Graphing a function of 4 variables

Question

I have a function of 4 variables: i, r, s, and p. This is the function:

$$f(i,r,s,p) = s*p(\frac{r} {1+i-r}) $$

D=$\mathbb{R}$

Is there a way for me to graph this function? Is it somehow possible to modify a 3D graph to express 4 variables if I were to combine $s*p$ into $m$, for example? Can/how would that work? Would a 4D graph be suitable? If a 4D graph would be the one, what would it look like? How can I make it?

I've read a bunch of blog posts talking about 4D graphs and stuff like that, but none of them really answer my question.

Thanks a lot for any responses! All are greatly appreciated!

EDIT:

As was pointed out to me, because I have 4 inputs, I actually want a 5 dimensional graph, not 4. This makes total sense as a 3D graph takes takes inputs $x$ and $y$, and outputs $z$. My real question is how I would graph a 5D graph. Sorry for my mistake!

Answer 1

First off, I agree that the correct approach is to conflate the two variables $s$ and $p$ into one variable, say, $m=s\times p$. This effectively allows you to reduce the problem by one dimension. Thus the problem becomes how to visualize the equation $$m\times r = c\times(1+i-r).$$ I emphasize the generic term visualize rather then the more precise term $graph$ because, as mentioned, the graph lives in 4D space - one dimension for each variable. There are other ways to visualize an equation, though. One common technique to visualize an equation involving three variables is to plot the surface consisting of all points where the equation is true. If $c=0$ in the above equation, for example, we get the equation $m\times r=0$, which is true when either $m=0$ or $r=0$. The graph in the $(m,r)$-plane consists of just the $m$ and $r$ axes together. When extended into $(m,r,i)$ space, this consists of all points of the form $(0,r,i)$ or $(m,0,i)$, which looks like so:

To visualize your equation involving four variables, we sketch several representative contour plots from the family of equations for various choices of $c$. We might place them all together in one plot:

or in several:

Finally, if you find yourself facing a genuine function of five variables with no easy way to conflate two of them, you could always sketch a 2D grid of 3D contours (rather than the 1D grid above). Similarly, this function of four variables could be visualized as a 2D grid of 2D contours:

Answer 2

I have a better idea. Simulate 3 variable color and the fourth variable is time. So draw this three-dimensional space as a colorful chart that is changeable over time.