Extending a ternary plot to a tetrahedron (using ListPlot)

Question

I have been making ternary plots of data that is a list of sets of 3 numbers, where the numbers in each set sum to one (call this "DATA"). To do this, I first transform DATA onto the 2D surface with the following:

transf[{a_, b_, c_}] := {b + c/2, (Sqrt[3] c)/2}

Map[transf,DATA].

This gives a list of sets of 2 numbers that map onto a simplex (call this DATA2). This is how I've plotted DATA2:

Show[{
  RegionPlot[Sqrt[3] a1 - a2 < 0, {a1, 0, .5}, {a2, 0, 1}, 
   DisplayFunction -> Identity, PlotStyle -> White, 
   BoundaryStyle -> None],
  RegionPlot[Sqrt[3] a1 + a2 > Sqrt[3], {a1, .5, 1}, {a2, 0, 1}, 
   DisplayFunction -> Identity, PlotStyle -> White, 
   BoundaryStyle -> None],
  RegionPlot[a2 < 0, {a1, 0, 1}, {a2, -.2, 1}, 
   DisplayFunction -> Identity, PlotStyle -> White, 
   BoundaryStyle -> None],
  ListPlot[{{0, 0}, {1, 0}, {.5, Sqrt[3]/2}, {0, 0}}, Joined -> True, 
   PlotStyle -> {Black, Thickness[.015]}],
  ListPlot[{DATA2}, Joined -> True,  
   PlotMarkers -> Automatic]},
 AspectRatio -> Automatic, Frame -> False, 
 PlotRange -> {{0, 1}, {0, 1}}] 

Example:

Question: Now, I have an extended data set that is made up of sets of 4 numbers that sum to one. I'd like to plot the data in an analogous way, but now on the corresponding tetrahedron. However, I'm not sure how to transform the data or how to draw the appropriate tetrahedron.

Can anyone help with this, please?

Answer 1

Is this the sort of thing you want?

data = #/Total[#, {2}] &@Log@RandomReal[1, {10000, 4}];
pts = data.PolyhedronData["Tetrahedron", "VertexCoordinates"];
Graphics3D[Point[pts, VertexColors -> RGBColor @@@ data],  Axes -> True]

Based on Ray Koopman's answer to Uniformly distributed n-dimensional probability vectors over a simplex, which I used in my answer to Random reals according to conditions for a similar purpose.

Updated -- Below is a plot the tetrahedron added and the points joined and colored in order:

data = #/Total[#, {2}] &@Log@RandomReal[1, {20, 4}];
pts = data.PolyhedronData["Tetrahedron", "VertexCoordinates"];
Graphics3D[{
  GraphicsComplex[pts,
   {PointSize[Medium],
    Point[Range@Length@pts, VertexColors -> Automatic], 
    Line[Range@Length@pts, VertexColors -> Automatic]},
   VertexColors -> (ColorData["Rainbow"] /@ Rescale[Range@Length@pts])
   ], {Thick, PolyhedronData["Tetrahedron", "Edges"]}}, Axes -> True]

Other color schemes than "Rainbow" may be used in the VertexColors option:

VertexColors -> (ColorData["Rainbow"] /@ Rescale[Range@Length@pts])

This GraphicsComplex option associates a color with each point in pts, that are used when the setting VertexColors -> Automatic occurs in Point, Line, or Polygon (not present here). Unfortunately Arrow does not accept the VertexColors option, so if arrows are used, each arrow would have to be colored indvidually (programmatically).

If more than one data set is to be colored, it would probably be best to use separate GraphicsComplexes and color them with sufficiently distinct color schemes.