Visualising set of inequalities

Question

Is there any way to visualise a set of inequalities when there are 4 variables at play?

Context: if we have inequalities involving two variables only, one can highlight the ensemble of points that satisfy the inequalities on a 2D plane. For example x+y<4. Similarly one can imagine for 3 variables in 3D. But if we have four variables with the following set of inequalities: $1<a<5$ and $1<b<5$ and $1<c<5$ and $1<d<5$ and $a\neq c.$ Is there a meaningful way of performing an embedding the solutions onto a 3D or 2D plane?

If visualisations are not possible, is there other ways Mathematica can help in giving more insight into the solutions of the above set?

Answer 1

Actually this is very simple. Think about it.

We live in 3 spatial dimensions. But we experience another one:

Time.

So lets use this. Lets make time our fourth dimension to apply our conditions.

Lets imagine your given set of inequalties and a fourth inequaltiy I produced to make this more interesting:

$$\begin{cases}1<x<5 \\1<y<5\\1<z<5\\x<w<z\end{cases}$$

Where $w$ is our fourth coordinate we want to change in time.

So lets use Manipulate:

Manipulate[
 RegionPlot3D[
  1 < x < 5 && 1 < y < 5 && 1 < z < 5 && x < w < z,
 {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, PlotPoints -> 50]
 , {w, -5, 5}]

This can be made with pretty strange conditions:

Manipulate[
 RegionPlot3D[
  Cos[x] > Sin[z] && Cos[z]^2 > y && 
   x > RiemannSiegelZ[w*10]*2 > y, {x, -5, 5}, {y, -5, 5}, {z, -5, 5},
   PlotPoints -> 20], {w, 0, 5}]

I also suggest, that you add a numeric indicator in your animations when you export them.

An example of a single condition with 4 variables (Which looks very reasonable):

Manipulate[
 RegionPlot3D[
  x^2 + y^2 + z^2 < 5^2*Sin[w]^2, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, 
  PlotPoints -> 20], {w, 0, Pi}]