Understanding Numpy dimensions of arrays

Question

I started looking into Numpy using a 'Python for data analysis'. Why is the array dimension for arr2d is "2", instead of "3". Also why is the dimension for arr3d "3", instead of "2".

I thought the dimension of the array is based on the number of rows? Or this doesn't apply to higher dimensional and multidimensional arrays?

arr2d = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

arr2d.shape

Output: (3, 3)

arr2d.ndim

Output: 2

arr3d = np.array([[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]])

arr3d.shape

Output: (2, 2, 3)

arr3d.ndim

Output: 3

Answer 1

Numpy stores its ndarrays as contiguous blocks of memory. Each element is stored in a sequential manner every n bytes after the previous.

(images referenced from this excellent SO post)

So if your 3D array looks like this -

np.arange(0,16).reshape(2,2,4)

#array([[[ 0,  1,  2,  3],
#        [ 4,  5,  6,  7]],
#
#       [[ 8,  9, 10, 11],
#        [12, 13, 14, 15]]])

Then in memory its stores as -

When retrieving an element (or a block of elements), NumPy calculates how many strides (of 8 bytes each) it needs to traverse to get the next element in that direction/axis. So, for the above example, for axis=2 it has to traverse 8 bytes (depending on the datatype) but for axis=1 it has to traverse 8*4 bytes, and axis=0 it needs 8*8 bytes.

With this in mind, let's understand what dimensions are in numpy.

arr2d = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
arr3d = np.array([[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]])
print(arr2d.shape, arr3d.shape)

(3, 3) (2, 2, 3)

These can be considered a 2D matrix and a 3D tensor respectively. Here is an intuitive diagram to show how this would look like.

A 1D numpy array with (ndims=1) is a vector, 2D is a matrix, and 3D is a rank 2 tensor which can be imagined as a cube. The number of values it can store is equal to - array.shape[0] * array.shape[1] * array.shape[2] which in your second case is 2*2*3.

Vector (n,)       -> (axis0,)                #elements
Matrix (m,n)      -> (axis0, axis1)          #rows, columns
Tensor2 (l,m,n)   -> (axis0, axis1, axis2)
Tensor3 (l,m,n,o) -> (axis0, axis1, axis2, axis3)

Answer 2

well see basically the dimension of the array is not based on the number of rows basically it is based on the brackets i.e [] that you entered in np.array() method

see

arr2d = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

in arr2d there are 2 brackets([[]]) or there are 2 opening brackets([[) or its has 2 closing brackets(]]) so its an 2D array of (3,3) i.e 3 rows and 3 columns

similarly

arr3d = np.array([[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]])

in arr3d there are 3 brackets([[[]]]) or there are 3 opening brackets([[[) or or its has 3 closing brackets(]]]) so its an 3D array of (2,2,3) i.e its has 2 arrays of 2 rows and 3 columns