Lewis-Williams is a South African archaeologist who studied the rock art produced by San hunter-gatherers. He applies ideas from there to the cave art of Upper Palaeolithic Europe.

There are many good things about this lovely book, the colour plates, the large number of line drawings. There are interesting sections on the differences between Homo sapiens hunter-gatherers and Neanderthals, on organisation of the caves and where parietal art appears within caves.
Shamanism was general in hunter-gatherer communities. Within each hunting group there were seers, who entered trance by sleep deprivation or using hallucinating substances, so they saw/ imagined entopic images and ultimately hallucinations. Lewis- Williams suggests hallucinations were remembered in producing the rock and cave art, accepting that the quality of the work is too good itself to have been produced in trance.
The “tiered universe” is a vision of living in the landscape with the heavens above and the spirit world below. He suggests such a vision is hard wired in Homo sapiens. It explains claims for shaman “psychic-flying” and that San rock face art and late-Pleistocene cave art symbolise contact with the spirit world within the rock and below the surface.
The thesis is interesting and, as an explanation, seductive. However it is unproven and essentially unprovable. It may just be that caves were the only European environment facilitating survival of art from so long ago.
I remain unconvinced by the book’s central thesis. There are other simpler explanations for cave art, in particular its connection with hunting. Lewis- Williams may be partly right , and his “tiered universe” an explanation of the emergence of religious belief rather than cave art.