The best way to visualize a function of many variables is to use the method of displaying series of 2-d cross-sections and 3-d control volumes; the details are discussed at

https://math.stackexchange.com/questions/1454730/plotting-functions-of-many-variables

It is also useful to find a set of critical points of a function to be visualized. For the general description of the problem I would like to suggest the classical text by Morse

https://projecteuclid.org/download/pdf_1/euclid.bams/1183493097

Application of the theoretical methods described by Morse could be very useful in finding which cross-sections and control volumes are the most representative for the investigated function.

You may alo consider multidimensional scaling (MDS) or principal components analysis (PCA).

http://www.cs.haifa.ac.il/~rita/uml_course/lectures/PCA_MDS.pdf

You can do it between the Xi variables and the inner product.

When Y is relative to X, the sum of the inner product of Xi is equal to 0.

It's going to be the 2/3pai four-dimensional coordinate system. Xi is just going to be 6 vectors.

The statistics package Data desk (www.datadesk.com) can display rotating plots in up to 10 dimensions. The visualization is 3-D, but you can select any 3 dimensions and continue the rotation. It is relatively easy to, for example, locate directions of greatest variation or to identify subsets and groups. You can color points and relate them to other displays dynamically.

@Paul: "It is relatively easy to, for example, locate directions of greatest variation or to identify subsets and groups." - if this was the task then I would go for PCA.