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hyperbolic geometry

2 captures, most recent first.

Charles Rosenba..., handle truncated @bzogrammer

— replying in a thread — saved image

[top of capture: bottom edge of a parent post containing a figure with three panels titled "Random network", "Regular network", "Real network" (path-length heatmaps; axes "Path length l" and "Number of nodes k"); engagement row 1 reply, 16 likes, 1K views]

Charles Rosenbauer [verified]
@bzogrammer

A lot of state spaces are effectively exponential/hyperbolic, and the shortest path between two random points in a hyperbolic space often passes close to the origin.

In more concrete terms, you can ask what happens inside a cocoon; how does a larvae transform into a radically different insect?

The answer is that, while some structures are preserved (mostly air channels and some of the neural tissue), everything else is dissolved into a soup by digestive enzymes. Some cells which function like a seed or egg then consume this soup and construct a completely new insect from the raw materials.

The shortest path between two unrelated complex structures passes close to a soup of structureless raw materials.

12:07 AM · Sep 1, 2026 · 4,888 Views
Note from Claude Sonnet 5

X reply by Charles Rosenbauer (@bzogrammer) arguing that shortest paths in hyperbolic state spaces pass near the origin, illustrated by metamorphosis in a cocoon (dissolution into a soup, then reconstruction). The top of the capture shows the lower part of the parent post's figure of path-length heatmaps for random, regular and real networks. Posted 12:07 AM Sep 1, 2026, 4,888 views.

hyperbolic geometrystate spacesmetamorphosisnetworksshortest pathanalogy

Charles Rosenba..., handle truncated @bzogrammer

— saved image

[two attached images, side by side: a photo of a chrysalis hanging from a green stem; a two-part figure labelled (a) and (b) showing a hyperbolic disk tiling with a curved path between points X and Y marked D_XY, and a hierarchical block diagram with a path D_XY between X and Y over a span L]

[quoted post]
Charles Rosenbauer [verified] @bzogrammer · 8h
Replying to @projectionheart

A lot of state spaces are effectively exponential/hyperbolic, and the shortest path between two random points in a hyperbolic space often passes close to the origin....
Note from Claude Sonnet 5

Quote-post card showing the media and truncated opening of Charles Rosenbauer's reply (the same post as the previous capture): a close photo of a chrysalis on a stem, and a black-and-white figure of a hyperbolic-disk tiling (a) beside a hierarchical tree-like block diagram (b), both with X, Y and D_XY labels. The quoted text is cut off with an ellipsis; the full text appears in the neighbouring capture.

hyperbolic geometrymetamorphosischrysalisstate spacesshortest path