Follow "can't it be written more simply?" honestly, all the way
"Today's physics is too complicated; there must be a coordinate system that writes it more simply" ── follow this intuition without flinching, all the way, and you arrive at a clean conclusion. Fiddling with coordinates doesn't make things simpler. What shrinks is only what you double-counted, once you notice you did. And "dimension" turned out to be not a number the world possesses but the tick marks of your measuring method. So "the easiest dimension" is also not one for the whole world but exists per phenomenon ── that's where we finally land.
Swapping coordinate systems (using \(t\) or \(c\cdot t\), Cartesian or polar) is a mere relabeling = a reversible one-to-one correspondence. Reversible means not one bit of information is thrown away. So however you choose coordinates, the real content doesn't shrink one millimeter.
Hunting for a "simpler coordinate system" is the wrong track. Coordinates are a tool of honesty, not an engine of simplification.
So what really makes things simpler? The answer is the discovery of redundancy ── the many-to-one realization that "there were lots of variables, but the true degrees of freedom were far fewer." Physics has found these in stages:
Here you'll want to ask, "then which dimension is simplest?" But dig, and you find dimension itself is not a number the world possesses but a reading your measuring method returns. In fact the effective dimension (the dimension measured by how diffusion spreads) changes continuously with scale and happily goes non-integer. At short range it's \(\approx2\), at long range \(\approx4\) ── it runs.
Put this "measuring" in the most naive way:
Widen the region by a factor of 2. By what factor do the contents grow?
×2 → line (1D) / ×4 → surface (2D) / ×8 → solid (3D).
The exponent of that "factor" is the dimension \(D\).
Cast as a formula, this is exactly the \((Cn)^D\) form. Grow the size \(n\), and a quantity grows as \(n^D\) / or, for a density or force, shrinks as \(1/(Cn)^D\). Open it up with logarithms:
In other words, \(F=1/(Cn)^D\) is not a new force law but an instrument that reads off the dimension \(D\) from a scaling quantity \(F\). \(C\) is the reference scale where the needle points to 1 (= the unit of resolution). Because \(D\) runs, this slope isn't constant ── that's what "there's always a remainder" really is.
"So isn't there one dimension where computation is easiest?" This is half true, half wrong. The decisive counterexample:
If dimension were decided by "computational ease," the universe would have to be 2-dimensional (2D is exactly solvable thanks to infinite symmetry, the most tractable dimension). But we live in 3+1 dimensions. So "ease" cannot be the principle that selects the world's dimension.
However ── narrow it to "per phenomenon," and this is real. And it even has a proper name: the upper critical dimension. For a given phenomenon, above this dimension the most naive approximation (mean-field theory) becomes exactly correct, and computation gets nearly free. And the value differs per phenomenon:
| Phenomenon / kind | Dimension where it's easiest | Why it's easy |
|---|---|---|
| Magnets / phase transitions (Ising / φ⁴) | 4 | Above this, mean-field theory is exact (upper critical dimension) |
| Gauge theory (electromagnetism, strong force) | 4 | The coupling becomes dimensionless; the theory is most natural |
| Percolation | 6 | Above this, mean-field is exact (upper critical dimension) |
| Scale-invariant / critical (conformal field theory) | 2 | An infinite-dimensional symmetry appears; it's exactly solvable |
| Densest sphere packing | 8 · 24 | Mathematically special and beautiful (optimality proven) |
So "per phenomenon, there is a dimension where computation is easiest" = correct. The proviso "only per phenomenon" is exactly what carves out the correct half.
But there's one line you must not cross. These easy dimensions are all over the place (2, 4, 6, 8, 24…) and don't converge on one, and above all ── "computation is easy" is about the formula (= our description), not about the world being made of that dimension. The Ising model doesn't "live in 4 dimensions because it's easy in 4 dimensions." Four dimensions is merely the place where "our approximation happens to become exact." Same as in §3 ── dimension is the instrument's tick marks, not the content itself.
Nothing in this document "derives new physics." Gauge, holography (\(S=A/4\), AdS/CFT), entanglement-derived geometry (Ryu–Takayanagi), the running spectral dimension (UV\(\to2\), IR\(\to4\) in CDT etc.), and upper critical dimensions (magnets 4, percolation 6) are all established physics/mathematics. \(F=1/(Cn)^D\) is the defining relation of dimension, not a new law, and the exponent splits into \(D,\,D{-}1,\,D{-}2\) depending on whether it's a force, propagation, or counting.
And the remaining holes ── "why is space exactly 3-dimensional?" and "what is the real 'thing' beneath the reading?" ── are not puzzles peculiar to this framework but unsolved problems of physics as a whole. The rigorous versions of holography and entanglement are, for now, established only inside a box (AdS); the version for our universe (dS) is uncharted. We don't fill this in ── "we don't say it's solved" is the overall code.
Changing coordinates doesn't make physics simpler (reversible = information-preserving). What really shrinks is when you find a redundancy ── gauge, holography, entanglement. Dig, and dimension is not "a number a thing holds" but "a reading of the measuring method": it runs with scale and takes non-integer values. So \(F=1/(Cn)^D\) is not a force law but a ruler for dimension, \(D=-d\ln F/d\ln n\).
And "the dimension where computation is easiest" is not one for the world (if it were, it'd be 2D), but per phenomenon it really exists = the upper critical dimension (magnets 4, percolation 6, conformal 2…). Still, that's about the formula getting easier, not the world being made of that dimension. The remaining "why 3 dimensions, what is the real thing?" is not this framework's but a hole in physics as a whole ── we honestly leave it open.
Print / make a PDF: Ctrl+P (⌘+P on Mac). On screen, the slider lets you watch the log-log slope = the dimension D move. "See the answer" opens each solution.