Measures 

Compute the arc length of a 1D mesh, the area of a 2D mesh, or the volume of a 3D mesh.

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X mesh1d = DiscretizeRegion[Circle[]]; mesh2d = DiscretizeRegion[Disk[]]; mesh3d = DiscretizeRegion[Cuboid[], MaxCellMeasure -> \[Infinity]];

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X Grid[{{mesh1d, mesh2d, mesh3d}, {ArcLength[mesh1d], Area[mesh2d], Volume[mesh3d]}}]

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Compute the centroid of any mesh.

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X Grid[{{Show[mesh1d, Graphics[{Red, Point[RegionCentroid[mesh1d]]}]], Show[mesh2d, Graphics[{Red, Point[RegionCentroid[mesh2d]]}]], Show[HighlightMesh[mesh3d, Style[2, Opacity[0.3]]], Graphics3D[{Red, PointSize[Large], Point[RegionCentroid[mesh3d]]}]]}}]

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Compute the volume of different levels of a procedurally generated mesh.

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X ArrayPatternMesh[a_?ArrayQ] := Module[{mp, pos, fpos, dims = Dimensions[a], cdims}, mp = RegionProduct @@ Map[MeshRegion[Partition[N[Range[0, #]]/#, 1], Line[Range[1, # + 1]]] &, dims]; pos = ArrayRules[a][[1 ;; -2, 1]]; cdims = Reverse[FoldList[Times, 1, dims[[2 ;; -1]]]]; fpos = 1 + (pos - 1).cdims; MeshRegion[MeshCoordinates[mp], MeshCells[mp, {Length[dims], fpos}]]]; drill[s_, t_] := Block[{sp, d = ArrayDepth[t], r = Length[s]/Length[t]}, If[r == 1, Mod[t s, 2], sp = Partition[s, Dimensions[s]/Dimensions[t]]; Flatten[MapThread[If[#1 == 0, #2 0, drill[#2, t]] &, {t, sp}, d], Transpose[Partition[Range[2 d], d]]]]]; SierpinskiArray[t_, k_] := Module[{s, dims = Dimensions[t]}, s = ConstantArray[1, dims^k]; drill[s, t]];

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X t = {{{1, 0}, {1, 1}}, {{0, 0}, {1, 0}}}; meshes = ArrayPatternMesh[SierpinskiArray[t, #]] & /@ Range[6]; Grid[{Part[meshes, 1 ;; 3], Part[meshes, 4 ;; 6]}]

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X vols = Volume[#] & /@ meshes

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Find a function that describes the sequence of volumes.

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X FindSequenceFunction[Rationalize[vols], k]

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