Are there such things as 3-dimensional (and higher) analogues of matrices, and if so, do they have any applications?

A matrix is a group of numbers arranged in a rectangle. I wonder, has anyone studied 3-dimensional and higher analogues of matrices? For example, there could be such a thing as a 2 by 2 by 2 3d matrix, whose entries are all equal to 1. Has anyone else defined these entities, and more importantly, are they used in mathematics?

Matrices in the mathematical sense are maps - linear transformations - between two vector spaces - and are therefore two dimensional. The mathematical object describing a multilinear map is called a tensor. A tensor has much more structure than a matrix. Tensors are extensively studied and used, especially in physical applications of mathematics.

If you are looking for a simple extension to the matrix concept, most programming languages support multi-dimensional matrices in data structures called arrays.

Another interesting example that’s unrelated to linear or multilinear algebra …

When modeling a 3D object, it’s common to use a 3-dimensional array of cells called “voxels”. The cell value is “true” if the cell’s center is inside the 3D object being represented. This is exactly analogous to the idea of using a 2D array of pixels to represent a bitmap image.