What complex mathematical feature did Ptolemy and others employ in the geocentric model to explain retrograde motion?

Answer

Epicycles, or circles moving upon other circles

The ancient Greeks, including Ptolemy, resorted to complex systems of epicycles to mathematically predict the seemingly erratic retrograde motion of the outer planets within the Earth-centered framework.

What complex mathematical feature did Ptolemy and others employ in the geocentric model to explain retrograde motion?
Galileoastronomyscientistuniverseheliocentrism