Awhile ago, Martin Gardner introduced Scott Kim's symmetric 2-pire map. There are 12 empires, each with two regions. All twelve empires share a border. This is a 2-pire solution for the Empire Coloring Problem.
In 3D, the 2-pire map becomes a dodecahedron with each face split.
For 3-pires, an 18 empire solution by Taylor is known. It's not symmetric.
Is there a symmetric 18 empire 3-pire map?