Achilles was a mythological Greek soldier, who lived around the 12th or 13th Century BC. He was the King of Pthia and leader of the Myrmidons who fought with him at Troy on the side of the Greeks against the Trojans in the Trojan War. He was the chief protagonist of the Iliad by Homer. He also features in Greek tragedy, such as Iphigenia at Aulis by Euripides.
The overarching theme of the Iliad, according to its opening line, is the "wrath of Achilles". This arose in the final year of the Trojan War when Agamemnon, the leader of the Greek forces, took the captive girl Briseis from the hero. Feeling slighted, Achilles withdrew from the war, leaving the Greeks short-handed against the Trojans. It was only when his companion Patroclus was killed in battle while wearing his armour that Achilles returned in a fury to exact vengeance on the Trojan prince and champion Hector, whose funeral marks the end of the epic.
In Greek mythology, when Achilles was an infant, his mother Thetis held him by the heel and dipped him into the river Styx to give him invulnerability. Every part of his body touched by the river became invulnerable except the place where she was holding him; thus he could only be killed by a wound to his heel. According to tradition (though the incident is never described explicitly by Homer) Paris killed him with a poisoned arrow which hit him in the heel. This myth has given rise both to the expression "Achilles' heel" to mean a weak or vulnerable spot, and the name "Achilles' tendon" for the large tendon which joins the calf muscle to the heel bone.
One of Zeno's paradoxes concerns "Achilles and the tortoise." It concerns an imaginary race between the swift warrior Achilles and a slow-moving tortoise, which is given a head start. Although Achilles runs much faster than the tortoise, Zeno argued that Achilles can never actually catch the tortoise. In order to catch him, Achilles must first reach the place where the tortoise started. But by the time Achilles reaches that place, the tortoise will have moved. Even if Achilles has run a hundred feet while the tortoise has moved only one foot, Achilles has not yet caught the tortoise. To catch him, Achilles must again reach the place where the tortoise is now. But, again, by the time he reaches that place the tortoise will have moved. Even though the tortoise moves a smaller distance each time, it seemed to Zeno that Achilles could never actually catch him. In mathematics, the paradox is resolved by observing that the times it takes Achilles to perform each iteration form a geometric series, which has an infinite number of terms but a finite sum.