Alexandria
The first state-funded research city.
Kernel
Alexandria is the first city in human history founded specifically to be a research center. Founded by Alexander in 331 BCE on the Mediterranean's deepest natural harbor, it housed the Mouseion (a state-funded institute) and the Library — at peak holdings perhaps 500,000 scrolls. Within a century it had produced Euclid's Elements, Eratosthenes's measurement of the earth, Aristarchus's heliocentric model, and the engineering tradition that gave us Heron's automata. For seven centuries it was the center.
Why it rose
Three coincidences: a Macedonian general with imperial appetite, an Egyptian grain surplus large enough to feed a research city, and a Mediterranean harbor at the exact point where Greek shipping met Egyptian agriculture. Add a deliberate Ptolemaic policy — the king will buy any scroll that arrives in port — and the world's first concentration of working scientific knowledge became state policy.
Doctrine
Pay scholars to live together; require nothing of them but their work. The model is later copied at Baghdad's House of Wisdom, Renaissance Florence's Medici court, and modern Bell Labs. Alexandria was the patient-capital research city's prototype.
What it gave the world
The Elements (Euclid). Earth's circumference (Eratosthenes, ~1% accurate). Heliocentrism (Aristarchus). The almost-modern catalog (Ptolemy). The translation infrastructure that put Hebrew, Egyptian, Persian, and Indian thought into Greek for the first time. The intellectual cargo Alexandria put on the road shaped Rome, Baghdad, and ultimately Europe.
Why it declined
Slow decline through Roman imperial neglect, gradual loss of the Library's holdings (the "single fire" story is a myth; the holdings dispersed through centuries of war, fire, and bureaucratic decay), and the 642 CE Arab conquest that re-routed the Mediterranean's center of gravity to Damascus and then Baghdad. The institution died; the texts traveled.