Visualization of binomial coefficients to the 4th power

Question

I have been reading about binomial coefficients in Wikipedia.
Where there is a visualization of binomial expansion up to the 4th power:

I do not understand the sequence for the 4th dimension, i.e.: $(a+b)^4$
My brain simply does not get it...

For example, take $4a^3b$ and and the orange sketch.
$6a^2b^2$ and the green one.
$4ab^3$ and the light blue one.
How do they map to each other?
Could someone give some hints how to get that?

Answer 1

Well, the coefficients form Pascal's triangle. The general formula to go from one row to the next one is $${n\choose k} = {n-1\choose k-1}+{n-1\choose k}.$$ For instance, $${4\choose 2} = {3\choose 1}+{3\choose 2} = 3 + 3 = 6.$$

Answer 2

For $(a+b)^4$ you can either use the binomial expansion, $$\sum_{k=0}^{4}{n\choose k}a^{4-k}b^k$$

or, and this is a much better alternative, pascals triangle. Write a 1 somewhere at the top of your paper, then branch out two more 1s. From that point on, just add those branches to form the next row, enclosed in 1s. You'll form a triangular shape where the outermost left and right sides are just ones. The counting numbers are the degrees of the expansion, and across are the terms.

I recommend watching this for Pascal's triangle: https://www.youtube.com/watch?v=OMr9ZF1jgNc

Edit: If your issue is conceptualizing the geometry of the expansion, then you can just look up animations online. There's not much help you'll get here for that. If your issue is deriving the sequence, then you'll be fine.

Answer 3

When increasing the degree, segments become squares, then cubes then tesseracts (and similarly with rectangles).

For the 4D case, you must think of all elements from the 3D case as getting a new dimension, in all possible ways.