All animations · In the chapter · Open full MP4
Animation 11
A function representation is linear
symmetry-translation-function-linearity.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 · 6.5 s · 156 frames · 12 samples
MP4 SHA-256 0d4cfa151e7bb58d86655ba109f676b5ea411c53809f88a8f1862f6e06eaa503
Frame manifest · Context markdown · Download complete review packet
Chapter context
Section: Translations and Function Representation. Excerpts are verbatim; line numbers refer to the included chapter markdown.
Caption
A function representation is linear
Image description
Function translation preserving linearity
Before the animation
Chapter lines 549–549
The zero function is also fixed:
Chapter lines 551–553
After the animation
Chapter lines 561–561
This makes sense. Translational symmetry needs an object to translate, just as $D_3$ symmetry needs a triangly thing to translate. We can think of the function as a shape. If a system possesses translational symmetry, that shape is preserved.
Chapter lines 563–563
This describes how the function transforms. Whether the physical system it describes behaves the same way after translation is a separate question.
Generation source
Run the main script without arguments. It renders two separate movies: this function-linearity example using draw_function_linearity_frame, and the point-translation example using its other renderer. Frames are drawn with Pillow and encoded with FFmpeg; no local Python imports or input movie are used.
Mapping evidence and limits
main() pairs the exact symmetry-translation-function-linearity stem with draw_function_linearity_frame. Constants specify 1280 x 720, 24 fps, 156 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_translation_linearity_animations.py · GitHub at 974f9de35254
SHA-256 4f89641cae5f356ac6ad2f3606ec88eaf23611baeb0c98f81dcf3fbb920c436b
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
