Tensor Diagram Videos

Making tensor-diagram derivation videos

The product: a 15–40s silent 1080p60 video in which a Matrix-Cookbook-style identity is derived, not asserted. The standard was set by three videos in this folder: trace_delete.mp4, kronecker_trace.mp4, softmax_jacobian.mp4, all rendered from scenes in advanced_rules.py.

The one-sentence philosophy: the animation is only a proof if every frame is a well-formed diagram and every mobject has an ancestry. Nothing may appear, vanish, flip, or morph without a semantic reason.

The book's notation (non-negotiable)

Check paper/chapters/functions.tex before inventing anything. Known rules:

• Function application: the function is a plain text node (no circle); the input edge carries an arrow pointing INTO the function. Solid arrow for vector→vector functions (softmax), densely dotted for elementwise ones (exp, pow_k) — and for elementwise functions the data wire stays on the argument (exp ←··· z ——, the free wire belongs to z).
• Sum = wire terminated by a copy-dot. diag(v) = 3-edged copy-dot with v hanging off. 1/x = pow₋₁ applied by arrow — not a fraction — though converting pow↔fraction on screen is a good beat (go full circle: open with the fraction, work in pow form, return to fractions at the end).
• No brackets when arrows already disambiguate. Scalar-argument functions (pow on a sum) still get their application arrow.
• A matrix is a node with two edges; vectors have one (a and b in aᵀXb are closed by the trace — count edges before drawing!).
• Covariance: a named edge exits the same side in every term and every frame. Two free edges never cross incidentally — after the twist convention, a crossing MEANS a swap. Rotating a node 180° IS transposition (so a turned glyph legitimately reads as Aᵀ — relabel by spinning it upright while the ᵀ fades in).
• Claim only what is derived (the softmax video says "Jacobian of softmax", not "Hessian of cross-entropy", because the CE connection is never shown).

Visual style

• White background, config.frame_height = 5.6 (set in __init__ before super().__init__), same pixels → everything ~40% bigger. Title top (to_edge(UP, buff≈0.4)), gray caption bottom (Tex … scale(0.62), to_edge(DOWN, buff=0.18)), one clause per beat — the captions ARE the voiceover script.
• Semantic color, constant per object across formula AND diagram: variables #C03B2B (red), function names #1F6FB2 (blue), recognized results #188A54 (green), derivative apparatus #B07000 (amber), structural wires/dots/signs black. The amber thread is pedagogy: whisker → dangling stub → final edge are visibly the same object.
• Uniform glyph sizes. No shrunken denominators or mini-copies; retune layout instead.
• Glyph halos (set_background_stroke(color=WHITE, width=6)), never BackgroundRectangle masks — rectangles clip other wires passing behind (visible the moment the scene moves). A curve's clearance around its own riding label must be a real parameter gap in the curve, sized from the local radius.
• Every node gets clearance — edges stop short of glyph ink, including operator glyphs like flatten triangles (stubs may start under a white-filled node so the fill covers the join).
• Arrows must clear both endpoints: tips short of subscripts (pow₋₂), tails outside the source glyph. When in doubt, widen the gap between the nodes.
• Finish: Indicate when a result lands, Circumscribe the final formula, settle the finished diagram up under its equation, hold ≥2.5s.

The animation grammar

1. Every frame is a valid diagram. Right edge count per tensor, no transient crossings, invariants hold mid-animation, not just at endpoints (e.g. delayed growth so dangling edges never cross).
2. Glyphs keep identity. A symbol persists (and may glide, with path_arc to hop over others), appears, or disappears — it never smears into a different symbol. The one sanctioned morph is a semantic renaming (exp absorbing its normalizer to become softmax). Titles: build the formula in sync with the diagram (X → AXB → Tr(AXB) → ∂·/∂z); keep it centered early and slide left as = … grows; hand-assemble fractions so the ∂ exists as its own animatable glyph; early title stages sit at the equation midline, not the numerator slot.
3. Count-match every transform. One wire → one wire. If topology changes, split the target at the join (de Casteljau at a bezier apex; closed loops split into quadrants) so correspondence is 1:1. A mismatch makes manim silently duplicate → ghost arcs.
4. Direction-match: anchored end first. Transforms map start→start. A target curve parameterized from its free end makes the source pivot/flip about the wrong end (this bug shipped four times: trace closure, A–X edge, j1f, jw2). Build every replacement wire starting from the end that stays attached.
5. Transform keeps the SOURCE mobject alive (wearing the target's shape); ReplacementTransform swaps the target in. Corollaries: never FadeOut/group/animate the target of a plain Transform (ghost loops, stray minus signs, unmoved edges — this bug shipped five times); after a Transform, later code must reference the source name.
6. Transform preserves z-order; ReplacementTransform re-layers to top. Use Transform when a morphing wire must stay under labels.
7. Copies spawn from their parent. Product rule, chain rule: .copy() superimposed on the original, then transform the copies out to the new rows/groups. Factor swaps arc over each other (animate(path_arc=…)).
8. The whisker is the promised edge. One whisker per derivative loop (per new edge); it survives distribution (loop→loop, dot→dot, whisker→whisker at hand-offs) and finally becomes the new edge. New edges are born already pointing in their final direction and never reverse (mirror sub-layouts if needed to keep all j's on one side).
9. Amber circle = pending derivative; evaluation spends it. Circling a function is applying ∂; when you evaluate (exp′=exp, pow₋₁′=−pow₋₂) the circle fades — and the relabel is the semantic event (the minus is BORN at pow₋₁′=−pow₋₂: keep ONE sign object centered between terms and flip it + → − at that moment). A chain-rule link through a scalar carries no d-edge at all.
10. Dots terminate their edges. A sliding sum-dot drags its wire (shorten in step). Dot-absorption is articulated: slide → merge → collapse, with the index-level receipt (Σₖδⱼₖ…) as the caption.
11. Static↔tracker hand-offs must be seamless both ways: animate the outgoing static to exactly the incoming always_redraw's initial state (labels glide with relaxing wires; freeze redraws to statics before a group shift).
12. Bundles are offset curves of ONE center path (foot-curl → dome → foot-curl), strands at ±offset along numeric normals — guaranteed parallel, and the top strand takes the inside of every turn. Draw the pair at zero gap until the bundling operator is resolved; the gap opening IS the resolution. Deforming closed curves = one parametric curve with keyframed shape parameters (never piecewise arcs), labels riding a fixed parameter point through a real gap.
13. More steps beat magic. Rewrite big jumps as chains of small ones, each with a caption: quotient → product rule with explicit factors; pow↔fraction conversions; collect terms onto one line (swapping factors so covariance holds) before the final recognition; every = in the classical derivation gets a diagram move, and the object mapping in each move is reasoned (z's persist, arrows persist, machinery is absorbed into the name it justifies, split wires turn rather than respawn).

Workflow

1. Read the relevant chapter first; the book may already contain the exact decomposition/figure (softmax) or pose the identity as an exercise.
2. Storyboard the beats; the user may supply frame-by-frame sketches — follow them panel by panel.
3. Build the scene in advanced_rules.py (helpers there: glyph halo text, farrow dotted/solid application arrows, wire, cdot, dcircle, dloop (whisker options), node, caption closure, smgroup-style builders). Fixed coordinate slots per beat; builders return dicts of named parts so transforms can be per-piece.
4. Iterate at -ql, extracting frames at beat boundaries: ffmpeg -ss T -i video.mp4 -frames:v 1 out.png (get duration via ffprobe), and Read them. Check mid-transition frames, not just settled ones.
5. Final review at 1080p (-qh) — delivery resolution. 480p anti-aliasing hides exactly the shippable-defect class: 1–2px gaps, wires grazing glyphs, mask edges. Use PIL crops for pixel-level checks of junctions.
6. Ship: copy .py + .mp4 to paper/animations/, commit with explicit pathspec, push. One commit per review round.
7. Expect ~5–15 user review rounds per video. Each note usually generalizes: fix the instance, then apply the principle everywhere (e.g. "circle is spent" came from one pow note; uniform font sizes from one screenshot).

Tooling pitfalls

• Patch scripts: assert old in s for EVERY replacement; a failed assert before write means the file is untouched and the subsequent render shows the OLD video — check which. Region-scope all s.index anchors to the class (base = s.index("class Foo"); s.index(anchor, base)) — a global index can match an earlier scene and splice duplicated chunks into the file.
• Multiple scenes share advanced_rules.py; anchor text ("# ---- title ----", "self.cap = None") repeats across scenes.
• The scratchpad can be wiped between sessions; the repo copy is the source of truth — restore, then re-apply.
• manim: MathTex(a, b, c) part-splitting enables per-glyph color/animate; set_background_stroke for halos; FadeOut(m, target_position=…) fades while translating (absorption); FadeIn(m, target_position=…) arrives from there; Rotate(m, about_point=…) for in-place glyph spins; DashedLine(...).add_tip(...) for dotted arrows; always_redraw closures over ValueTrackers for deforming geometry (keyframe param arrays + np.interp); Date-like nondeterminism absent but LaTeX compile cache makes re-renders fast.
• Render one scene: python -m manim render -qh --disable_caching advanced_rules.py SceneName (venv: .venv/bin/python; latexmk not needed — manim compiles its own tex).