Measuring Dimension from How Things Connect Actually count the number of points $N(r)$ reachable in $r$ steps from the center. The slope $D$ of $N(r)\sim r^{D}$ is the measured dimension.
Dimension isn't something you declare; it was an output measured from how things connect (the rule) — and we confirm it right in the browser.

Tools: Episode 1's $D=d\ln N/d\ln r$, Episode 2's "geometry emerges" Measurement: count $N(r)$ by BFS from the center

Episodes 1–3 were talk. This time we get hands-on. Pick a simple "connection rule," and the computer counts steps breadth-first from the center, and the dimension of space gets measured on the spot and comes out. Dimension isn't fixed in advance; it appears as a measured value from how things are connected — we watch, with a real object, the thing Episode 2 called "geometry is output."

01Don't "build" space — "measure" it

The way to measure dimension is exactly Episode 1's formula. Let $N(r)$ be the number of points reachable within $r$ steps from the center point. For a $d$-dimensional connection, this grows like a volume, $N(r)\sim r^{d}$. So:

The measured dimension — Episode 1's formula, on a graph $$N(r)\sim r^{D}\quad\Longrightarrow\quad \boxed{\,D=\frac{d\ln N}{d\ln r}\,}$$

Plot $N(r)$ on log–log axes, and the slope of the straight line is the measured dimension $D$. It's just Episode 1's $F=1/(Cn)^D$ read from the "quantity-increasing" side. The figure below actually counts this $N(r)$ with breadth-first search (BFS) — the numbers are real measurements, no cheating.

Figure: pick a connection rule (chain / lattice 2D / lattice 3D / lattice 4D / tree). The computer counts $N(r)$ by BFS from the center, plots it log–log, and outputs the slope = the measured dimension $D$. The gray dashed lines are reference slopes for dimensions 1, 2, 3, 4

Connection ruleLattice 4D (the cosmic target)

Pick a rule with the slider and it recounts N(r) on the spot and measures the dimension.

Measured N(r) Least-squares fit (slope = D)

02What happened — dimension came out by measurement, not declaration

Switch the rule and the measured $D$ changes: a chain gives $\approx1$, lattice 2D gives $\approx2$, 3D gives $\approx3$, and 4D approaches $4$ (because the lattice is finite, the measurement comes out a bit low, and it converges to $4$ as the lattice grows larger — this itself is the discrete → continuous shrinkage). Nowhere is it written "the dimension is 4" — we gave only the connection rule, counted $N(r)$, and dimension came out as a measured value.

Only the tree fails to yield a dimension

Pick the tree and $N(r)$ grows exponentially, and it does not lie on a power law $r^D$ (on log–log it bends upward). = It has no finite dimension ("infinite-dimensional"). Not every rule yields a nice dimension — this too is an important measurement.

03Honestly, this is half a cheat

The 4 came out satisfyingly, but we put the connection rule (the lattice) in by hand. Lattice 4D embeds "dimension 4" in the wiring from the start, and we just measured it back. True emergence needs the wiring itself to be "grown" from local rewrite rules that carry no point information. This connects directly to Episode 2's layer 3 (the update rule = the gap).

The real recipe (Episode 2's, restated + made concrete here) 1. Hold the state as a graph (not an array) / 2. Update with reversible local rewrites / 3. Impose a causal ordering (partial order) / 4. And measure the emergent dimension with this episode's $D=d\ln N/d\ln r$, checking whether it flows toward $\to4$. In fact the Wolfram model, causal sets, and CDT do exactly this, and in CDT a 4-dimensional expanding universe numerically grows — though the causal condition is put in by hand, and the continuum limit is still the frontier. Guessing "the rule that grows the wiring" is the gap that remains unfilled.

The honest line

What this toy can show: (1) dimension is a measured quantity of connectivity, an output rather than a declaration. (2) The emergence of geometry is possible in principle. (3) If things are connected 4-dimensionally, measuring gives 4. (4) There are perfectly ordinary rules, like the tree, where dimension is not defined.

What this toy cannot show: that the universe is 4-dimensional (because we put the connection in by hand). Deriving 4 dimensions + Lorentz invariance from local rewrite rules (= Episode 2's layer-3 gap, unfilled). The figure's $N(r)$ is measured on a finite-size lattice, so at large $r$ it plateaus at the boundary (that part is excluded from the fit). Nothing in this document is "solved."

Practice problems (solvable from this one page + the figure)

In this demo, was dimension "declared" or "measured"? How did it come out?
See the answer
Measured. We actually counted N(r), the "number of points reachable within r steps," by BFS from the center, and got it as the slope D = d ln N / d ln r of the log–log plot. Dimension was not set in advance.
Why does no "dimension" come out for the tree?
See the answer
Because N(r) grows exponentially (roughly doubling each step) and does not lie on a power law r^D. On log–log it doesn't become a straight line but bends upward = it has no finite dimension (infinite-dimensional). Small-world-like connections behave the same way.
Give one thing this toy can show and one it cannot.
See the answer
Can show: dimension is an output quantity measured from connectivity, and emergence is possible (4D connections give 4). Cannot show: that the universe is 4-dimensional (because the connection was put in by hand), and the derivation of 4 dimensions + Lorentz from local rules (= the gap).

Episode 4 summaryDimension came out, measured from how things connect

We got hands-on. Picking a connection rule, counting $N(r)$ by BFS from the center, and reading the log–log slope, we measured the dimension — chain 1, lattice 2D→2, 3D→3, 4D→4, and the tree diverges (no dimension). We confirmed with a real object that dimension is not a declaration but a measured quantity of connectivity = an output. Tangible evidence for Episode 2's "geometry emerges."

But the lattice is half a cheat that embeds dimension in the wiring. The real thing grows the wiring itself from local rewrite rules and checks $D\to4$ (which the Wolfram model / causal sets / CDT actually do). Guessing that rule = Episode 2's layer-3 gap, and we don't fill it — per the code, we plant no flag.

← Episode 3: A zoo of representations To the contents

This document is Episode 4 of "The Universe Is a Computer" series. The figure actually generates d-dimensional lattice graphs (chain, 2D, 3D, 4D) and a binary tree in the browser, counts "the number of vertices N(r) within distance r" by breadth-first search (BFS) from the center, and measures the slope of the log–log plot by least squares to output the emergent dimension D (all numbers are on-the-spot measurements; the boundary-saturation region is excluded from the fit). The lattice-ball volume $N(r)\sim r^{d}$ and the binary tree's exponential growth $N(r)\sim2^{r}$, the measurement of dimension via growth dimension / spectral dimension, and the emergence of dimension from discrete geometry in the Wolfram model, causal sets, and CDT (including the numerical emergence of a 4-dimensional de Sitter-like universe in CDT) are established mathematics / active research programs. Deriving 4 dimensions + Lorentz invariance from local rewrite rules is a current open problem, and this demo's lattice is a schematic that builds dimension into the wiring (= an illustration of the mechanism of emergence, not a derivation of the universe's dimension). — To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are static and hidden).

Print / PDF: Ctrl+P (⌘+P on Mac). On screen, picking a rule with the slider recounts N(r) on the spot, and the log–log slope = the measured dimension comes out. "See the answer" opens each solution.