Absolute Compass

Celsius zero is where water freezes. That’s useful if you’re a chemist or a plumber, but it’s arbitrary — pegged to one substance’s behaviour at one atmospheric pressure. Fahrenheit zero is even worse: the coldest temperature Daniel Fahrenheit could make with a salt-ice bath in 1724. Both scales pick a human-convenient reference point and measure from there.

Kelvin fixes this. Zero kelvin is the absence of thermal energy — no molecular motion at all. It’s lethal, and the third law of thermodynamics says you can never actually reach it. But it’s the only structurally honest origin for measuring the quantity. Every temperature is a positive distance from that floor.

Political “centrism” is Celsius zero. It’s calibrated to whatever feels normal right now, in this country, in this decade. The American centre in 1955 looked nothing like the American centre in 2024, and neither resembles the Swedish centre. Move between cultures and the reference point slides with you. That’s fine for casual use, but it’s a terrible origin for a coordinate system.

the absolute compass

What’s the political equivalent of absolute zero? Zero state coercion — no taxation, no regulation, no military, no police, no courts, no impositions of any kind. Anarchy in the strict sense. Some people have a catastrophic interpretation of anarchy but in this view it is a “zero power” state of being, sort of like zero temperature. And to be sure this state of being exists in the world sometimes. International relations can be anarchic, sovereign states cooperating and competing without a higher authority. Your friendships are anarchic. Pre-state societies lived near zero for millennia. Wild animals are usually in this state. The apocalyptic version of anarchy people imagine — sudden collapse, fighting in a power vacuum, warlords filling the vacuum — isn’t actually zero. A warlord is coercion. That’s moved away from the origin. Zero is just the natural floor of the quantity being measured: how much coercive power is exerted on members of the group, and in what direction?

Measure from there and familiar things look different. Libertarianism isn’t at the origin — it advocates for property rights, defence, courts. Call it 30 kelvin. A Scandinavian social democracy sits further out. The Soviet Union and Nazi Germany are both at high radius, out near the periphery, pointing in different directions but at similar magnitude.

That’s the horseshoe. People have argued about horseshoe theory for decades — Jean-Pierre Faye coined the image in 2002, and critics rightly point out it conflates structural similarity with ideological equivalence. Stalinism and fascism aren’t the same thing. But in a radial model, you don’t need them to be. They’re at the same distance from the origin, pointing different directions. The geometry captures the structural convergence (both employ massive state power) without erasing the ideological distinction (they employ it toward different ends).

Write a political position as a vector: $\mathbf{p} = r\hat{\theta}$, where $r$ is the magnitude of state power and $\theta$ is direction — the ideological orientation of that power. The origin is anarchy, and distance from it is force. The angle tells you what the force is for.

the basis problem

But “state power” isn’t one thing. A government can regulate markets tightly while leaving personal life alone. Singapore is economically interventionist and socially conservative, but with a thriving private sector in certain domains. Or the reverse — a country with a generous welfare state and permissive social norms but a modest military. Each type of coercion has its own natural zero and its own scale.

Economic regulation, social control, military force, information control — each is a Kelvin-like axis where zero means the state doesn’t touch that domain and magnitude increases outward. Together they form a vector space. A country’s political position isn’t a single number — it’s a point in this multidimensional space, with a component along each axis.

The question every compass designer eventually hits: what’s the minimum set of independent axes you need? Poole and Rosenthal built NOMINATE scores for US congressional voting and found two dimensions explain about 85% of the variance — a liberal-conservative economic axis and a cross-cutting social one. That sounds tidy, but it’s partly an artifact of two-party competition compressing a higher-dimensional space into a binary choice. Bayesian analysis of voter survey data finds six to ten real dimensions in the US and UK. Trade policy, immigration, institutional trust, populism — these all form distinct axes that don’t reduce to left-right.

Here’s what I find most interesting. Malka, Lelkes, and Soto surveyed 99 countries in 2019 and found the correlation between economic conservatism and social conservatism is slightly negative globally. The US bundling — where “conservative” means both free-market and traditional-values — is the exception, not the rule. In much of the world, the two pull in opposite directions. The basis vectors of political space rotate depending on how a given political system bundles its issues. What looks like a natural axis in Washington is a historical accident when viewed from Warsaw or Jakarta.

This means any fixed compass — including this one — is a simplification. The axes I’m sketching here (economic, social, military, informational coercion) are a starting point, not a final decomposition. The real basis might rotate between countries the way a crystal’s axes rotate under stress. But the origin stays fixed. Zero coercion is zero coercion everywhere.

political embeddings

Maybe the problem is that we keep trying to design the axes. Name them up front — economic, social, military — and argue about whether the list is right. Every political compass since the Nolan Chart in 1969 has done this, and every one has been criticized for what it leaves out.

There’s a different approach. In machine learning, word embeddings map words into a high-dimensional vector space — not by hand-labelling dimensions like “noun-ness” or “formality” but by learning from co-occurrence patterns. “King” and “queen” end up near each other not because someone tagged them as royalty but because they appear in similar contexts. The dimensions are unnamed — they emerge from the data.

Political traditions could work the same way. Instead of deciding that “economic regulation” is axis one and “social control” is axis two, you’d take the full policy output of every government — thousands of dimensions covering everything from tariff rates to speech laws to conscription policy — and let dimensionality reduction find the structure. Principal component analysis, or something like it. The axes that fall out wouldn’t have clean labels — they’d be whatever independent directions best explain the variance in how governments actually use power.

This reframes the question. Instead of “is this a 2D compass or a 4D one?”, it becomes: how many independent components does the coercive apparatus of a state actually have? The Malka finding suggests the answer depends on where and when you look. But the method stays the same: learn the embedding, don’t design it.

And embeddings give you tools the old compass doesn’t. Cosine similarity between two political traditions measures how similarly they orient state power, regardless of magnitude. Two governments might both be high-coercion (large $\|\mathbf{p}\|$) but differ completely in direction. Or two might aim state power the same way but at different intensities — same angle, different radius. Cosine similarity captures the first distinction; Euclidean distance captures both.

$\text{sim}(\mathbf{p}_1, \mathbf{p}_2) = \frac{\mathbf{p}_1 \cdot \mathbf{p}_2}{\|\mathbf{p}_1\| \, \|\mathbf{p}_2\|}$

The horseshoe, in this framing, is a prediction: high-magnitude political vectors will tend to cluster in cosine similarity even when their stated ideologies diverge. Authoritarian regimes converge operationally because the machinery of coercion has its own logic. Concentration of power demands secret police, censorship, loyalty tests, command structures — regardless of whether the flag is red or black. The embedding would show this as a high-radius cluster, ideologically scattered but operationally dense.

And if the Malka rotation finding holds up, training this embedding on different countries would produce rotated versions of the same space. The dimensionality stays roughly constant, but the axes reorient depending on how each political system bundles its conflicts. Washington and Warsaw aren’t operating in different political spaces — they’re rotated embeddings of the same one.

plotting a few positions

These are pins on a map, not borders.

Classical liberalism sits at low radius — limited state, individual rights, courts and contracts. Near the origin but not at it, because it still wants those contracts enforced. A social democrat is further out on the economic axis (redistribution, public services) but might sit closer to the origin on the military one. A libertarian and a social democrat can share a position on information freedom while diverging sharply on economic intervention. That’s hard to see on a flat grid, but it’s obvious in a vector space.

An authoritarian nationalist pushes outward on military, social control, and information simultaneously — high radius in every dimension. Which is why authoritarian regimes of every ideology start to resemble each other operationally, even when their flags look nothing alike.

The traditional 2D compass declares the centre to be wherever the axes cross. This model says: there is no centre in the conventional sense. There’s the origin, and varying distances from it in varying directions. What people call “the centre” is whatever cluster of positions happens to be most common in their country right now. A statistical mode, not a geometric one. And it moves.

where this breaks

“Degree of force” is doing a lot of work here. Taxation and imprisonment are both state coercion, but they’re qualitatively different — one takes a percentage of your income, the other takes years of your life. Putting them on the same scale loses something important. A better model would weight different kinds of coercion by severity, but I don’t have that weighting, and I suspect nobody does.

The origin has a libertarian aesthetic, even if it’s not a libertarian argument. Putting anarchy at zero and measuring outward from there feels like it privileges minimal-state positions. I don’t think it does — the origin is descriptive, not prescriptive, and plenty of functional systems operate near it. But I understand the objection.

And there’s a deeper limit. No finite set of axes can fully represent political ideology, because ideologies don’t just answer the same questions differently — they propose different questions entirely. A Marxist and a libertarian aren’t disagreeing about where to set the tax rate. They disagree about what “property” means. The vector space model captures positions within a shared framework but can’t capture the framework disagreements themselves. Even word embeddings hit a version of this — “bank” the financial institution and “bank” the riverbank occupy the same token, and the embedding can’t fully separate them without context. Political homonyms work the same way. “Freedom” means something different in every tradition that claims it.

I still find it useful. The model explains why the “centre” keeps moving, why horseshoe convergence keeps showing up, and why adding more axes to a flat grid never quite solves the problem. The move isn’t more dimensions on a designed compass — it’s a better origin and a learned basis.

update: take the test

Added February 2026. Everything above is abstract, so here’s a concrete version you can try. Nine questions, each measuring one dimension of state coercion from zero to ten. Your answers form a nine-dimensional vector — a point in the political space this post describes. The tool projects it down to two dimensions using PCA and plots it alongside nine political traditions, from anarchy at the origin to fascism at the periphery. Magnitude is distance from zero. Direction is what the force is for.