The Pyrihios (Pyrrhichios) or Pyrrhike dance was the best known war dance of the Greeks. It was probably of Dorian origin and practiced at first solely as a training for war. According to ancient sources it was an armed dance.
Plato (Leges, 815a) describes it as imitating by quick movements the ways in which blows and darts are to be avoided and also the modes in which an enemy is to be attacked. It was dance to the sound of the aulos; its time was quick and light, as is also shown by the metric foot called pyrrhic.
It was described by Xenophon in his work the Anabasis. In that work he writes that the dance was performed at a festival held in Trapezus to celebrate the arrival of the troops in the city. The following is the part in which the pyrrhic dance is mentioned:
A Mysian who saw that they were amazed, retorted by persuading one of the Arcadians who had acquired a dancing girl to dress her in the finest costume he could, fit her with a light shield and bring her on to give a graceful performance of the 'Pyrrhic' dance. Thereupon there was a roar of applause, and the Paphlagonians asked if the Greek women also fought side by side with their men. The Greeks answered that these were the very women who had routed the king from his camp.
Homer refers to the Pyrihios (Pyrrichios) and describes how Achilles danced it around the burning funeral of Patroclus.
The dance was loved in all of Greece and especially by the Spartans who considered it light war training. This belief led the Spartans to teach the dance to their children while they were still young. Pyrrhic Dance (Πυρρίχιος) - Athens Olympics 2004
Athenian youth performed the dance in the palaestra as part of training in gymnastics.The dance was also performed in the Panathenaic Games. There were three classes of competitors: men, youth and boys.
Pyrrhic Dance - Athens Olympics 2004
The Pyrrhic dance is an ancient war dance of the Greeks of Pontus...the Pontian people have maintained it to this day
From πύρριχος "red", itself from πυρρός "blazing red" (cf. πῦρ "fire") from Proto-Greek *purwo- from Proto-Indo-European *peh2-ur "fire" (see R. S. P. Beekes, Etymological Dictionary of Greek, Brill, 2009, pp. 1260 and 1264).
The Cambridge Series for Schools and Training Colleges: Xenophon, Anabasis VI with vocabulary