Gödel, Escher, Bach is a very large book by Douglas Hofstadter. Some of the ideas covered in the book are:
- self-reference
- acquired meaning
- logical systems
- recursion
- infinity
- layers of reality
- paradoxes
- strange loops (an idea that ties together many of these concepts)
It relates these ideas to subjects such as music, computer science, mathematics, and art.
Introduction: A Musico-Logical Offering 1
Bach
GEB begins with a story about Bach’s composition of a six-voice fugue Ricercar a 6.
A fugue is like a more complicated canon. A canon uses a single theme played against transformations of itself:
- Translation over time
- Scaling of pitch, speed
- Inversion
These are all isomorphisms - reversible (information-preserving) transformations. Music has at least these dimensions, and probably more:
- Scale
- Frequency
- Pitch
- Volume
The last canon he describes is an endlessly rising canon, where 3 voices create a cycle that sounds as though it rises without end. It’s not simply a spiral, but maybe like a toroidal spiral? In taiji, it’s like the path power takes in the Positive and Negative Circles. He calls this a Strange Loop: where an upwards traversal of the levels in a hierarchical system takes one back to one’s starting point.
Escher
M.C. Escher visualizes strange loops in his drawings, as the viewer is led in a climb through imagined dimensions that loop without reversing. Across all his drawings, there arises the notion of the tightness of the paradoxical loop: how many discrete steps taken before the loop completes. Although, there is some ambiguity in how to count steps. Interestingly, the viewer is also a dimension in the paradox.