Castalia, in Greek myth, was a fetching maiden whom Apollo, after pursuing her and failing to capture her attentions, transformed  into a fountain at Delphi  or, as others tell us,  to escape Apollo….she threw herself into a spring on Mt Parnassus.

The Story of Castalia
https://www.academia.edu/38910016/The_Story_of_Castalia_Understanding_Understanding_

Perhaps as a bit of feminism,  the Nine Muses, all devoted handmaidens to Apollo, came to revere the one who escaped as the 'muse of all muses"

As her waters streamed beneath the famed Temple at Delphi,  pilgrims come  from all over to be inspired by either drinking or merely being in range of its sounds.    Some, such as competitors at the first Greek Olympics) would bathe in the waters prior to contest to assure victory.

The Temple was the seat of the oracular wisdom dispensed by  priestesses  speaking in riddles inspired by wisdom bubbling up from from beneath. Kings and Emperors, including Alexander, would come for cryptic counsel

Recently, it was found that the flowing waters under the temple emitted a hallucinogenic vapor likely to have been the actual source of inspiration of the Oracle's perplexing counsel.

Castalia, for me, was a name chosen before I knew of the Greek legend.
In Hesse's "Magister Ludi" ("The Glass Bead Game") that mystical "bead game" took place in a republic of the future, which he named "Castalia" 

In this utopia they managed to distill all available knowledge into an elaborately coded game. It was of course never described, but was an extension of games as chess or 'Go'. The player who surpassed all others rose to the rank of 'Magister Ludi", the master of the game.

“This Game, writes Hesse, "has developed into a kind of universal speech, through the medium of which the players are able to express values in lucid symbols and to place them in relation to each other . A game can originate,  ..from a given astronomical configuration, a theme from a Bach fugue, a phrase of Leibniz or from the Upanishads, and the fundamental idea awakened can be built up or enriched through assonances to relative concepts. While a moderate beginner can, through these symbols, formulate parallels between a piece of classical music and the formula of a natural law, the Master can lead the opening theme into the freedom of boundless combinations.”

As he put it, " The Bead Game is thus a mode of playing with the total contents and values of our culture; it plays with them as, say, in the great age of the arts a painter might have played with the colors on his palette. All the insights, noble thoughts, and works of art that the human race has produced in its creative eras, all that subsequent periods of scholarly study have reduced to concepts and converted into intellectual property—on all this immense body of intellectual values the master player plays like the organist on an organ. And this organ has attained an almost unimaginable perfection; its manuals and pedals range over the entire intellectual cosmos; its stops are almost beyond number. Theoretically this instrument is capable of reproducing in the Game the entire intellectual content of the universe."

We soon realized that we  had always had a passion for that game...we just did not know its name....nor all the rules....nor, for that matter, the implications of devoting oneself to the adventure of its play. Yet it seemed impossible to encompass the vast knowledge required to play.

Perhaps Hesse chose "Castalia as his muse, because that Temple at Delphi was known as a center of Heraclitean wisdom, where the aim  was "understanding", which has its roots as "standing in the midst of" rather than "knowledge", a concept which haunted by being a 'standing apart" from what is presumably known.

“There is truth," they discovered in Castalia, as Hesse put it, " the doctrine you desire, absolute, perfect dogma that alone provides wisdom, does not exist. Nor should you long for a perfect doctrine, The deity is within you, not in ideas and books. Truth is lived, not taught."

Although the words "know thyself" are thought to have come from Plato's Socrates, they were etched over the doors of Castalia's Temple at Delphi. Their meaning is founded in the wisdom of Heraclitus and not the pedantry of the peculiar Platonic "knowing'

The key to achieving the resonant harmony of mind of which Hesse wrote is to be found not in the ingenious manipulations of our various symbolic expressions, but only in another mode of expression. The Game is not about "everything" but about only 'one thing:  an "intuition" or 'nous" rooted within. 

That is the game, as we now finally have come to understand it, that we are here to play.  We hope you join us.
Address: Shelton, Connecticut, United States