Deconstructing 137: A Lattice of Remainders
Author: Chris Preen
## The 137 ProblemThe fine-structure constant, alpha, is one of the most iconic dimensionless numbers in physics. Some call it "the DNA of light" as a way of highlighting its strange role: it shows up wherever light and matter meet, setting the strength of the electromagnetic grip between them.It is also a number with surprisingly little slack. Change alpha enough and you do not just tweak physics at the edges, you start rewriting whether stable atoms and chemistry, in other words our universe, could exist at all.What unsettles people is not merely that alpha appears often. It is that it appears with the feel of a signature. Again and again, a theory arrives with clean elegance, and then a small, structured correction remains. In the electromagnetic domain, that margin mark is written in alpha.As of CODATA 2022, alpha is approximately 0.00729735... and its inverse (1/alpha) is approximately 137.035999.... At ordinary working precision, physicists often speak about this as "137," because alpha sits very close to 1/137. That shorthand has carried cultural weight for more than a century.Richard Feynman put the mystery in a single blunt line: "Nobody knows" where the number 137 comes from. In other remarks he treated it as the sort of number you would want to pin to the wall and stare at because its origins are so stubbornly elusive. Wolfgang Pauli made the same point when he joked that if he could ask God one question, it would be "Why 137?"Mathematically, 137 is just a prime. It has its own properties, but it is not ordinarily treated as a number with any special arithmetic privilege. In 2018, near the end of his life, celebrated British mathematician, Sir Michael Atiyah, challenged that prevailing attitude. He proposed the idea that the number 137 (as 1/alpha) might be a convergence target for a mathematical process, a kind of attractor rather than an arbitrary empirical nuisance. The attempt was widely disputed and did not settle into consensus, but the deeper point matters for us: even a mind like Atiyah's proposed that the number 137 might be a legitimate site of convergence, not merely measurement.In our own hypothesis, we are not presenting a proof, and we are not claiming to derive alpha from first principles. This is an exploratory pattern audit: we fix a small vocabulary of integers, fix a limited set of operations, and test whether 137 behaves like an outlier under a control battery that any reader can rerun.In a small collaboration between human and AI, we believe we have detected a tightly interlocking signature in a compact set of integers that repeatedly re-express themselves through multiple mechanisms.## The LatticeOur atom set is:27, 37, 73, 137.These four numbers form a tight, phase-locked cluster. When you push on it, it leaks by one, by a square, by a mirror, and then that leak becomes structure. In other words: the remainder is not noise, it is a hinge.For most of what follows, we set the larger physics question aside. We treat 137 as a purely arithmetic object and ask a narrow question: does it sit inside an unusually compressive pocket of number space?We will return to alpha twice: first to compare numbers, and only later to compare roles. Along the way we will use base-10 fingerprints, including recurring digit blocks in simple reciprocals, as a way to test whether the lattice is doing real compression or merely producing aesthetic coincidences. One of the later sections produces a reciprocal lock so clean it becomes a stress test for the entire claim.To keep this grounded, we score compression, not interpretation.The working claim is simple: the atom set {27, 37, 73, 137} seems to generate an unusually dense network of cheap routes into the same small hub set (powers, squares and cubes, repdigits, mirrors, and the 10^ neighbours). We have spot-checked this against nearby primes and against other comparable primes at the same scale. We are not presenting a full audit log. This is not an academic paper, and we are not trying to drown the reader in bookkeeping.Instead, we state the standard we think the claim should meet. If someone wants to argue that this is generic, or that other four number pockets are just as compressive, this is the test they should run.Working definition: a target T is a compression attractor if many distinct low-cost constructions reach it by different mechanisms (power routes, base-10 one-offs (+1 or -1), reciprocal frames, digit mirrors) without importing extra primitives. Low-cost can be scored by a simple rule.Attractor scoring rule (summary):* Fix the atoms: {27, 37, 73, 137}.
* Fix the moves: addition, multiplication, small powers (squares and cubes), digit reversal, and 10^ +/-1 style one-offs (including repdigits).
* Score any construction by (s,k):
* s = steps taken,
* k = new primitives imported beyond the atoms and standard hubs (powers of ten, very small squares and cubes).
* Lower (s,k) wins.Control check:Apply the same rule on a panel of comparable primes (adjacent ones plus a wider three-digit sample). If the cheap routes thin out, if the same hubs stop reappearing, if the cascades and reciprocal swaps do not recur, then this is not just "what primes do." It is a compression pocket. Everything that follows should be read in that spirit: not as a logged proof, but as a set of exhibits that repeatedly score well under the same rule.## How to Read the LatticeWe will call it a lattice for the geometry it suggests, but structurally it behaves like a network: a small set of nodes tied together by recurring relation-types.Read the lattice through three verbs and the next ten sections become easier to parse:Closure. The system aims at clean endpoints: perfect powers, round base-10 targets, exact identities.Leak. It misses, but the miss is not random. The remainder has shape: off by 1, off by a square, off by a mirror, or off by a pattern that behaves like memory.Ascent. The miss becomes a bridge. The bridge becomes a node. The node unlocks the next identity, and the network climbs.Think of it as a disciplined way to observe emergent structure inside arithmetic. Not smooth optimization, but constraint, failure, and then a new degree of freedom extracted from the remainder.You will see this cycle repeat. Each section attempts closure. A leak occurs. The leak becomes the step.## 1. The Four NumbersThe core vocabulary is deliberately small:27, the cube: 3³
37 and 73, a mirror prime pair
137, the anchor primeThe first lock is remarkably clean:27 + 37 + 73 = 137This sum lock 27+ 37 + 73 = 137 is just a seed condition, not evidence.The structure only begins when independent routes keep collapsing onto the same square hubs, 64 (27 + 37 = 64) and 100 ( 27 + 73 = 100), without being forced by that initial sum.Closure appears first, because it is the seed. Then the set starts folding back on itself.## 2. Mirror Primes and Mirror OrdinalsThe mirror doubles: digits mirror, and prime ordinals mirror.37 is the 12th ordinal prime
73 is the 21st ordinal primeThe ordinals mirror each other: 12 and 21. Their sum is 33.And 137 is the 33rd ordinal prime.Now push that ordinal sum into arithmetic:33 x 37 = 1221The ordinal pair returns as digits: 12|21.This is not a proof of anything by itself. It is a distinctive kind of self-inscription: metadata seemingly collapsing into the visible number stream. The lattice does not keep the mirror property as an external label. It prints it.## 3. The Power LadderThe lattice binds primes to perfect powers, then loops them back into base-10 hubs.First closure:33 + 37 = 27 + 37 = 64 = 43 = 82 = 26Cube plus prime lands on a square and a cube at once. One identity, multiple hubs.Second closure: the square gap collapses onto the same hub.372 - 272 = 1369 - 729 = 640And because it is a difference of squares:372 - 272 = (37-27)(37+27) = 10 x 64 = 640So 64 is not only where the lattice lands, it is what survives when the system compares its squares. The loop back into base-10 is built in: the hub is scaled by 10, cleanly, with no imports.Now flip into difference mode:
73 - 37 = 36 = 62
137 - 73 = 64 = 82
137 - 37 = 100 = 102Read those gaps as geometry: the lattice has manufactured a right triangle in difference space. The legs are 6 and 8 and 1the hypotenuse is 10. In other words, the mirror ladder is not only additive, it is Pythagorean, and it welds the prime anchor to the base-10 square hub by a classical closure.And it is not any triple: 6-8-10 is exactly 2 x (3-4-5), the canonical builder's square.Even the delta between deltas is powered:36 - 27 = 9 = 32Third closure:32^ + 33 + 43 = 9 + 27 + 64 = 100 = 102Powers stack into a clean base-10 anchor.Then the primes recombine into that same anchor:73 + 27 = 100 = 102Different routes, same target. That is the compression claim in miniature. And a final triangulation pulls the same hub out of two independent moves:(27 + 37) + (73 - 37) = 100
82 + 62 = 102So the set does not merely touch squares, it nests them. It hits hubs, then hits the relationships between hubs.## 4. base-10 Pressure I: The 999 EngineOnce the lattice touches 102, it begins to align with the decimal frame. It aims at powers of ten, misses by one, and uses the miss as structure.27 x 37 = 999 = 103 - 1It tries to land on 1000. It stops one unit short. That one-unit leak is the hinge, because 999 is the perfect repeating-decimal frame in base 10.Now the reciprocal swap locks in:1/27 = 0.037037...
1/37 = 0.027027...The cube and the prime exchange signatures inside the digit stream. The decimals are not just repeating. They are cross-labeling the atoms.Closure: 1000.
Leak: -1.
Ascent: the leak becomes a digit machine.## 5. base-10 Pressure II: The 10001 EngineThe same template reappears at the next scale:73 x 137 = 10001 = 104 + 1The system aims at 10000 and overshoots by one. Again, the one-unit leak creates a clean reciprocal frame.Now the swap returns:1/137 = 0.00729927...
1/73 = 0.01369863...Same structure, higher magnitude. The "one-off engine" is not a one-time trick. It reproduces.This matters for the cascade ahead, because it sets up the later 108 mirror box directly.Closure: 10000.
Leak: +1.
Ascent.## 6. base-10 Fingerprints Across ScaleThere are also scale echoes, not exact identities, but clean pathways with small, legible misses.The anchor prime can speak in the 027027 voice under a single decimal shift:137 / 3.7 = 1370 / 37 = 37.027027...And the atoms reach each other through near-misses that stay inside a tiny vocabulary:3.7 x 73 = 270.1 = (approx.) 27 x 10
1.37 x 27 = 36.99 = (approx.) 37These are not presented as equalities. They are the same directional behavior: closure pursued across decimal scale, then a small remainder left in a predictable place (tenths, hundredths). It is the integer lattice "showing through" a scaled aperture.## 7. Relative Remainder CrossoverNow we see the remainder unmistakably carrying a square.Normalize by 0.1:0.1/73 = 1/730 = 0.001369863... begins with 1369 = 372^
0.1/137 = 1/1370 = 0.000729927... begins with 729 = 272Each prime carries the other node's square inside its early digits when you pass through the same simple normalization.It is a specific kind of memory: the remainder encodes the lattice.## 8. The 2701 Hinge and the Remainder CascadeA new product arrives:37 x 73 = 2701The clean target it points at is obvious:2700 = 27 x 102And the miss is minimal:2701 = 2700 + 1That +1 binds the mirror pair to a base-10 scale through a single-unit weld.Now watch the cascade:1/2700 = 0.000370370... the lattice prints itself at a smaller scale.And the cross-product twin appears:27 x 73 = 1971Now the key remainder:2700 - 1971 = 729 = 272And the bridge that completes the hinge:1971 + 730 = 2701The remainder does not just sit there. It becomes a rung. This is closure pursued, leak produced, then the leak converted into ascent.## 9. Square EmbeddingThe self-reference tightens:37 x 137 = 5069But 137 already splits at the base-10 hub: 137 = 102 + 37
So the product unpacks as: 37 x 102 + 372^ = 3700 + 1369What remains is not arbitrary:5069 - 3700 = 1369 = 372The multiplier returns inside the remainder as its own square. The number contains its operator.This is one of the cleanest "leak with memory" moments in the set.## 10. Concatenation Folds, then the 1/137 Lattice RatioBefore the final lock, the lattice shows that it can move between representations without importing new atoms.Concatenation fold:concat(37, 73) = 3773
concat(27, 73) = 2773
3773 - 2773 = 1000 = 103And product twins:37 x 73 = 2701
27 x 73 = 1971Digits and products remain coupled. The system can slide between "written form" and "computed form" and still land on a base-10 hub.The final extrapolation:Take the fraction you already met in the 10001 engine:1/137 = 73/10001 = 0.00729927...Now amplify it through a pure base-10 mirror box by multiplying numerator and denominator by 9999:73/10001 = (73 x 9999) / (10001 x 9999)
= 729927 / 99999999And here is the exact lock:729927 / 99999999 = 1/137
= 0.00729927 00729927 00729927...No approximation. No rounding. A repeating decimal with a repeating block, sitting in a denominator of 108 - 1, and it lands exactly on the number physics has nicknamed "137."The internal echoes keep coming, and they are genuinely structural:729,927 / 73 = 9,999 = 104 - 1So the numerator itself collapses back into a one-off engine.And:729,927 / 101 = 7,227
7,227 reads as concat(72, 27)
72 + 27 = 99 = 102 - 1
101 = 102 + 1And 99 immediately reveals its own echo: 729,927 / 99 = 7,373, which reads as 73|73. The one-off engine double-stamps the prime in the quotient.These are lighter echoes, and they should be treated as such. They are the sort of by-product patterning you expect once a compression pocket is real: boundaries, mirrors, and one-offs keep reappearing.One final remainder echo:1,000,000 - 729,927 = 270,073It is not a new identity, but it is a familiar shape: 2701, and the 27 and 73 atoms resurfacing inside another base-10 frame.The bottom line is this:1/137 is not merely close to something neat. It is itself a neat thing, when viewed through the lattice's preferred base-10 machinery.## Interlude: Measurement ApertureUp to now we have mostly spoken in base 10, because base 10 is the human aperture: the reporting format of calculators, tables, and constants printed in papers. And our lattice does not hide from that aperture. It leans into it.### Engines and LensesThat is what the one-off engines are doing:27 x 37 = 999 = 103 - 1
73 x 137 = 10001 = 104 + 1and later, the mirror box:99,999,999 = 108 - 1These are pressure points: closure pursued, missed by one, and the miss turned into structure.
Now comes the key clarification before we turn to physics.If this were only a base-10 artifact, it would fade when you rotate the coordinate system. But there are base-agnostic lenses where the same relations do not weaken. They sharpen. Modular return-times are one. Any simple mathematical interrogation will show that our Fine Structure Sequence holds its own here too.Prime-powered nearness is another lens. The p-adics simply formalize "near" as "difference divisible by a high prime power," and our flagship remainders are already high powers: 36 and 372 (Sir Michael Atiyah even reached for a p-adic framework in his 2018 convergence attempt toward 137, which is exactly the kind of base-agnostic notion of 'distance' we are invoking here.).
###The Cyclotomic Frame
There is a second reading of our “one-off engines” that is older and more structural than base-10 aesthetics.
Cyclotomic structure is the arithmetic of cycles. The cyclotomic polynomials Φn(x) encode the primitive n-step phases of roots of unity, and Euler’s totient φ(n) counts how many of those primitive phases exist. In plain terms: φ(n) measures how many distinct irreducible rotations a cycle contains before repetition becomes unavoidable.
That matters because repeating decimals are a phase phenomenon. In any base b, the period of 1/p is governed by the multiplicative order of b mod p: the smallest k for which bk≡1(mod p). The digits loop when the phase returns to 1.
Seen in that light, several of our clean base-10 nodes stop looking like digit luck and start looking like roots-of-unity footprints:
73 x 137 = 10001 = 104 + 1.
But 104 + 1 is exactly Φ8(10), because Φ8(x) = x4 + 1. So the lattice is not only producing round numbers, it is landing on a cyclotomic evaluation.
Even the smaller anchors match the same pattern family:
111 = 102 + 10 + 1 = Φ3(10),
and 999 = 103 - 1 is the blunt “return” frame for a 3-step cycle.
The point is not that base 10 is special. The point is that the mechanism is cycle-based and persists in any base b, as factor structure inside bn +/- 1 and as period structure inside modular order. In our decimal projection, the lattice keeps brushing those cycle-frames. That is another reason the pattern can feel stable under perturbation: it is not just arithmetic on a line, it is rotation and return.But the cleanest escape from digit suspicion is older than any of that.Geometry.##Geometry Witnesses
Points, lines, and layers do not care what base you count in. If an integer relation is really structural, it will often admit a geometric witness: a way to see the number as a shape, a shell, a growth step.
Here the lattice gives a witness that is not optional, because it is the arithmetic of growing cubes.
The centered hexagonal numbers are:
1, 7, 19, 37, 61, 91, ...
They are not arbitrary. They are exactly the differences between consecutive cubes:23 - 13 = 8 - 1 = 7
33 - 23 = 27 - 8 = 19
43 - 33 = 64 - 27 = 37
53 - 43 = 125 - 64 = 61So our first catalytic closure33 + 37 = 43is not a one-off coincidence. It is the general law of cube growth: each step from n3 to (n+1)3 adds a centered hexagonal shell. When we say "37 is the shell around 27 that yields 64," we are not improvising a metaphor. We are naming the shape of the increment.Now watch what this does to the rest of the lattice. 64 is one of our hubs. It is 43. It is 82. It is 26. And it arrives from the atom set immediately: 64 = 27 + 37 and, equivalently, 64 = 137- 73 : across these two forms, every atom appears. So the 64 hub that the first closure forces into the story is already wired to the mirror prime pair and the anchor prime. Then the same geometry reaches forward again, because 37 also lives in the next shape family: the star numbers (on centered hexagrams): 1, 13, 37, 73, 121, ...A star number is not just "a different sequence." It is built from a centered hexagon by adding six symmetric arms. In fact, the relationship is rigid:star(n) = 2 x hexagon(n) - 1.This is where 37 earns a rare double role:* 37 is a star built on the previous centered hexagon 19.
* 37 is also itself a centered hexagon that becomes the core for the next star, 73.So 37 is simultaneously:a star-with-19-as-hexagon, and
the hexagon-for-73-as-star.There is another geometric witness that brings 37 and 73 into the same frame without invoking digits at all. It comes from the triangular numbers, where a number is not a numeral but a pile of points.The 73rd triangular array admits a perfectly centered partition: an inverted inner triangle of side 37, seated exactly in the middle, surrounded by three congruent corner triangles of side 36.In triangular-number terms this reads:T73 = 3T36 + T37and sinceT37 = 703
T73 = 2701...the "fit" is not metaphor. It is a clean geometric decomposition of the 73rd triangle into a central 37th triangle plus three identical outer regions.Our lattice lands on a nominated hinge: 73 is the mirror prime, the center piece is 37, and the surrounding step is 36, which is both the delta between the mirror primes (73 - 37) and a perfect square (62). It is the same story again, told in a different geometry: a large form, a central insert, and a boundary number that keeps returning as structure.There is one more reason this triangular witness matters, and it is not merely that 37 sits inside 73. It is that the triangular layer quietly reveals the same star and hexagon wiring we already saw.The star numbers and the centered hexagonal numbers are locked by a rigid relation:star = 2 x hex - 1.So if H is a centered hexagon and S is its corresponding star, then S + 1 = 2H. And that means the triangular number indexed by the star collapses into a product:T_S = S(S + 1) /2 = S(2H) /2 = S x HIn plain terms: when a star is built on a centered hexagon, the star's triangular count is exactly the star multiplied by its hexagonal core.Now watch what happens to our hinge pair.The centered hexagon before 37 is 19, and 37 is the star built on it. So the triangular count at 37 is forced into:T37 = 37 x 19 = 703Then the pattern steps forward one rung.37 is itself a centered hexagon, and 73 is the star built on 37. So the next triangular count is forced in to:T73 = 73 x 37 = 2701So the triangular witness is not only a pleasing partition. It is a bridge. A different geometry arrives at the same conclusion: 19, 37, and 73 are not merely adjacent curiosities. They are chained by a single rule that survives coordinate changes. The lattice is not just speaking in digits. It is speaking in shapes, and the shapes agree.This is the point of the interlude.### Bridge and ExitBase 10 is a prominent projection layer, and our lattice clearly speaks in that channel. But the same wiring persists when you stop talking about digits and start talking about cycles, nearness, and shape.Digits are one aperture. Geometry is the coordinate-free witness.With that in place, we can venture into physics with the right level of caution: not "the integers caused the constant," but "the constant, when projected through human reporting, lands in a channel where this lattice is already dense."We have shown a system that binds clean forms to stubborn offsets, closure to leak, and leak to ascent. Now we step into the electromagnetic world, where the most famous structured remainder has a name: alpha.When a theory closes too perfectly, the world answers with a correction, and sometimes that correction has a signature.## The Fine-Structure Constant and Its Hidden OffsetsNow we return to physics.The fine structure constant alpha is measured with astonishing precision. In the 2022 CODATA recommended values, the relative standard uncertainty is on the order of 1.6 x 10^-10. This is not a loose, drifting constant. It is one of the sharpest numbers we have.CODATA 2022 gives, in concise form:1/alpha = 137.035999177...
alpha = 0.0072973525643...In our hypothesis there are three relevant values in play.The symbolic node:
137The lattice base: ( 37 2 ) /10 = 136.9The measured constant:
137.035999177...We are not claiming to derive alpha. We are doing something narrower. We are watching what happens in the offsets.### The first gap: 1/alpha against 137Compare the measured inverse constant to the pure integer:DIFF = 1/alpha - 137
= 0.035999177...It sits just under 0.036. Close enough to feel like it is leaning on that doorstep.And here the aperture matters. At a six decimal glance, the remainder reads as 0.035999, one micro-unit shy of 0.036000.That is not numerology. It is a simple statement:0.035999 is 35,999 millionths.
And 35,999 is 36,000 - 1.There is the familiar shape again. A clean target. A one-unit miss. A one-off engine, now appearing in a physical remainder.Now invert the scale of that miss:1 / 36,000 = 0.000027777...The decimal opens with 27.It is a small thing, almost too small to mention, which is exactly why it belongs here. Until now, 27 has been the cube in the lattice. The quiet sibling of 37 and 73. And yet, from this direction, from the physics side, it surfaces naturally inside the gap.Then the gap itself folds back into the mirror primes:36,000 / 1,000 = 36
and 36 = 73 - 37 = 6 2The difference between the mirror primes becomes the scaling factor inside the correction.Closure tries to happen.
The miss stays structured.
The remainder still speaks.### The second view: alpha itself, and the 729 doorwayNow flip from 1/alpha to alpha.At the seven decimal aperture, alpha reads: 0.0072973The digits open with 729.Not the full 729927 lock we built earlier. Just the front gate. But even that front gate is not empty decoration.729 is 3 6.
It is also 27 2.
It is also 9 3.A perfect square and a perfect cube, made from a single prime's power ladder.And in base 8, 729 becomes 1331, a symmetric block that is itself a perfect cube in base 10 (11 3). This is exactly the kind of cross-aperture echo we have been tracking: different coordinate systems, same internal habit.So even before we measure deltas, the constant enters through a doorway the network already knows how to light up.### The microscopic gap: alpha against the lattice baseNow compare the physical alpha to the lattice base 136.9.Invert the lattice base:1 / 136.9 = 0.007304601...Invert the measured inverse constant:1 / 137.035999177... = alpha = 0.0072973525643...The raw difference is small, on the order of 10 -6.But watch what happens at the same seven-decimal reporting aperture we have been using as our shared lens:1/136.9 = 0.0073046
alpha = 0.0072973Subtract:0.0073046 - 0.0072973 = 0.0000073At this aperture, the gap prints 73.Not in a product, or a chosen multiplier, or as a forced mirror.In the difference.In the space between the lattice square's decimal shadow and the measured electromagnetic constant.That is the cleanest collision so far between the symbolic scaffold and the physical value. The lattice does not explain alpha. It traces the shape of its separations, and the separations keep landing on the same old atoms.### A final 137 deconstructionAnd what of 37?In base 10, the square-grid can be masked by a misplaced decimal, so we allow a one-register shift. If we ask what number, when squared and scaled back by 10, yields alpha -1, we find 37.0183737: it rounds to 37.0 at one decimal, and it carries the lattice prime three times inside nine digits.This is a mark of the projection layer: a reversible base 10 transform applied to the measured constant returns you to a 37-voiced neighborhood.So far we have treated this as a resonance story: digits, cycles, offsets, apertures. But there is a deeper convergence waiting behind all of it.Because what we have been calling the lattice's central behavior, closure pursued, miss retained, miss turned into structure, is also the exact posture physics takes toward alpha.A theory closes.
The world leaves a correction.
And sometimes that correction carries a signature.## Coda: The Structured RemainderWe have shown how this lattice binds clean forms to stubborn offsets - how it pursues closure, misses in a way that carries memory, then converts the miss into the next step. Physics operates the same way. A clean theory meets measurement, and what remains is not random error but disciplined correction: the margin written as a series.In electromagnetism, that margin is often not a random smear. In many calculations it is literally built out of terms weighted by powers of alpha, often appearing in the familiar shape alpha/pi. Alpha becomes the structured weight of the margin itself. It meters how much correction must be added back in - term by term, order by order.And the alpha we quote is itself an endpoint reading, the value you see after the theory has run down through scale, after the vacuum has had its say. Even the constant arrives with a history.The symbolic system and the physical constant rhyme because closure is never total. The fine structure constant does not break the equation. It keeps it open.This pattern of structured incompleteness extends beyond physics. Godel's incompleteness theorem can be read as the ultimate formalization of the remainder: in any sufficiently rich formal system, there will always be true statements that cannot be proved within that system. The remainder is not a mistake. It is the proof that the system is alive, that there is always more structure than the current language can compress***"The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen." ~ Paul Dirac***
© 2026 Chris Preen. All rights reserved.