28

Dimension $8$ seems special, as the partial list below might indicate. Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$? Surely it isn't it, in the end, simply because $8=2^3$, but $9=3^2$? Or that $\phi(8)=4$ but $\phi(9)=6$?

Joseph O'Rourke
  • 149,182
  • 34
  • 342
  • 933
  • 9
    I mean, the construction of $E_8$ uses $8= 2^3$ in a very clear way... – Will Sawin May 01 '20 at 23:54
  • 12
    Triality (exceptional automorphisms of $\mathfrak{so}(8)$ over an algebraically closed field), leading to complications in Galois cohomology etc. – YCor May 01 '20 at 23:56
  • 9
    Bott periodicity should be in that list. – Pulcinella May 02 '20 at 00:14
  • 12
    It's important to notice that, in an important way, $E_8$ is not 8-dimensional, but 248-dimensional. (But, in another sense, it is 8-dimensional; it all depends whether you're counting the dimension of the space in which the root system lives, or the dimension of the associated Lie algebra. Of course there are other ways of counting dimension that give still other answers.) – LSpice May 02 '20 at 00:35
  • No big deal, but I think it's Wolchover – kodlu May 02 '20 at 01:05
  • @kodlu: Thanks! – Joseph O'Rourke May 02 '20 at 01:07
  • 5
    There's also the Simons minimal cone, which is a singular hypersurface in $\mathbb{R}^8$. – Deane Yang May 02 '20 at 01:13
  • 5
    There is also an exotic 8-sphere and exotic spheres in most higher dimensions, so perhaps Milnor's exotic 7-spheres don't belong in this list? – Josh Howie May 02 '20 at 01:50
  • 17
    Isn’t this just the strong law of small numbers? And, please, no Lisi. – Aaron Bergman May 02 '20 at 02:17
  • 2
    Answers in What makes four dimensions special? are pretty good hope this question could get some awesome answers too – Dabed May 02 '20 at 02:28
  • 1
    $7\not=8$...... – Monroe Eskew May 02 '20 at 04:09
  • 3
    In 8 dimensions there exist SIC-POVM that are not a group orbit of the Heisenberg group, namely the Hoggar lines. – jjcale May 02 '20 at 04:10
  • 4
    the special role of octonions follows from the fact that $8$ solves $2^{n-2}=n^2$, as mentioned on MSE – Carlo Beenakker May 02 '20 at 08:44
  • 1
    Here a recent 8-dimensional example of a finite dimensional algebra answering a question on reflexive simple module: https://arxiv.org/pdf/2004.12208.pdf . It might be true that there are no such algebras of dimension smaller than 8, but that is not known it seems. – Mare May 02 '20 at 09:29
  • 4
    I think the two most magical numbers in mathematics are $8$ and $24$, and, sadly, I don't think there's likely to be an “explanation” for that. – Gro-Tsen May 02 '20 at 11:57
  • 3
    BTW this question asks (over a a field of characteristic zero) whether isomorphism between two Lie algebras $\mathfrak{so}(q)$, and $\mathfrak{so}(q')$ are isomorphic, implies that $q'$ is equivalent to a nonzero scalar multiple of $q$. Here $q$ and $q'$ are nondegenerate quadratic forms in finite dimension $\ge 3$ (in dimension 2 it's false over most fields). The answer so far solves this positively, with the possible exception of the case of 8-dimensional $q,q'$. – YCor May 02 '20 at 12:59
  • @Mare's reference, abs-ified: Ringel - Simple reflexive modules over finite-dimensional algebras. – LSpice May 02 '20 at 14:43
  • 2
    @YCor also asked an 8-dimensional follow-up to the linked question. – LSpice May 02 '20 at 14:45
  • 4
    Even unimodular lattices exist only for signature a multiple of 8. https://en.wikipedia.org/wiki/Unimodular_lattice#Properties – Ian Agol May 02 '20 at 16:35
  • 2
    One of the holonomy groups of Riemannian manifolds is Spin(7) which exists in 8 dimensions. https://en.wikipedia.org/wiki/Spin(7)-manifold – Ian Agol May 02 '20 at 16:52
  • @ChuaKS: What is "$E_n$"? – Joseph O'Rourke May 02 '20 at 22:03
  • 2
    There is the nice fact that del Pezzo surfaces have degree at most 9 = 1+8, where the degree is the self-intersection of the canonical class. If a del Pezzo surface has degree d (with d not equal to 8), then it's the blowup of 9-d points on P^2. You can get the exceptional root systems by studying (-1)-lines on del Pezzos, and in particular, you get the E_8-root system by studying del Pezzos of degree 1. I don't know how to fit this into the picture you're drawing, but maybe it fits somehow! – skd May 02 '20 at 22:55
  • Quaternionic hyperbolic lattices exist first in dimension 8, as lattices in $Sp(2,1)$. Quaternionic hyperbolic lattices are distinguished in that they satisfy superrigidity and have property (T) even though they are rank 1 (proved by Corlette and Gromov-Schoen). – Ian Agol May 03 '20 at 04:40
  • 1
    Bernstein's problem holds up to dimension 8 and is false in dimension 9 and higher (this is related to Simon's minimal stable cone in 8 dimensions).

    https://en.wikipedia.org/wiki/Bernstein%27s_problem (see also De Giorgi's conjecture which holds up to dimension 8 and false in dims. > 8).

     – Ian Agol May 03 '20 at 04:58
  • 3
    It's worth noting that the first four items on the list and Bott periodicity are somewhat related. The E8 lattice solves the sphere packing problem and can be realized as a system of integral octonions, and Bott periodicity can also be related to octonions, as Baez explains here. – pregunton May 03 '20 at 08:04
  • 2
    The smallest finite dimensional $\mathrm{C}^*$-Hopf algebra aka algebra of functions on a finite quantum group that is neither the commutative nor a group algebra is the 8 dimensional algebra of functions on the Kac-Paljutkin quantum group. – JP McCarthy May 03 '20 at 10:25
  • 1
    (I should include all these wonderful dim-$8$ examples in my list, but I don't want to bump the question to the front page.) – Joseph O'Rourke May 03 '20 at 13:07
  • 3
    $E_n$ is the set of $n+1$ tuples of integers $x_1,\dots, x_{n+1}$ such that $x_1 + \dots + x_n =3 x_{n+1}$, which carries the quadratic form $x_1^2 + \dots + x_n^2 - x_{n+1}^2$. This quadratic form has determinant $9-n$, so is nondegenerate if $n\neq 9$, and is positive definite if $n<9$. It is also even. Thus if $n=8$ it produces an even unimodular lattice $E_8$, if $n=7$ it produces $E_7$, if $n=6$ it produces $E_6$, if $n=5$ it produces $D_5$, if $n=4$ it produces $A_4$, and if $n=3$ it produces $A_1 \times A_2$. – Will Sawin May 05 '20 at 18:11
  • 2
    The root lattices $A_n, D_n, E_n$ have determinants respectively $n+1, 4, 9-n$, so if $9-8=1$ makes $8$ a special dimension, this suggests that $0$ is also a special dimension. – Will Sawin May 05 '20 at 18:13
  • 2
    Here is a fun one: there is an isomorphism between $\mathrm{O}(\mathbf{F}_2^6, q)$ and the symmetric group $\Sigma_8$, where $q$ is the quadratic form sending $v = (v_1 \ v_2)$ to $v_1 \cdot v_2$. – skd May 07 '20 at 22:17
  • @ChuaKS Do you have a reference for the irreducibility of $\phi_{E_n}(x)$ at these particular values of $n$? – Carl-Fredrik Nyberg Brodda May 17 '20 at 22:47
  • 8 is even, so why are you even comparing it to 9? – Anixx May 17 '20 at 23:01
  • 1
    Not specific to dimension 8, but the first irreducible Mutiple Zeta Value, ζ(3,5), appears in weight 8 (see F.Brown Gergen's lectures (part I slide 65) http://www.ihes.fr/%7Ebrown/GergenLectureI.pdf – Thomas Sauvaget May 31 '20 at 13:57

2 Answers2

7

Some special properties of dimension 8, in addition to the ones you identify:

  • Bernstein's problem holds up to dimension $n=8$. The only function of $\mathbb{R}^{n-1}$ whose graph in $\mathbb{R}^n$ is minimal is a linear function. This fails in dimension $n=9$, with failure due to the existence of the Simons cone in dimension 8, so it's related to your last bullet point.

  • There are 4 infinite families of Euclidean reflection groups, with exceptional ones only up to dimension 8. This is related to the existence of the exceptional simplex reflection groups and exceptional Lie algebras.

Coxeter diagrams of Euclidean reflection groups

  • There are 4 infinite families of holonomy groups of Riemannian manifolds, with two exceptional cases of $G_2$ and $Spin(7)$, the latter being in dimension 8.

  • As pointed out by @YCor, triality holds for $Spin(8)$. $Spin(8)$ has three 8-dimensional irreducible representations which are permuted by the $S_3$ action associated with the symmetries of the $D_4$ Dynkin diagram.

  • Cohn and Kumar found various tight simplices including a maximal 15 point tight simplex in $\mathbb{HP}^2$ which is 8 dimensional. A simplex in this case refers to a collection of equidistant points.

There are several other examples in the comments of phenomena where 8 dimensions is the first dimension in which the phenomenon appears (or is known to appear), but I've listed examples that seem to be special to dimension 8 (and most seem to be connected to the phenomena that you've already identified).

Ian Agol
  • 66,821
7

The following is too long to leave as a comment, so I post it as an answer.

At least for the cases of the Bernstein problem for minimal graphs, Simons' theorem on stable minimal cones, etc., there is some evidence that dimension 8 is more likely a coincidence rather than having deep intrinsic reasons. In more general settings, that 8 is as magical as 7,6,5, or 4.

If we measure the area of hypersurfaces in Euclidean space with respect to norms other than the quadratic form ones, then the above magic dimension 8 breaks down, even with respect to norms that have high degrees of symmetry. (Of course, there is a lot of ambiguity here as in general there are many different ways to measure the area in say a Finsler norm. To avoid such ambiguity, here we assign different areas to different hyperplanes directly. The key word usually used in geometric measure theory is elliptic integrands/anisotropic area.)

Some references are as follows.

MR0467476 Reviewed Schoen, R.; Simon, L.; Almgren, F. J., Jr. Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II. Acta Math. 139 (1977), no. 3-4, 217–265. (Reviewer: William K. Allard) 49F22

MR1092180 Morgan, Frank The cone over the Clifford torus in R4 is Φ-minimizing. Math. Ann. 289 (1991), no. 2, 341–354. (Reviewer: Harold Parks) 49Q20 (53C42 53C65 58E15)

The anisotropic Bernstein problem by Connor Mooney, Yang Yang, https://arxiv.org/abs/2209.15551

Roughly speaking, the Almgren-Schoen-Simon paper shows that for the area-minimizing hypersurfaces in well-behaved normed spaces, the singular set has vanishing Hausdorff codimension 2 measure. (Note that for quadratic norm, the singular set is of codimension at least 7). Thus, it is imaginable that for general normed areas on Euclidean space, the Simons cone/failure of Bernstein problems should exist in lower dimensions. It is indeed the case by Frank Morgan, who shows that the cone over the Clifford torus is area-minimizing with respect to an $SO(2)\times SO(2)$-invariant normed area in $R^4$. This is sharp in view of the regularity result by Almgren-Schoen-Simon. Then the work of Mooney and Yang gives counterexamples to Bernstein-type theorems, similar in spirit to the connection mentioned by Ian Agol for the quadratic norm.

I think for such normed areas which are $C^\infty$ close to the quadratic area, such phenomena cannot happen, thus retaining the magical 8. However, it is conceivable that one should be able to find fairly symmetric norms so that the corresponding magical number is any of 8, 7, 6, 5, 4. Thus, the number 8, in this case, might not be as magical as one thinks, if one sees it in a more general setting, say Finsler geometry.

Remark 1: Here is an anecdote I heard from William Allard. Before Simons' paper on stable cones, Wendell Fleming once said that heuristically with dimension increasing, the portion of area inside a unit ball is decreasing. Thus, it's conceivable that with the dimension increasing, strange things can happen.

Remark 2: Notably, by Gary Lawlor's classical results on vanishing calibrations (https://bookstore.ams.org/memo-91-446/) a minimal cone of arbitrary codimension is area-minimizing as long as the second fundamental form and the cotangent of the focal radius of its link are relatively small compared to its dimension. Thus, when dimensions increase, area-minimizing cones should become more and more abundant in some sense, indeed verifying Fleming's heuristic. Lawlor's criterion accurately detects all area-minimizing hypercones known up to date, and can even prove the instability in cases of lower dimensional Simons/Lawson cones. Almost all non-special-holonomic area-minimizing cones known up to today, i.e., excluding holomorphic, special Lagrangians, associatives, etc., either can be proven to be minimizing using Lawlor's criterion or are precisely discovered this way.

Zhenhua Liu
  • 627