Lets say you have a 2 × 5 matrix.

Using the traditional way of specifying matrix sizes, that’d mean you had a matrix with 2 rows and 5 columns. Okay, fine.

But what if you had a 2 × 3 × 4 matrix?

In python with numpy, this might look something like (with a ton of whitespace added for clarity):

arr = numpy.array([ [ [1,2,3,4], [2,3,4,5], [3,4,5,6] ], [ [4,5,6,7], [5,6,7,8], [6,7,8,9] ] ])

The way that we’re specifying the array in numpy actually helps add some clarity here. The 2 × 3 × 4 matrix is really just a list of lists of lists of elements.

*I thought you said it added clarity?*

Well, it does. Rows are made up of elements, 2D matrices are made up of rows, and 3D matrices made up of 2D matrices.

Anyway, the point of this post is, what do you call the measure of that third dimension? If the measure of the first dimension (the row) is called “columns”, the measure of the second dimension “rows”, what should the measure of the 3rd dimension be called?

This thread in the /r/math subreddit makes the following suggestions, in order of upvotes:

- Aisles
- Pages
- Slices
- Tensors
- Layers
- Depth(s?)

I’m not a huge fan of “aisles”, despite it getting the most upvotes. It just doesn’t seem intuitive enough. When I first think about an aisle, I’m thinking about the narrow passage ways in the stores you walk down, not the sides of those that contain the product (which is where I think that commenter was headed.) That could just be me, however.

I personally also rule out “slices” and “tensors” out, slices having somewhat specific meanings in python, and tensors having specific meanings in linear algebra and mechanics (which don’t align completely to the concept of the measure of the 3rd dimension).

I also rule out “depth”, if only because its not easily pluralized like row → rows, column → columns. A phrase like “number of columns” has some issues when you move to “number of depths”. That being said, the “depth” of a 3D matrix makes a lot of sense to me. Just as you might gain a “depth” perception when you go from a “flat” 2D Cartesian coordinate system to a 3D one.

So to that end, “layers” and “pages” both fit my intuitive concept of the measure of the 3rd dimension. You have a stack of “pages”, each with a 2D matrix printed on it — fine.

Between the two, I think “layers” is probably the better term — it seems more general.

However, there is an interesting “natural” relation that springs up if you go with “pages”. Consider:

- Columns
- Rows
- Pages
- Books
- Series

“Series” is pretty weak, I admit, but the first four degrees are pretty solid. Nevertheless, I’m still going with “layers”.

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