Visualize a vector. For example, the vector
So the numbers 1 through 5, stored in sequence in memory. Got it? rand will do something similar.
rand(1,5)
ans =
0.5599 0.1491 0.3358 0.0136 0.0369
5 numbers, stored in the vector V.
Do you see that a vector is just a set of scalars, stacked together into a one-dimensional list of numbers?
Next, what is a 2 dimensional array?
A2d = rand(2,3)
A2d =
0.9718 0.3620 0.6652
0.4545 0.6057 0.3576
2 rows, 3 columns. But it is just a set of 6 numbers here, stored with an associated shape. In this case, a rectangular shape. We can think of it as 3 column vectors, each of length 2, as as 2 row vectors, each of length 3, if you prefer.
Got it?
Next, visualize a cube. A cube has three dimensions, right? We can think of that cube as squares, packed on top of each other. But each square can be thought of as 2 dimensional arrays, and we already understand 2 dimensional arrays. (Well, I hope we do.)
So if we extend the idea of a 2-dimensional array as just a set of 1-dimensional vectors, stacked next to each other, or on top of each other, then a 3 -dimensional array is no different. I'll do this one using a vector of integers first, so you can predict what numbers will be where.
A3d = reshape(1:12,[2 2 3])
A3d =
A3d(:,:,1) =
1 3
2 4
A3d(:,:,2) =
5 7
6 8
A3d(:,:,3) =
9 11
10 12
What do we see there? The first "plane" of that array is a 2x2 matrix. We saw it displayed above, but we can extract it like this:
Again, think of a higher dimensinal array as just lower dimensional arrays, stacked up.
R = rand(2,3,4)
R =
R(:,:,1) =
0.0669 0.7084 0.4073
0.5308 0.4514 0.4074
R(:,:,2) =
0.7872 0.5268 0.1943
0.7919 0.1478 0.6438
R(:,:,3) =
0.9757 0.3458 0.6599
0.5446 0.9618 0.4549
R(:,:,4) =
0.9378 0.4052 0.7834
0.1438 0.1839 0.2659
So a 2x3x4 array.
Finally, we can stack three dimensional arrays, into a 4-dimensional array. We can see tham all listed below.
b = rand(2,2,2,2)
b =
b(:,:,1,1) =
0.4676 0.0003
0.3202 0.1176
b(:,:,2,1) =
0.6656 0.8099
0.9382 0.7427
b(:,:,1,2) =
0.3774 0.8263
0.2789 0.0461
b(:,:,2,2) =
0.7243 0.2364
0.0349 0.2116
Unfortunately, our minds live in a 3-dimesinal world. They are not constructed to think in higher dimensions than 3. But surely you can follow the abstract idea of stacking lower dimensional things, to create something with one more dimension?
