Arts and Ideas doesn't really go into the history of Alexander the Great, which kicked off the Hellenistic period. I know something about it from reading Peter Green's Alexander of Macedon last year, so here is a brief summary. Alexander was a Macedonian king who succeeded his father Philip at the age of 20 and over the next nine years conquered everything between Greece and Pakistan, a feat that more or less overnight spread Greek culture farther east than it had ever been:
However, his empire lasted approximately three and a half seconds after his death. According to one account, on his deathbed he bequeathed his empire "to the strongest", which is totally apocryphal but demonstrates the attitudes of his generals. His empire was quickly split up between his most ambitious generals: Ptolemy was the first to formally break off, declaring himself Pharaoh and setting up the Ptolemaic dynasty, the 32nd and last, in Egypt. Seleucus killed Perdiccas, who had briefly tried to rule the entire empire, and staked out his own claim in in Mesopotamia and Syria (his kingdom was really the last bastion of Hellenism against the Romans; it lasted until the 60s BC). Meanwhile, Lysimachus took Thrace (northeast of Greece on the Black Sea) and parts of Asia Minor and Antigonus Cyclops made himself king of Macedonia. These generals, called the Diadochi or successors, settled the borders amongst themselves in 320 BC, three years after Alexander's death. I found a translation of a Byzantine summary of the agreement here, and you can see even from the fragment how completely the empire had collapsed. The chaos didn't stop there, but the point is that Alexander's empire only lasted a few years, which makes the spread of Hellenistic culture all the more amazing. Despite the political fragmentation, Greek styles of art and architecture quickly superseded their Achaemenid equivalents in the old Persian empire. For example, the Pharos and Library of Alexandria are both thought to have been Hellenic-style works of architecture, and the Library contained mostly Greek texts. The Acropolis of Pergamon (a formerly Persian city), which is described at length in Fleming, was built by the successors to Lysimachus. Greek architecture would almost certainly never have come to dominate the city without Alexander.