EYE TO EYE

What makes i^i interesting is the four real numbers it generates. (The numbers are +.2078… , -.2078… , +4.8104… , and -4.8104… .)

Can anyone give a geometric reason why an imaginary number raised to the power of an imaginary number generates four real numbers and no imaginary ones? What does \sqrt{-1}^{\sqrt{-1}} even mean? Is there anyone who can visualize a reason why the answers make sense? Are all the answers even correct? Or is only one correct, as any calculator that can do the calculation will tell you?

Complex numbers are two-dimensional numbers that are made by raising the number “e” to the power of an imaginary number—called “i”—times an angle in radians. Complex numbers lie on a circle in the complex number plane. Unless “e” is preceded by a number that stretches or shrinks it, the numbers lie on a unit circle like the one in the picture. Recall that “i” is the square root of minus one. When “i” is raised to the power of “i”, the result collapses onto the real number line—in one of four possible places. Which one? The numbers aren’t found on the unit circle. The process can be demonstrated mathematically, but any physical intuition about why imaginary numbers with imaginary exponents behave the way they do can be elusive.

Abstract math that hides no model that anyone can visualize makes results startling, even unnerving. It’s a lot like the quantum mechanics of entanglement or the physical meaning of gravity. They can be mathematically described and their effects accurately predicted, but no one can explain why.   

Mathematics alone can sometimes describe (or at least approximate) realities of the universe and how it seems to work, but as often as not when humans dive deep into the abyss of ultimate knowledge, math is unable to provide a picture that anyone can understand. How can that be? Things seem to happen that cannot be thought about except by playing around with numbers and being taken by surprise. Intuition is difficult, if not impossible.

Here is the solution of  i^i. Perhaps clues exist in the math that I’ve overlooked. If a model exists in the mind of a reader somewhere, I hope they will share it with me.


(1)       i^i = e^{\ln(i^i)} = e^{i\ln(i)} = cos (ln i) + i sin (ln i)

Now:  e^{i\frac{\pi}{2}} = i

Also:  ln ^{(e^{i\frac{\pi}{2})}} = ln i

Therefore:     ln i =  i (\frac{\pi}{2})

By substitution into line (1):    i^i = cos (i\frac{\pi}{2}) + i sin (i\frac{\pi}{2})

By half angle formulas:             i^i  = (\sqrt\frac{1 + cos (i\pi)}{2}) + i (\sqrt\frac{1 - cos (i\pi)}{2})

Convert 2nd term i to  \sqrt -1 :     i^i  = (\sqrt\frac{1 + cos (i\pi)}{2}) + \sqrt -1 (\sqrt\frac{1 - cos (i\pi)}{2})

(2)     Simplify the 2nd term:     i^i  = (\sqrt\frac{1 + cos (i\pi)}{2}) + (\sqrt\frac{cos (i\pi)-1}{2})

Euler’s cosine identity is:   cos θ =  \frac{e^{i\theta} + e^{-i\theta}}{2}

Therefore:                          cos (iπ) =  \frac{e^{i(i\pi)} + e^{-i(i\pi)}}{2}

(3)     Simplifying:               cos (iπ) =  \frac{e^{-\pi} + e^{\pi}}{2}

Substitute line (3) into line (2) and simplify:
i^i  = \sqrt{{\frac{1}{2} + \frac{e^{-\pi} + e^{\pi}}{4}}} + \sqrt{{\frac{e^{-\pi} + e^{\pi}}{4}} - \frac{1}{2}}

Now it’s just a matter of pulling out an old calculator and punching the keys.

e^{-\pi} = .043214;  e^{\pi} = 23.140693. I rounded off both numbers, because they seem to go on forever like π and “e”; they probably are irrational, because they don’t seem to be formed from ratios of whole numbers. Using these values will enable anyone to compute {i^i} who has a calculator with a square root key.

When square roots are calculated the answers can be positive or negative. Two negatives make a positive, right? So do two positives. So doing the math gives four numbers. See if your numbers match mine: .2078… , -.2078… , 4.1084… , and -4.1084… .  

I don’t know why. The answers aren’t intuitive. Who would guess that imaginary numbers raised to powers of imaginary numbers yield real numbers?—not a solitary number like anyone might expect, but four. Pick one. In nature a unique answer can be arbitrary—determined by chance, most likely.

In this case, no.

It feels to me like the imaginary fairies flying around in complex space are destined to collapse onto the real number line for no good reason, except that the math says they must collapse (maybe from exhaustion?) in at least one of four places. Can anyone make sense of it? 

The ln i is well known. It is i\frac{\pi}{2}, which equals (1.57078… i ). The ln of i^i can be rewritten by the rules of logarithms as i ln i, which is i times (1.57078…i ), which equals -1.57078… (a real number). Right? The ln of the correct answer must equal this number. Only one of the four results listed above has the right ln value: .2078… .

It seems odd that a set of equations I know to be sound should return a set of results from which only one can be validated by back-checking. Maybe there is something esoteric and arcane in the mathematics of logarithms that I missed during my education along the way.

Then again square roots can be messy; there are two square roots in the final equation, each of which can be evaluated as positive or negative. Together they produce four possible answers, but just one result is the right one.

Adding the four numbers is kind of interesting. They sum to zero. That is so like the way the universe seems to work, isn’t it? When everything is added up, physicists like Stephen Hawking claim, there’s really nothing here. Everything is imaginary. Some philosophers agree: everything that is real is at its core imaginary.   

Are there clues in the pictures and models of complex number space that would ever make anyone think? Sure, I totally get it. Yeah, I’ve got this. Real numbers cascading out of imaginary powers of imaginary numbers make perfect sense—like snowflakes falling from a dark sky.

A mathematician told me, Rotating and scaling is all it is. The base must be the imaginary “i” alone; “i” is the key that unlocks everything. The power of the key can be any imaginary number at all; “i” is why the result of every imaginary power of “i” becomes real. 

The explanation calms me; but it seems somehow incomplete; it’s missing something; in my gut I feel like it can’t be entirely right, though it purports to persuade what the math insists is truth. We are being asked to believe, for now at least, and move on.

Billy Lee 

What is e exp (-i π) ?

What is e^{-i\pi} ?

I posted a long answer on Quora.com where it sort of didn’t do well. Answers given by others were much shorter but they seemed, at least to me, to lack geometric insights. After two days my answer was ranked as the most read, but for some reason no one upvoted it. It did receive a few positive replies, though.

I can’t help but believe that there must be nerds in cyberspace who might enjoy my answer. Why not post it on my blog? Maybe someday one of my grandkids will get interested in math and read it. Who knows?

I added some graphics here that wouldn’t post on Quora. The site lacks graphics functionality, apparently. Either that or I’m too dense to figure out how to insert something.


NOTE: 24 Oct 2017: Today Billy Lee finally figured out how to post media on Quora.com. After admonishment and chastisement by the staff, he added a pic and a GIF to his “answer.” Unfortunately, Billy Lee never did get the GIF to run right, so he took it down. Otherwise, Billy Lee is doing good. He really is. THE EDITORIAL BOARD


Anyway, this pic and a working GIF below make a big difference in understanding, I hope. And anyone who doesn’t understand something can always click on a link for more information. (No one ever clicks on links, but I spend a lot of time adding them—maybe so I can click on them myself during times when remembering my name or where I live seems to lie just outside my intellectual skill-set.)

Here is the drawing I added and the answer:

This diagram is excellent but contains a mystery point not on the unit circle—{i^i}. The point is shown at .2078… on the real number line. It has 3 other values on the horizontal line that are not shown: -.2078… and +/- 4.8104… . Its solution can be reviewed in this video link, which Billy Lee has carefully reviewed and confirmed to be sound. An imaginary number raised to the power of an imaginary number yields a result that is four real numbers. How can that be? It’s something to ponder; something to think about. The Editorial Board 

What is e^{-i\pi} ?

The expression evaluates to minus one; the answer is (-1). Why?

Numbers like these are called complex numbers. They are two-dimensional numbers that can be drawn on graph-paper instead of on a one-dimensional number line, like the counting numbers. They are used to analyse wave functions—i.e. phenomenon that are repetitive—like alternating current in the field of electrical engineering, for example.

“e” is a number that cannot be written as a fraction (or a ratio of whole numbers). It is an irrational number (like π, for instance). It can be approximated by adding up an arbitrary number of terms in a certain infinite series to reach whatever level of precision one wants. To work with “e” in practical problems, it must be rounded off to some convenient number of decimal places.

Punch “e” into a calculator and it returns the value 2.7182…. The beauty of working with “e” is that derivatives and integrals of functions based on exponential powers of “e” are easy to calculate.

What is “e” raised to the power of (-iπ) ?

A wonderful feature of the mathematics of complex numbers is that all the values of expressions that involve the number “e” raised to the power of “i” times anything conveniently lie on the edge (or perimeter) of a circle of radius one. This happy fact makes understanding the expressions easy.

I should mention that any point in the complex plane can be reached by adding a number in front of e^{i\theta} to stretch or shrink the unit circle of values. We aren’t going to go there. In this essay “e” is always preceded by the number one, which by the familiar convention is never shown.

The number next to the letter “i” is simply the angle in radians where the answer lies on the circle. Draw a line from the center of the circle at the angle specified in the exponent of “e” and it will intersect the circle at the value of the expression. What could be easier?

For the particular question we are struggling to answer, the number in the exponent next to “i” is (-π), correct?

“π radians” is 180 degrees, right? The minus sign is simply a direction indicator that says to move clockwise around the unit circle—instead of counter-clockwise if the sign was positive.

After drawing a unit circle on graph paper, place your pencil at (1 + 0i)—located at zero radians (or zero degrees)—-and trace 180 degrees clockwise around the circle. Remember that the circle’s radius is one and its center is located at zero, which in two dimensional, complex space is (0 + 0i). You will end up at the value (-1 + 0i) on the opposite side of the circle, which is the answer, by the way.

[Trace the diagram several paragraphs above with your finger if you don’t have graph paper and a pencil. No worries.]

Notice that +π radians takes you to the same place. The value you land on is (-1 + 0i), which is -1. The answer is minus one.

Imagine that the number next to “i” is (π/2) radians. That’s 90 degrees, agreed? The sign is positive, so trace the circle 90 degrees counter-clockwise. You end at (0 + i), which is straight up. “i” in this case is a distance of one unit upward from the horizontal number line, so the number is written (0 + i)—-zero distance in the horizontal direction and “plus one” distance in the “i” (or vertical) direction.

So, the “i” in the exponent of “e” says to “look here” to find the angle where the value of the answer lies on the unit circle; on the other hand, the “i” in the rectangular coordinates of a two-dimensional number like (0 + i) says “look here” to find the vertical distance above or below the horizontal number line.

When evaluating “e” raised to the power of “i” times anything, the angle next to “i”—call it “θ”—can be transformed into rectangular coordinates by using this expression: [cos(θ) + i sin(θ)].

For example: say that the exponent of “e” is i(π/3). (π/3) radians is 60 degrees, right? The cosine of 60 degrees is 0.5 and the sine of 60 degrees is .866….

So the value of “e” raised to the power of i(π/3) is by substitution (0.5 + .866… i ). It is a two-dimensional number. And it lies on the unit circle.

The bigger the exponent on “e” the more times someone will have to trace around the circle to land at the answer. But they never leave the circle. The answer is always found on the circle between 0 and 2π radians (or 0 and 360 degrees) no matter how large the exponent.

It’s why these expressions involving “e” and “i” are ideal for working with repetitive, sinusoidal (wave-like) phenomenon.

In this essay Billy Lee uses θ in place of the Greek letter φ shown in this GIF. “r” equals “one” in a unit circle. The Editorial Board

Some readers might wonder about what radians are. A radian is the radius of a circle, which can be lifted and bent to fit perfectly on the edge of the circle. It takes a little more than three radius pieces (3.14159… to be more precise) to wrap from zero degrees to half-way around any circle of any size. This number—3.14159…—is the number called “π”. 2π radians are a little bit more than six-and-a-quarter radians (radius pieces) and will completely span the perimeter (or circumference) of a circle.

A radian is about 57.3 degrees of arc. Multiply 3.1416 by 57.3 to see how close to 180 degrees it is. I get 180.01… . The result is really close to 180 degrees considering that both numbers are irrational and rounded off to only a few decimal places.

One of the rules of working with complex numbers is this: multiplying any number by “i” rotates that number by 90 degrees. The number “i” is always located at 90 degrees on the unit circle by definition, right? By the rule, multiplying “i” by “i” rotates it another 90 degrees counter-clockwise, which moves it to 180 degrees on the circle.

180 degrees on the unit circle is the point (-1 + 0i), which is minus one, right?

So yes, absolutely, “i” multiplied by “i” is equal to -1. It follows that the square root of minus one must be “i”. Thought of in this way, the square root of a minus one isn’t mysterious.

It is helpful to think of complex numbers as two dimensional numbers with real and imaginary components. There is nothing imaginary, though, about the vertical component of a two-dimensional number.

The people who came up with these numbers thought they were imagining things. The concept of two-dimensional numbers was too radical at the time for anyone to believe that numbers could exist on a plane as naturally as they do on a simple number line of one dimension.

Visit my website for an illustrated and concise explanation of these topics titled, “What is Math?” It is easily found by making a simple entry into the search box on theBillyLeePontificator.com .

Comments
J.G.
Thanks, I was confused about the -i part, so it’s not – sqrt(-1)? My calculator gives me an answer of -1.

S.X.
Good answer, I approve 🙂

B.L.
Thanks!

Billy Lee

Fine-Structure Constant

fine-structure-constant-formulaWhat is the fine-structure constant?

Many smart physicists wonder about it; some obsess over it; a few have gone mad. Physicists like the late Richard Feynman said that it’s not something any human can or will ever understand; it’s a rabbit-hole that quantum physicists must stand beside and peer into to do their work; but for heaven’s sake don’t rappel into its depths. No one who does has ever returned and talked sense about it.

I’m a Pontificator, not a scientist. I hope I don’t start to regret writing this essay. I hope I don’t make an ass of myself as I dare to go where angels fear to tread.

My plan is to explain a mystery of existence that can’t be explained—even to people who have math skills, which I am certain most of my readers don’t. Lack of skills should not trouble anyone, because if anyone has them, they won’t understand my explanation anyway. 

My destiny is failure. I don’t care. My promise, as always, is accuracy. If people point out errors, I fix them. I write to understand; to discover and learn.

My recommendation to readers is to take a dose of whatever medicine calms their nerves; to swallow whatever stimulant might ignite electrical fires in their brains; to inhale, if necessary, doctor-prescribed drugs to amplify conscious experience and broaden their view of the cosmos. Take a trip with me; let me guide you. When we’re done, you will know nothing about the fine-structure constant except its value and a few ways curious people think about it.

Oh yes, we’re going to rappel into the depths of the rabbit-hole, I most certainly assure you, but we’ll descend into the abyss together. When we get lost (and we most certainly will)—should we fall into despair and abandon our will to fight our way back—we’ll have a good laugh; we’ll cry; we’ll fall to our knees; we’ll become hysterics; we’ll roll on the soft grass we can feel but not see; we will weep the loud belly-laugh sobs of the hopelessly confused and completely insane—always together, whenever necessary.

spelunkers-caving-rabbit-hole-fine-structure
We will get lost together. This rabbit-hole is the Krubera Cave of Abkhazia land. It is the deepest cave in the world. Notice the tiny humans, for scale.

Isn’t getting lost with a friend what makes life worth living? Everyone gets lost eventually; it’s better when we get lost together. Getting lost with someone who doesn’t give a care; who won’t even pretend to understand the simplest things about the deep, dark places that lie miles beyond our grasp; that lie beneath our feet; that lie, in some cases, just behind our eyeballs; it’s what living large is all about, isn’t it?

Anyway, relax. Don’t be nervous. The fine-structure constant is simply a number—a pure number. It has no meaning. It stands for nothing—not inches or feet or speed or weight; not anything. What can be more harmless than a number that has no meaning?

Well, most physicists think it reveals, somehow, something fundamental and complicated going on in the inner workings of atoms—dynamics that will never be observed or confirmed, because they can’t be. The world inside an atom is impossibly small; no advance in technology will ever open that world to direct observation by humans.

What physicists can observe is the frequencies of light that enormous collections of atoms emit. They use prisms and spectrographs. What they see is structure in the light where none should be. They see gaps—very small gaps inside a single band of color, for example. They call it fine structure.

The Greek letter alpha (α) is the shortcut folks use for the fine-structure constant, so they don’t have to say a lot of words. The number is the square of another number that can have (and almost always does have) two or more parts—a complex number. Complex numbers have real and imaginary parts; math people say that complex numbers are usually two dimensional; they must be drawn on a sheet of two dimensional graph paper—not on a number line, like counting numbers always are.

Don’t let me turn this essay into a math lesson; please, …no. We can’t have readers projectile vomiting or rocking to the catatonic rhythms of a panic attack. We took our medicines, didn’t we? We’re going to be fine.

I beg readers to trust; to bear with me for a few sentences more. It will do no harm. It might do good. Besides, we can get through this, together.

Like me, you, dear reader, are going to experience power and euphoria, because when people summon courage; when they trust; when they lean on one another; when—like countless others—you put your full weight on me; I will carry you. You are about to experience truth, maybe for the first time in your life. Truth, the Ancient-of-Days once said, is that golden key that unlocks our prison of fears and sets us free.

Reality is going to change; minds will change; up is going to become down; first will become last and last first. Fear will turn into exhilaration; exhilaration into joy; joy into serenity; and serenity into power. But first, we must inner-tube our way down the foamy rapids of the next ten paragraphs. Thankfully, they are short paragraphs, yes….the journey is do-able, peeps. I will guide you.

The number (3 + 4i) is a complex number. It’s two dimensional. Pick a point in the middle of a piece of graph paper and call it zero (0 + 0i). Find a pencil—hopefully one with a sharp point. Move the point 3 spaces to the right of zero; then move it up 4 spaces. Make a mark. That mark is the number (3 + 4i). Mathematicians say that the “i” next to the “4” means “imaginary.” Don’t believe it.

They didn’t know what they were talking about, when first they worked out the protocols of two-dimensional numbers. The little “i” means “up and down.” That’s all. When the little “i” isn’t there, it means side to side. What could be more simple?  

Draw a line from zero (0 + 0i) to the point (3 + 4i). The point is three squares to the right and 4 squares up. Put an arrow head on the point. The line is now an arrow, which is called a vector. This particular vector measures 5 squares long (get out a ruler and measure, anyone who doesn’t believe).

The vector (arrow) makes an angle of 53 degrees from the horizontal. Find a protractor in your child’s pencil-box and measure it, anyone who doubts. So the number can be written as (553), which simply means it is a vector that is five squares long and 53 degrees counter-clockwise from horizontal. It is the same number as (3 + 4i), which is 3 squares over and 4 squares up.

The vectors used in quantum mechanics are smaller; they are less than one unit long, because physicists draw them to compute probabilities. A probability of one is 100%; it is certainty. Nothing is certain in quantum physics; the chances of anything at all are always less than certainty; always less than one; always less than 100%.

multiply-complex-numbers-fine-structure
To multiply the vectors Z and W, add their angles and multiply their lengths. The vector ZW is the result; its overall length is called its amplitude. When both vectors Z and W are shorter than one square in length, the vector ZW will become the shortest vector, not the longest (as it is in this example), because multiplying fractions together always results in a fraction that is less than the fractions that were multiplied. Right? To calculate what is called the probability density, simply multiply the length of the amplitude vector by itself, which will shrink it further, because its length (called its magnitude) is always a fraction that is less than one in quantum probability problems. This operation is called squaring the magnitude of an amplitude; it results in a one-dimensional unit-less number, which is—as we said before—a probability. Experiments with electrons and photons must be performed to reveal interaction amplitude values; when these numbers are squared, the fine structure constant is the result. The probability density is a constant. That by itself is amazing.

Using simple rules, a vector that is less than one unit long can be used in the mathematics of quantum probabilities to shrink and rotate a second vector, which can shrink and rotate a third, and a fourth, and so on until the process of steps that make up a quantum event are completed. Lengths are multiplied; angles are added. The rules are that simple. The overall length of the resulting vector is called its amplitude.

Yes, other operations can be performed with complex numbers; with vectors. They have interesting properties. Multiplying and dividing by the “imaginary” i rotates vectors by 90 degrees, for example. Click on links to learn more. Or visit the Khan Academy web-site to watch short videos. It’s not necessary to know how everything works to stumble through this article.

The likelihood that an electron will emit or absorb a photon cannot be derived from the mathematics of quantum mechanics. Neither can the force of the interaction. Both must be determined by experiment, which has revealed that the magnitude of these amplitudes is close to ten percent (.085424543… to be more exact), which is about eight-and-a-half percent.

What is surprising about this result is that when we multiply the amplitudes with themselves (that is, when we “square the amplitudes“) we get a one-dimensional number (called a probability density), which, in the case of photons and electrons, is equal to alpha (α), the fine-structure constant, which is .007297352… or 1 divided by 137.036… . 

Get out the calculator and multiply .08524542 by itself, anyone who doesn’t believe. Divide the number “1” by 137.036 to confirm.

From the knowledge of the value of alpha (α) and other constants, the probabilities of the quantum world can be calculated; when combined with the knowledge of the vector angles, the position and momentum of electrons and photons, for example, can be described with magical accuracy—consistent with the well-known principle of uncertainty, of course, which readers can look up on Wikipedia, should they choose to get sidetracked, distracted, and hopelessly lost.  

Magical” is a good word, because these vectors aren’t real. They are made up—invented, really—designed to mimic mathematically the behavior of elementary particles studied by physicists in quantum experiments. No one knows why complex vector-math matches the experimental results so well, or even what the physical relationship of the vector-math might be (if any), which enables scientists to track and measure tiny bits of energy.

To be brutally honest, no one knows what the “tiny bits of energy” are, either. Tiny things like photons and electrons interact with measuring devices in the same ways the vector-math says they should. No one knows much more than that. And no one knows the reasons why. Not even the late Richard Feynman knew why the methods of quantum chromodynamics (QCD) and the methods of quantum electrodynamics (QED)—which he invented and for which he won a Nobel Prize in 1965—worked.

There used to be hundreds of tiny little things that behaved inexplicably during experiments. It wasn’t only tiny pieces of electricity and light. Physicists started running out of names to call them all. They decided that the mess was too complicated; they discovered that they could simplify the chaos by inventing some new rules; by imagining new particles that, according to the new rules, might never be observed; they named them quarks.

By assigning crazy attributes (like color-coded strong forces) to these quarks, they found a way to reduce the number of elementary particles to seventeen; these are the stuff that makes up the so-called Standard Model. The model contains a collection of neutrons and muons; and quarks and gluons; and thirteen other things—researchers made the list of subatomic particles shorter and a lot easier to organize and think about.

Some particles are heavy, some are not; some are force carriers; one—the Higgs—imparts mass to the rest. The irony is this: none are particles; they only seem to be because of the way we look at and measure whatever they really are. And the math is simpler when we treat the ethereal mist like a collection of particles instead of tiny bundles of vibrating momentum within an infinite continuum of no one knows what.

feynman-diagram
Feynman diagrams help physicists think about what’s going on without getting bogged down in the mathematical details of subatomic particle interactions. Click this link to learn how to interpret and draw them.

Physicists have developed protocols to describe them all; to predict their behavior. One thing they want to know is how forcefully and in which direction these fundamental particles move when they interact, because collisions between subatomic particles can reveal clues about their nature; about their personalities, if anyone wants to think about them that way.

The force and direction of these collisions can be quantified by using complex (often three-dimensional) numbers to work out between particles a measure during experiments of their interaction probabilities and forces, which help theorists to derive numbers to balance their equations. These balancing numbers are called coupling constants.

The fine-structure constant is one of a few such coupling constants. It is used to make predictions about what will happen when electrons and photons interact, among other things. Other coupling constants are associated with other unique particles, which have their own array of energies and interaction peculiarities; their own amplitudes and probability densities; their own values. One other example I will mention is the gravitational coupling constant.

Despite their differences, one thing turns out to be true for all coupling constants—and it’s kind of surprising. None can be derived or worked out using either the theory or the mathematics of quantum mechanics. All of them, including the fine-structure constant, must be discovered by painstaking experiments. Experiments are the only way to discover their values. Here’s the mind-blowing part: once a coupling constant—like the fine-structure alpha (α)—is determined, everything else starts falling into place like the pieces of a puzzle.

The fine-structure constant, like most other coupling constants, is a number that makes no sense. It can’t be derived—not from theory, at least. It appears to be the magnitude of the square of an amplitude (which is a complex, multi-dimensional number), but the fine-structure constant is itself one-dimensional; it’s a unit-less number that seems to be irrational, like the number π.

For readers who don’t quite understand, let’s just say that irrational numbers are untidy; they are unwieldy; they don’t round-off; they seem to lack the precision we’ve come to expect from numbers like the gravity constant—which astronomers round off to four or five decimal places and apply to massive objects like planets with no discernible loss in accuracy. It’s amazing to grasp that no constant in nature, not even the gravity constant, is a whole number or a fraction.

Based on what scientists think they know right now, every constant in nature is irrational. It has to be this way.

Musicians know that it is impossible to accurately tune a piano using whole numbers and fractions to set the frequencies of their strings. Setting minor thirds, major thirds, fourths, fifths, and octaves based on idealized, whole-number ratios like 3:2 (musicians call this interval a fifth) makes scales sound terrible the farther one goes from middle C up or down the keyboard.

Jimi Hendrix, a veteran of the US Army’s 101st Airborne Division, rose to mega-stardom in Europe several years before 1968 when it became the American public’s turn to finally embrace him after he released his landmark album, Electric Ladyland. Some critics today say that he remains the best instrumentalist who has ever lived. Mr. Hendrix achieved his unique sound by using non-intuitive techniques to tune and manipulate string frequencies. Some of these methods are described in the previous link. It is well worth the read. Click the following YouTube link to view his iconic version of the Star Spangled Banner.

No, in a properly tuned instrument the frequencies between adjacent notes differ by the twelfth root of 2, which is 1.059463094…. . It’s an irrational number like “π”—it never ends; it can’t be written like a fraction; it isn’t a ratio of two whole numbers.

In an interval of a major fifth, for example, the G note vibrates 1.5 times faster than the C note that lies 7 half-steps (called semitones) below it. To calculate its value take the 12th root of two and raise it to the seventh power. It’s not exactly 1.5. It just isn’t.

Get out the calculator and try it, anyone who doesn’t believe.  

[Note from the Editorial Board: a musical fifth is often written as 3:2, which implies the fraction 3/2, which equals 1.5. Twelve half-notes make an octave; the starting note plus 7 half-steps make 8. Dividing these numbers by four makes 12:8 the same proportion as 3:2, right? The fraction 3/2 is a comparison of the vibrational frequencies (also of the nodes) of the strings themselves, not the number of half-tones in the interval.

However, when the first note is counted as one and flats and sharps are ignored, the five notes that remain starting with C and ending with G, for example, become the interval known as a fifth. It kind of makes sense, until musicians go deeper; it gets a lot more complicated. It’s best to never let musicians do math or mathematicians do music. Anyone who does will create a mess of confusion, eight times out of twelve, if not more.]  

Click this link for a better explanation, some might think, by Minute Physics.

An octave of 12 notes exactly doubles the vibrational frequency of a note like middle C, but every note in between middle C and the next higher octave is either a little flat or a little sharp. It doesn’t seem to bother anyone, and it makes playing in large groups with different instruments possible; it makes changing keys without everybody having to re-tune their instruments seem natural—it wasn’t as easy centuries ago when Mozart got his start.

The point is this: music sounds better when everyone plays every note a little out of tune. It’s how the universe seems to work, too. Irrationality is reality. It works just fine.

As for gravity, it works in part because space seems to curve and weave in the presence of super-heavy objects. No particle has ever been found that doesn’t follow the curved space-time paths that surround massive objects like our Sun.

Notice the speed of the hands of the clocks and how they vary in space-time. Clocks slow down when they are accelerated or when they are immersed in the gravity of a massive object. 

Even particles like photons of light, which have no mass (or electric charge, for that matter) follow these curves; they bend their trajectories as they pass by heavy objects, even though they lack the mass and charge that some folks might assume they should have to conduct an interaction.

Massless, charge-less photons do two things: first, they stay in their lanes—that is they follow the curved currents of space that exist near massive objects like a star; second, they fall across the gravity gradient toward these massive objects at exactly the same rate as every other particle or object in the universe would if they found themselves in the same gravitational field.

Measurements of star-position shifts near the edge of our own sun have helped to prove that space and time are curved like Einstein said and that Isaac Newton‘s gravity equation gives the most accurate results only when the curvature of space-time is added into the computations. Einstein once told a science reporter that space and time cannot exist in a universe devoid of matter and its flip-side equivalent, energy. Today, all physicists agree.

The coupling constants of subatomic particles don’t work the same way as gravity. No one knows why they work or where the constants come from. One thing scientists like Freeman Dyson have said: these constants don’t seem to be changing over time.

Evidence shows that these unusual constants are solid and foundational bedrocks that undergird our reality. The numbers don’t evolve. They don’t change. Confidence comes not only from data carefully collected from ancient rocks and meteorites and analyzed by folks like Denys Wilkinson. French scientists examined the fossil-fission-reactors located in Gabon in equatorial Africa. The by-products of these natural nuclear reactors of yesteryear provide incontrovertible evidence that the fine-structure constant, at least, hasn’t changed in the last two-billion years. Click on the links to learn more.  

Since this essay is supposed to describe the fine-structure constant named alpha (α), now might be a good time to ask: What is it, exactly? Does it have other unusual properties beside the coupling forces it helps define during interactions between electrons and photons? Why do smart people obsess over it?

We are going to answer these questions, and after we’ve answered them we will wrap our arms around each other and tip forward, until we lose our balance and fall into the rabbit hole. Is it possible we might not make it back? I suppose it is. Who is ready?

Alpha (α) (the fine-structure constant) is simply a number that is derived from a rotating vector (arrow) called an amplitude that can be thought of as having begun its rotation pointing in a negative (minus or leftward direction) from zero and having a length of .08524542…. . When the length of this vector is squared, the fine-structure constant emerges.

It’s a simple number—.007297352… or 1/137.036…. It has no physical significance. The number has no units (like mass, velocity, or charge) associated with it. It’s a unit-less number of one dimension derived from an experimentally discovered, multi-dimensional (complex) number called an amplitude.

We could imagine the amplitude having a third dimension that drops through the surface of the graph paper. No matter how the amplitude is oriented in space; regardless of how space itself is constructed mathematically, only the absolute length of the amplitude squared determines the value of alpha (α). 

Amplitudesand probability densities calculated from them, like alpha (α)—are abstract. The fine-structure constant alpha (α) has no physical or spatial reality whatsoever. It’s a number that makes interaction equations balance no matter what systems of units are used.

Imagine that the amplitude of an electron or photon rotates like the hand of a clock at the frequency of the photon or electron associated with it. Amplitude is a rotating, multi-dimensional number. It can’t be derived. To derive the fine structure constant alpha (α), amplitudes are measured during experiments that involve interactions between subatomic particles; always between light and electricity; that is, between photons and electrons.

I said earlier that alpha (α) can be written as the fraction “1/137.036…”. Once upon a time, when measurements were less precise, some thought the number was exactly 1/137. 

The number 137 is the 33rd prime number after zero; the ancients believed that both numbers, 33 and 137, played important roles in magic and in deciphering secret messages in the Bible. The number 33 was Christ’s age at his crucifixion. It was proof, to ancient numerologists, of his divinity.

The number 137 is the value of the Hebrew word, קַבָּלָה (Kabbala), which means to receive wisdom. In the centuries before quantum physics, during the Middle Ages, non-scientists published a lot of speculative nonsense about these numbers. When the numbers showed up in quantum mechanics during the twentieth century, mystics raised their eyebrows. Some convinced themselves that they saw a scientific signature, a kind of proof of authenticity, written by the hand of God.

Numerology is a rabbit-hole in and of itself, at least for me. It’s a good thing that no one seems to be looking at the numbers on the right side of the decimal point. 036 might unglue the too curious by half. I’m going to leave it there. Far be it for me to reveal information that might drive the innocent and uninitiated insane.

fine-structure-constant-triangle
People’s fascination with the fine-structure constant has led to many unusual insights, such as this one, found during an image search on the web. The hypotenuse is 137.036015… .

The view today is that, yes, alpha (α) is annoyingly irrational; yet many other quantum numbers and equations depend upon it. The best known is:

e = the square root of (2hcεα) or  e=\sqrt{2hc\epsilon\alpha} .

What does it mean? It means that the electric charge of an electron is equal to the square root of a number. What number? Well… it is a number that is two times the Planck constant (h); times the speed of light constant (c); times the electric constant (ε); times the fine-structure constant (α). 

Why? No one knows. These constants (and others) show up everywhere in quantum physics. They can’t be derived from first principles or pure thought. They must be measured. As our technology improves, we make better measurements; the values of the constants become more precise. These constants appear in equations that are so beautiful and mysterious that they can sometimes raise the hair on the back of a physicist’s head.

The equations of quantum physics tell the story about how small things we can’t see relate to one another; how they interact to make the world we live in possible. The values of these constants are not arbitrary. Change their values even a little, and the universe itself will pop like a bubble; it will vanish in a cosmic blip.

How can a chaotic, quantum house-of-cards depend on numbers that can’t be derived; numbers that appear to be arbitrary and divorced from any clever mathematical precision or derivation? What is going on? How can it be?

The inability to solve the riddles of these constants while thinking deeply about them has driven some of the most clever people on earth to near lunacy—the fine-structure constant (α) is the most famous nut-cracker, because its reciprocal (137.036…) is so very close to the numerology of ancient alchemy and the kabbalistic mysteries of the Bible.

What is the number alpha (α) for? Why is it necessary? What is the big deal that has garnered the attention of the world’s smartest thinkers? Why is the number 1/137 so dang important during this modern age, when the mysticism of the ancient bards has been largely put aside?

Well, two reasons come immediately to mind. Physicists are adamant; if α was less than 1/143 or more than 1/131, the production of carbon inside stars would be impossible. All life we know is carbon-based. The life we know could not arise.

The second reason? If alpha (α) was less than 1/151 or more than 1/124, stars could not form. With no stars, the universe becomes a dark empty place. We got lucky. The fine-structure constant (α) sits smack-dab in the middle of a sweet spot that makes a cosmos full of stars and life possible; perhaps inevitable.

fundamental-constants
These are the values of some of the fundamental constants mentioned in this essay. Plug them into formulas to confirm they work, any reader who enjoys playing with their calculator. It’s clear that these numbers make no precisional sense; their values don’t correspond to anything one might find on any list of rational numbers. It’s possible that they make no geometric sense, either. If so, then God is not a mathematician. Without mathematics, humans have no hope of understanding the universe. Yet, here we are wrestling against all the evidence; against all the odds that the mysteries of existence will forever elude us. We cling to hope like a drowning sailor at sea, praying that the hour of rescue will soon come; we will blow our last breath in triumph; humans can understand. Everything is going to fall into place just as we always knew it would.

It might surprise some readers to learn that the number alpha (α) has a dozen explanations; a dozen interpretations; a dozen main-stream applications in quantum mechanics.

One explanation that seems reasonable on its face is that the magnetic-dipole spin of an electron must be interacting with the magnetic field that it generates as it rushes around its atom’s nucleus. This interaction, when added to the hopping of energy states that produce photons, juggles the emitted photon frequencies slightly. This juggling (or hopping) of frequencies causes the fine structure in the colors seen on the screens and readouts of spectrographs and in the bands of light that flow through prisms. OK… it might be true. It’s possible. Nearly all physicists accept some version of this model.

Beyond this idea and others, there are many unexplained oddities—peculiar equations that can be written, which seem to have no relation to physics, but are mathematically beautiful.

For example: Euler’s number, “e” (not the electron charge we referred to earlier), when multiplied by the cosine of (1/α), equals 1 — or very nearly. (Make sure your calculator is set to radians, not degrees.) Why? What does it mean? No one knows. 

What we do know is that Euler’s number shows up everywhere in statistics, physics, finance, and pure mathematics. For those who know math, no explanation is necessary; for those who don’t, consider clicking this link to Khan Academy, which will take you to videos that explain Euler’s number. 

What about other strange appearances of alpha (α) in physics? Take a look at the following list of truths that physicists have noticed and written about; they don’t explain why, of course; indeed, they can’t; many folks wonder and yearn for deeper understanding:

  • One amazing property about alpha (α) is this: every electron generates a magnetic field that seems to suggest that it is rotating about its own axis like a little star. If its rotational speed is limited to the speed of light (which Einstein said was the cosmic speed limit), then the electron, if it is to generate the charge we know it has, must spin with a diameter that is 137 times larger than what we know is the diameter of a stationary electron—an electron that is at rest and not spinning like a top. Digest that. It should give pause to anyone who has ever wondered about the uncertainty principle. Physicists don’t believe that electrons spin. They don’t know where their electric charge comes from.

 

  • The energy of an electron that moves through one radian of its wave process is equivalent to its mass. Multiplying this number (called the reduced Compton wavelength of the electron) by alpha (α) gives the classical (non-quantum) electron radius, which, by the way, is about 3.2 times that of a proton. The current consensus among quantum physicists is that electrons are point particles—they have no spatial dimensions that can be measured. Click on the links to learn more.

 

  • The physics that lies behind the value of alpha (α) demands that the maximum number of protons that can coexist inside an atom’s nucleus must be 137. Uranium is the largest naturally occurring element; it has 92 protons. Physicists have created another 26 elements in the lab, which takes us to 118. When 137 is reached, it will be impossible to create more. Plutonium—the most poisonous substance known—has 94 protons; it is man-made; one isotope (the one used in bombs) has a half-life of 24,000 years. Percolating plutonium from rotting nuclear missiles will destroy all life on the earth someday; it is only a matter of time. It is impossible to stop the process, which has already started with bombs lost at sea and damage to power plants like the one at Fukushima, Japan.

 

  • When sodium light (from certain kinds of street lamps, for example) passes through a prism, its pure yellow-light seems to split. The dark band is difficult to see with the unaided eye; it is sodium-lamp-spectrumbest observed under magnification. The split can be explained by the value of the fine-structure constant alone. It is an exact relationship. It is this “fine-structure” that Arnold Sommerfeld noticed in 1916, which led to his nomination for the Nobel Prize; in fact Sommerfeld received eighty-four nominations for various discoveries. For some reason, he never won.

 

  • graphene-matrixThe optical properties of graphene—a form of carbon used in solid-state electrical engineering—can be explained in terms of the fine-structure constant alone. No other variables or constants are needed.

 

  • The gravitational force (the force of attraction) that exists between two electrons that are imagined to have masses equal to the Planck-mass is 137.036 times greater than the electrical force that tries to push the electrons apart at every distance. I thought the relationship should be the opposite until I did the math.

    It turns out that the Planck-mass is huge—2.176646 E-8 kilograms (the weight of the egg of a flea, according to a source on Wikipedia). Compared to neutrons, atoms, and molecules, flea eggs are heavy and huge. The ratio of 137.036 / 1.000 (G force vs. e force) is hard to explain, but it seems to suggest a way to form micro-sized black holes at subatomic scales. Of course, once black holes get started their appetites can become voracious. 

The good thing is that no machine so far has the muscle to make Planck-mass morsels. Alpha (α) has slipped into the mathematics in a non-intuitive way, perhaps to warn folks that should anyone develop and build an accelerator with the power to produce Planck-mass particles, they will have, perhaps inadvertently, designed a dooms-day device that could very well devour the universe.  

 

 

  • The Standard Model of particle physics contains 20 or so parameters that cannot be derived; they must be experimentally discovered. One is the fine-structure constant (α), which is one of four constants that help to quantify interactions between electrons and photons.

 

  • The speed of light is 137 times greater than the speed of “orbiting” electrons in hydrogen atoms. The electrons don’t actually “orbit.” They do move around, though, and alpha (α) describes the ratio of their velocities to the cosmic speed limit of light.

 

  • The energy of a single photon is precisely related to the energy of repulsion between two electrons by the fine-structure constant alpha (α), when the distance between the electrons is identical to the wavelength of the photon. The Planck relation and Planck’s law can provide additional insights for readers who want to know more.  

 

  • The charge of an electron divided by the Planck charge—the electron charge defined by natural units, where constants like the speed of light and the gravitational constant are set equal to one—is equal to   \sqrt\alpha . This strange relationship is another indicator that something fundamental is going on at a very deep level, which no one has yet grasped.

Now, dear reader, I’m thinking that right now might be a good time to share some special knowledge—a reward for your courage and curiosity. We’ve spelunked together for a while now. We seem to be lost, but no one has yet complained.

Here is a warning and a promise. We are about to descend into the deepest, darkest part of the quantum cave. Will you stay with me? I  know the way. Do you believe me? Do you trust me to bring you back alive and sane?

In the Wikipedia article about α, the author writes, In natural units, commonly used in high energy physics, where ε = c = h/2π = 1, the value of the fine-structure constant is:
\alpha=\frac{e^2}{4\pi}

Every quantum physicist knows the formula. In natural units e = .302822…. α, οf course, doesn’t change. Its value is still 1/137.036…. What physicists don’t know for certain is why. What is the number 4π about? Why, when 4π is stripped away, does there remain only “α“—the mysterious number that seems to be helping to quantify the coupling value of two electrons? 

Well… electrons are fermions. Like protons and neutrons they have increments of 1/2 spin. What does 1/2 spin even mean?

It means that under certain experimental conditions when electrons are fired through a polarized disc they project a visible interference pattern on a viewing screen. When the polarizing disc is rotated, the interference pattern on the screen changes. The pattern doesn’t return to its original configuration until the disc is rotated twice—that is, through an angle of 720 degrees, which is 4π radians.

Since the polarizer must be spun twice, physicists reason that the electron must have 1/2 spin (intrinsically) to spin once for every two spins of the polarizer. Yes, it makes no sense. It’s crazy.

What is more insane is that an irrational, dimensionless number that cannot be derived by logic or math is all that is left. We enter the abyss when we realize that this number describes the interaction of one electron and one photon of light, which is an oscillating bundle of no one knows what (electricity and magnetism, ostensibly) that has no mass and no charge. 

All photons have a spin of one, which reassures folks (because it seems to make sense) until they realize that all of a photon’s energy comes from its so-called frequency, not its mass, because light has no mass.

Frequency is the part of Einstein’s energy equation that is always left out because, presumably, teachers feel that if they unveil the whole equation they won’t be believed—if they are believed, their students’ heads might explode. Click the link and read down a few paragraphs to explore the equation.

In the meantime, here is the equation: E=\sqrt{m^2c^4+(hf)^2} . When mass is zero, energy equals the Planck constant times the frequency. It’s the energy of photons. It’s the energy of light.  

Photons can and do have any frequency at all. A narrow band of their frequencies is capable of lighting up our brains, which have a strange abilty to make sense of the hallucinations that flow through them. 

Click on the links to get a more detailed description of these mysteries.

What do physicists think they know for sure?

When an electron hops between its quantum energy states it can emit and absorb photons of light. When a photon is detected, the measured probability amplitude associated with its emission, its direction of travel, its energy, and its position is related somehow to the square root of a one-dimensional number—the fine-structure constant alpha (α)—that is, everything is related to the square root of a probability density of a measured vector quantity called an amplitude

When amplitudes are manipulated by mathematics, terms emerge from these complex numbers, which can’t be ignored. They can be used to calculate the interference patterns in double-slit experiments, for one thing, performed by every student in freshman physics.

The square root of the fine-structure constant matches the experimentally measured magnitude of the amplitude of electron/photon interactions—a number close to .085. It means that the vector that represents the dynamic of the interaction between an electron and a photon gets “shrunk” during an interaction by almost ten percent, as Feynman liked to describe it.

Because amplitude is a complex (multi-dimensional) number with an associated phase angle or direction, it can be used to help describe the bounce of particles in directions that can be predicted within the limitations of the theory of quantum probabilities. Square the amplitude, and a number (α) emerges—the one-dimensional, unit-less number that appears in so many important quantum equations: the fine-structure constant.  

Why? It’s a mystery. It seems that few physical models that go beyond a seemingly nonsensical vision of rotating hands on a traveling clock can be conjured forth by the brightest imaginations in science to explain why or how.

The fine-structure constant, alpha (α)—like so many other phenomenon on quantum scales—describes interactions between subatomic particles that seem to make no intuitive sense. It’s a number that is required to make the equations balance. It just does what it does. The way it is—for now, at least—is the way it is. All else is imagination and guesswork backed by some very odd math and unusual constants.

I’m thinking that right now might be a good time to leave this rabbit hole and get on with our lives. Anyone bring a flashlight?

Follow me.

Billy Lee