Short answers, pulled from the story.
Hero of Alexandria was a Greek mathematician and engineer who was active in Alexandria during the Roman era. Scholarly estimates for his dates range from 150 BC to 250 AD, though a lunar eclipse referenced in his work Dioptra best matches one that occurred in 62 AD, suggesting he was active in the 1st century AD.
The aeolipile, also called Hero's engine, was a steam-powered reaction device in which steam escaped through nozzles causing the vessel to spin. Vitruvius had mentioned a version of the device before Hero, but Hero published the most widely recognized description of it.
Hero of Alexandria described the first known coin-operated vending machine in his book Mechanics. A coin inserted through a slot landed on a pan attached to a lever; the lever opened a valve to dispense a fixed amount of water, then a counter-weight closed the valve once the coin slid off the pan.
Heron's formula calculates the area of a triangle using only the lengths of its three sides, without requiring the height or any angle measurements. Hero described it in his work Metrica.
Hero of Alexandria built a windwheel that operated an organ, which stands as the earliest documented instance of wind powering a machine on land.
Hero formulated the principle that light traveling between two points within the same medium follows the shortest possible path. Ibn al-Haytham extended this principle to reflection and refraction in the Middle Ages, and Pierre de Fermat restated it in 1662.