# What is the volume of the intersection of four cylinders of equally radius equally spaced?

I saw that question at the certain forum and answered it with help of Mathematica 13.1 in such a way. The angles between the unit vectors {0, 0, 1}, {-Sqrt[2/9], -Sqrt[2/3], -1/3}, {-Sqrt[2/9], Sqrt[2/3], -1/3} and {Sqrt[8/9], 0, -1/3} are equal because all the pairwise inner products equal -1/3. The first cylinder is defined as cyl1 = ImplicitRegion[x^2 + y^2 <= 1, {x, y, z}];. The second cylinder is made from the first one by the rotation around the origin which transforms {0, 0, 1} to {Sqrt[8/9], 0, -1/3}}:

 cyl2 = TransformedRegion[r1, Function[p, RotationMatrix[{{0, 0, 1}, {Sqrt[8/9], 0, -1/3}}] . {p[[1]],  p[[2]], p[[3]]}]]


ParametricRegion[{{-(x/3) + (2 Sqrt[2] z)/3, y, -((2 Sqrt[2] x)/3) - z/3}, x^2 + y^2 <= 1}, {x, y, z}]

As we see, cyl2 is parametrically defined. Next, we find the intersection and volume of these cylinders:

intersect1= RegionIntersection[cyl1, cyl2]; Volume[intersect1]


4 Sqrt[2]

Now we consider the third cylinder

 cyl3 = TransformedRegion[r1, Function[p, RotationMatrix[{{0, 0, 1}, {-Sqrt[2/9], Sqrt[2/3], -1/3}}] . {p[[1]], p[[2]], p[[3]]}]];


and its intersection with intersect1

intersect2 = RegionIntersection[intersect1, cyl3];


and the volume

Volume[intersect2]


4.72311

We see Mathematica fails to produce an exact result (which is of no need in the most of cases).

At last, the fourth cylinder

cyl4 = TransformedRegion[r1,Function[p, RotationMatrix[{{0, 0, 1}, {-Sqrt[2/9], -Sqrt[2/3], -1/3}}] . {p[[ 1]], p[[2]], p[[3]]}]]


, its intersection with the previous ones

intersect3 = RegionIntersection[intersect2, cyl4]


and the final volume

 Volume[intersect3]


4.54725

Let us visualize

 RegionPlot3D[intersect2, PlotRange -> {{-3/2, 3/2}, {-3/2, 3/2}, {-3/2, 3/2}},
PlotPoints -> 100, PlotTheme -> "Monochrome"]


and

RegionPlot3D[intersect3, PlotRange -> {{-3/2, 3/2}, {-3/2, 3/2}, {-3/2, 3/2}},
PlotPoints -> 100, PlotTheme -> "Monochrome"]


Is it possible to obtain an exact result for Volume[intersect3] with Mathematica? The symmetry of intersect3 and that answer encourage.