Based on the fact that:
There are 120 non-cyclical permutations of {1,2,3,4,5,6}
Each permutation could be represented by a hexagon with 6 triangles of different colours; e.g.,
Is it possible to write a formula or algorithm (function) to arrange (in a tessellation) these 120 hexagons in such a way that each triangle of colour is completely separated from its own colour i.e. it is surrounded by 12 triangles of colours that are not it's own colour. e.g., like this:
So that the result would be 720 Triangles composing 120 hexagons with unique non-cyclical permutations where no triangle touches (either at the point or at the side) a triangle of the same colour.
To clarify - If one colouring is a rotation of another then it is not part of the 120 permutations that are permitted to compose the final pattern in which all unique permutations of colours are allowed to be used and should be used only once.
Can the above pattern be generated ad infinitum? Can we frame this in terms of colouring a 12 regular infinate graph?