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There is no connection between the Aztec Empire and the Persian Empire. They did not know of each other's existence.
Iran
There were no counties in the days of the Aztecs. Perhaps you would like to repost your question
From the Levant to Central Asia
5,000 km = 3200 miles
The term Persian Empire is loosely used to cover several periods. The real empire of the Persians was 550-321 BCE. Other peoples took over Persia, starting with Alexander the Great.
around 1,677 miles with 111 rely stations
It depends on the time period. At some point their empire stretched from western india, the northern parts of africa, and the lower halves of europe. They persians conquered Spain, and stayed in control for a while. You have to look at specific Persian rulers to see how much land they owned.
Miletus - its Persian-appointed Greek tyrant Aristagoras persuaded the Persian provincial governor to participate in a conquest of the Greek island of Naxos. The expedition failed, and Aristagoras, knowing Persian retribution was coming, stirred up the Ionian cities to revolt to cover himself.
Rome did not have a globalising empire because this empire did not cover the whole globe.
The Greek city-states, used to independence, rebelled. The spark was Aristagoras the Greek tyrant of Miletus, who had been appointed by Persia, had persuaded the Persian provincial governor to get involved in a failed takeover of the island of Naxos, and expecting retribution, conned the Greek cities into revolt to cover himself.