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Animation 10
Shifting a number line is not a linear operation
symmetry-translation-point-linearity-failure.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 8e71aef46dd223bb43ca609758f270c3867484b936e193b4f31d6cc60093123c
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Chapter context
Section: Translations and Function Representation. Excerpts are verbatim; line numbers refer to the included chapter markdown.
Caption
Shifting a number line is not a linear operation
Image description
Point translation failing linearity
Before the animation
Chapter lines 505–511
Chapter lines 513–513
These are not equal unless $a=0$. Translation does act sensibly on a number line, but it is not a linear operation when acting on this space.
After the animation
Chapter lines 521–521
On the other hand, the space of functions on a number line does form a linear representation. Instead of translating the point $x$, we translate a function by shifting its argument:
Chapter lines 523–525
Generation source
Run the main script without arguments. It renders two separate movies: this point-translation example using draw_point_linearity_frame, and the function-linearity 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-point-linearity-failure stem with draw_point_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
