# [GIF] Omnes Pro Uno (Mercator projection of level sets of dot product sum)

Posted 5 months ago
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 Omnes Pro UnoAs Correlations was for the vertices of the octahedron and Power Surge was for the vertices of the tetrahedron, this is for the vertices of the triangular bipyramid: the Mercator projection of the level sets of the function which takes the sum of the absolute values of the dot products of a point on the sphere with the given set of points. In this case, I'm actually showing half the sphere twice: the longitude runs from $-3\pi/2$ to $3\pi/2$, which depicts the "front" half of the sphere twice.As in the previous animations, I'm using ContourPlot[], so for each point $(x,y)$ in the plane, I'm applying inverse Mercator projection and then summing the dot products of the resulting point on the sphere with the vertices of the triangular bipyramid. InverseMercator[{x_, y_}] := {Sech[y] Cos[x], Sech[y] Sin[x], Tanh[y]}; bipyramidverts = Normalize /@ PolyhedronData[{"Dipyramid", 3}, "VertexCoordinates"]; BiPyramidTotalDotProduct[{x_, y_}, θ_] = FullSimplify[ Total[ Abs[ InverseMercator[{x, y}].RotationTransform[θ, {0, 1, 0}][#]] & /@ bipyramidverts], -π < x < π && -π < y < π && 0 < θ < 2 π, TimeConstraint -> 1]; Then it's just a matter of choosing a color scheme and animating: Manipulate[ ContourPlot[BiPyramidTotalDotProduct[{x, y}, θ], {x, -3 π/2, 3 π/2}, {y, -π, π}, AspectRatio -> 2/3, Frame -> False, ImageSize -> 540, ContourStyle -> None, Contours -> Range[1.8, 2.8, .1], PlotRangePadding -> -0.01, PlotPoints -> 50, ColorFunction -> (ColorData["SunsetColors"][1 - #] &)], {θ, 0, π}]