Tumbolia, viewed by many readers as one of Hofstadter's more intriguing inventions, occurs only four times in the book. The concept is introduced in the dialogue "Little Harmonic Labyrinth" (based on the piece of the same name by J.S.Bach) with the words quoted above. In the later dialogue "A Mu Offering" (named after Bach's Musical Offering), the Tortoise gets rid of a knot in a string by tying a second one, and both disappear to Tumbolia; apparently, this is "The Law of Double Nodulation" (a parody of the law of double negation). The return of the two knots from Tumbolia prompts the speculation that some "layers of Tumbolia" are more accessible than others; this is the only information we are given about what Tumbolia itself may be like. It is mentioned that "pushing" or "popping" potion can be used (drunk by the characters) to navigate up and down the various levels of Tumbolia.
In Chapter 9, Hofstadter compares Tumbolia to the Zen view of life after death, using the image of a snowflake, a self-contained subsystem of the universe, dissolving into "the larger system which once held it". Finally, in the book's last dialogue, Hofstadter (himself appearing as a character) tells us that Tumbolia is where dreamed characters go when the dreamer wakes up.