A column about curious stories from the worlds of science, technology, and film. Today — about how a 240-year-old riddle, in which a majority sometimes votes against itself, acquired an exact geometric form: three mathematicians proved that a voting cycle is shaped like a Klein bottle, a surface with no inside or outside — and from that shape, Arrow’s theorem on the impossibility of ideal elections emerged anew, now as a statement about geometry.
On October 26, Quanta Magazine reported on the work of Ori Livson (University of Sydney), Mikhail Prokopenko, and Siddhartha Pritam (Mathematical Institute in Chennai): the Condorcet paradox — the oldest “malfunction” of majority rule — had received a complete topological model, and that model turned out to be a Klein bottle. What is fresh here is not even the fact itself, but the genre: Arrow’s theorem, the central “failure” result of social choice theory, is usually retold as a list of axioms after which the reader’s attention runs out. Here, impossibility has for the first time been taken as an object: it has rims, a direction of travel, and a rigorously defined way of being glued together.
The non-obvious side is this: the familiar logic of “the results were counted fairly — so everything is fine” does not survive an encounter with topology. In this story, ballots can be as honest as you please, the ballot box transparent, and the answer will still go in circles, because the space of preferences itself has no consistent side. And there is a framing device worthy of Greek tragedy: the man who devoted his life to the arithmetic of fairness was killed by the machinery of terror that he himself had sought to put into arithmetic.
In 1785, the Marquis Marie Jean Antoine Nicolas de Condorcet published Essay on the Application of Analysis to the Probability of Majority Decisions. He was secretary of the French Academy of Sciences and wanted one thing: a fair procedure for electing new members. From this thoroughly кабинетная task grew a discovery that outlived both the academy and France itself.
Take three voters and three candidates. The first ranks them Alice, Bob, Carol; the second — Bob, Carol, Alice; the third — Carol, Alice, Bob. Now compare the candidates pairwise, as happens in any runoff. Alice beats Bob: the first and third voters back her. Bob beats Carol: the first and second back him. Carol beats Alice: the second and third back her. Every ballot is perfectly rational, every count impeccable — yet the result runs in a circle, and no winner of the head-to-head contests exists at all. That is the Condorcet paradox.
The man who discovered it is one of the most wrenching figures of the Enlightenment. Mathematician, philosopher, economist; he advocated equal rights for women, public education, and the abolition of slavery. For this, he was called “the last witness of the Enlightenment.” He greeted the Revolution with enthusiasm — which makes the ending all the bitterer: in 1793 he criticized the Jacobin constitution, the Convention issued an order for his arrest, and on March 29, 1794, he was found dead in a prison cell, two days after being detained. The circumstances of his death remain disputed to this day — accounts range from exhaustion to suicide.
The irony is arranged so perfectly that there is no need to invent it: Condorcet believed rational analysis could calculate a fair collective decision — while in political science curricula, what remains of that faith is mostly a proof that a simple majority sometimes chooses nothing at all.
Skepticism about majoritarian arithmetic is easy to sharpen with a simpler example. Suppose that in an election Alice receives 35%, Bob 34%, and Carol 31%. Alice wins. But if Carol’s supporters rank Bob above Alice, then 65% of voters preferred Bob. Every ballot has been counted fairly — the thing is, what counts as winning in the first place?
Condorcet proposed a criterion: the true winner is the person who beats every rival in a one-on-one contest. As his own paradox showed, such a candidate may not exist. For two hundred years, engineers of electoral systems tried one patch after another: runoffs, ranked-choice voting, point systems. Change the rule — and the winner changes, even when the ballots remain exactly the same.
In 1951, economist Kenneth Arrow put an end to the search for patches. He was looking for a procedure that aggregates voters’ rankings into a ranking for society and rests on three modest requirements: the result is not dictated by one person; if everyone prefers Alice to Bob, society does too; the relative order of Alice and Bob does not depend on where Carol appears on the ballot. Plus freedom: voters may rank candidates however they like. The verdict: with three or more alternatives, no procedure satisfying all the conditions at once exists. “The will of the people quite often simply does not exist,” Oxford political scientist Iain McLean later concluded. Changing the electoral system changes only which condition will be violated, while mathematics merely tells us that some violation is unavoidable. Which one, precisely — that is no longer mathematics’ question; it is a question of values.
Standard proofs of Arrow’s theorem are algebraic: three pages of manipulations with preference profiles, and there you have it. The idea of looking at elections another way came from elsewhere. In 1980, mathematician and economist Graciela Chichilnisky (Columbia University) published “Social choice and the topology of spaces of preferences” in Advances in Mathematics — the first systematic attempt to measure the space of all possible preferences with topological tools. In a footnote, a conjecture flashed by: the Condorcet paradox was, apparently, a problem in topology.
Topology is the branch of mathematics in which figures are considered the same if one can be continuously deformed into the other without tearing or gluing. A cup and a doughnut are one object to a topologist: both have exactly one hole. The question “what shape is the space of the Condorcet paradox?” sounds exotic, but it is professionally framed: if impossibility has a geometry, it can be classified just as surfaces are classified.
In 1993, Yuliy Baryshnikov gave this program its first major victory: his paper “Unifying Impossibility Theorems: A Topological Approach” built a geometric model of the ballot and derived from it a new proof of Arrow’s theorem. The model is elegant to the point of transparency. One ballot with three candidates is a triangle: its vertices correspond to the three pairwise comparisons. There are exactly six ballots that produce different rankings; the triangles are glued along shared edges into a strip, and the strip closes into a cylinder. The entire space of conceivable collective preferences is a hollow tube. Baryshnikov went around its rims: along one edge of the tube, preferences form the cycle Alice-Bob-Carol-Alice; along the other, they run in reverse. The rims were the Condorcet paradox, but they played no part in his proof.
In April 2025, Ori Livson, a PhD student at the University of Sydney, returned to the rims. His question was this: cycles occur not only in elections — rock-paper-scissors is cyclical too, and nobody finds that troubling. But in a game, the cycle is permitted by the rules, whereas in elections transitivity is mandatory: if Alice is better than Bob, and Bob is better than Carol, society must rank Alice above Carol. Can one build a geometric model that distinguishes these two worlds?
The key observation sounds almost frivolous. Every contradictory cycle among three alternatives says the same thing: every candidate is simultaneously better than every other candidate — a claim equally absurd in all its variations. And in topology, as Livson puts it, “when you realize that two objects are one and the same, — you can glue them together.” So the rims of the cylinder on which contradictory cycles live must be glued to one another. All that remained was to understand how.
Gluing them head-on produces a doughnut — a torus. Beautiful and useless: with that kind of gluing, the edge corresponding to the order “Bob, Alice, Carol” sticks to the edge “Bob, Carol, Alice,” and the model begins asserting incompatible things at once. The only permissible option is to glue the rims with a twist, turning the preference order on one of them by half a rotation. The paper proves something stronger: there is no other way; the twist is mandatory.
This is where it is worth slowing down and listening to how topology sounds. Glue a paper ring with a twist, and you get a Möbius strip: a one-sided surface with an edge. Glue a tube that way, and you get a Klein bottle: a one-sided surface with no edge at all. To walk along a Klein bottle is to travel far enough and find yourself “on the other side,” although there is no other side. In three-dimensional space, it has to be drawn with a self-intersection — an ugly glass figure passing through its own wall. In four dimensions, it closes without a single seam.
The point of the model lies in an exact morphism. Every small patch of a Klein bottle is entirely normal: it has two sides, like any piece of surface. But an attempt to color the bottle globally in two colors — “inside” and “outside” — fails across the whole surface at once: the local order of sides cannot be assembled into a global one. That is exactly what happens with voting: every ballot is locally impeccable, but the attempt to assemble them into one global order breaks against a cycle. Rationality in pieces and meaninglessness as a whole — the same mechanism in politics and in geometry.
The new paper goes beyond a pretty picture and lays out the entire zoo across four surfaces. If cycles are permissible and unrealized, the result is a cylinder. If they are permissible and realized, a sphere. If they are contradictory and unrealized, a Klein bottle. If they are contradictory and realized, a projective plane. The diagnostic rule is short: a surface is non-orientable if and only if the model describes contradictory cycles. Topology independently distinguishes the harmless cyclicality of a game from the fatal cyclicality of elections — a distinction usually blurred in writing about the paradox.
The final move is a reformulation of Arrow’s theorem itself. The paper proves that the statement “there exists no social choice function satisfying unanimity, independence of irrelevant alternatives, non-dictatorship, and an unrestricted domain” is equivalent to the statement “there exists no such function whose social choices are orientable.” Arrow’s impossibility turns out to be a theorem of non-orientability: if you want to preserve all the axioms, accept that the space of collective preferences is structured like a Klein bottle, and that it contains no global “side of society.” The only loophole is to restrict the domain: forbid voters from bringing profiles that generate cycles to the ballot box. But the price is that such a system no longer aggregates everyone’s will — only the will of configurations filtered in advance.
The practical layer of this story is more modest than the headlines. Topology does not offer a new electoral system and does not save democracy; it shows that the cost of any design is structural. A system can be built on runoffs, ranked ballots, or Condorcet contests — one simply has to choose honestly which of Arrow’s conditions to sacrifice. As McLean concludes, this is “not a mathematical dispute, but a dispute about values.” Mathematics gives us the price list; we must choose the rate.
There is one more thing I like about Livson and his colleagues’ work — the direction of the shot. The same group had earlier compared Arrow’s theorem with Gödel’s first incompleteness theorem: in that parallel, Gödel sentences play the role of profiles aggregating into a contradictory cycle. Now the family of “circular” impossibilities — self-reference, preference cycles, non-orientable surfaces — has acquired a common geographical map, and the question “why are impossibilities so similar to one another?” has for the first time become formal. Perhaps we are looking at the same obstacle, caught in different disciplines.
And the beginning of the story remains the most human part of it. Condorcet died in the prison of a revolution he considered the arithmetic continuation of reason, never knowing that he had left behind not a computational error but the first sketch of a map of terrain where “the will of the people” does not exist as an object. Drawing that map took 240 years, four surfaces, and one bottle that cannot be stitched together without a twist. And the most satisfying thing about this story is that the twist is the only honest way to glue it. Everything else would have been a torus with a lie sewn inside it.