That is a myth.
Archimedes did not conquer anything. He was a scientist, not a general or a politician.
The Persian empire.
His influence was only in the eastern Mediterranean where he established his empire over Greece and the Persian Empire. He slaughtered opposition, and tried to sow Greek culture in the ex-Persian Empire.
After its violent establishment, it tried to provide a stable, secure and prosperous environment for the people within it.
The Ionian Revolt from 499 BCE in which the Greek city-states in Asia Minor within the Persian Empire tried to throw off Persian rule.
They were appalled by their constant fighting of each other, and this spilling over into the Persian Empire, harming its peace and prosperity. Therefore they tried to bring the Greek city-states inside the Persian Empire to be able to control their disruptions.
Persia tried to bring the Greek city-states under control to prevent their incessant warfare with each other spilling into and disrupting the Persian Empire.
The wars and killing to establish the Empire wasn't too good. However when it was established, the Persian Empire tried to promote peace and prosperity. That's where the Greeks came to the fore and promoted disruption leading to fifty years of war between Persia and the Greek city-states.
The Persian empire.Note: Athens was also fighting with the Spartans, see the Pelopponnesian war.
They tried to bring security, peace and prosperity to their empire while leaving habitual local government to remain in place.
Persia returned to running its empire, promoting stability and prosperity. Athens converted the anti-Persian Delian League into an empire of its own and tried to extablish dominance in the Greek world, leading to the destructive 27-year Peloponnesian war against and alliance led by Sparta.
It grew, it tried to defeat the Greek city-states, failed and made peace with them, and was eventually taken over by Macedonia led by Alexander the Great.