The same sentence that governs a rotated glyph describes how a group holds an identity across upheaval. Taken precisely, though, the mathematics cuts against the most common political use of it.
In political terms, “identity emerges from invariance” says that a group's identity solidifies around fixed points of reference — core principles, traditions, historical narratives — that hold while policies and societies churn around them. So far, so familiar. But the full claim is invariance across allowable transformations, and that word does the real work.
In the mathematics, the set of allowable transformations is chosen first; the invariants are simply whatever survives it. The political analog of that set is the menu of changes a community decides it can absorb and still call itself the same community. That menu is contested and constructed — not handed down. Which is exactly the explainer's sharpest point, from the identity-boundary section: the boundary is not in the glyph; it is learned, and meaning decides where it sits.
Four mechanisms — three that fix an identity, and one that deliberately moves it — each with a place where the math complicates the story it tells about itself.
The demonstrations on the main page are not decorations here — several are precise models of an identity dynamic.
Civic vs. ethnic nationalism are two geometries of the same nation. Enlarge the group of allowable transformations and the shared identity grows thinner but reaches further; shrink it and the identity thickens but travels less. Every coalition negotiates exactly this trade.
The dogmatist's dilemma as a conservation law. The more a movement fixes (large stabilizer), the fewer forms it can take (small orbit). Adaptability and purity are formally in tension; you cannot maximize both.
An identity grounded in its own intrinsic features (language, practice, law) is stable like the cross-ratio. An identity grounded in opposition is grounded on a moving point — which is why affective polarization feels so escalatory and unstable.
The boundary of who counts as “us” is contested, not given — and the same strokes can read as a different letter. Recognition struggles are fights over where, exactly, the community will draw its invariant.
An identity robust to continuous change — one that survives gradual reform — genuinely conserves something. An identity that tolerates only the zero-change “identity transformation” conserves nothing dynamically; it is not stable, merely frozen. Caveat: genuinely continuous symmetries are rare in politics — most change is discrete — so this is the most inspirational of the analogs, not a literal mechanism.
Everything above is descriptive. The very same machinery underwrites a minority's cultural survival and an exclusionary ethnonationalism; orthodox Marxism and constitutional originalism; an independence movement and a purge. Invariance explains how an identity is held together — it does not certify that holding it together is right, or that the chosen invariants are the ones worth keeping.
That leaves two pathologies, one on each side. Over-fixing: treating contingent features as eternal, multiplying purity tests, losing the capacity to adapt — the brittle orbit of size one. Under-fixing: no durable reference at all, an identity that dissolves under the first pressure. A living political identity, like a good representation, stores the invariants that carry meaning and lets the rest vary.
The cleanest test case: a literally fixed text, where the entire fight is over which invariant of its meaning to preserve — the semantic meaning, the original expected application, or the underlying purpose. Those are three different transformation groups over the same words, so two honest originalists can reach opposite results while each insists they are conserving “the” fixed meaning. Living constitutionalism is then just Erlangen's grow the group: admit more allowable transformations, and the text supports a wider orbit of readings. The disagreement is not about whether to be faithful to an invariant — it is about which invariant the document even has. Put concretely: Scalia-style textualism, original-expected-application, and purposivism are literally three different choices of transformation group acting on the same constitutional “glyph.”
Identity-by-negation at national scale: the invariant becomes “not the other side.” Because that is a relative invariant, it ratchets — each side's identity is pinned to a moving point, so movement by either party deforms both. Solidarity hardens precisely as the shared external reference is lost.
Revolutionary and independence identities persist for generations by fixing a small set of reference points — a founding text, a martyr, a date, a wrong (e.g. a declaration of independence or a constitution) — and treating everything else as negotiable. Case studies in keeping the stabilizer small enough to manoeuvre and fixed enough to endure.
Each generation inherits the invariant through a different medium — oral, print, broadcast, network. A society can enlarge its allowable transformations across generations while trying to keep the core fixed; the transmission channel changes, and the live question is what survives the handoff.
Meme communities revise their allowable transformations almost weekly — the in-group invariant is whatever survives this month's churn. A fast group, a thin invariant.
Held by intrinsic markers — language, rite, cuisine — it travels; held only against a host culture, it bends whenever the host does. Intrinsic versus negation, lived.
The EU and similar unions are a deliberate enlargement of the transformation group: a thinner shared identity, traded for far wider reach. Erlangen, at the scale of treaties.
Pick a political movement. What is its stabilizer — the elements it will not let change? What transformations is it currently allowing, and which is it resisting? The answers usually locate it precisely on the over-fixing ↔ under-fixing axis.
This page is analytic, not advocacy. The framework applies across the entire political spectrum, and naming a dynamic is not endorsing anyone who uses it. Where it touches contested politics, treat it as a lens for seeing the move — never as a verdict on who is right.