I'd like to plot the TetrahedronFiveCompound as in this image from https://mathworld.wolfram.com/Tetrahedron5-Compound.html. If possible without the black edges and choosing each tetrahedron's color. I can't find the way to do that in the options for PolyhedronData.

An extended comment...

The notebook that you can download from the MathWorld Tetrahedron5-Compound page includes the following code:

Off[General::spell1];
Needs["GraphicsPolyhedra"]
vertices = Vertices[Dodecahedron];
Triangles[{f1_, f2_, f3_, f4_}] :=
{{f1, f2, f3},
{f1, f3, f4},
{f1, f4, f2},
{f4, f3, f2}}
tetrapoints = {{1, 9, 12, 20}, {2, 10, 13, 16},{3, 6, 14, 17}, {4, 7, 15, 18}, {5, 8, 11, 19}};
poly = (vertices[[#]] &) /@ Flatten[ Triangles /@ tetrapoints, 1];
linearcombination[r_, a_, b_] := (1 - r)*a + (r)*b
HollowTriangle[t_] :=

Block[{a = t[], b = t[], c = t[], aa, bb, cc},
aa = linearcombination[1/8, a, (b + c)/2];
bb = linearcombination[1/8, b, (c + a)/2];
cc = linearcombination[1/8, c, (a + b)/2];
{{a, b, bb, aa}, {b, c, cc, bb}, {c, a, aa, cc}} ]
On[General::spell1]
tetrapict = Show[Graphics3D[ Polygon /@ Flatten[ HollowTriangle /@ poly,1]],ViewVertical -> {.52, .38, .85}, Boxed -> False]

Which would produce: Except that more current versions of Mathematica, no longer support the Vertices[] function.

I didn't find an obvious direct equivalent of Vertices[] in the new functionality, but someone here can (hopefully) direct you to it.

Maybe someone more familiar with GraphEmbedding (see documentation) or the integration of the earlier Combinatorica package can help.

All of the above said, this would only get you an image of the

TetrahedronFiveCompound

You (or others on this site) then need to figure out how to get to the specifics of what you want to achieve, plotting a similar image,

If possible without the black edges and choosing each tetrahedron's color.

I'll give this some more thought as I can.

• I didn't realize I could download the code from Mathworld! I thought it was some kind of default drawing mode, but it seems it is more manual than that. I'll see if I can tweak the code. Thank you! May 21 '21 at 18:24
data=PolyhedronData["TetrahedronFiveCompound", "Polyhedra"];
Graphics3D[
Riffle[ColorData /@ Range,
data]] • Thank you, I meant without faces, but this is nice too. May 21 '21 at 18:21