Why was Pauli mesmerised by the Sommerfield constant?

Pauli studied under Sommerfield at university. He was mesmerised by the Sommerfield constant more correctly known as the fine-structure constant, because it could answer an ago-old mystery, or at least, that is how Pauli brilliantly made use of Sommerfield's finding that the constant determines the strength of the electromagnetic interaction.

Pauli's understanding of Jungian theory added to Sommerfield's Platonic explanations of the constant meant that now scientists could embrace Pauli’s pointing to the World Clock and the Cosmological Circle; these archetypal images of space with the circle, square, triangle, and the cycles of time could now be understood in terms of the fine-structure constant.