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Animation 12
Translation of functions preserves their shape
symmetry-function-translation-shape.mp4
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1280 × 620 · 24 fps · 5.5 s · 132 frames · 12 samples
MP4 SHA-256 70b781ea9a0259674c356828596b2f59ecf49ed5e6daee2fff5b9d0fe5bfe166
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Chapter context
Section: Translations and Function Representation. Excerpts are verbatim; line numbers refer to the included chapter markdown.
Caption
Translation of functions preserves their shape
Image description
Translation preserving a function shape
Before 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.
After the animation
Chapter lines 571–571
Indeed, this representation is completely analogous to our 3 dimensional representation of $D_3$. There a vertex contained a component value, here an input value does. Indeed, the $D_3$ representation is itself a function representation, with a domain that contains only three members and is cyclic.
Generation source
Run the main script without --poster-only to draw and encode the translated-function frames. FFmpeg is resolved from configuration, the local tool installation, PATH, or imageio-ffmpeg. No local Python imports or input movie are used.
Mapping evidence and limits
main() explicitly writes symmetry-function-translation-shape.mp4; constants specify 1280 x 620, 24 fps, 132 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_function_translation_shape_animation.py · GitHub at 974f9de35254
SHA-256 b5196f7ba9384160fa221e5d85e30a9602ffecdab222a9dafc58ab4b412fd5dd
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
