# Visualising set of inequalities

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?

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}]


• this is most interesting, thanks a lot. Can we actually use this to also compute the area (more correctly the Lebesgue measure) of the solution set? – user21766 May 27 '17 at 18:43
• This shouldn't be that hard with plain Integrate or NIntegrate as long as you have conditions which form a border or represent a Region – Julien Kluge May 28 '17 at 17:55