By the way Alice Through the Looking Glass is based very precisely on the Tesseract. Just as Sylvie and Bruno is based on the Mobius Strip.
Lewis Carroll was influenced by Hinton like Tolkien was influenced by Dunne.
In 3 dimensions we have five platonic solids as seen in Euclid, and explored for their synergies by B. Fuller.
In 4 dimensions there are six platonic solids, that is one extra with no corresponding figure in three space.
In all higher dimensions there are only three platonic solids.
So the six platonic solids in four space are its fundamental structures. Also there are others beyond these that are native to the fourth dimension.