Why 137?

“Why 137?” was the question Nobel Prize winning physicist Wolfgang Pauli (1900-1958) was preoccupied with throughout his life. You see the fine structure constant, a dimensionless fundamental constant of physics, has a value nearly equal to 1/137. Here’s an overview:

In his Nobel lecture delivered in Stockholm on 13 December 1946, Pauli expressed his goal was to establish a theory “which will determine the value of the fine-structure constant and will thus explain the atomistic structure of electricity, which is such an essential quality of all atomic sources of electric fields actually occurring in nature.”

The charge of an electron is equal to the square root of (fine structure constant x 4π) in natural units.

Pauli famously quipped, “When I die my first question to the Devil will be: What is the meaning of the fine structure constant?” Unfortunately Pauli died without accomplishing his goal in the Red Cross Hospital of Zurich in Room 137—and he was aware of that synchronistic irony before he died.

The great psychotherapist Carl Jung helped Pauli in his quest to find a mathematical basis for the fine structure constant by interpreting more than 400 of Pauli’s dreams, published in Jung’s Psychology and Alchemy. Jung believed that Pauli unconsciously comprehended “some grand cosmic order.” Arthur Miller wrote a book about their collaboration entitled, 137: Jung, Pauli, and the Pursuit of a Scientific Obsession.

The brilliant Nobel Prize winning physicist Richard Feynman had this to say about the fine structure constant:

“It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to π or perhaps to the base of natural logarithms [known as e]? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the ‘hand of God’ wrote that number, and ‘we don’t know how He pushed his pencil.’ We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly!” -Richard Feynman QED: The Strange Theory of Light and Matter (1985)

Belshazzar's Feast by Rembrandt (1635) — slightly altered by the author

Belshazzar’s Feast by Rembrandt (1635) — slightly altered by the author

Curiously, the Hebrew word קבלה, meaning Kabbalah, takes a Gematria value of 137. The Bible says Ishmael lived to the age of 137. The Koran says Ishmael is buried next to the Kaaba in Mecca.

Symbolically, this is the threshold between the physical dimension and the utterly spiritual dimension. In other words, at the boundary line of the physical world, the number 137 emerges…

Physicist Leon M. Lederman numbered his home near Fermilab 137 based on the significance of the number to those in his profession. Lederman expounded on the significance of the number in his book The God Particle: If the Universe Is the Answer, What Is the Question?, noting that not only was it the inverse of the fine-structure constant, but was also related to the probability that an electron will emit or absorb a photon. He added that it also, “contains the crux of electromagnetism (the electron), relativity (the velocity of light), and quantum theory (Planck’s constant). It would be less unsettling if the relationship between all these important concepts turned out to be 1 or 3 or maybe a multiple of π. But 137?”

-Wikipedia article on 137 (number)

Cosmologist Robert L. Oldershaw reasons that 137 is the relationship of the strength of the unit electromagnetic interaction compared with the strength of the unit gravitational interaction. That sounds pretty fundamental to me.

Strangely, approximately 137 x 10,000,000 years have elapsed since the Big Bang.

In the Bohr atomic model, the innermost electron of a hypothetical atom with atomic number 137 would be orbiting just below the speed of light, and the next heaviest element would be impossible because its electron would have to exceed c. Atoms close to the theoretical limit of 137 are unstable and not found in the universe.

The fact is, the fine structure constant is not exactly 1/137. The currently accepted value is 1/137.035999074 (CODATA 2010). The denominator of the constant is called α (alpha) and it is slightly larger than 137.

In order of increasing accuracy, here are various thinkers’ attempts to discover the math behind 1/α.

The quest for a mathematical basis of 1/α

The quest for a mathematical basis of 1/α

I learned about most of these discoveries from an outstanding paper called Dimensionless Physical Constant Mysteries by Ke Xiao.

My illustration of the golden angle below is coincidentally exactly equal to Bohr’s formulation of alpha, 360/(Ф^2). The golden angle controls plant patterning of repeated biological units (called phyllotaxis) and governs the geometrical optics of rainbows. More info on these subjects is available in The Divine Proportion. It is worth pondering whether phyllotaxis and/or rainbows are ultimately governed by the golden angle or alpha? They are 99.7% equivalent.

The golden angle, from The Divine Proportion

The golden angle, from The Divine Proportion

The equation by D. Beere (1973) above is actually from his book USP-A: Physics for Flying Saucers. I had to work out the math to find the hidden formula for alpha and it turns out to be an elegant one, which comes out of comparing electrical inductance in a loop with the fluid mechanics of a vortex. It is fascinating that putting the math together from these two different but analogous phenomena (magnetic vortex vs. material vortex) yields an approximation of alpha. This might be an avenue for more research.

W. A. Harrington contacted me a few days ago because of my work on the Divine Proportion and shared some of his complex equations related to the fine structure constant (one of his is shown above). What little I understand of them inspired me to dig deeper and attempt to answer Pauli’s question myself, “Why 137?” and this blog post (and my whimsical formulation of 1/α) is the result.

W. R. Mellen’s equation is elegant because it is simply the hypotenuse of a Pythagorean triangle. However, it doesn’t tell us anything about where 137 comes from.

The hypotenuse is α

The hypotenuse is α

In my final formulation of 1/α (revised Jan 2016), the repetition of digits was the key and it resonates with the fact that 137 is the 33rd prime. I was able to figure it out because one of W. A. Harrington’s formulas for 1/α has a value of 137.037037037… and I recognized that repeating pattern.

When I cracked Jules Verne’s code in Quantification (20,000 Leagues Under the Sea was a journey of 111,111 km), I realized that 33.333…^3 = 37037.037037037… and this led to the following equation, with error in parts per million. The bars above 3 of the digits indicate eternally repeating digits.


An approximation of the ratio of a circle’s circumference to its diameter can be expressed in terms of alpha and repeating threes.


The way I exactly expressed Ф and its reciprocal in The Divine Proportion was another repetition of digits.


Feynman called Euler’s Identity “our jewel” and “the most remarkable formula in mathematics.” It normally takes the form you can read about on Wikipedia but I came across an alternate formulation of Euler’s Identity in Xiao’s paper that I think is even more beautiful because it relates e, i (√-1), π, Ф, and ϕ, fundamental constants others have used to express α:

The most beautiful equation in mathematics

The most beautiful equation in mathematics

The above equation is a kind of mathematical poetry connecting 4 of the most important constants. The poetry is beautiful because of this elegant interconnection. Having 1 and 0 in the standard formulation is better than -1 but I think that the most pleasing formulation includes the golden ratio Ф (≈1.618) and its reciprocal ϕ (≈0.618) because this expresses the unique nature of the divine proportion whilst simultaneously forming a elegant relationship with e, i and π.

Astrophysicists measured variations in the fine structure constant in 2011. WTF!

“While the data from just one telescope seemed to suggest that alpha varies in time, data from the two telescopes show that alpha also seems to vary in space. Such a discovery could have major implications, starting with shattering the basic assumption that physical laws are the same everywhere in the universe. The results also violate the Einstein Equivalence Principle, and suggest that the universe may be much larger than currently thought – or even infinite in size.

If alpha were just 4% bigger or smaller than it is, stars wouldn’t be able to make carbon and oxygen, which would have made it impossible for life as we know it to exist.

One intriguing possible implication is that the fine-structure constant is continuously varying in space, and seems fine-tuned for life in our neighborhood of the universe.” –PhysOrg.com

Fortunately for us the amount of variation is 4.1σ – 4.2σ, which means we only have confidence in α to 99.99%.

The mystery about α is actually a double mystery. The first mystery – the origin of its numerical value α ≈ 1/137 has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized. —Malcolm H. Mac Gregor, The Power of Alpha (2007)

It will be harder to elicit the true math behind the fine structure constant until we understand precisely how such a fundamental constant changes.

“There is no need to suppose that all the laws of nature sprang into being fully formed at the moment of the Big Bang, like a kind of cosmic Napoleonic code, or that they exist in a metaphysical realm beyond time and space…If we want to stick to the idea of natural laws, we could say that as nature itself evolves, the laws of nature also evolve, just as human laws evolve over time. But then how would natural laws be remembered or enforced? The law metaphor is embarrassingly anthropomorphic. Habits are less human-centred. Many kinds of organisms have habits, but only humans have laws.” –Rupert Sheldrake

The reason 137 has obsessed so many thinkers to try and find the math behind it is clear. Einstein wrote, “In a reasonable theory there are no numbers whose values are only empirically determinable.”

“It is impossible for human minds to construct something as infinitely rich as mathematics, that so successfully describes our universe down to the finest detail. Given that it is impossible for anything else to describe our world so accurately, it is also impossible to conclude that the universe is not mathematical.

It’s often said that the whole of biology is explained by chemistry, the whole of chemistry by physics and the whole of physics by mathematics, the queen of the sciences. If all physics is contained within mathematics, it has the most radical consequences. Advocates of the Multiverse seek to argue that life in our universe is an accident, an outcome of a random version of physics that applies to our universe. If, however, there’s only one physics, defined entirely by ontological mathematics (which is a single, interlocked edifice) then the fact that there is life in our universe is no accident at all. It implies nothing less than that our universe is alive. It is designed for life because it IS life.

The fundamental constants of physics are finely tuned for life precisely because they are manifestations of a living system. [emphasis my own]

A living universe is a universe of mind. The ideology of scientific materialism is thereby refuted. Many scientists are so blindly wedded to materialism that they would rather invoke infinite universes and infinite versions of physics and infinite sets of varying fundamental constants than embrace mind and life. Yet infinity does not help them. How can infinity convert the inanimate into the animate? How can consciousness come from mindless atoms? These are impossibilities, pure and simple. EVERYTHING makes sense as soon as it’s accepted that the universe, in its most basic aspect, is living not dead, and minded not mindless.” –Mike Hockney


Yours is the eye perceiving atop the pyramid, the answer to the question, “Why 137?”

32 thoughts on “Why 137?”

  1. Congratulations on your fine equation. I’m slowly beginning to see how physics arises from the behavior of number. It’s a beautiful mystery of consciousness – thank you for being a leading explorer!

    1. Thanks Michael. Can you imagine a situation where physics contradicts or has qualities independent of mathematics? No way! “Radical consequences” indeed.

      1. Wouldn’t you say that the infinity symbol is simply a label we ascribe to the failure of the human mind to comprehend the abyss? Empirical observations point to an infinite universe – in other words, an infinite physical aspect of reality. The use of infinity in an equation is nonsensical, therefore, the observable nature of the universe itself contradicts mathematical limits. I’m probably missing something.

        1. The cosmic microwave background is thermal radiation left over from the big bang, propagating in a sphere approximately 46 billion light years away from us. We can’t see beyond that because there is no light beyond the wavefront of when light was created. The only energy possibly beyond it are neutrinos, which are very hard to detect. Infinity is a slippery concept for our senses because the symbol ∞ points to the abstract concept of boundlessness. The universe seems to be finite in its overall size (not considering possible infinite inflation) but there are infinities within any finite size. Take for example, the fact that there are an infinite number of numbers between 0 and 1.

          1. When you say, “we can’t see beyond that”, it implies the existence of reality’s infinite somethingness. The immeasurable aspects of nature, such as infinity, contradict our present grasp of mathematics. The infinitely small and the infinitely large are expressed as an abstract symbol, whereas mathematical symbols can be transmuted into a finite objective reality. We interplace infinity with math as a temporary solution, but I am not aware of a field unifying the finite and the infinite. Man will never fully comprehend the universe – many of its mysteries will remain out of our intellectual grasp. You asked the question, “Can you imagine a situation where physics contradicts or has qualities independent of mathematics?” Isn’t the origin of 137 a situation where physics can’t be explained by mere mathematics? Thank you for your great work. I just received Anthology as a gift.

    1. These words are Aramaic:
      MENE, a mina (from the root meaning “to count”),
      TEKEL, a spelling of shekel (from the root meaning “to weigh”),
      PERES, half a mina (from the root meaning “to divide”).

      Counting, weighing, and dividing are necessary skills to comprehend the “grand cosmic order,” so the writing on the wall is sage advice for anyone trying to answer the question, “Why 137?”

  2. Michael C Grasso commented on Facebook: Scott, 6/5 also references one of the fundamental patterns of the base 10 system, the number 12 (6 / 5 = 1.2), which is half of the full pattern of 24 and as far as the number 360, the “Matrix” will show it to be a fundamental (non-arbitrary) pattern embedded as well in the decimal system. A circumference of 432 (360 + 36 + 36) units (432 also = 108 + 108 + 108 + 108, 108 = Phi angle) has a diameter equal to 137.51. 1 / 137.51 = 0.007272198…, which is 99.66% of the Fine-Structure Constant of 0.007297352… Close enough to not seem just a coincidence, considering all other correspondences, but more research in this direction will surely produce interesting results.

  3. This last comment, concerning 1.2, brings up something i´ve found in the metrology. From Glastonbury Tor to Cergy the distance is 222.00 NM to the column where the axis turns. A bit further down the grand Cergy axis the distance is 222.1 NM (there is a small square there in the axis), and a line from Glastonbury Tor to the entrance of the Karnak temple at Luxor is exactly ten times this 222.1 NM distance.
    Further, a line from the Tor to the Kaaba is in feet 16 192 480 feet, and from the Tor to just ahead of this small square on the main axis of Cergy we have the same distance in inches, ergo 1/12 the distance from Tor to Kaaba.
    There is then a possible connection from Glastonbury Tor to the Karnak Temple via Cergy with 1/10, and from the Tor with 1/12 to the Kaaba via Cergy.

    1. Intriguing… perhaps this will assist you. 360 degrees in a sphere. 1.618 and .618 are the “Golden Proportions”. 137.5 is the most profound degree in nature.

      .618 x .618 = .382
      .618 + .382 = 1

      360 x .618 = 222.48
      360 x .382 = 137.52

  4. ♡love♡ the art of communication. The mathematical craftsmanship (regarding “why 137”) is so beautiful I wonder if it was translated into music or a sound some how would it sound like an “om” or what? I look forward to your connections or studies on dreams considering your interest in the great mind. If I simply missed it please direct or link me? Your work is greatly appreciated, Scott! Could I have signed copies of all of your books or the ones I must have for $216?

    1. Thanks for your kind comments! I haven’t written about dreams before but thanks for the suggestion. My books are printed on demand and shipped out immediately so unfortunately I don’t get a chance to sign them.

  5. I have an alternate theory that requires the speed of light in meters to work with 137.

    If you write the speed of light in meters and remove the digits 1 3 and 7 a message can be read in reverse if we translate the numbers to english. It would read “He Big I”.

    Now if this idea was important it should be encoded into some structure either directly written or in the dimensions. I propose the second largest pyramid of giza is this structure. Khafre has a royal cubit dimension of 411 X 274 if we take 2xside/height and side-height we get 3 or c(speed of light) and 137.

    If you are interested in a larger explanation of he big I you can find it on YouTube video Hidden message of the Reverse Great Seal.

    Here is the link: https://youtube.com/watch?v=sVzoChQAvfw

  6. Scott,
    since the number 137 is indeed the 33rd prime number, I wonder then why the number 33 occurs so often?

    33 bones make up the human vertebrae
    33° is the highest order of scottish rite freemasonry
    the great spiritual teachers died at age 33 (Jesus, Krishna)
    10 * pi + golden ratio = 33.033960439

    I’m certain that if we were to look closely we’ll see more occurrences/co-incidences of relations between 137 and 33.

  7. C. G. Jung worked with W. Pauli, publishing his interpretation of Pauli’s dreams in his book, Psychology and Alchemy. I am very interested in physics and this number in particular as a doorway into the mysteries of psyche and matter. Perhaps Pauli’s dreams hold the mystery of this profound number. Thank you for this very interesting article, Scott.

  8. In botany, phyllotaxis or phyllotaxy is the arrangement of leaves on a plant stem (from Ancient Greek phýllon “leaf” and táxis “arrangement”).[1] Phyllotactic spirals form a distinctive class of patterns in nature.The term phyllotaxis means “leaf arrangement” in Greek and was coined in 1754 by Charles Bonnet, a Swiss naturalist (Livio “Story,” 109). In the 1830s, a pair of scientist brothers found that each new leaf on a plant stem is positioned at a certain angle to the previous one and that this angle is constant between leaves: usually about 137.5 degrees. “That is, if you look down from above on the plant and measure the angle formed between a line drawn from the stem to the leaf and a corresponding line for the next leaf, you will find that there is generally a fixed angle, called the divergence angle” (Tung 13).

    We now turn our discussion to the sunflower, in which one can observe two families of spiral patterns: one winding clockwise and the other counterclockwise. The quantity of spirals in each family are always two consecutive Fibonacci numbers. This effect is the result of closely packing points separated by 137.5 degrees in tight spirals. This implies that the a golden-ratio based phyllotaxis allows not only for optimal sun exposure but also for maximal packing in horizontal space.

  9. 137 was known to King Harald Bluetooth, as is evident from the fact that his famous rune stone in Jelling was placed right in the middle between the two mounds. The distance is precisely 137,5 m.
    The figure on the rune stone – now to be seen in all Danish passports – is ho Pantokratôr = 70+1822 = 1892, which is 137,5 x 13,76.
    Moreover, the Jelling rhombus (google) is 4 x 360 = 1440. The hexagon inscribed in the 1440 circle is preciisely 1375.
    Hence, Q (as I call 137,5) is by no means a modern discovery. It is yet another case of an ancient secret in plain sight.
    One of king Harald´s main sources is the Revelation of John. If I have counted correctly, the figure 7 occurs 137 times in that celebrated book, which is also the main source of ho Pantokratôr, the Almighty 1892.
    One of the main sources of John´s Revelation is the Buddhist gospel known as Saddharmapundarîkasûtram, the number of which is 2059, i.e. 1,5 x 1372,666.
    This Buddhist text is often mentioned in the New Testament, including Revelation 13,18 (with the famous 666).
    Dr. Christian Lindtner (Denmark)

  10. 13 at last supper. Christ 7th ressurects age 33.
    6 glands with two hemispheres (12 both sides) 7th gland (13) atop 33 vertebrae pineal gland undivided at centre of brain. 137 is pure alchemy. Thus crucial to kabbalah & mysticism.

  11. Hi Scott, here’s some new research for your extensive article.

    Fine-structure constant from Sommerfeld to Feynman
    Michael A. Sherbon


    The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant.
    Journal of Advances in Physics, 16(1), 335-343 (2019). https://doi.org/10.24297/jap.v16i1.8402

    1. Thank you for sharing your paper Michael. It is full of interesting information. Anyone interested in the fine structure constant and its connections with the golden ratio should read it!

  12. Pingback: The number 137 – Symbolism of Mathematics

  13. (137^2 -1)/20 = 938.4 : The inverse of fine structure constant to the second power minus 1 divided by 20 equals the Mev of the Proton. See my lectures on youtube if this interests you. I’ve discovered many new nuclear equations by using the 6 axis metric of synergetcs.

  14. Greetings,

    I am no mathematician or physicist. I am a simply an intuitive, metaphysical and philosophical author, poet. The number of times a day that the number 137 and or is parts in unison or repetition show in my day would astound the average person to the point I would be said to be mentally ill. Example 2 days ago on a trip I said to a family member “Everyone says I make this stuff up, or make it happen. If that where the case I would have had to know exactly how long it would have taken us on this road trip to come to this exact point at this exact time” The mile post marker was #137 and the time was 11:37 I got a bit of a blank look from her. A few minutes later we made a stop for fuel. A side transaction for some soft drinks and snacks came to a total of $11.73. Again I got a funny look from family. That kind experience in my life is constant and a regular event. All I can say is that my personal and intuitive mind tells me that the representation of that number, or the number itself is somehow liked to the connection of the physical and non-physical realms of our universe and the transfer of energy between them. Unique that a frequency analysis ( http://rumkin.com/tools/cipher/frequency.php ) of my Pen Name Archer is 1.7333 My best to all and with thanks for all of the info I find here. “Love life unconditionally and life will unconditionally love you” Archer De Angelo

  15. Sufi wisdom: the One Primal Effulgence (God, Yahweh or Allah) created only after two other aspects emerged from eons of meditation on the Divine Self. The two were called Light (Nur) and Wisdom (Qutb). They sent out 7 rays of consciousness which became the phenomenal universes.
    Thus 1 became 3 then became 7.

  16. Hi Scott

    Let me see your powers of 3 and raise you… this equation is based off of an approximation I found for the royal cubit.

    1/α ≈ √(137² + ((ρ^9)/4)²)

    which is similar to the 137²+π² version, but uses a better value in place of the π.
    ρ is the plastic ratio/number, the solution to ρ+1=ρ³

    The WolframAlpha version : √(137² + ((plastic number^9)/4)²)

    Answer comes out as 137.0359992970725102075551820936909082242952278384186387671…

    which translates to 99.9999998 % correct …..

    100/137.0359992970725102075551820936909082242952278384186387671 / (fine structure constant)

    FWIW, in my analysis of Giza, apart from Khafre referencing 137, there are also a lot of times where it references 3. Curious.

    Cheers, Ian

  17. Pingback: Approximating the Fine Structure Constant | IanDoug

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