Four circles cannot do it

Three overlapping circles is the picture everybody has seen. It is also, in a way that is easy to miss, exactly the right size.

Three sets have $2^3 = 8$ possible membership patterns: in all three, in the first two only, in none, and so on. The picture shows eight regions. Every pattern appears exactly once, which is what makes the diagram useful — point at a pattern and there is somewhere to point.

The natural next thought is that four circles will do the same for four sets. It will not, and the reason has nothing to do with arranging them cleverly.

What the picture is being asked to do #

Before the counting, it is worth being clear about what job the diagram has, because the failure is a failure to do that job and not a failure to look right.

A diagram of this kind is a catalogue of possibilities. It is not a picture of three particular collections of things; it is a picture of every way membership in three collections could fall out. That is why it has to show the region for “in $B$ only” even when there happens to be nothing in $B$ alone — the region stands for a possibility, and a possibility with nowhere to be drawn is a possibility the diagram cannot discuss.

Once that is the job, the diagram becomes a piece of machinery rather than an illustration. A statement about the three sets is a statement about which regions are empty. All $A$ are $B$ says the region inside $A$ and outside $B$ has nothing in it. Some $A$ is $B$ says the region inside both has something. Every categorical statement is a claim about regions, and a diagram with a region for every pattern can express every such claim.

That is the whole reason it matters whether all $2^n$ regions are present. A diagram missing two of its sixteen regions is not merely untidy: there are two statements about four sets that it has no way to express, and the reader has no signal that anything is absent. The picture looks complete. That is the failure this essay is about, and it is a good example of a defect whose symptom is absence — the hardest kind to notice, because nothing on the page is wrong.

The requirement, stated exactly #

A Venn diagram for $n$ sets is a family of $n$ closed curves such that:

• every one of the $2^n$ membership patterns occurs somewhere in the plane, and
• each pattern occupies a single connected region.

Both conditions matter and they are usually run together. The first says nothing is missing. The second says nothing is duplicated — a pattern spread over two disconnected pieces is a diagram that has drawn the same set twice, and a reader shading “in $A$ and $B$ but not $C$” would have to remember to shade both.

What four circles actually produce #

Fourteen. The two that never appear are the ones for the two diagonal pairs: the region inside $A$ and $C$ but outside $B$ and $D$, and the region inside $B$ and $D$ but outside $A$ and $C$. Opposite circles do meet — every pair crosses — but the lens where two opposite circles overlap lies entirely inside the other two, so there is nowhere in it that is outside them.

The instinct at this point is to move the circles. Make them bigger, smaller, differently placed. It does not help, and the reason it does not help is a count that does not depend on where the circles are.

Euler’s formula settles it #

Take $n$ circles in general position, meaning every pair crosses at exactly two points and no three circles pass through one point. The arrangement is a graph drawn on the sphere, and the count that every corner pays for applies to it.

Vertices. Each pair of circles contributes two crossings, so $V = 2 \binom{n}{2} = n(n-1)$.

Edges. Each circle is cut by the other $n-1$ circles at $2(n-1)$ points, and those points divide it into $2(n-1)$ arcs. So $E = 2n(n-1)$.

Faces. Euler’s formula gives $F = E - V + 2 = 2n(n-1) - n(n-1) + 2$, which is $n^2 - n + 2$.

That is the whole argument. For $n = 3$ it gives $3^2 - 3 + 2 = 8$, and eight is $2^3$, so three circles have exactly the room required. For $n = 4$ it gives $4^2 - 4 + 2 = 14$, and $2^4 = 16$. Fourteen regions cannot hold sixteen patterns, so no arrangement of four circles is a Venn diagram — not the arrangement above, not any other.

The figures do not take the formula on trust. Each one samples the plane, finds the connected pieces of equal membership pattern by flood fill, and asserts that the measured count equals $n^2 - n + 2$. Two ways of getting the same number, one from Euler’s formula and one from counting the drawing, agreeing.

Why it is circles that fail, and not diagrams #

The failure is specific and it is worth being precise about what is responsible, because the wrong lesson is easy to draw.

The step that costs everything is $V = n(n-1)$: two circles cross at most twice. That is a fact about circles — two distinct circles meet in at most two points, because subtracting their equations leaves a line — and it caps the number of vertices, which caps the edges, which caps the faces.

A curve that can cross another more often gets more vertices and therefore more faces, and the requirement is met. So the fix is not to place circles better; it is to stop using circles.

Ellipses do work, and only just #

An ellipse can cross another ellipse four times rather than twice, which doubles the vertex count and buys the room. Venn found such a figure himself in 1880, and it has been redrawn many times since.

The configuration above is not the classical one; it was found by search, and what the search rejected is more instructive than what it accepted. Several arrangements produce all sixteen patterns and fail the second condition: two of the sixteen come out as two separate pieces each, touching at a pinch so narrow that a measurement of the drawing reads sixteen regions at one sampling resolution and eighteen at the next. An arrangement whose verdict depends on how finely it is sampled is not evidence of anything, and three otherwise attractive candidates were discarded before this one, which returns sixteen patterns in sixteen pieces at every resolution from 130 samples across to 1,000, and whose smallest region is a third of a percent of the frame rather than a thousandth.

That is the difference between the two conditions doing real work. “All sixteen patterns appear” is easy to hit by accident; “each in one piece” is the one that costs.

Five, and the shape of the general answer #

The gap grows fast, because $n^2 - n + 2$ is a polynomial and $2^n$ is not. At six sets, circles offer 32 regions against 64 patterns; at ten, 92 against 1,024. Whatever the fix is, it cannot be a mild improvement to circles.

The general fix is Venn’s own and it is constructive. Given a diagram for $n$ sets, add an $(n+1)$-th curve that threads through every existing region exactly once, entering and leaving each — a curve that snakes around the whole diagram. Every region it passes through is split in two, one half inside the new curve and one outside, so the count doubles from $2^n$ to $2^{n+1}$ and every new pattern is realised.

Such a curve exists for every $n$, so Venn diagrams exist for every number of sets. They are simply not made of circles, and past four they are not made of anything anyone would draw by hand: the snaking curve for six sets crosses the others 62 times and looks like a scribble.

There is a beautiful special case worth naming. A Venn diagram with $n$-fold rotational symmetry exists exactly when $n$ is prime. Henderson showed in 1963 that primality is necessary; Grünbaum drew one for five sets in 1975; and the construction for every prime was completed by Griggs, Killian and Savage in 2004.

The necessity half is a counting argument and it is short. Under a rotation of order $n$, the $2^n - 2$ patterns other than “all” and “none” fall into orbits of size $n$, so $n$ must divide the number of patterns of each size — that is, $n \mid \binom{n}{k}$ for every $k$ strictly between $0$ and $n$. That holds precisely when $n$ is prime.

Two other things four circles get wrong #

The missing regions are the headline, and there are two smaller failures in the same picture that are worth seeing because they are what a reader would notice first if they noticed anything.

The regions have wildly different sizes. In the three-circle picture the eight regions are of comparable size and the diagram reads evenly. With four, some regions are large lobes and others are slivers, and the sliver regions are exactly the ones a reader will fail to point at. Nothing in the construction controls this and nothing can: the shapes are determined by the circle placement, and the placement is already fully used up by the requirement that every pair crosses.

The pattern for a region is not readable off the picture. With three circles a reader can see which circles a point is inside. With four, the innermost regions are bounded by arcs of all four circles and telling them apart takes tracing. The diagram has stopped being a thing to look at and become a thing to work out, which is most of its value gone even where it is correct.

Both of these get worse with $n$ faster than the region deficit does, which is part of why the honest answer for large $n$ is not a better Venn diagram but a different representation entirely — a table of patterns, or the cube of assignments, neither of which is trying to be a picture of the plane.

What the diagram is for, and what it is not #

Two things get expected of these pictures that they cannot deliver, and both come from the same confusion.

A Venn diagram is not a picture of particular sets. It is a picture of the pattern space — of all the ways membership could fall out. That is why every region has to be present even when it is empty for the sets in question: a diagram in which the empty combinations are simply left out is an Euler diagram, which is a different and often more useful object, and which does not have this problem because it is not trying to show everything.

Region areas mean nothing. Nothing in the construction makes the region for “in $A$ only” proportional to anything. Reading area as quantity is the standard misuse and the diagrams are silent about it — there is a whole small literature on area-proportional Venn diagrams, and the interesting result there is that three circles cannot generally achieve given areas either, for reasons of the same kind: not enough freedom in the shape.

What the diagram is for is the thing this field will use it for repeatedly. It turns a statement about sets into a statement about which regions are empty, and once that translation is made, a claim about all possible situations becomes a claim about a finite picture with a finite number of regions that are either shaded or not. Two hundred and fifty-six arguments become a decidable question exactly because three sets have eight regions and $2^8$ ways of being occupied.

The diagram that gives up the guarantee #

There is a second family of pictures that solves the problem by refusing it, and it is the one most real diagrams in the world actually are.

An Euler diagram draws only the regions that are non-empty for the particular sets in hand. Three sets where $A$ and $B$ are disjoint get two separated circles; three sets where $A$ is contained in $B$ get one inside the other. No region is required to exist merely because it could, so the counting obstruction never arises and Euler diagrams exist for any number of sets in any configuration.

What is given up is the guarantee. A Venn diagram is a fixed template: its shape is the same whatever the sets are, and every claim about them is a shading of it. An Euler diagram’s shape is the claim, so two Euler diagrams for the same three sets under different assumptions look nothing alike, and a reader cannot tell by looking whether a region is absent because it is empty or absent because the draughtsman ran out of room.

That trade is worth naming because it is the same one the field makes over and over. A representation that shows every possibility is uniform, complete and does not scale; a representation that shows only what is the case is compact, readable and carries no information about what it left out. Venn wanted the first, drew circles, and found at four that the first is not always available.

Why an impossibility this small is worth the space #

The result is modest: a particular kind of drawing does not work past a particular size. Nobody is inconvenienced.

What earns it a place is the shape of the proof, which is the one this field keeps using. There is no search over arrangements and no cleverness about circle placement. There is a quantity — the number of faces — that is determined by the combinatorics regardless of the arrangement, and a requirement that the quantity be at least $2^n$, and at $n = 4$ the two numbers can be compared and one is smaller.

That is an impossibility proof of the cheapest available kind, and its cheapness is the point. It rules out infinitely many attempts at once without examining any of them, in the same way that the count of angles at a vertex rules out a sixth regular solid and a parity argument rules out a walk over the bridges. In each case the invariant is small enough to compute and rigid enough not to care about the details it is being asked to rule out.

The habit worth carrying forward: when a construction is failing repeatedly, stop improving the construction and look for a number it cannot change.