Six lines all at one angle

How many lines can pass through a single point so that every pair of them meets at the same angle? In the plane the answer is three, and everybody has drawn them: the three lines through opposite corners of a regular hexagon. In ordinary space the answer is six, which is much less obvious, and the six are the long diagonals of a regular icosahedron. Beyond that the question gets strange. Four dimensions allow no more than three do, five allow ten, six allow sixteen, and seven allow twenty-eight, after which the number refuses to move for seven more dimensions and then climbs to 276 in dimension twenty-three.

No formula produces that list, and for most dimensions above twenty nobody knows the right value. Yet there is one clean ceiling, proved in a few lines, which is reached four times in the first twenty-three dimensions and possibly never again. The proof is a small piece of pure inner-product reasoning, and it depends on replacing each line by a matrix so that a nuisance disappears.

Why a line is an awkward thing to measure #

A vector has a length and a direction, and two unit vectors $u$ and $v$ have an inner product, $\langle u, v\rangle$, which the essay on the dot product read as the length of one’s shadow on the other. A line has no such convenient description. It is a vector with its direction forgotten, so a line is really the pair $\{u, -u\}$, and the inner product of two lines is known only up to sign. Saying that lines meet at equal angles means that $|\langle u_i, u_j\rangle|$ is one fixed number $\alpha$ for every pair, while the signs of the inner products are whatever they happen to be.

The table beside the icosahedron shows what that looks like. All fifteen pairs have cosine of size $1/\sqrt 5$, which is an angle of 63.43 degrees, and the signs form a pattern that has no obvious order. Worse, the pattern is not even well defined, since replacing $u_3$ by $-u_3$ flips the third row and column and describes exactly the same six lines. Any argument that treats those signs as data is working with something the geometry does not contain.

That is what makes the problem awkward and what suggests the way out. It would help to have an object attached to a line that does not change when the direction is reversed, and that still has an inner product.

Three lines, and why not four #

The plane shows the idea in its smallest form, and it is worth doing there first because it can be seen. A line through the origin at angle $t$ is unchanged by adding 180 degrees to $t$, so its natural coordinate is the doubled angle $2t$, which runs once round a circle as the line turns once. Doubling sends each line to a point of a unit circle, and it sends the angle between two lines to the angle between two points, doubled again.

Now the question has changed shape. Equal angles between lines have become equal distances between points, and this is a question with an obvious answer: three points can be equally spaced on a circle, and four cannot, because four mutually equidistant points are the corners of a regular tetrahedron and do not lie in a plane at all. So the plane allows three lines. Trying four, with the natural choice of 45-degree steps, gives pairs at 45 degrees and pairs at 90 degrees, which is the picture on the right.

That doubling looks like a trick special to the plane, since it uses the fact that a direction there is a single angle. Its general form is a change of object. The doubled angle is nothing other than the description of a line by the matrix that projects onto it, and the matrix makes sense in every dimension.

The matrix a line becomes #

For a unit vector $u$, the matrix $P = uu^{\mathsf T}$ sends a vector to its nearest point on the line through $u$. Reversing $u$ changes nothing, because the two minus signs cancel. So $P$ belongs to the line and not to a choice of direction. And $P$ is symmetric, so it lies in the space of symmetric $d \times d$ matrices, which has its own inner product, the trace of the product, under which those matrices form a space of dimension $d(d + 1)/2$.

One line of algebra gives the inner product of two projectors:

$$\operatorname{tr}(P_i P_j) = \operatorname{tr}\!\left(u_i u_i^{\mathsf T} u_j u_j^{\mathsf T}\right) = \langle u_i, u_j\rangle^2.$$

The sign has gone, because it has been squared. Equiangular lines therefore become unit vectors in a larger space whose pairwise inner products are all exactly the same positive number $\alpha^2$. That is a far more rigid thing than a table of cosines with unknown signs: it is a regular simplex in disguise, a set of points all mutually equidistant.

The table of these inner products is $G = (1 - \alpha^2)I + \alpha^2 J$, where $J$ is the matrix of all ones, and its eigenvalues can be read straight off. For the six lines of the icosahedron, $\alpha^2 = 1/5$, and the eigenvalues are $0.8$ five times and $2$ once. None of them is nought, which is the whole point, since a Gram matrix with no zero eigenvalue means the matrices it describes are linearly independent. Unless $\alpha = 1$, which is one line repeated, this holds for any number of lines at any angle.

Independent elements of a space cannot outnumber its dimension. So:

The number of equiangular lines in $d$ dimensions is therefore at most

$$\frac{d(d + 1)}{2}.$$

This is Gerzon’s bound, given in a 1973 paper by Lemmens and Seidel, who credit Michael Gerzon. In the plane it gives three, matching the hexagon. In space it gives six, matching the icosahedron, which proves at a stroke that no seventh line can be added to those six, and no cleverer arrangement can beat them. The icosahedron was already known to be special for other reasons, but nothing about the list of regular solids suggests this. The argument never mentions a solid. It belongs to the family of proofs in which a counting bound falls out of independence, the same style of reasoning that bounded a cap set by the dimension of a space of polynomials.

From geometry to a table of signs #

The bound says how many lines might fit. To find out how many do, the signs have to come back. Fix a direction for each line and let $S$ be the matrix with nought on the diagonal and, off the diagonal, the sign of $\langle u_i, u_j\rangle$. Then the Gram matrix of the original vectors is

$$U = I + \alpha S.$$

This is the observation that J. J. Seidel made central. A Gram matrix must have no negative eigenvalues, and its rank is the dimension the vectors span. So $S$ needs a least eigenvalue no smaller than $-1/\alpha$, and the dimension needed is $n$ minus the multiplicity of that eigenvalue when $-1/\alpha$ equals it. Taking $\alpha$ as large as the matrix allows, which is the reciprocal of minus its least eigenvalue, uses the smallest possible number of dimensions. Everything geometric has been turned into a sign pattern. Lines exist exactly when a symmetric table of plus and minus signs has a least eigenvalue with high multiplicity.

Flipping one line’s direction flips one row and column of the table, so the first row can be made all plus without loss, and what remains is a finite list. For $n$ lines there are $2^{(n-1)(n-2)/2}$ tables, and they can simply all be examined, one of them being the degenerate all-plus table which describes one line repeated.

The result of that search is the start of the known sequence, found from nothing. Three lines need two dimensions. Four, five and six lines need three, and only twelve of the 1,023 tables on six lines manage it. Then comes a jump: seven lines cannot be fitted into four dimensions, and they need five, as the empty cell in the last row shows. Not one of the 32,767 sign patterns on seven lines has a least eigenvalue of multiplicity three.

So four dimensions hold exactly six lines, no more than three dimensions do. That is a striking failure of the natural expectation that an additional dimension must buy something. The bound $d(d + 1)/2$ would allow ten in four dimensions, and it allows ten in no way that can be realised. The reasons are arithmetic. If a Seidel matrix has only two different eigenvalues, as the sets meeting the bound do, its eigenvalues are forced by its size and by the trace identities $\operatorname{tr} S = 0$ and $\operatorname{tr} S^2 = n(n - 1)$, and Lemmens and Seidel’s analysis shows that the ones that matter must be odd whole numbers. So a dimension either has a matrix with suitable arithmetic or it has not, and the sizes are erratic in the way that integer conditions on spectra so often are.

Twenty-eight lines in seven dimensions #

A set meeting the bound looks accidental until it is written down. Take eight coordinates and every arrangement of the vector $(-3, -3, 1, 1, 1, 1, 1, 1)$, one for each choice of the two places holding $-3$. There are 28 of these vectors. Each has entries that sum to nought, so they all lie in the seven-dimensional space perpendicular to the all-ones vector. Two of them either share one position of $-3$ or share none, and in both cases the inner product is $\pm 8$ against a squared length of 24. So all 378 pairs meet at cosine $1/3$, and 28 lines lie in seven dimensions, exactly $7 \times 8/2$.

The table’s eigenvalues are 9, seven times, and $-3$, twenty-one times. With $\alpha = 1/3$ the least eigenvalue is exactly $-1/\alpha$, and its multiplicity leaves seven dimensions. The calculation is complete and short. The surprise is what these lines are. Reading them in the other direction, these 28 lines are the 56 weights of the smallest representation of the exceptional structure called $E_7$, taken in opposite pairs, and they share their symmetry group with the 28 bitangents of a plane quartic, the lines that touch a generic curve of degree four at two points. A nineteenth-century problem in the geometry of curves and a question about angles between lines turn out to have one group between them, and no calculation with angles would ever suggest it. The same structure, one step up the exceptional series, gives the 240 directions of the eight-dimensional lattice.

That the icosahedron’s six lines and these twenty-eight both meet the bound is also more than coincidence. There is a condition that the sign table must satisfy: it must have two eigenvalues, so that $S^2$ is a combination of $S$ and $I$, and a table with that property is called a regular two-graph, a structure with links to the strongly regular graphs and their failure to be identified by their spectra. The search for sets meeting the bound therefore becomes a search for such graphs of very particular sizes.

The list the bound leaves #

The known values, from eight dimensions upward taken from the literature rather than from any search here, are plotted against the bound. They form an odd shape.

After 7 the value sits at 28 all the way through dimension 14, even though the bound rises from 28 to 105. Adding seven dimensions buys nothing. That plateau has an explanation, in that any further sets at cosine $1/3$ are limited: Lemmens and Seidel proved that with $\alpha = 1/3$ and $d$ at least 15, the most lines is exactly $2(d - 1)$, a merely linear count. To get more, the angle must change, and cosine $1/5$ is the next odd reciprocal that serves. It produces 276 lines in dimension 23, whose symmetry group is Conway’s sporadic group $\mathrm{Co}_3$. So there the bound is reached a last time, as far as anyone knows.

Above 23, the bound requires an arithmetic coincidence. The dimension must be $k^2 - 2$ for an odd number $k$, since the relevant eigenvalue must be $k$. The first candidates, after simple arithmetic tests, are 47, 79 and 167. Each requires a strongly regular graph of one particular size. The first two have been excluded, 47 by Alexander Makhnev in a long argument about the one graph it needs, and Eiichi Bannai, Akihiro Munemasa and Boris Venkov showed by arithmetic that infinitely many others fail too. Others remain undecided. Whether another example exists, or whether the four known sets are the entire list, is not known.

For large dimensions there is an even more striking result, and it shows how little the bound may mean at scale. Fixing the angle and letting the dimension grow, Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao proved in 2021 that the most equiangular lines at a fixed angle grows only linearly in $d$, with a slope set by the spectral radius of graphs. A quadratic number of lines therefore needs an angle that shrinks with the dimension. Dominique de Caen found in 2000 a family with about $\tfrac{2}{9}d^2$ lines, so the quadratic order of the bound itself is correct, and this was a real surprise when it appeared, since many had expected the opposite.

The complex version, and a very different failure #

The same argument works over the complex numbers, and it gives a different and more famous problem. A complex line’s projector is a Hermitian matrix, and the Hermitian matrices have real dimension $d^2$, so the bound becomes $d^2$. A set of $d^2$ equiangular lines in complex $d$-space is called a SIC, for symmetric informationally complete measurement, and it plays a role in quantum theory: measuring a quantum system against such a set gives enough statistics to reconstruct its state.

Gerhard Zauner conjectured in his 1999 dissertation that such a set exists in every dimension. In the real case the bound is met four times; in the complex case it appears to be met always. Numerical solutions exist in every dimension that has been tried up to the hundreds, and exact solutions in many of them, with a connection to units in the real-quadratic number fields that is very far from what the question appears to be about. A general construction has not been found, and nobody can prove that one exists. The two versions of the question differ by one small letter and have opposite outcomes on the evidence, and the same short argument gives both bounds. No figure here reaches that case, because a line in complex space is not something a page can draw.

How far a drawing reaches #

A diagram of lines through a point is convincing in two and three dimensions and useless after that. The search above gives something a drawing cannot, since it decides which dimensions work up to seven lines. But it decides nothing about larger sets, because the number of tables grows like two to the power $n^2/2$. At eight lines it is already two million, at twenty-eight it is $2^{351}$, and known results depend on arithmetic restrictions rather than enumeration.

There is also an important distinction between the two kinds of result. The upper bound is a proof, and it applies in every dimension uniformly. The specific constructions are proofs for particular dimensions. Between them, in dimensions such as 17 to 20, what is known is only a range of two or three possible values, and those ranges were reduced by lengthy case analysis, by Gary Greaves and his collaborators among others, not by any general theorem. The eigenvalue condition can exclude a value and can produce one, but it does not produce the answer in general.

What was computed and what was quoted #

The six lines, their cosines, the Gram matrix of projectors and its eigenvalues were computed and checked to rounding error. Doubling in the plane is a proof, and so is the bound. The sign-pattern search is complete for its range: 33,861 tables on three to seven lines with the first row fixed, the degenerate one excepted, each with its eigenvalues found numerically to twelve places and the eigenvalue gaps checked by that tolerance. The floating-point step means it is a computation and not a formal proof, though the integer tables involved have well-separated eigenvalues. Dimension 5 for seven lines agrees with the literature’s value.

The twenty-eight lines were built and all 378 pairs verified. The values from eight dimensions upward, the plateau theorem, de Caen’s construction, the 2021 theorem, the exclusion of 47, the link to $E_7$ and the bitangents, and everything about SICs are quoted from the literature and have not been recomputed here.

Still open: whether a fifth set meets the ceiling #

Is there an equiangular set meeting Gerzon’s bound in any dimension above 23? The dimension has to be $k^2 - 2$ with $k$ odd, and each candidate is a question about one specific strongly regular graph. Those graphs have enormous sizes, and the arithmetic tests known for them, while strong, do not settle them all. The same question arises across spectral problems about graphs, and it has the same character: it is a finite and exact question about one object, and there is no prospect of searching for it.

In small dimensions the unsettled ones are few and precise. Dimensions 17, 18, 19 and 20 each have a known range and no exact value. A value could be raised by exhibiting a single sign table, and could be lowered only by an argument that excludes a whole class of them. For complex lines, the open question is more basic: does a set of $d^2$ equiangular lines exist in complex $d$-space for every $d$? Numerical solutions in every dimension tried say yes, and no proof exists at all.

Why the signs had to go #

A line has no direction, so its inner products with other lines are known only up to sign; squaring it into the matrix that projects onto it removes the sign, and then equal angles become mutually equidistant points, which must be independent. That single change of object yields the ceiling of $d(d + 1)/2$, and it is what proves that six lines in space cannot be beaten.

Then the signs return, as the table whose least eigenvalue decides the dimension. Lines that seemed to be about geometry are revealed to be about arithmetic: a searchable table up to seven lines, a sharp jump from six lines to seven, an exact construction in seven dimensions from nothing but $-3$ and $1$, and a list of possible dimensions that may already be complete. That a question this easy to state should stop being answerable at dimension 17 is the measure of how much the sign pattern hides.