Thinking in Waves Episode 30 ／ Part XII · Moving the dimension
Change exactly one character in the equation
Move the dimension and the order moves by halves ──
only 3 dimensions gives \(\alpha=0\)
In the wave equation we change the single letter \(d\). And then ── in even dimensions waves leave a tail, in odd ones they pass cleanly.
★ The order of the tail is \(\alpha=(d-3)/2\). Dimension and order are one ladder, and one dimension is exactly half an order.
★★ And Episode 26's hypothesis gets tested in a completely different field: the geometry of empty space.
Episode 29 found that "second order means both directions". This time we barely change the equation at all. Made spherically symmetric, \(d\) appears in exactly one place ── the coefficient \((d-1)\). We move that.
01Line up the exact solutions ── retarded Green's functions
The response to a point source is known in each dimension. With \(c=1\) and \(s=t-r\) (time since the front):
| \(d\) | Green's function | Behind the front | Order \(\alpha\) |
|---|---|---|---|
| 1 | \(\tfrac12\theta(s)\) | ★ a step that never leaves | −1 |
| 2 | \(\dfrac{1}{2\pi\sqrt{t^2-r^2}}\) | ★ a tail \(s^{-1/2}\) | −1/2 |
| 3 | \(\dfrac{\delta(s)}{4\pi r}\) | ★ sharp (no tail) | 0 |
| 4 | \(\propto\partial_t\big[\theta(s)/\sqrt{t^2-r^2}\big]\) | ★ a tail \(s^{-3/2}\) | +1/2 |
| 5 | \(\propto\delta'(s)/r^3+\delta(s)/r^2\) | sharp but differentiated | +1 |
| 6 | ── | ★ a tail | +3/2 |
The order is defined by \(G\propto D^\alpha\delta(s)\) (\(D=d/dt\)). A bare \(\delta\) is \(\alpha=0\); \(\theta(s)\) is its integral, so \(-1\); and \(s^{-1/2}\) is a half-order integral — the same shape as Episode 2's Warburg — so \(-1/2\).
02Measuring the front exponent numerically
If \(G\propto s^\lambda\), then \(\lambda=-\alpha-1\). Measured from the exact solutions:
| \(d\) | \(s=t-r\) | \(G\) | Measured \(\lambda\) | Predicted |
|---|---|---|---|---|
| 1 | \(10^{-6}\) | 5.0000e−1 | 0.00000 | 0 |
| 2 | \(10^{-6}\) | 1.1254e+2 | −0.50000 | −0.5 |
| 4 | \(10^{-6}\) | 3.5355e+8 | −1.50000 | −1.5 |
The heart of §02
Converting with \(\alpha=-\lambda-1\) gives \(-1,\ -1/2,\ +1/2\) ── matching \((d-3)/2\).
★ And \(d=3\) alone has no \(s^\lambda\) form at all (a bare \(\delta\), zero width) ── that is Huygens' principle.
03★ Where the half order comes from ── remove one dimension, lose half an order
This is the heart of the episode: Hadamard's method of descent ── the solution in \(d\) dimensions is the solution in \(d+1\) dimensions integrated along one direction. A 2D wave is what you see from a line of sources laid uniformly along \(z\) in 3D.
\(G_3\) is a sharp \(\delta\), yet integrating over \(z\) produces a tail
| \(r\) | \(t\) | Descent integral | Exact 2D solution | Rel. error |
|---|---|---|---|---|
| 1.0 | 2.0 | 0.09188888 | 0.09188815 | 8.0e−6 |
| 1.0 | 5.0 | 0.03248739 | 0.03248737 | 7.1e−7 |
| 2.0 | 6.0 | 0.02813490 | 0.02813488 | 5.9e−7 |
Why \(1/\sqrt{\ }\)? Count the arrival delays from the points along \(z\).
| \(s\) | \(dz/ds\) (numeric) | \(\sqrt{r/2s}\) | Ratio |
|---|---|---|---|
| \(10^{-3}\) | 22.377447 | 22.360680 | 1.000750 |
| \(10^{-5}\) | 223.608475 | 223.606798 | 1.000008 |
The heart of §03 ── ★★★ Episode 26's hypothesis, tested in another field
Episode 26 [rewritten hypothesis]: a fractional order appears when the coarse-grained degrees of freedom are distributed scale-invariantly (as a power law).
★ Here: the removed degrees of freedom are "the line along \(z\)", and their delays are distributed as the power law \(s^{-1/2}\). The order that came out is exactly \(-1/2\). Just as the hypothesis says.
★ And it does not conflict with Episode 26's counterexample (the uniform spring chain, integer order) ── the \(z\) axis is uniform as space, but becomes a power law once converted to delays. What matters is "uniform in what". The geometry of the light cone is what made the power law.
04★ Where the tail comes from ── the order appears as a \(1/r^2\) potential
There was only one term containing \(d\). Let us remove it.
A 1D wave equation plus a \(1/r^2\) potential. \(d\) survives only inside \(l\)
★★★ And \(l=(d-3)/2\) is exactly the order \(\alpha\) from §01 ── \(\alpha=l\). The order has reappeared as the strength of a \(1/r^2\) potential.
| \(d\) | \(l=\alpha\) | \(l(l+1)\) | Episode 13's \(\lambda=-l(l+1)\) | What it is |
|---|---|---|---|---|
| 1 | −1 | 0 | 0 | ★ no potential |
| 2 | −1/2 | −0.25 | 0.25 | ★★ exactly Episode 13's critical \(\lambda=1/4\) |
| 3 | 0 | 0 | 0 | ★ no potential |
| 4 | +1/2 | 0.75 | −0.75 | repulsive |
| 5 | +1 | 2 | −2 | repulsive |
| 7 | +2 | 6 | −6 | repulsive |
The heart of §04 ── ★★ back to Episode 13
Only \(d=1\) and \(d=3\) make the potential vanish identically. There the equation is just the 1D wave equation ── there is nothing for a tail to come from.
★★★ And \(d=2\) lands exactly on Episode 13's critical \(1/r^2\).
\(l(l+1)\) has its minimum \(-1/4\) at \(l=-1/2\), so \(\lambda=-l(l+1)\le1/4\) ── no dimension ever exceeds critical. Equality holds only at \(l=-1/2\), i.e. \(d=2\).
Episode 13 showed the exponents turning complex for \(\lambda>1/4\) (imaginary order, Efimov). ★ The dimension of space cannot take you there ── \(d=2\) is the very edge.
Running this form numerically (absorbing boundaries, outgoing pulse from \(r=10\), receiver at \(r=30\)):
| \(d\) | \(l=\alpha\) | +2 s / peak | +10 s | +20 s | Fate of the tail |
|---|---|---|---|---|---|
| 1 | −1 | 3.7e−6 | 2.7e−13 | 5.1e−12 | ★★ exactly zero |
| 2 | −1/2 | 8.7e−3 | 9.6e−3 | 1.2e−2 | ★ stays |
| 3 | 0 | 3.7e−6 | 2.7e−13 | 5.1e−12 | ★★ exactly zero |
| 4 | +1/2 | 2.5e−2 | 2.3e−2 | 1.3e−2 | ★ stays |
| 5 | +1 | 6.6e−2 | 4.3e−2 | 2.2e−3 | fades |
| 6 | +3/2 | 1.2e−1 | 4.6e−2 | 1.2e−2 | ★ stays |
★ \(d=1\) and \(d=3\) sit at \(10^{-13}\) ── the rounding-error level. The tail really is absent. Everywhere else a tail appears, and in even dimensions (half-integer \(l\)) it stays, while in odd ones (integer \(l\)) it fades.
Stated honestly
This reduced 1D computation is not a proof of Huygens' principle. The origin has been replaced by an absorbing boundary, so the only dimensions where "exactly zero" is actually demonstrated are \(d=1\) and \(d=3\).
★ The evidence for sharpness at \(d\ge5\) is the exact Green's functions of §01–02.
05★ Why 3 dimensions is the special one
| \(d\) | \(\alpha=(d-3)/2\) | Behind the front | How it sounds |
|---|---|---|---|
| 1 | −1 | a step that never leaves | ringing forever |
| 2 | −1/2 | a tail \(s^{-1/2}\) | pond ripples that never die |
| 3 | 0 | ★ a bare \(\delta\) | ★ undistorted, and gone once past |
| 4 | +1/2 | a tail \(s^{-3/2}\) | a tail |
| 5 | +1 | \(\delta'\) (differentiated) | sharp but the waveform is distorted |
| 7 | +2 | \(\delta''\) (twice differentiated) | sharp but distorted further |
The heart of §05
Sharpness (no tail) holds only in odd dimensions \(d\ge3\). But even among odd ones, \(d\ge5\) has \(\alpha\ge1\) ── the waveform gets differentiated and distorted.
★★★ \(d=3\) is the only dimension with \(\alpha=0\) exactly.
★ So three dimensions is the only one where a wave arrives undistorted and then is gone. That is why conversation works and why music does not turn to mud. In two dimensions, every word you speak would keep piling up in the room.
06★ The habit learned in Episode 29 ── is this \(\alpha\) that \(\alpha\)?
Episode 29 corrected Episode 15 for mixing two different \(\alpha\)s. So we check ourselves here too. There is a tempting coincidence: Episode 29's passivity band was \(\beta\in[-1,+1]\) (width 2), and here \(\alpha=(d-3)/2\) lies in \([-1,+1]\) for \(1\le d\le5\). The same width-2 band.
| \(d\) | \(\alpha\) | \(\cos(\alpha\pi/2)\) | Passivity says | What is actually true |
|---|---|---|---|---|
| 3 | 0 | 1.00000 | passive | ── |
| 5 | +1 | 0.00000 | passive (edge) | ── |
| 7 | +2 | −1.00000 | ★ active ＝ forbidden? | ★ but a free field in 7D conjures nothing |
The heart of §06 ── ★★ rejected
This is not a meaningful coincidence.
Passivity, \(\operatorname{Re}Z\ge0\), is a condition on the impedance a medium has. But \(\alpha=(d-3)/2\) is an exponent from the geometry of empty space, where there is nothing to dissipate into. A free field in 7 dimensions merely distorts; it does not conjure energy.
★ Width 2 showed up in both because both count "one order either side of \(\delta\)" ── not because they come from the same constraint.
── ★★ The habit learned in Episode 29, applied immediately to ourselves: when you see the same number, first check what it was divided by.
Exercises
- What would conversation be like in 2 dimensions? Argue from \(\alpha=-1/2\).
Show the answer
Sound would never leave; it would pile up. \(G_2\propto1/\sqrt{t^2-r^2}\) decays only like \(t^{-1}\) behind the front, so every utterance leaves a reverberation.
And the tail \(s^{-1/2}\) is a half-order integral ── the same shape as Episode 2's Warburg. ★ A two-dimensional room rings as a half-order filter.
★ Ripples on water (2D) do linger this way. Though water waves are also dispersive, so this is not the only cause. - Inverting gives \(d=2\alpha+3\), so \(\alpha=-1/2\) corresponds to \(d=2\). Is Episode 2's Warburg therefore two-dimensional?
Show the answer
No. Warburg's \(-1/2\) comes from semi-infinite diffusion, an entirely different origin from spatial dimension.
★ This is the same caution as §06 ── the same \(-1/2\) can come from "the \(\sqrt t\) of diffusion" or from "the geometry of the light cone". Equal numbers are not evidence of equal causes.
★ But the mechanism is the same: both are the result of removing degrees of freedom distributed as a power law (Episode 26). Different causes, same machinery ── that is this episode's real yield. - Since \(d=2\) sits exactly at Episode 13's critical point, is there a "dimension" corresponding to imaginary order (\(\lambda>1/4\))?
Show the answer
There is none. With \(\lambda=-l(l+1)\) and \(l(l+1)\) minimised at \(-1/4\) (at \(l=-1/2\)), \(\lambda\le1/4\) always. Equality only at \(l=-1/2\), i.e. \(d=2\).
★ Even allowing real-valued dimensions, you cannot cross critical.
★ Efimov states lived at \(\lambda>1/4\) in Episode 13 because that potential came from an attraction between particles, not from geometry. The same \(1/r^2\), but the one geometry can build has a ceiling.
Summary Dimension and order are one and the same ladder
We moved the single letter \(d\) in the wave equation.
★ The order of the tail is \(\alpha=(d-3)/2\). Measuring the front exponent from the exact solutions gives \(\lambda=0,\ -1/2,\ -3/2\) for \(d=1,2,4\) ── exactly the prediction \(\lambda=-\alpha-1\). Only \(d=3\) has no \(s^\lambda\) form: a bare \(\delta\) (Huygens' principle).
★★ The half order turned out to be Hadamard's descent. \(G_2=\int G_3\,dz\) matches the exact solution to \(10^{-7}\), and the delays from points along \(z\) are distributed as \(dz/ds=\sqrt{r/2s}\propto s^{-1/2}\) ── the removed degrees of freedom followed a power law.
★★★ That is an independent test of Episode 26's hypothesis. Not electrochemistry, not viscoelasticity, but the geometry of the vacuum produced the same half order. And it does not conflict with Episode 26's counterexample ── the \(z\) axis is uniform as space, but a power law once converted to delays. What matters is "uniform in what".
★ The origin of the tail also came out. With \(v=r^{(d-1)/2}u\) the equation becomes 1D plus a potential \(l(l+1)/r^2\), with \(l=\alpha\) ── the order reappears as the strength of the potential. It vanishes identically at \(d=1\) and \(d=3\) (numerically \(10^{-13}\)), and ★★ \(d=2\) lands exactly on Episode 13's critical \(\lambda=1/4\) ── and no dimension ever exceeds it (\(l(l+1)\ge-1/4\)).
★ Only \(d=3\) gives \(\alpha=0\) — undistorted, and gone once past.
Three dimensions is the only one where a wave arrives clean and then leaves.
Finally, we applied Episode 29's lesson to ourselves ── the match between the passivity band (width 2) and this episode's band (width 2) is a coincidence. A free field has nothing to dissipate into, so they are different objects. When you see the same number, first check what it was divided by.
★ Next time we go digital ── the \(z\) transform and the unit circle. The derivative \(D\) becomes \((1-z^{-1})/T\), and \(D^\alpha=(1-z^{-1})^\alpha\) is precisely the Grünwald–Letnikov fractional derivative.
★★ And it connects straight back to this episode ── no tail = FIR, a tail = IIR. "Odd dimensions are FIR, even dimensions are IIR."
This document is Episode 30 of the "Thinking in Waves" series. The computations are in kenshou/jigen.js. On the numbers here: the front-exponent measurements, the descent integral, the distribution of delays, and the reduced 1D wave computation were all done for this article. The retarded Green's functions in each dimension, Huygens' principle (the strong form holding in odd \(d\ge3\)), Hadamard's method of descent, and the reduction to a \(1/r^2\) potential via \(v=r^{(d-1)/2}u\) are all well-known results and not discoveries of this article ── what this article did was place them on a single axis, the order \(\alpha\), make \(\alpha=(d-3)/2\) and \(\alpha=l\) explicit, and use them to test Episode 26's hypothesis. The numerical experiment in §04 replaces the origin with an absorbing boundary, so it is not a numerical proof of Huygens' principle (exact zero is demonstrated only for \(d=1,3\)). The Green's function for \(d=6\) is not written out (only its order). "Sound piles up in two dimensions" is a qualitative consequence of the form of \(G_2\); real two-dimensional systems such as water waves are also dispersive and are not explained by this calculation alone. ── To print, use your browser's Print → Save as PDF.