"The universe is 3-dimensional" ── with which ruler, and at what scale, is that a reading?
"How many dimensions is the universe?" ── you'll want to answer plainly, "three (plus time)." But as we saw in Episode 1, dimension is not a number the world possesses; it's a reading your ruler returns, \(D=-d\ln F/d\ln n\). If so, then "how many dimensions is the universe" has no single answer either ── a number only appears once you decide what, and at what scale, you measure. In this episode we line up several "theories" using one ruler. Let me give the conclusion up front ── space as a container is ~3. But the structure actually built inside it (the web of galaxies) is not 3.
The flattest way to measure dimension: inside a ball of radius \(r\), how much stuff fits? The exponent of how it grows is the dimension.
Double the radius and the contents go up by \(2^D\). A line gives \(D=1\), a surface 2, a solid 3. \(D\) can be non-integer, and it runs with scale (Episode 1). "How many dimensions is the universe" breaks down into the question of what you apply this \(D\) to and where.
First, "space itself." The dimension of space can be measured by how force dilutes (Gauss's law). In \(d\)-dimensional space, the field lines from a point source dilute over a sphere of radius \(r\), \(\propto r^{d-1}\):
Both our gravity and our electric force are inverse-square = \(d=3\). This is the substance of "the universe is 3-dimensional," and the inverse-square law has been tested precisely down to below a millimeter (~50 µm) (Episode 11). But ── this is about the container (how space spreads out). What is inside it, and how, needs a different ruler.
Now let's look at the structure actually built inside that container. Galaxies do not fill space uniformly. Filaments and walls, and enormous voids ── a sparse web, the cosmic web. Measure it with the ruler of §01, and you get less than 3.
Two-point correlation function (how much galaxies cluster)
$$\xi(r)=\Big(\frac{r}{r_0}\Big)^{-\gamma},\quad \gamma\approx1.8,\ r_0\approx5\,h^{-1}\text{Mpc}$$The dimension measured from the galaxy count within radius r
$$D_2=\frac{d\ln N(\lt r)}{d\ln r}\approx 1.2\text{–}2\quad(\text{small–medium scales, }\ \lt\text{ tens of Mpc})$$Because galaxies "cluster," widening the radius doesn't grow the contents as fast as a solid (\(r^3\)) ── so \(D_2<3\). The web of filaments and voids is, in dimensional terms, between a surface and a line (about 2).
But if you widen the radius even more, \(D_2\) approaches 3. Beyond a certain size, the universe looks the same density everywhere (the cosmological principle = homogeneity). That boundary is the homogenization scale \(R_H\approx70\text{–}100\,h^{-1}\)Mpc (the "End of Greatness"; 71–81 in WiggleZ). Let's play with this run in the figure below.
Conversely, dive to a really small scale (quantum gravity = toward the Planck length), and this time the dimension of spacetime itself runs. The spectral dimension \(d_s\), measured by diffusion (the heat kernel), goes from 4 on large scales to about 2 near the Planck scale ── dimensional reduction.
Causal Dynamical Triangulations (CDT) gives \(d_s=4.02\pm0.10\to1.80\pm0.25\); asymptotic safety and Hořava gravity give the same ── several independent methods all return "≈2 in the UV." What runs is the spectral dimension measured by diffusion; the Hausdorff dimension measured by volume stays about 4 (these two are easily confused). Episode 1's "\(D\) is a running reading" is happening even in the deepest layer of spacetime.
Make "what you count" into information, and yet another number comes out. A region's degrees of freedom are proportional not to volume but to area (holography):
A solid's worth of information (\(r^3\)) won't fit ── only the boundary's worth (\(r^2\)) fits. So the dimension measured by information is 2 (already noted in Episode 1). 't Hooft and Susskind.
The numbers so far are 3, 2, 2 ── if anything, 3 or less. So where does "bigger than 3" become right? ── only inside the hypothesis that the underlying spacetime has curled-up extra dimensions.
| Theory | Spacetime dimension | Basis |
|---|---|---|
| String theory | 10 (9 space + 1 time) | Critical dimension (cancellation of the Weyl anomaly) |
| M-theory | 11 | The maximal dimension for supergravity |
| Bosonic string | 26 | Critical dimension |
The extra dimensions are curled up small (compactified) and invisible in daily life. But ── if gravity alone leaks into the extra dimensions, the inverse-square law deviates at short range. This is exactly Episode 11's "hidden dimensions are the signature of \(1/r^{2+n}\)":
For large extra dimensions (ADD), \(M_{\rm Pl}^2=M_*^{\,2+n}R^n\). The torsion balance (Eöt-Wash) has tested inverse-square down to \(52\,\mu\)m and ruled out a gravitational-strength deviation for \(\lambda>38.6\,\mu\)m ── no sign so far. "Bigger than 3" is still an untested hypothesis.
Let's line them all up. It's a difference in what the same ruler \(D=-d\ln F/d\ln n\) is applied to, and at what scale.
| Theory / what's measured | Formula | Dimension \(D\) | Is it 3? |
|---|---|---|---|
| Container: space | \(F\propto1/r^{d-1}\) (inverse-square) | 3 | Nearly 3 (container) |
| Contents: the cosmic web | \(N(\lt r)\propto r^{D_2}\) | ≈1.2–2 →3 | Not 3 (→3 at homogenization) |
| Quantum-gravity UV: spectral dimension | \(P(\sigma)\propto\sigma^{-d_s/2}\) | 4→2 | 2 in the UV |
| Information: holography | \(S\le A/4\ell_P^2\) | 2 | 2 (area law) |
| Critical phenomena (per phenomenon) | Upper critical dimension \(d_c\) | 4 · 6 · 2 | Per phenomenon |
| Extra dimensions (string, hypothesis) | \(1/r^{2+n}\), \(M_{\rm Pl}^2{=}M_*^{2+n}R^n\) | 9–10 | Bigger than 3 (hypothesis) |
| Complex dimension (Episode 8) | \(D=a+i\beta\) | complex | Non-integer |
The numbers are 2, 3, 4, 6, 9, 10… ── they don't converge on one. This isn't a contradiction; it's the very conclusion of Episode 1 ── dimension is not a number the world possesses; it's a reading returned by the ruler and the scale. So "how many dimensions is the universe" has no single answer.
The 3 of the container (the spread of space), the ~2 of the contents (the galaxy web), the 2 of the UV (spacetime's deepest layer), the 2 of information (the area law), and the 9–10 of the extras (a hypothesis about the underlying layer) are separate answers to separate questions. They only share the word "dimension"; the thing being measured differs.
And the answer to your question ── "the structure of the universe is not 3" is correct (the cosmic web of the contents is fractal, about 2). "Bigger than 3" is correct only in the reading for extra dimensions (a hypothesis). They are two different questions, and neither is "exactly 3" ── on that point, your intuition is right.
The homogenization of the cosmic web (\(D_2\to3\)) is consistent with standard cosmology (ΛCDM, the cosmological principle), and the "infinitely self-similar fractal universe" theory has been ruled out. The specific value of \(D_2\) varies with the sample and the definition (correlation dimension vs. box-counting, and the scale range) (\(\approx1.2\)–\(2.5\)). The spectral dimension \(4\to2\) is a quantity internal to the theory that several quantum-gravity methods return; it is not something measured directly in our universe. Extra dimensions are currently a hypothesis with zero sign. The rigorous version of holography is established in anti-de Sitter (a box); the de Sitter version for our universe is uncharted (noted in Episode 1).
And the biggest hole ── "why is the container-space exactly 3-dimensional?" ── is not specific to this framework but an unsolved problem of physics as a whole (Episodes 1 and 11). What this episode did is show, at cosmic scale, "there is no single number; it's decided by the ruler and the scale" ── not "we solved why it's 3." As the code goes, we plant no flag.
"How many dimensions is the universe" has no single answer (Episode 1, "dimension is a reading"). The container-space is ~3 (inverse-square). But the contents' cosmic web is not 3 ── with a fractal dimension \(D_2\approx2\), it only approaches 3 beyond the homogenization scale (\(\sim100\,h^{-1}\)Mpc). Further, the quantum-gravity UV is 2 (spectral dimension 4→2), information gives 2 (the area law), and extra dimensions give 9–10 (a hypothesis, being hunted via \(1/r^{2+n}\)).
The numbers don't converge on one (2, 3, 4, 6, 9, 10…) ── that is the conclusion. Even under the same word "dimension," the thing being measured differs, so don't mix them. "The structure of the universe is not 3" is correct as the contents' fractal, and "bigger than 3" is correct only as extra dimensions (a hypothesis). The remaining "why is space exactly 3?" we leave open ── we plant no flag.
Print / make a PDF: Ctrl+P (⌘+P on Mac). On screen, changing the measuring radius with the slider shows how the cosmic web's dimension \(D_2\) runs from 2 to 3. "See the answer" opens each solution.