Is there a way to graphically imagine smash product of two topological spaces?

To eloborate on what Arnav Tripathy said in a comment: if $X$ and $Y$ are compact, then $X \wedge Y$ is the one-point compactification of $(X \setminus \{x_0\}) \times (Y \setminus \{y_0\})$.

Here's a picture of a smash product that I drew for this talk, as far as I can tell it's what "Qfwfq" is describing in the middle paragraph.

As far as I remember, the smash product $X\wedge Y$ of two (pointed) spaces $(X,x_0)$ and $(Y,y_0)$ is obtained by taking the product of the two spaces $X\times Y$ and collapsing both the vertical "line" $\{x_0\}\times Y$ and the horizontal "line" $X\times\{y_0\}$ to a point.

So, if you (very loosely) imagine $X$ and $Y$ as segments, you can imagine $X\wedge Y$ as a square handkerchief which is shrinked along the central vertical and horizontal line, and the result is four "overhangs" coming out from the base point. [Of course the 4 overhangs are just an "artefact" of your simplified mental picture of the spaces $X$ and $Y$ as segments, it's not something general!]

Was I "graphical" enough? :)