Word to describe a slice of multi-dimensional space Ported from English Language & Usage
I'm in the market for a mathematical (or otherwise) term to describe a slice of a hypercube.
Tensor is out of the running as that's the name of the object I am slicing.
The second I could use a hand with is a term to describe an index (or access point) that spans more than a single point in each dimension.
Though about using Ranged Index as a collection of Ranged Dimension, but I'm really hoping there's a more concise and explicit alternative.
For example:


*

*Regular index [1, 2, 1] would access index 1 in dimension-one, index 2 in dimension-2 and index 1 in dimension-3 

*Spanning Index (or whatever) [3->4, 1, 4->9] would access all elements between indices 3 and 4 of dimension-1, index 1 of dimension-2, etc...


More about Index:
It's programming terminology for a address to a section of data. I have a Tensor: a finite set of data points collected into a hypercube with each datapoint being accessible by a unique "address" using its position along each axis of the hypercube.
For a matrix: $\begin{bmatrix}5 & 3\\2 & 9\end{bmatrix}$
The value at:


*

*matrix[0,0] is 5

*matrix[1,0] is 2

*matrix[0,1] is 3

*matrix[1,1] is 9


with [0,0],[1,0],... being indices. For a cube of data one would access using [a,b,c]. If one wanted to specify a range of dimensions (i.e. cut a matrix out of a cube) They could index [0, 1->4, 2->5] extracting a 4x4 matrix from the cube.
Anyways, if indexing is out of scope and/or my explanation lacks clarity, simply ignore it.
 A: You shouldn't use the term "hypercube" here, because that's not what you're talking about. Instead, you are talking about a multi-dimensional table, which you could think of as the vertex set of a refinement of the 1-dimensional skeleton of a hypercube.
It is important to understand what you need this term for, though. For example, are you writing an article? If so and your target are computer scientists, then slice is actually the best term, because in various programming languages used in the scientific community (e.g. Python or MATLAB) it refers exactly to what you call "spanning index" in your question. On the other hand, if your target are mathematicians then you could choose almost anything, by you should define the terms you use. Both slice and "partial section" would be nice choices — note that I coined the second term, but a mathematician shouldn't have trouble understanding what you mean by it (but don't rely on this).
By the way, If you're talking to CS people index is a fine term, but if you're talking to mathematicians then you should use either entry if you're talking about a table, vertex if you're talking about a graph, or point if you're talking about a geometrical object. Confusingly enough, the objects that form what you call an index, e.g. $1$, $2$, and $1$ in $[1,2,1]$, are called indices in the first case and coordinates in the other two cases.
A: In math, a tuple used to index a multi-dimensional array is often called a multi-index. In programming, an index that spans a range of values is often called a slice or a slice index. I would therefore suggest a term like multi-slice or slice multi-index.

By the way, you're using the words "tensor" and "hypercube" in a way that could be confusing to many mathematicians.
In typical mathematical parlance, just as a matrix is a way of representing a linear map, a multi-dimensional array is a way of representing a tensor. You're dealing with multi-dimensional arrays, rather than the tensors they represent.
Although a matrix might be described informally as a square of numbers, it could be confusing to refer to a matrix as a square. Referring to a multi-dimensional array as a hypercube is liable to cause similar confusion.
