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In programming I've come across the idea of arrays containing arrays containing arrays etc., and as it's pretty intuitive to think of an array of arrays as a matrix, it seems like a reasonable idea to think of higher structures as higher-dimensional matrices. Unfortunately, trying to find out about n-dimensional matrices is difficult, as searching for '3d' gives $3\times3$ matrices, and any other reference to dimension gives rise to information concerning the height and width of a regular matrix.

So are higher-dimensional matrices used in maths for anything in particular? I'm assuming they might have some links to mathematical physics, or some kind of modelling where there are many constraints/equations in a system. Even if not, could somebody provide some links to where I can read a bit more about the idea, or (if possible) give me some interesting ideas, theorems, proofs, applications about them? I am more interested in the pure side of things rather than simply lists of how they might be used, but any information would be lovely. Thanks in advance!

J. M. ain't a mathematician
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Tim
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An $n$-dimensional array filled with numbers may (or may not) represent a tensor (of order $3$, if it has three indices). One often encounters tensors in the form of a tensor field, which is an assignment of a tensor to every point of a space. A prominent example is the Riemann curvature tensor which has order $4$. The Ricci curvature tensor and the Riemannian metric tensor both have order $2$, so they can be interpreted as matrix fields.

Christoffel symbols have three indices, but do not form a tensor field, because under the coordinate transforms they do not behave like a tensor field would. The point is that not every $3$-dimensional array represents a tensor of order $3$: the rules of transformations under coordinate changes are important. Another array that is not a tensor is the Levi-Civita symbol.

There must be some natural examples of order $3$ tensors, but I can't think of any. There's even a thread Tensors of order 3 but the question there was edited, and the answers address a different question.

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  • The difference between two Christoffel symbols, for two different metrics on the same space, is a tensor. And, of course, a volume form on a $3$-fold is a tensor. Those are the best differential geometry examples I can think of. – David E Speyer Jun 13 '13 at 13:43