RegionUnion issues with many Regions

Question

The new RegionUnion[] function is just what I needed if only I could get it to work. I have many non-overlapping regions that I will need to use as plotting domains and integration domains. The following simple code illustrates my problem.

Make a simple $n$-gon

Clear[ngon]
ngon[r_, n_, c_: {0, 0}] := 
Polygon[Table[
c + r {Cos[q], Sin[q]}, 
{q, 0, 2 Pi, (2 Pi)/n}]];

Make a ring of hexagons:

 n = 4;
 R = 3;
 polys = Table[
    ngon[1, 6, R {Cos[q], Sin[q]}], 
   {q, 0, 2 Pi, (2 Pi)/n}];

This will display them:

 Graphics[{polys}]

Now use them as regions in a simple Plot3D[]

 Plot3D[1, {x, y} \[Element] RegionUnion[polys]]

If you have been running this code you will see that everything works as expected. But when I increase the number of regions to n=9 my computer works and works until the kernel runs out of memory. My real problem will be much more complex than what I have shown here.

Is there a better way to combine regions? Am I doing things poorly?

Mathematica Ver: 10.0.1 running on Ubuntu Linix

Answer 1

DiscretizeRegion makes it is easier to deal with Regions:

Manipulate[
 Module[{p = 
    Table[ngon[1, n, r {Cos[q], Sin[q]}], {q, 0, 2 Pi, (2 Pi)/n}], ps},
  ps = RegionUnion[DiscretizeRegion /@ p];
  Plot3D[1, {x, y} \[Element] ps, Mesh -> False]], {r, 1, 6}, {n, 
  Range[4, 12]}]