{"query": "The angles inside a hexagon", "count": 20, "results": [{"id": "card_works_polygon", "title": "The angles inside a hexagon", "shelf": "the-works", "surface": "secular", "snippet": "The interior angles of any n-sided polygon sum to (n−2)·180°. For a hexagon that is 720° — the reason six equilateral triangles, or a honeycomb, tile the plane.  Worked & sealed by the engine — (6−2)·", "authority_tier": "verified", "source": "The Works — worked & sealed", "readable": false, "generated": false}, {"id": "card_theory_honeycomb_conjecture", "title": "Optimal packing & the honeycomb conjecture (Hales)", "shelf": "theories", "surface": "secular", "snippet": "The hexagonal grid is the most efficient way to divide a surface into regions of equal area with the LEAST TOTAL PERIMETER. Stated by Pappus of Alexandria around 300 AD as an observation about bees, i", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_girih_quasiperiodic", "title": "Girih tiles — quasi-periodic design in medieval Islamic architecture", "shelf": "theories", "surface": "secular", "snippet": "Girih are the polygonal tiles — decagon, pentagon, hexagon, bowtie, rhombus — carrying strapwork lines, used across medieval Islamic architecture. Peter Lu and Paul Steinhardt showed in Science (2007)", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_span_477756f02cfb", "title": "OPEN-AIR  POULTRY  HOUSES", "shelf": "practical", "surface": "secular", "snippet": "ing  quarters,  tliere  is  no  roof,  and  at  the  same  time  to  protect  the \nfowls  from  coyotes,  owls,  and  other  night  marauders. \n\nMr.  Stoddard  says  that  the  heat  of  southwest  Te", "authority_tier": "primary_pd", "source": "Open-air poultry houses for all climates", "readable": true, "generated": false}, {"id": "card_n_48f380838b48", "title": "The perfect numbers — six days and a month", "shelf": "science", "surface": "secular", "snippet": "6 is the first perfect number — equal to both the sum AND the product of its parts\n(1+2+3 = 1×2×3), sealed — and the days of creation's work. The second perfect number is 28\n(1+2+4+7+14), sealed — a m", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "readable": false, "generated": false}, {"id": "card_src_etym_hexagon", "title": "hexagon", "shelf": "etymology", "surface": "secular", "snippet": "hexagon: etymology (Webster 1913) — n.: [L. hexagonum, Gr. six) + (Geom.) Defn: A plane figure of six angles. Regular hexagon, a hexagon in which the angles are all equal, and the sides are also all e", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_src_word_hexagon", "title": "hexagon", "shelf": "dictionary", "surface": "secular", "snippet": "A plane figure of six angles. Regular hexagon, a hexagon in which the angles are all equal, and the sides are also all equal.", "authority_tier": "reference", "source": "webster-1913", "readable": false, "generated": false}, {"id": "card_src_openstax_contemporary_mathematics_key_terms_992ca346", "title": "Key Terms — Contemporary Mathematics", "shelf": "reference", "surface": "secular", "snippet": "Key Terms\n\n10.1\n\nPoints, Lines, and Planes\n\nline segment\n\nplane\n\nunion\n\nintersection\n\nparallel\n\nperpendicular\n\n10.2\n\nAngles\n\nvertex\n\nright angle\n\nacute angle\n\nobtuse angle\n\nstraight angle\n\ncomplementa", "authority_tier": "reference", "source": "OpenStax: Contemporary Mathematics (CC-BY 4.0)", "readable": false, "generated": false}, {"id": "card_src_food_174649", "title": "Cereals Ready-To-Eat, Ralston Crispy Hexagons", "shelf": "nutrition", "surface": "secular", "snippet": "Cereals ready-to-eat, RALSTON Crispy Hexagons — a food in the USDA FoodData Central reference. Per 100 g: 379 kcal, 5.94 g protein, 0.95 g fat, 86.78 g carbohydrate.", "authority_tier": "reference", "source": "USDA FoodData Central (public domain)", "readable": false, "generated": false}, {"id": "card_src_pron_hexagon", "title": "hexagon", "shelf": "pronunciation", "surface": "secular", "snippet": "hexagon: pronounced (ARPABET) HH EH1 K S AH0 G AA2 N. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_src_word_regular_hexagon", "title": "regular hexagon", "shelf": "dictionary", "surface": "secular", "snippet": "regular hexagon: (noun) a hexagon with six sides of equal length", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "readable": false, "generated": false}, {"id": "card_src_etym_six", "title": "six", "shelf": "etymology", "surface": "secular", "snippet": "six: etymology (Webster 1913) — a.: [AS. six, seox, siex; akin to OFries. sex, D. zes, OS. & OHG. sehs, G. sechs, Icel., Sw., & Dan. sex, Goth. saíhs, Lith. szeszi, Russ. sheste, Gael. & Ir. se, W. ch", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_src_word_hexagonal", "title": "hexagonal", "shelf": "dictionary", "surface": "secular", "snippet": "hexagonal: (adjective) having six sides or divided into hexagons — syn: hexangular", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "readable": false, "generated": false}, {"id": "card_src_word_hexangular", "title": "hexangular", "shelf": "dictionary", "surface": "secular", "snippet": "hexangular: (adjective) having six sides or divided into hexagons — syn: hexagonal", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "readable": false, "generated": false}, {"id": "card_theory_discrete_vs_continuous", "title": "Discrete versus continuous (the oldest dividing line)", "shelf": "theories", "surface": "secular", "snippet": "Discrete versus continuous (the oldest dividing line) — an engine domain that can touch it: number_theory. Calibration: map-only — a foundational distinction; specific cardinality results compute. Som", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_julian_day_number", "title": "Julian Day Number (continuous astronomical time)", "shelf": "theories", "surface": "secular", "snippet": "Julian Day Number (continuous astronomical time) — an engine domain that can touch it: calendar_time. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it,", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_stochastic_processes", "title": "Stochastic processes & Ito calculus", "shelf": "theories", "surface": "secular", "snippet": "Stochastic processes & Ito calculus — an engine domain that can touch it: mathematics. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn from it, and seal them.", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_discrete_structures_in_life", "title": "Discrete units in living things (codon, gene, species, phoneme)", "shelf": "theories", "surface": "secular", "snippet": "Discrete units in living things (codon, gene, species, phoneme) — an engine domain that can touch it: genetics. Calibration: seals — codon tables, allele frequencies and inventory counts compute. Biol", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_kolmogorov_probability_axioms", "title": "Kolmogorov probability axioms", "shelf": "theories", "surface": "secular", "snippet": "Kolmogorov probability axioms — an engine domain that can touch it: probability. Calibration: seals. Three axioms (1933): probabilities are non-negative, the whole sample space has probability 1, and ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_mendelian_inheritance", "title": "Mendelian inheritance", "shelf": "theories", "surface": "secular", "snippet": "Mendelian inheritance — an engine domain that can touch it: genetics. Calibration: seals — ratios and Punnett squares compute exactly. Traits are carried by discrete factors, one inherited from each p", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_works_polygon"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}