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FourToSixSymmetry[FourWayImage_] := Module[
{
Hex = Graphics[Polygon[CirclePoints[6]], ImageSize -> 720],
Quarter = Graphics@Polygon[{{-1, -1}, {-1, 1}, {0, 0}, {1, 1}, {-1, 1}}],
Equalateral, SixWay
},
Equalateral = ImageResize[ImageSubtract[FourWayImage, Quarter], {360, Sqrt[3]/2 * 720 }];
SixWay = Table[ImageRotate[Equalateral, i * Pi/3], {i, -3, 2}];
ImageAdd[Prepend[SixWay, Hex]]
]
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