Final answer:
Persian architecture had a lasting influence both in regions previously under Persian control and in places like Greece and Macedonia, where building practices were inspired by the wealth of Persian cities. The innovations of Persian architecture, notably in Persepolis, were disseminated and adapted in various architectural traditions following the Empire's fall. The Persian cultural resurgence during the Islamic Golden Age further ensured the longevity of its architectural imprint.
Step-by-step explanation:
After the fall of the Persian Empire, Persian architecture continued to have a significant influence across a wide geographic area. The architectural traditions and innovations of the Persians, including the use of the terraced plane and lightweight roofing in monumental complexes such as Persepolis, had a lasting impact. In the aftermath of the Empire's demise, areas previously under Persian control, such as those in Central Asia, Egypt, and Asia Minor, saw remnants of Persian architectural styles woven into their own. Additionally, the conquests by Alexander the Great spread not only the influence of Greek culture but also allowed for a two-way exchange where Persian architecture influenced the building practices in Greece and Macedonia. This led to a renaissance of building back in these regions, funded by the wealth acquired from Persian cities.
During the rise of Islam, Persian contributions to the arts and architecture saw a revival, with Iranian arts, including the intricate carpet-weaving traditions reflecting the refined aesthetic principles of Persian design. The Persian language and culture, despite various conquering powers, remained intact and influential. Moreover, the round arch, invented in the Mesopotamian region, which informed Persian building practices, went on to influence world architecture significantly, exemplified by structures such as the Ishtar Gate.