Is there an easier/analogous explanation of geometric mean, similar to that of arithmetic mean? My simple understanding of arithmetic mean is that the arithmetic mean represents a "central point" (central tendency) where all others numbers converge as the accumulative linear distance from the left equals the accumulative linear distance from the right.
While I know (memorize) the definition of geometric mean, I cannot "interpret/visualize" it (maybe I should not?) in a way analogous to that of the arithmetic mean. I suspect that it may have to do with the fundamental concept of "root", which I fail to grasp, (I did try to read up on the concept of root but have yet found anything beyond its definition and application) but still I cannot find a satisfying answer how taking an nth root would yield a "mean" that reasonably represents the group.
Is there an easier way to "understand" geometric mean? (I tried to look for a similar question in this forum but have not found one so far).

As Jordan Ellenberg nicely shows in his book Shape, the geometric mean of $a_1$ and $a_2$ is the length of the side of a square having the same area as a rectangle with side $a_1$ and $a_2$. This generalizes to the geometric mean of more numbers as well. For more numbers/higher dimensions, the geometric mean is the length of the side of the regular shape having the same multi-dimensional volume.
