# Granularity of rounding affects whether coordinates make a Prism region

Given the following six 3D coordinate triples,

    theCoordinates =
{
{4., 8., 12.},
{4., 8., 16.},
{5.75, 10.98956518577535, 14.},
{3.5684934338081673, 8.252590742770462, 12.},
{3.5684934338081673, 8.252590742770462, 16.},
{5.318493433808167, 11.242155928545811, 14.}
};


For some reason:

RegionQ@Prism[theCoordinates]
False


And likewise:

RegionQ@Prism[Round[theCoordinates, .001]]
False


And again:

RegionQ@Prism[Round[theCoordinates, .00001]]
False


And yet... !

RegionQ@Prism[Round[theCoordinates, .0001]]
True


Is there something obvious I am failing to see here?

• After experimenting some more with Prisms used as Regions, I am wondering whether a Prism is in general a usable Region when some of the coordinates are not exact numbers. A Mathematica Prism (a polyhedron that I would call a triangular prism) has two triangular faces and three rectangular faces. There seems to be a difficulty about whether the 4 vertices of each rectangular face are coplanar. It occurs to me that coplanarity of four points in three dimensions is tricky to decide unless the 12 numbers in the 4 coordinate triples are all exact (or symbolic). The problem is ill conditioned (?). – Ralph Dratman Jan 2 '16 at 3:31

This is not an answer but rather an extended comment.

theCoordinates = {{4., 8., 12.}, {4., 8., 16.}, {5.75,
10.98956518577535, 14.}, {3.5684934338081673, 8.252590742770462,
12.}, {3.5684934338081673, 8.252590742770462,
16.}, {5.318493433808167, 11.242155928545811, 14.}};


Even when the Prism does not evaluate to True with RegionQ, it may or may not be drawn with RegionPlot3D

Table[
{Graphics3D[
p = Prism[Round[theCoordinates, 10.^-n]],
PlotLabel -> StringForm["With n = , ReqionQ is ", n, RegionQ[p]]],
RegionPlot3D[p]},
{n, 2, 5}] // Grid


• Thanks for your extended comment. I see a message there about non-coplanar vertices, which I did not get, who knows why. Do you think that might be the problem with RegionQ etc? . . . . Incidentally, do you think your non-answer answer might lead others to think the question has been taken care of? – Ralph Dratman Jan 2 '16 at 1:42
• @RalphDratman - No. Since you have not accepted it, it will continue to show as open. Further, the first sentence states that it is not an answer. This is somewhat common on this site. – Bob Hanlon Jan 2 '16 at 1:58