# Show, if possible, small isolated “islands” in the unit cube using RegionPlot3D

I have a certain three-dimensional constraint

-(1/3) < t3 < 1/3 && -(1/3) < t2 < 1/3 && 9 t1^2 < (1 - 3 t3)^2 && 9 t1^2 < (1 + 3 t3)^2 && t1^2 t2^2 t3^2 > 1/23328  .


If I perform an integration over the unit cube $$[-1,1]^3$$, I obtain (using the GenericCylindricalDecomposition and FullSimplify command) the result

(1/(54 Sqrt))(8 Sqrt[2 - Sqrt] - ArcTanh[Sqrt[1 - 1/Sqrt]] (8 + Log) + Log[2 - Sqrt[4 - 2 Sqrt]]^2 - Log[2 + Sqrt[4 - 2 Sqrt]]^2 - 4 PolyLog[2, 1/4 (2 - Sqrt[4 - 2 Sqrt])] + 4 PolyLog[2, 1/4 (2 + Sqrt[4 - 2 Sqrt])]


which evaluates to approximately 0.00221357, while, of course, the unit cube has a much larger volume, that is, $$2^3=8$$.

I believe--on the basis of prior ("bound-entanglement" probability) considerations, that the regions in which the constraint is satisfied are disjointed and form an "archipelago" of islands.

However, my limited attempts (using various ranges of coordinates for the variables $$t1, t2, t3$$) to show these presumed regions using RegionPlot3D just return vacuous plots.

RegionPlot3D missed the regions of interest when using the default number of plot points because they are a very small fraction of the entire plotting volume. You can increase the PlotPoints used to get a result:

RegionPlot3D[
-(1/3) < t3 < 1/3 && -(1/3) < t2 < 1/3 && 9 t1^2 < (1 - 3 t3)^2 &&
9 t1^2 < (1 + 3 t3)^2 && t1^2 t2^2 t3^2 > 1/23328,
{t1, -0.6, 0.6}, {t2, -0.6, 0.6}, {t3, -0.6, 0.6},
PlotPoints -> 200, Mesh -> None
] Note that I restricted the plotting range to emphasize the regions of interest. You can use the full $$(-1,1)$$ range and it works the same. The Mesh -> None option is simply to produce a more readable plot, but it has no functional effect.

• I've tried upping the number of PlotPoints--but the resulting plots seem to stay exactly the same. I strongly suspect that these islands have a craggy/rough nature. Is it possible to reveal that as well? As it stands, they seem to have a rather smooth character. Maybe one would have to zero in on particular islands to achieve that. – Paul B. Slater Dec 18 '19 at 20:11
• @Paul I haven't zoomed very far in, but from a quick glance at the result of Reduce[yourconditions, {t1, t2, t3}] the boundaries of these lunettes seem to be fourth-order polynomials, so they may actually be relatively smooth. – MarcoB Dec 18 '19 at 23:14