program, and for humanity overall. Hopefully, both lander and rover will remain operational and able to fulfill their scientific objectives.

Connections

Aviation, Health, History, Physics, Politics No Responses »

Jan 012019

Today, I answered a question on Quora about the nature of $c$, the speed of light, as it appears in the one equation everyone knows, $E=mc^2.$

I explained that it is best viewed as a conversion factor between our units of length and time. These units are accidents of history. There is nothing fundamental in Nature about one ten millionth the distance from the poles to the equator of the Earth (the original definition of the meter) or about one 86,400th the length of the Earth’s mean solar day. These units are what they are, in part, because we learned to measure length and time long before we learned that they are aspects of the same thing, spacetime.

And nothing stops us from using units such as light-seconds and seconds to measure space and time; in such units, the value of the speed of light would be just 1, and consequently, it could be dropped from equations altogether. This is precisely what theoretical physicists often do.

But then… I commented that something very similar takes place in aviation, where different units are used to measure horizontal distance (nautical miles, nmi) and altitude (feet, ft). So if you were to calculate the kinetic energy of an airplane (measuring its speed in nmi/s) and its potential energy (measuring the altitude, as well as the gravitational acceleration, in ft) you would need the ft/nmi conversion factor of 6076.12, squared, to convert between the two resulting units of energy.

As I was writing this answer, though, I stumbled upon a blog entry that discussed the crazy, mixed up units of measure still in use worldwide in aviation. Furlongs per fortnight may pretty much be the only unit that is not used, as just about every other unit of measure pops up, confusing poor pilots everywhere: Meters, feet, kilometers, nautical miles, statute miles, kilograms, pounds, millibars, hectopascals, inches of mercury… you name it, it’s there.

Part of the reason, of course, is the fact that America, alone among industrialized nations, managed to stick to its archaic system of measurements. Which is another historical accident, really. A lot had to do with the timing: metric transition was supposed to take place in the 1970s, governed by a presidential executive order signed by Gerald Ford. But the American economy was in a downturn, many Americans felt the nation under siege, the customary units worked well, and there was a conservative-populist pushback against the metric system… so by 1982, Ronald Reagan disbanded the Metric Board and the transition to metric was officially over. (Or not. The metric system continues to gain ground, whether it is used to measure bullets or Aspirin, soft drinks or street drugs.)

Yet another example similar to the metric system is the historical accident that created the employer-funded healthcare system in the United States that American continue to cling to, even as most (all?) other advanced industrial nations transitioned to something more modern, some variant of a single-payer universal healthcare system. It happened in the 1920s, when a Texas hospital managed to strike a deal with public school teachers in Dallas: For 50 cents a month, the hospital picked up the tab of their hospital visits. This arrangement became very popular during the Great Depression when hospitals lost patients who could not afford their hospital care anymore. The idea came to be known as Blue Cross. And that’s how the modern American healthcare system was born.

As I was reading this chain of Web articles, taking me on a tour from Einstein’s $E=mc^2$ to employer-funded healthcare in America, I was reminded of a 40-year old British TV series, Connections, created by science historian James Burke. Burke found similar, often uncanny connections between seemingly unrelated topics in history, particularly the history of science and technology.

Merry Christmas

Personal, Religion, Space No Responses »

Dec 242018

A quote from 50 years ago is the most appropriate one tonight, considering that our world is just as troubled as the world of 1968:

“And from the crew of Apollo 8, we close with good night, good luck, a Merry Christmas – and God bless all of you, all of you on the good Earth.“

Early winter

Canada, Local Ottawa, Personal, Weather and Climate No Responses »

Nov 162018

I know, I know, it’s not for the weather that we love Canada.

Still… plunging into winter in the middle of November?

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