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I met pi in school. You probably met pi that way too. It is that number used to calculate the circumference of a circle. Pi is shown symbolically as:


Pi is the ratio of the circumference of a circle to its diameter. It is an “irrational number” which means its exact value is inherently unknowable.

Using computers, we have calculated billions of digits of pi, starting with 3.14159265358979323…   –  but no recognizable pattern emerges. So strange. The digits of pi continue to infinity. Does anyone really understand infinity?

Ancient mathematicians did not like irrationality because it didn’t work with the concept of an omniscient God.

Recently I read about another pi connection which is also strange. In 1996, the UK earth scientist Hans-Henrik Stølum published a paper announcing that pi explains the seemingly chaotic paths of rivers in a mathematically predictable pattern.

This is called a river’s sinuosity. By dividing the river’s actual meandering length by the length of the direct line drawn from source to sea.

Of course, some rivers flow pretty straight from source to mouth , so they have small meandering ratios. Some rivers wander all over the place and have high meandering ratios.

But the average meandering ratio of rivers seems to be pi. Good old 3.14.

Albert Einstein used fluid dynamics and chaos theory to show that rivers tend to bend into loops.

If a river has a curve that will generate faster currents on the outer side of the curve. Those currents will cause erosion and so a sharper bend. That will eventually make the loop tighten. I have read that then chaos will eventually cause the river to double back on itself and form a loop in the other direction.

I did some more research on this river connection and found that this claim may not be accurate.

Someone put up a website at one point to crowdsource river data. The site at seems to be dead now. People could put in the coordinates of the mouth and the source of a river, and the length of the river (from Google Maps and Wikipedia probably) to calculate the sinuosity of a river. That study looked at 258 rivers and found an average sinuosity of an un-Pi-like 1.94.

Hmmm. Maybe it is another mathematical constant, like the golden ratio (phi) which we often find in nature. That value is 1.618. Nope.

What about if you look at pi/phi? You get 1.94. Okay, that’s a strange “coincidence.”  Or something more than coincidence?

I need to be careful with all this, because I saw the film titled Pi. I saw this science fiction film when it was released in 1998. It is a difficult film to label. It is surrealist, psychological, thriller, that delves into religion, mysticism, the relationship of the universe to mathematics and number theory. It was written and directed by Darren Aronofsky in his directorial debut.

I read it as a cautionary tale. It is about a genius oddball mathematician, Max, who has been working for a decade trying to decode the numerical pattern beneath ordered chaos. The ordered chaos he studies is the stock market.

Max’s belief that there is some mathematical “code” underlying everything compares in my mind with Einstein trying to find that theory that explains it all. That quest frustrated Einstein through the end of his life.

Beware of that quest.

After decades of research, in September 2015 the twin Laser Interferometer Gravitational-wave Observatory (LIGO) detectors measured ripples in the fabric of spacetime. That ripple is known as gravitational waves. They arrived at the Earth from a cataclysmic event in the distant universe. The new detectors had just been brought into operation for their first observing run when the very clear and strong signal was captured.

Back in 1916, Albert Einstein predicted the existence of gravitational waves. These miniscule ripples in the fabric of spacetime are generated by unfathomably powerful events.

In Black Hole Blues and Other Songs from Outer Space, Janna Levin says that if those ripples and other vibrations could somehow be recorded, we could observe our universe through sound. What might we hear? The hissing of the Big Bang, the songs of collapsing stars, the low rumblings of merging galaxies, the smash of two black holes collapsing into one.

Spacetime takes the concepts of time and three-dimensional space and fuses them together. In classical mechanics – think of Isaac Newton – time is separate from space. In special relativity – think of Einstein – time and space are fused together into a single 4-dimensional “manifold” called spacetime.

Can you really grasp that concept? I think I do, but ask me to explain it and I go blank.

Many things about space and time are at a scale that really is incomprehensible to most of us. Based on the observed signals, the LIGO scientists estimate that the black holes for the event they detected were about 29 and 36 times the mass of the sun. I can’t imagine that. The event took place 1.3 billion years ago. I also can’t imagine that.

For this event, about 3 times the mass of the sun was converted into gravitational waves in a fraction of a second. The peak power output would have been about 50 times that of the whole visible universe. This energy is emitted as a final strong burst of gravitational waves.

This past October, Rainer Weiss, Kip Thorne, and Barry Barish won the Nobel Prize in physics for directly detecting gravitational waves.



“Imagination is more important than knowledge.” ― Albert Einstein

Knowledge versus imagination. Einstein spent the latter part of his life pursuing a “single, all encompassing theory of the universe.”  He wanted to able to describe all of nature’s forces – to explain it all. He didn’t find it.

James Taylor sings in “Secret of Life

Now the thing about time
Is that time isn’t really real.
It’s just your point of view,
How does it feel for you?
Einstein said he
Could never understand it all.
Planets spinning through space,
The smile upon your face,
Welcome to the human race.

That Einstein quote at the top of this article continues “…for knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.” Imagination is often the pathway to increasing knowledge.

It is interesting that astronomy experiments now might test an idea of Einstein’s that he proposed almost exactly a century ago. It has been a longstanding question of why the Universe is expanding at an accelerated rate. Calculations in a new study could help to explain whether dark energy, as required by Einstein’s theory of general relativity, or a revised theory of gravity are responsible.

Einstein wasn’t a big fan of a lot of the physics that came at the end of his life, and would probably not be a fan of string theory.

Brian Greene is a professor of mathematics and physics at Columbia University who is probably best-known to the public for his NOVA television specials. He is one of the best “explainers” of this deep science. He explains string theory and I can understand it – until he stops explaining it and I have to tell someone else what he meant. The idea of minuscule filaments of energy vibrating in eleven dimensions that make up the “fabric of space.. and create every particle and force in the universe” is not easy to understand or accept.

String theory fills in the gaps of Newtonian physics, especially regarding how gravity works, and Einstein’s Unification Theory depends on the existence of extra dimensions, which contain these filaments and some string theorists posit that there are at least eleven dimensions. For all of us used to living in four dimensions, that is tough to imagine.

James Gates is known for work on supersymmetry, supergravity, and superstring theory. When he was asked about Einstein’s statement that “imagination is more important than knowledge,” he said“For a long time in my life, imagination was the world of play. It was reading about astronauts, and monsters, and traveling in galaxies, all of that kind of stuff, invaders from outer space on earth. That was all in the world of the imagination. On the other hand, reality is all about us. And it’s constraining, and it can be painful. But the knowledge we gain is critical for our species to survive.”


Brain Greene on string theory


Photo credit: NASA/JPL-Caltech, D. Figer (Space Telescope Science Institute/Rochester Institute of Technology), E. Churchwell (University of Wisconsin, Madison) and the GLIMPSE Legacy Team

In 1947, eight years before his death, Einstein wrote to a friend that he could not seriously believe in quantum mechanics because “physics should represent a reality in time and space, free from spooky actions at a distance.”Albert Einstein came up with the phrase “spooky action at a distance” because he had a problem with the completeness of quantum mechanics, particularly as it conflicted with his own special theory of relativity.

This is most famously seen in the “EPR paradox” of 1935 named after its inventors Einstein, Boris Podolsky, and Nathan Rosen which concerns how a pair of particles are “strangely” linked.

The newer term for this is “entanglement,” which is when two particles are “so deeply linked that they share the same existence.”

That’s more physics than I can really grasp , and it involves mathematical relationships such as a wavefunction.

We think of space – be it ten feet between two people or thousands of miles – as the medium that separates things. Quantum entanglement makes you question that apparent truth. Entanglement means that quantum connections between two particles can persist even if the two particles are on opposite sides of the universe.

What Einstein found “spooky” was that when these entangled particles become widely separated in space, action upon one of them (such as measurement) immediately influences the other. Distance is irrelevant.

And Einstein found this impossible according to special relativity, so, quantum mechanics must be wrong, or at least incomplete, was his conclusion.

The paradox had a solution, but it came in the decade after Einstein’s death. A physicist at CERN, John Bell,  in 1964 described entanglement as an entirely new kind of phenomenon, which he termed  “nonlocal.”

What I find interesting about this, as it applies more to my non-physics world, is that Bell’s theory is more concerned with the transfer of information. Physicists are still experimenting on this theory, but it fits this information age we live in.

As Brain Greene said on a NOVA episode, quantum entanglement brings to mind voodoo, but the scientific evidence that it exists is overwhelming.

Einstein was trying at the end of his life to find a single theory that wrapped all of this and energy and time into a beautiful package. He didn’t find it. He would not be pleased that modern physicists reject the notion that quantum mechanics requires general relativity to be consistent.



EPR Before EPR: A 1930 Einstein-Bohr Thought Experiment Revisited


The general theory of relativity conceives of gravity as a result of curved space-time. (Public domain image by NASA.)

If non-science people have heard of one modern theory of physics, it is likely to be that of relativity. In 1905, Albert Einstein determined that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum was independent of the motion of all observers. This was the theory of special relativity. Einstein then spent ten years trying to include acceleration in the theory and published his theory of general relativity in 1915. In it, he determined that massive objects cause a distortion in space-time, which is felt as gravity.

One hundred years later, Sarah Howe wrote a sonnet titled “Relativity” for a commission by Britain’s National Poetry Day. It was to be a poem on light. She wrote, paradoxically, about its absence. The poem is about black holes and is dedicated to Stephen Hawking. It begins:

When we wake up brushed by panic in the dark
our pupils grope for the shape of things we know.

That is the light we know in our world of gravity. She is also writing about light at the level of quantum physics where photons behave like particles and also as waves in a mysterious duality. Howe has said of the poem that she was also thinking about how scientist Galileo and poet Milton in their blindness “came to rely on other sorts of eyes.”

Last November was the 100th anniversary of Albert Einstein unveiling the key equations of general relativity, which he did in four lectures at the Prussian Academy of Sciences, and by the end of the month, he had arrived at the ten equations that physicists still use today.

One of his theories was of “spacetime.” Einstein’s theory viewed space and time not as two separate elements, but interwoven. A change in one produces an effect on the other. (His former professor, Hermann Minkowski actually came up with the space-time continuum  but Einstein elaborated on it.

One of the parts of this that you may have heard of is that as the rate of speed goes up, the rate of time must go down and vice versa. For an object moving slowly through space, time is passing quickly and for an object moving at a very high rate of speed, time actually slows down. It is not something that we can observe firsthand, and since we don’t get to travel at anything close to the speed of light, we don’t “go back in time.”

But the theory has been tested many times in experiments sending the most accurate atomic clocks for orbits in rockets, and when they return to Earth the clocks on the rockets are just slightly behind their earthly counterparts.

He also theorized that light curves because gravity pulls at the fabric of spacetime. Einstein thought that curve should be visible during an eclipse, and in 1919, photographs of a solar eclipse proved that the deflection of the sunlight matched Einstein’s prediction.

Isaac Newton had said that gravity was a universal force always pulling on one body on another. We learned that in school and plenty of people know only that about gravity. Einstein argued that there was no “force” of gravity at all.

His concept of space and time is often compared to a stretched fabric or trampoline that can warp and bend because of the presence of massive objects, like our sun. Objects like Earth or us on it move as straight as they can, flowing through curved space-time.

If you are curious about how politics also shaped Einstein’s theory of general relativity, check out this article from

Albert Einstein’s birthday just passed (March 14) and although I have read many books and articles about him, I still discover stories that I hadn’t heard. A birthday post on the brought me a few new stories and led me down the Internet rabbit hole to find out more about them.

It is pretty common knowledge that Albert was slow to begin speaking, and was an unremarkable student. (This always gives hope to other unremarkable students and their parents.)

He did get a teaching degree from the Swiss Federal Polytechnic, but with average grades and a lot of missed classes. If something didn’t interest him, he didn’t work on it. Imagine him in a school today… lazy, unmotivated, possibly ADHD or with a learning disability, a bored but gifted child. He was the only member of his college class not offered a teaching job at the institute.

After two years trying unsuccessfully to find a permanent teaching position, he took a job as a technical assistant at the Swiss patent office in Bern.


Einstein in his office in Berlin. AP IMAGES via

At the patent office, he worked six days a week, eight hours a day at a lectern, reviewing applications of inventions to see if they were worthy of receiving patents.

That sounds worse than school, but he said of the job that “Working on the final formulation of technological patents was a veritable blessing for me. It enforced many-sided thinking and also provided important stimuli to physical thought.” Einstein called the patent office “that worldly cloister where I hatched my most beautiful ideas.”

Skepticism was encouraged. Don’t be taken in by the assumptions of the would-be inventors, he was told. He was a tough grader and surprisingly efficient. He was able to get a day’s work done in just a few hours, which left him the rest of the day to pursue his own ideas.

He had a drawer in his desk which he called his “theoretical physics department” where he kept his notes. Those notes were about physics and also about philosophy.

Always to be a scientist without a laboratory, he conducted his gedankenexperimenten, or thought experiments, about electric light, power, clocks, and electromagnetism standing at his lectern and gazing out the window.

Around 1902, he had advertised himself as a physics tutor to make extra money. A Romanian philosophy student named Maurice Solovine went to Einstein to hire him and they talked for hours. Einstein soon decided this tutor-student relationship would not be as useful as just getting together to talk as peers, and the two expanded to include others. They called themselves the Olympia Academy and met at Einstein’s apartment, where they ate sausages, cheese, and fruit, and debated big ideas.

In his amazing year of 1905, this patent clerk, published four papers that changed the field of physics. The papers were about 1) his particle theory of light 2) determining the size of molecules suspended in liquid 3) and how to determine their motion and 4) special relativity, which included the one equation all students remember relating energy and matter: E=mc².

Then came fame.

In 1913, he finally got that university teaching job. Max Planck and Walther Hermann Nernst came to Berlin and offered him the chance to be, at age 34, the youngest member of the Prussian Academy.

He received many honors, including a rather-belated 1921 Nobel Prize for his early work on the photoelectrical effect. But he really did not make another significant contribution to the ongoing life of the physical sciences. Not that what he had already done wasn’t enough!

He devoted himself, some say that he became obsessed with, a quest to find “a mathematically unified field theory in which the gravitational field and the electromagnetic field are interpreted only as different components or manifestations of the same uniform field.”

He was never comfortable with quantum theory, which was the big news after his own theories were accepted and for the remainder of his lifetime. Quantum theory had too much uncertainty, too many paradoxes or him.

He said in 1912 that “The more successes the quantum theory enjoys, the sillier it looks.” He certainly could not adapt his own theoretical foundation of physics to quantum science.

He wrote that his belief that there was a unified theory of all the fields went all the way back to his childhood fascination with magnets and a sense that “something deeply hidden had to be behind things.”

John Updike tells in a review of Walter Isaacson’s “thorough, comprehensive, affectionate new biography,” Einstein: His Life and Universe, that in 1931, while visiting America for the second time, he and his second wife, Elsa, attended a California séance at the home of  Upton Sinclair. My, that would have been an interesting event to attend!

Einstein probably was skeptical of this other form of “spooky action at a distance” and Mrs. Sinclair challenged his views on science and spirituality. We don’t know what Albert might have replied, but Elsa told their hostess, “You know, my husband has the greatest mind in the world.” Mrs. Sinclair replied, “Yes, I know, but surely he doesn’t know everything.”

On that same trip, Einstein had asked to meet Charlie Chaplin, and they had a brief conversation at the première of Chaplin’s film, City Lights.

Einstein stayed in Berlin until 1932, when the combination of rising Nazism and offers from America impelled him to leave Germany. He never returned to Germany or Europe.

Mercer St., Princeton - by Dmadeo, CC BY-SA 3.0,

112 Mercer St., Princeton – by Dmadeo, CC BY-SA 3.0

Robert A. Millikan, a physicist whose experiments had verified Einstein’s photoelectrical equation, was now the president of Caltech, and he very much wanted Einstein to come to Caltech. Another educator, Abraham Flexner, had worked for the Rockefeller Foundation and now had money donated by the Bamberger department-store family fortune (a store I frequented in Newark, NJ as a child) to create a new place for scholars, and he also courted Einstein.

The new Institute for Advanced Study was to be built next to but not affiliated with Princeton University. Flexner’s offer was accepted by Einstein. He originally intended to split his time between Europe and America and Princeton would be his American home, but WWII would change those plans.

He and Elsa moved there and eventually bought a the modest house at 112 Mercer Street, where Einstein lived until his death, in 1955. Elsa Einstein died in 1936 while living in this house.

Einstein on the porch of his Princeton home in fuzzy slippers – Photo: Gillett GriffinAlbert via

I have written before about the famous anti-quantum quote of Einstein that he apparently used a number of times and that people often point to as evidence of him having religious beliefs. “God does not play dice with the universe.” Less well known is his colleague Niels Bohr’s response: “Einstein, stop telling God what to do!”

Einstein and this “God” is enough of a topic that there are books written about this part of Einstein’s life and philosophy. In his own writing and speaking, Einstein would use terms like “the Almighty” or “the Old One” (der Alte) and yet he maintained that after a brief period of “deep religiousness,” at age 12, he had permanently distanced himself from any organized religion.

Of course, that doesn’t mean he distanced himself from a God. Perhaps, that God was the what he saw the unified theory as being, or it was his way of labeling that “something deeply hidden had to be behind things.”

There is a collection of his brief pieces published as The World As I See It (read online or as a book) in which you will find two paragraphs labeled as  “The Religiousness of Science.”

“You will hardly find one among the profounder sort of scientific minds without a peculiar religious feeling of his own. But it is different from the religion of the naïve man. For the latter God is a being from whose care one hopes to benefit and whose punishment one fears; a sublimation of a feeling similar to that of a child for its father, a being to whom one stands to some extent in a personal relation, however deeply it may be tinged with awe.

But the scientist is possessed by the sense of universal causation. The future, to him, is every whit as necessary and determined as the past. There is nothing divine about morality, it is a purely human affair. His religious feeling takes the form of a rapturous amazement at the harmony of natural law, which reveals an intelligence of such superiority that, compared with it, all the systematic thinking and acting of human beings is an utterly insignificant reflection. This feeling is the guiding principle of his life and work, in so far as he succeeds in keeping himself from the shackles of selfish desire. It is beyond question closely akin to that which has possessed the religious geniuses of all ages.”

Of course, growing up in New Jersey, I had to make a few pilgrimages to his home in Princeton. I spoke to the occupants once when I was freshman at Rutgers and visiting nearby (and yet so far away) Princeton University. It is not a museum and there isn’t anything to really see other than the house, and to imagine him sitting on the porch or wandering inside doing thought experiments and walking and riding his bike around the neighborhood.

People that actually went there and talked to Einstein have written about that house. One book focuses on a meeting there with Albert,  Bertrand Russell (logician, philosopher, humanist), Wolfgang Pauli (physicist) and Kurt Godel (another groundbreaking logician). Too bad we don’t have recordings or transcripts of these talks – and of those very early talks with his Olympia Academy group.

People view Einstein as this kind, grandfatherly, pacifist, eccentric genius, but I also see a sadness to his life.By the time I had made that college visit to his Princeton home, I had read enough to know that he was a terrible father and husband. That really bothers me.

I always find it hard to separate the person from their work. The writer or artist or scientist who was cruel, who cheated on spouses, mistreated family, destroyed him or herself with drugs or booze and perhaps ended as a suicide – I have trouble unifying those things with any great work  they accomplished.

Max Born said about Einstein, “For all his kindness, sociability and love of humanity, he was nevertheless totally detached from his environment and the human beings in it.”

When an old friend he knew from his youthful Zurich days, Michele Besso, died, he wrote a letter of condolence to the Besso family. He said that Michele’s most admirable trait had been being able to live harmoniously with a woman – “an undertaking in which I twice failed rather miserably.”

He also wrote that “Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion.”

Einstein died one month and 3 days after his friend, on April 18, 1955.


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