{"id": "card_works_euler_identity", "kind": "verified", "title": "Euler's identity, assembled from its parts", "body": "Often called the most beautiful equation in mathematics: it binds e, i, π, 1 and 0 in a single line. We hand the engine each piece and the whole, and it confirms every one.  Worked & sealed by the engine — cos(π) = −1 · sin(π) = 0 · e^{iπ} + 1 = 0.  [HOLDS; open the seal to re-check.]", "source": {"label": "The Works — worked & sealed", "url": "https://narrowhighway.com/s/809fbaa9498918cf30444c3e16a63c6be7eb2c2026d944d08c467ecc0a305421", "domain": "mathematics", "authority_tier": "verified"}, "shelf": "the-works", "box": "mathematics", "bands": ["the-works", "mathematics", "mathematics", "complex", "analysis"], "connections": [{"to_card_id": "card_spine_the-works", "relationship": "member_of", "evidence": "a member of the the-works shelf in the keeping"}, {"to_card_id": "card_k_floor_of_discovery", "relationship": "paves", "evidence": "A worked, engine-sealed demonstration — a paving-stone of the floor of reality."}, {"to_card_id": "card_v_23b7ec8074a4", "relationship": "demonstrates", "evidence": "A worked demonstration in the same field (mathematics)."}, {"to_card_id": "card_v_3f924f5023bc", "relationship": "demonstrates", "evidence": "A worked demonstration in the same field (mathematics)."}, {"to_card_id": "card_v_99e6939d3390", "relationship": "demonstrates", "evidence": "A worked demonstration in the same field (mathematics)."}, {"to_card_id": "card_works_calculus_chain", "relationship": "kindred", "evidence": "A sibling worked demonstration in mathematics."}, {"to_card_id": "card_works_trig_identities", "relationship": "kindred", "evidence": "A sibling worked demonstration in mathematics."}, {"to_card_id": "card_works_information", "relationship": "adjacent", "evidence": "Neighbouring worked demonstrations in the same part of the volume."}, {"to_card_id": "card_works_counting", "relationship": "adjacent", "evidence": "Neighbouring worked demonstrations in the same part of the volume."}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "subject": "Complex analysis", "extra": {"seal_hash": "809fbaa9498918cf30444c3e16a63c6be7eb2c2026d944d08c467ecc0a305421", "cite_url": "https://narrowhighway.com/s/809fbaa9498918cf30444c3e16a63c6be7eb2c2026d944d08c467ecc0a305421", "demonstration_id": "works_euler_identity", "verdict": "HOLDS"}, "presentation": {"glyph": "•", "kind_label": "mathematics", "by": "The Works — worked & sealed", "authority": "verified", "posted": "", "standing": "", "link": {"url": "https://narrowhighway.com/s/809fbaa9498918cf30444c3e16a63c6be7eb2c2026d944d08c467ecc0a305421", "host": "narrowhighway.com", "provider": "", "titled": "Euler's identity, assembled from its parts", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_spine_the-works", "title": "The Works — sealed, worked demonstrations", "relationship": "on the shelf of", "why": "a member of the the-works shelf in the keeping", "href": "/card/card_spine_the-works", "resolved": true}, {"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "paves", "why": "A worked, engine-sealed demonstration — a paving-stone of the floor of reality.", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_v_23b7ec8074a4", "title": "sin²x + cos²x = 1 · cos 2x = 1 − 2 sin²x", "relationship": "demonstrates", "why": "A worked demonstration in the same field (mathematics).", "href": "/card/card_v_23b7ec8074a4", "resolved": true}, {"id": "card_v_3f924f5023bc", "title": "cos(π) = −1 · sin(π) = 0 · e^{iπ} + 1 = 0", "relationship": "demonstrates", "why": "A worked demonstration in the same field (mathematics).", "href": "/card/card_v_3f924f5023bc", "resolved": true}, {"id": "card_v_99e6939d3390", "title": "d/dt (t³) = 3t²  (position → velocity) · d/dt (3t²) = 6t  (velocity → accelerati", "relationship": "demonstrates", "why": "A worked demonstration in the same field (mathematics).", "href": "/card/card_v_99e6939d3390", "resolved": true}, {"id": "card_works_calculus_chain", "title": "Differentiate twice: position → velocity → acceleration", "relationship": "kindred", "why": "A sibling worked demonstration in mathematics.", "href": "/card/card_works_calculus_chain", "resolved": true}, {"id": "card_works_trig_identities", "title": "From the unit circle to the double-angle law", "relationship": "kindred", "why": "A sibling worked demonstration in mathematics.", "href": "/card/card_works_trig_identities", "resolved": true}, {"id": "card_works_information", "title": "Measuring information: one bit, and a distance", "relationship": "adjacent", "why": "Neighbouring worked demonstrations in the same part of the volume.", "href": "/card/card_works_information", "resolved": true}]}