I have several questions.
Ditto all of Bruce's questions, plus a question about how you have enough explanatory power to model a 4 way interaction (or enough mental energy). Wow.
I would be splitting my sample and breaking down the sub-relationships in separate pictures rather than trying to put it all into one. No matter how much sense it makes to you, your reader won't have the patience to sort out a complicated graph. (IMHO).
An issue that will pop up is dimentionality of the data. If I have a simple (X, Y) relationship, I'm already at 2 dimensions. X1, X2 and Y, I'm at 3D. X1-X4 and Y, that 5D. There might be 10 dimensional beings. Unfortunately, we are not among them. I would thin the best you could do was break everything down like Julia said. Groups of 20-way and maybe 3-way interactions.
Adding to what Andrew & Julia have said, in the case of all categorical variables (e.g., factorial ANOVA model), the number of levels of each variable plays a role. For example, a 2*2*k interaction can be displayed as the 2*2 interaction at each of the k levels of the 3rd variable, and is not terribly difficult to understand, if you just think about the nature of the 2*2 interaction varying across the k levels of the other variable (e.g., cross-over in one panel, non-parallel lines meeting at one end of the graph in another panel, parallel lines in another, etc). But suppose it was a 5*5*k. This becomes considerably more difficult to interpret, because most of us are probably not able to grok differences in 5*5 interactions the way we often can for 2*2 interactions.
http://www.merriam-webster.com/dictionary/grok
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