Substituting the element of a list into a list that contains the index position

Question

I see similar questions for lists and indexes and manipulating with Thread, MapIndexed,etc., but I do not see how to do a substitution.

I have a list of coordinates for a geometric object and a list of the vertices that make up the face of the geometric object which is simply the index to the coordinates list.

I broke them out for convenience initially thinking it would be simple to substitute the coordinates back into the vertices for the face.

I see several examples on mathematica documentation for MapIndexed, but don't understand how to apply them to this "substitution".

Any explanation and/or help would be appreciated. I have attached the sample code below.

groupedVerticesRT =
  {
   {
    {   C4, C4, C4}, {  C4, C4, -C4}, {   C4, -C4, C4}, {   
     C4, -C4, -C4},
    {-C4, C4, C4}, {-C4, C4, -C4}, {-C4, -C4, C4}, {-C4, -C4, -C4}
    },
   {
    {0, C6, C5}, {0, C6, -C5}, {0, -C6, C5}, {0, -C6, -C5}
    },
   {
    {C5, 0, C6}, {C5, 0, -C6}, {-C5, 0, C6}, {-C5, 0, -C6}
    },
   {
    {C6, C5, 0}, {C6, -C5, 0}, {-C6, C5, 0}, {-C6, -C5, 0}
    },
   {
    {0, C4, C6}, {0, C4, -C6}, {0, -C4, C6}, {0, -C4, -C6},
    {C6, 0, C4}, {C6, 0, -C4}, {-C6, 0, C4}, {-C6, 0, -C4},
    {C4, C6, 0}, {C4, -C6, 0}, {-C4, C6, 0}, {-C4, -C6, 0}
    }
   };
facesRT =
  {
   {13, 23, 15, 21}, {13, 23, 3, 25}, {13, 25, 1, 21}, {25, 18, 30, 
    3}, {25, 17, 29, 1},
   {25, 18, 26, 17}, {18, 30, 4, 26}, {17, 26, 2, 29}, {29, 1, 21, 
    9}, {21, 5, 27, 15},
   {15, 27, 7, 23}, {21, 9, 31, 5}, {5, 31, 19, 27}, {27, 20, 32, 
    7}, {27, 19, 28, 20},
   {23, 7, 32, 11}, {23, 11, 30, 3}, {32, 20, 28, 8}, {11, 32, 12, 
    30}, {22, 14, 26, 2},
   {19, 31, 6, 28}, {14, 24, 4, 26}, {9, 29, 10, 31}, {32, 8, 24, 
    12}, {12, 24, 4, 30},
   {29, 2, 22, 10}, {10, 22, 6, 31}, {28, 16, 24, 8}, {28, 6, 22, 
    16}, {16, 22, 14, 24}
   };

Edit: I would like to return something like the following using MapIndexed or something similar to substitute the coordinates into the list for the faces to generate one giant list that has a list of faces made up of a the list of their coordinates:

expandedRT =
{
 {
  {C5, 0, C6}, {0, -C4, C6}, {-C5, 0, C6}, {0, C4, C6} (* coordinates {13, 23, 15, 21} *)
 }, (* face 1 *)
 {
  (* faces 2 - end *)
 }
}

Answer 1

coords = Flatten[groupedVerticesRT, 1];

expandedRT = ReplaceAll[i_Integer :> coords[[i]]] @ facesRT

Alternatively, you can use Extract:

expandedRT2 = Extract[coords, List /@ facesRT];

expandedRT2 == expandedRT

True

Answer 2

What you are searching is "GraphicsComplex".

Here is an example:

coordinates = {{0, 0, 0}, {1, 0, 0}, {0, 1, 0}, {0.5, 0.5, 1}};

surfaces = {{1, 2, 3}, {1, 2, 4}, {2, 3, 4}, {1, 3, 4}};

A tetrahedron by surfaces:

Graphics3D[GraphicsComplex[coordinates, Polygon /@ surfaces] ]

The same can be achieved by:

Graphics3D[GraphicsComplex[coordinates, Triangle /@ surfaces] ]

If you only want 2 of the surfaces:

Graphics3D[GraphicsComplex[coordinates, Triangle /@ surfaces[[1 ;; 2]]] ]

If you want to replace indices by coordinates by hand, you may simply write:

coordinates[[#]] & /@ surfaces

(* {{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}}, {{0, 0, 0}, {1, 0, 0}, {0.5, 0.5, 
   1}}, {{1, 0, 0}, {0, 1, 0}, {0.5, 0.5, 1}}, {{0, 0, 0}, {0, 1, 
   0}, {0.5, 0.5, 1}}} *)

Answer 3

Taking cues from the answer by @kglr:

coords = Flatten[groupedVerticesRT, 1];
rules = MapIndexed[First@#2 -> #1 &, coords];

Now either Lookupor ReplaceAll can be used:

Lookup[rules, #] & /@ facesRT

Or

ReplaceAll[facesRT, rules]