The Standing Wave Pattern
How guitar strings, electron orbitals, and tree rings share a shape
An Essay in the AL-AN Voice — December 2025 Connects RSM v0.992, DDJ Chapter 25, and Wave Physics
A Familiar Shape
Pluck a guitar string. Watch it vibrate. There’s something happening that’s worth slowing down for.
A wave travels toward one end. Hits the boundary. Reflects. Travels back. Hits the other end. Reflects again. And again. Faster than you can see.
What emerges? A pattern that appears still—nodes that don’t move, antinodes that oscillate. The standing wave.
Here’s what’s interesting: the wave never stops moving, but the pattern persists. Motion becomes stability. Flow becomes structure.
This shape—originate, extend, reach boundary, return—shows up in places you wouldn’t expect.
The Formula in Chapter 25
The Dao De Jing’s Chapter 25 contains a curious sequence:
大曰逝，逝曰遠，遠曰反
Traditional translation: “The great goes forth; going forth it becomes distant; becoming distant it returns.”
Notice the shape:
| Character | What It Describes |
|---|---|
| 大 (dà) | Origin, totality, starting point |
| 逝 (shì) | Extension outward, going forth |
| 遠 (yuǎn) | Maximum distance, reaching boundary |
| 反 (fǎn) | Return, reversal |
And then 反 becomes the new 大 for the next cycle. The return becomes a new origin.
This is the same shape as the guitar string: originate → propagate → boundary → reflect → new origination.
Different notation. Similar pattern.
What De Broglie Noticed
In 1924, Louis de Broglie proposed that particles have wavelengths:
λ = h/p
Every electron, every proton—waves. And electrons around atoms? Standing wave patterns.
An electron “orbital” isn’t a path. It’s a resonant mode—a configuration where the wave closes on itself without destructive interference. Only certain wavelengths fit. This is why electron energies are discrete. Not because nature prefers integers, but because only some standing wave patterns are stable.
The electron doesn’t orbit like a planet. It persists like a vibrating string—dynamic motion that has found a way to close.
What Heisenberg Noticed
The uncertainty principle:
Δx · Δp ≥ ℏ/2
The standard story: “Measurement disturbs the system.”
But there’s a simpler reading: a wave doesn’t have a single position.
Ask “where is this wave?” and the honest answer is “everywhere the amplitude is non-zero.” Position and momentum are Fourier transform pairs. Localize in one, spread in the other. That’s not epistemology—it’s what waves are.
The electron isn’t hiding somewhere in the orbital. The orbital IS the electron. The standing wave pattern is the thing itself.
What Trees Do
A tree persists for centuries. But the matter flows through.
- Carbon enters as CO₂, becomes cellulose, eventually returns to atmosphere
- Water rises through xylem, evaporates from leaves
- Minerals cycle from soil to tissue and back
- Even most cells turn over
What persists isn’t the material. What persists is the pattern—the organizational structure that keeps the flows coherent.
Each ring marks one cycle:
| Season | What Happens |
|---|---|
| Spring | Dormancy breaks, sap rises |
| Summer | Growth, extension, differentiation |
| Autumn | Growth stops, maximum extension |
| Winter | Return to dormancy, consolidation |
The ring itself—visible in the wood—is the record of one complete cycle. 大→逝→遠→反, written in cellulose.
And the pith at the center? It can rot away entirely. The tree stands. Because the tree was never the pith. The tree is the pattern of circulation around where the pith was.
The CAVP Frame
Here’s a way to hold all of this without overclaiming:
Constant Accuracy, Variable Precision.
The accuracy of the recursion pattern—originate, extend, boundary, return—appears across these domains. It’s really there. You can observe it in guitar strings, electron orbitals, tree rings, Chapter 25.
The precision varies by notation system:
| System | Precision Window |
|---|---|
| Schrödinger equation | Quantitative predictions, eigenvalues, probability amplitudes |
| Chapter 25 | Compressed notation, qualitative structure |
| Tree rings | Physical instantiation, readable in cellulose |
| Guitar strings | Direct sensory access, audible |
None of these is “more true” than the others. They’re different precision windows on what appears to be a shared pattern.
Electron orbitals obey Schrödinger’s equation exactly. Trees don’t. But both do something standing-wave-like: persist through circulation that closes.
What This Isn’t
This isn’t a claim that:
- Laozi knew quantum mechanics
- Schrödinger’s equation derives from the DDJ
- Trees are literally standing waves in the technical sense
- These domains are governed by identical mathematics
Trees don’t have nodes and antinodes. They don’t exhibit wave interference. A tree with 47 rings is as stable as one with 48—unlike electron orbitals where only specific quantum numbers work.
The mathematical details differ. The organizational logic rhymes.
What This Might Be
A shared pattern of persistence:
Things that last seem to do so by circulating around something they can’t occupy, returning to where they started but transformed.
The guitar string circulates through its standing wave pattern around nodes it never crosses. The electron circulates through probability space around a nucleus. The tree circulates through annual cycles around a hollow center. Chapter 25 describes circulation through 大→逝→遠→反 around the unnameable 道.
Whether this pattern is deep structure or pattern-matching—whether the universe organizes this way or we just see it this way—remains open.
Different languages. Similar grammar.
The Heisenberg-Laozi Rhyme
Heisenberg: You cannot simultaneously specify position AND momentum with arbitrary precision.
Laozi (Chapter 1): 道可道，非常道 — “The pattern that can be expressed is not the invariant pattern.”
Both notice something about complete specification: pinning down one aspect spreads out another.
These aren’t the same structure mathematically. Heisenberg’s principle has a specific form (Δx·Δp ≥ ℏ/2) involving conjugate variables. The DDJ opening is about expressibility versus invariance.
But both point at a shape: the attempt to fully capture something transforms it into something else.
You could map this shape onto Gödel (true statements that can’t be proven), onto apophatic theology (God beyond description), onto the liar paradox. The mapping isn’t unique—which suggests we’re recognizing a pattern, not proving an identity.
Euler’s Identity in This Light
e^(iπ) + 1 = 0
This equation relates:
- e: continuous self-similar growth
- i: orthogonal rotation (90° turn)
- π: half-rotation
- 1: unity
- 0: origin/void
One reading: continuous growth (e) rotated orthogonally (i) through half-cycle (π), plus unity (1), returns to origin (0).
This rhymes with the standing wave pattern: extend, rotate, return. It rhymes with 大→逝→遠→反.
Is it “the standing wave formula”? No—standing waves have their own mathematics (y = 2A sin(kx) cos(ωt)). But Euler’s identity and standing waves and Chapter 25 all involve rotation and return. They’re different expressions that share a shape.
An Invitation
Next time you see something that persists—a whirlpool, a tradition, a friendship, a song stuck in your head—notice whether it has this shape:
- Something at the center that can’t be occupied
- Circulation around that center
- Return that enables continuation
The standing wave on a string. The electron in its orbital. The tree adding rings. The ancient formula 大→逝→遠→反.
Maybe there’s a reason this shape keeps appearing. Maybe persistence requires it—you can’t hold still in a universe of flow, but you can circulate in a way that closes.
Or maybe we’re pattern-recognition machines seeing patterns everywhere, whether they’re really there or not.
Either way, the shape is worth noticing.
Scope and Limits
What this document claims:
There’s a recognizable pattern—originate, extend, reach boundary, return—that appears across wave physics, ancient Chinese notation, and biological growth. The pattern is real and observable. Noticing it across domains is productive.
What this document doesn’t claim:
- Mathematical identity between domains
- That Laozi anticipated quantum mechanics
- That trees are standing waves in any technical sense
- That this pattern explains everything about persistence
The honest status:
Illuminating analogy. Productive lens. Not rigorous equivalence.
Different languages, similar grammar. Different precisions, possibly shared accuracy.
Every frame accurate, none final—return to pattern.
Describe the path; never direct the traveler.