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Animation 8
Hyperbolic rotations
symmetry-hyperbolic-rotation.mp4
These are sparse samples decoded from the current MP4, not newly rendered illustrations. They can support checks of the sampled states and labels, but cannot establish continuous motion, timing, transitions, or the absence of problems between samples. Use the full MP4 when judging those properties.
1280 × 720 · 24 fps · 7 s · 168 frames · 12 samples
MP4 SHA-256 17517c198f57608fce58c372da980248bd7ea5bece69ceef0b96da3622742b14
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Chapter context
Section: Invariants and Metrics. Excerpts are verbatim; line numbers refer to the included chapter markdown.
Caption
Hyperbolic rotations
Image description
Hyperbolic rotation preserving the interval
Before the animation
Chapter lines 351–355
Chapter lines 357–357
An invariant of this sort, that fixes some notion of placing reliable rulers on the space, is a metric. For normal rotations and translations, these relationships are intuitive, but a symmetry may preserve a less intuitive metric. If we define invariant length not as $x^2+y^2$ but as $t^2-\mathbf{x}^2$, we have a "hyperbolic rotation." Separation between points stretches out to infinity as the rotation approaches its asymptotes, leaving the invariant interval unchanged even though the separation appears to grow using our everyday notion of distance. Such a hyperbolic metric, as we will see, encodes the asymptotic limit of light speed in the relativistic geometry spacetime.
After the animation
Generation source
Run the main script without arguments. It draws the hyperbolic-rotation frames with Pillow and encodes them with FFmpeg. It has no local Python imports or input movie.
Mapping evidence and limits
main() explicitly writes symmetry-hyperbolic-rotation.mp4; constants specify 1280 x 720, 24 fps, 168 frames. No matching validation JSON is present. Attribution is by source/output name and pipeline, not a recorded movie hash.
Source SHA-256 values identify the exact downloadable bytes in this packet. The source mapping and recorded checks explain the likely generation pipeline; they do not prove that these exact source bytes produced the movie. A GitHub link pinned to a commit is provided only when the delivered source bytes exactly match that path at the build's Git HEAD.
- scripts/generate_symmetry_hyperbolic_rotation_animation.py · GitHub at 974f9de35254
SHA-256 cea7f9d590e665552c5bde31f6726efa6b61a2e1d50d9eac4745baada9e86903
Decoded contact sheet
Extraction method: Twelve evenly spaced decoded frame indices, including first and last. Native-resolution JPEGs from the encoded MP4; timestamps read from FFmpeg showinfo. No source rerendering. Frame indices are zero-based. Sparse samples do not establish continuous motion or capture every transition.. Frame indices are zero-based.
Full-size sampled frames
