Tiger's eye is a fibrous quartz with a strong silky chatoyancy, and how it came to be that way is an open question with two answers in the literature.
The classic explanation, taught for more than a century, is pseudomorphic replacement. Crocidolite, a fibrous blue amphibole, is progressively replaced by silica while its fibrous structure is preserved, and the iron in the original mineral oxidises to give the golden brown colour. Blue material in which the replacement is incomplete is sold as hawk's eye, and the two occur together in the same deposits, which appeared to confirm the sequence.
Later petrographic work has argued otherwise: that the fibres formed by a crack-seal process, in which a vein opens repeatedly and quartz and amphibole crystallise together across the opening each time, so the two minerals grew at once rather than one replacing the other. Evidence cited includes the way the fibres relate to the vein walls and the presence of features that a replacement model does not predict.
The question has not been closed to everyone's satisfaction, and both accounts remain in print. The atlas records it as unsettled rather than choosing.
The principal source is the Northern Cape of South Africa, in banded ironstones of great antiquity, and the material has been quarried there in quantity since the nineteenth century. Roman-period examples are known from elsewhere, so the stone was worked long before the South African deposits were opened, though on nothing like the same scale.