{"schemaVersion":1,"studyVersion":1,"exhibitId":"topology","runId":"run-20260917003910210-62944-002","caseLabel":"Sphere · level 0 · scale (1.5, 0.75, 1)","modelVersion":"topology-1.0.0","parameters":{"model":"sphere","subdivision":0,"scaleX":1.5,"scaleY":0.75,"scaleZ":1,"triangulate":true},"numericalMethod":"Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.","softwareVersion":"0.1.0+0e6017e6a747.dirty.02665ba9098c","createdAt":"2026-09-17T00:39:10.215Z","observables":[{"label":"Vertices V","value":12,"precision":0},{"label":"Edges E","value":30,"precision":0},{"label":"Faces F","value":20,"precision":0},{"label":"Euler characteristic χ","value":2,"precision":0},{"label":"Boundary components","value":0,"precision":0},{"label":"Counted cells","value":"Triangular cells"},{"label":"Interior angle defect","value":12.566370614359172,"unit":"rad","precision":8},{"label":"Boundary turning","value":0,"unit":"rad","precision":8},{"label":"Total integrated curvature","value":12.566370614359172,"unit":"rad","precision":8},{"label":"Theorem prediction 2πχ","value":12.566370614359172,"unit":"rad","precision":8},{"label":"Absolute residual","value":0,"unit":"rad","precision":12,"description":"Measured angle sum compared with 2π times the independently counted Euler characteristic."}],"warnings":[],"sourceIds":["M1"],"result":{"exhibitId":"topology","modelVersion":"topology-1.0.0","parameters":{"model":"sphere","subdivision":0,"scaleX":1.5,"scaleY":0.75,"scaleZ":1,"triangulate":true},"geometry":{"vertices":[[-0.7885966681787004,0.63798810626403,0],[0.7885966681787004,0.63798810626403,0],[-0.7885966681787004,-0.63798810626403,0],[0.7885966681787004,-0.63798810626403,0],[0,-0.3942983340893502,0.85065080835204],[0,0.3942983340893502,0.85065080835204],[0,-0.3942983340893502,-0.85065080835204],[0,0.3942983340893502,-0.85065080835204],[1.27597621252806,0,-0.5257311121191336],[1.27597621252806,0,0.5257311121191336],[-1.27597621252806,0,-0.5257311121191336],[-1.27597621252806,0,0.5257311121191336]],"faces":[[0,11,5],[0,5,1],[0,1,7],[0,7,10],[0,10,11],[1,5,9],[5,11,4],[11,10,2],[10,7,6],[7,1,8],[3,9,4],[3,4,2],[3,2,6],[3,6,8],[3,8,9],[4,9,5],[2,4,11],[6,2,10],[8,6,7],[9,8,1]],"edges":[[0,11],[5,11],[0,5],[1,5],[0,1],[1,7],[0,7],[7,10],[0,10],[10,11],[5,9],[1,9],[4,11],[4,5],[2,10],[2,11],[6,7],[6,10],[1,8],[7,8],[3,9],[4,9],[3,4],[2,4],[2,3],[2,6],[3,6],[6,8],[3,8],[8,9]],"scalars":[0.6826389260384893,0.6826389260384893,0.6826389260384893,0.6826389260384893,0.7578195211964776,0.7578195211964776,0.7578195211964776,0.7578195211964776,1.7011342063548263,1.7011342063548263,1.7011342063548263,1.7011342063548263],"scalarLabel":"Integrated angle defect (rad)","signed":true,"paths":[]},"triangles":[[0,11,5],[0,5,1],[0,1,7],[0,7,10],[0,10,11],[1,5,9],[5,11,4],[11,10,2],[10,7,6],[7,1,8],[3,9,4],[3,4,2],[3,2,6],[3,6,8],[3,8,9],[4,9,5],[2,4,11],[6,2,10],[8,6,7],[9,8,1]],"triangleFaces":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19],"vertexAngles":[[{"triangleId":0,"faceId":0,"angle":1.3776461023344415},{"triangleId":1,"faceId":1,"angle":0.8428630191796467},{"triangleId":2,"faceId":2,"angle":0.8428630191796467},{"triangleId":3,"faceId":3,"angle":1.3776461023344415},{"triangleId":4,"faceId":4,"angle":1.1595281381129208}],[{"triangleId":1,"faceId":1,"angle":0.8428630191796467},{"triangleId":2,"faceId":2,"angle":0.8428630191796467},{"triangleId":5,"faceId":5,"angle":1.3776461023344415},{"triangleId":9,"faceId":9,"angle":1.3776461023344415},{"triangleId":19,"faceId":19,"angle":1.1595281381129208}],[{"triangleId":7,"faceId":7,"angle":1.1595281381129208},{"triangleId":11,"faceId":11,"angle":0.8428630191796467},{"triangleId":12,"faceId":12,"angle":0.8428630191796467},{"triangleId":16,"faceId":16,"angle":1.3776461023344415},{"triangleId":17,"faceId":17,"angle":1.3776461023344415}],[{"triangleId":10,"faceId":10,"angle":1.3776461023344415},{"triangleId":11,"faceId":11,"angle":0.8428630191796467},{"triangleId":12,"faceId":12,"angle":0.8428630191796467},{"triangleId":13,"faceId":13,"angle":1.3776461023344415},{"triangleId":14,"faceId":14,"angle":1.1595281381129208}],[{"triangleId":6,"faceId":6,"angle":1.279834615464023},{"triangleId":10,"faceId":10,"angle":0.7549149699122814},{"triangleId":11,"faceId":11,"angle":1.4558666152305002},{"triangleId":15,"faceId":15,"angle":1.279834615464023},{"triangleId":16,"faceId":16,"angle":0.7549149699122814}],[{"triangleId":0,"faceId":0,"angle":0.7549149699122814},{"triangleId":1,"faceId":1,"angle":1.4558666152305002},{"triangleId":5,"faceId":5,"angle":0.7549149699122814},{"triangleId":6,"faceId":6,"angle":1.279834615464023},{"triangleId":15,"faceId":15,"angle":1.279834615464023}],[{"triangleId":8,"faceId":8,"angle":1.279834615464023},{"triangleId":12,"faceId":12,"angle":1.4558666152305002},{"triangleId":13,"faceId":13,"angle":0.7549149699122814},{"triangleId":17,"faceId":17,"angle":0.7549149699122814},{"triangleId":18,"faceId":18,"angle":1.279834615464023}],[{"triangleId":2,"faceId":2,"angle":1.4558666152305002},{"triangleId":3,"faceId":3,"angle":0.7549149699122814},{"triangleId":8,"faceId":8,"angle":1.279834615464023},{"triangleId":9,"faceId":9,"angle":0.7549149699122814},{"triangleId":18,"faceId":18,"angle":1.279834615464023}],[{"triangleId":9,"faceId":9,"angle":1.0090315813430704},{"triangleId":13,"faceId":13,"angle":1.0090315813430704},{"triangleId":14,"faceId":14,"angle":0.9910322577384362},{"triangleId":18,"faceId":18,"angle":0.5819234226617471},{"triangleId":19,"faceId":19,"angle":0.9910322577384362}],[{"triangleId":5,"faceId":5,"angle":1.0090315813430704},{"triangleId":10,"faceId":10,"angle":1.0090315813430704},{"triangleId":14,"faceId":14,"angle":0.9910322577384362},{"triangleId":15,"faceId":15,"angle":0.5819234226617471},{"triangleId":19,"faceId":19,"angle":0.9910322577384362}],[{"triangleId":3,"faceId":3,"angle":1.0090315813430704},{"triangleId":4,"faceId":4,"angle":0.9910322577384362},{"triangleId":7,"faceId":7,"angle":0.9910322577384362},{"triangleId":8,"faceId":8,"angle":0.5819234226617471},{"triangleId":17,"faceId":17,"angle":1.0090315813430704}],[{"triangleId":0,"faceId":0,"angle":1.0090315813430704},{"triangleId":4,"faceId":4,"angle":0.9910322577384362},{"triangleId":6,"faceId":6,"angle":0.5819234226617471},{"triangleId":7,"faceId":7,"angle":0.9910322577384362},{"triangleId":16,"faceId":16,"angle":1.0090315813430704}]],"angleSums":[5.600546381141097,5.600546381141097,5.600546381141097,5.600546381141097,5.525365785983109,5.525365785983109,5.525365785983109,5.525365785983109,4.58205110082476,4.58205110082476,4.58205110082476,4.58205110082476],"defects":[0.6826389260384893,0.6826389260384893,0.6826389260384893,0.6826389260384893,0.7578195211964776,0.7578195211964776,0.7578195211964776,0.7578195211964776,1.7011342063548263,1.7011342063548263,1.7011342063548263,1.7011342063548263],"boundaryVertices":[],"boundaryEdges":[],"boundaryLoops":[],"V":12,"E":30,"F":20,"chi":2,"boundaryComponents":0,"closed":true,"decomposition":"triangles","totalAngleDefect":12.566370614359172,"totalBoundaryTurning":0,"totalCurvature":12.566370614359172,"expectedTotalCurvature":12.566370614359172,"curvatureResidual":0,"measurements":[{"label":"Vertices V","value":12,"precision":0},{"label":"Edges E","value":30,"precision":0},{"label":"Faces F","value":20,"precision":0},{"label":"Euler characteristic χ","value":2,"precision":0},{"label":"Boundary components","value":0,"precision":0},{"label":"Counted cells","value":"Triangular cells"},{"label":"Interior angle defect","value":12.566370614359172,"unit":"rad","precision":8},{"label":"Boundary turning","value":0,"unit":"rad","precision":8},{"label":"Total integrated curvature","value":12.566370614359172,"unit":"rad","precision":8},{"label":"Theorem prediction 2πχ","value":12.566370614359172,"unit":"rad","precision":8},{"label":"Absolute residual","value":0,"unit":"rad","precision":12,"description":"Measured angle sum compared with 2π times the independently counted Euler characteristic."}],"plots":[{"title":"Local integrated curvature","xLabel":"Mathematical vertex ID","yLabel":"Angle defect / turning (rad)","series":[{"label":"Angle defect","points":[[0,0.6826389260384893],[1,0.6826389260384893],[2,0.6826389260384893],[3,0.6826389260384893],[4,0.7578195211964776],[5,0.7578195211964776],[6,0.7578195211964776],[7,0.7578195211964776],[8,1.7011342063548263],[9,1.7011342063548263],[10,1.7011342063548263],[11,1.7011342063548263]]}]}],"diagnostics":{"warnings":[],"method":"Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.","tolerance":1.3928735076156922e-11,"residual":0},"csv":{"columns":["model","subdivision","decomposition","scale_x","scale_y","scale_z","vertex_id","x","y","z","boundary","incident_angles_rad","angle_sum_rad","integrated_curvature_rad","V","E","F","chi","boundary_components","interior_total_rad","boundary_total_rad","total_rad","theory_rad","absolute_residual_rad"],"rows":[["sphere",0,"triangles",1.5,0.75,1,0,-0.7885966681787004,0.63798810626403,0,0,"[{\"triangleId\":0,\"faceId\":0,\"angle\":1.3776461023344415},{\"triangleId\":1,\"faceId\":1,\"angle\":0.8428630191796467},{\"triangleId\":2,\"faceId\":2,\"angle\":0.8428630191796467},{\"triangleId\":3,\"faceId\":3,\"angle\":1.3776461023344415},{\"triangleId\":4,\"faceId\":4,\"angle\":1.1595281381129208}]",5.600546381141097,0.6826389260384893,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,1,0.7885966681787004,0.63798810626403,0,0,"[{\"triangleId\":1,\"faceId\":1,\"angle\":0.8428630191796467},{\"triangleId\":2,\"faceId\":2,\"angle\":0.8428630191796467},{\"triangleId\":5,\"faceId\":5,\"angle\":1.3776461023344415},{\"triangleId\":9,\"faceId\":9,\"angle\":1.3776461023344415},{\"triangleId\":19,\"faceId\":19,\"angle\":1.1595281381129208}]",5.600546381141097,0.6826389260384893,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,2,-0.7885966681787004,-0.63798810626403,0,0,"[{\"triangleId\":7,\"faceId\":7,\"angle\":1.1595281381129208},{\"triangleId\":11,\"faceId\":11,\"angle\":0.8428630191796467},{\"triangleId\":12,\"faceId\":12,\"angle\":0.8428630191796467},{\"triangleId\":16,\"faceId\":16,\"angle\":1.3776461023344415},{\"triangleId\":17,\"faceId\":17,\"angle\":1.3776461023344415}]",5.600546381141097,0.6826389260384893,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,3,0.7885966681787004,-0.63798810626403,0,0,"[{\"triangleId\":10,\"faceId\":10,\"angle\":1.3776461023344415},{\"triangleId\":11,\"faceId\":11,\"angle\":0.8428630191796467},{\"triangleId\":12,\"faceId\":12,\"angle\":0.8428630191796467},{\"triangleId\":13,\"faceId\":13,\"angle\":1.3776461023344415},{\"triangleId\":14,\"faceId\":14,\"angle\":1.1595281381129208}]",5.600546381141097,0.6826389260384893,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,4,0,-0.3942983340893502,0.85065080835204,0,"[{\"triangleId\":6,\"faceId\":6,\"angle\":1.279834615464023},{\"triangleId\":10,\"faceId\":10,\"angle\":0.7549149699122814},{\"triangleId\":11,\"faceId\":11,\"angle\":1.4558666152305002},{\"triangleId\":15,\"faceId\":15,\"angle\":1.279834615464023},{\"triangleId\":16,\"faceId\":16,\"angle\":0.7549149699122814}]",5.525365785983109,0.7578195211964776,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,5,0,0.3942983340893502,0.85065080835204,0,"[{\"triangleId\":0,\"faceId\":0,\"angle\":0.7549149699122814},{\"triangleId\":1,\"faceId\":1,\"angle\":1.4558666152305002},{\"triangleId\":5,\"faceId\":5,\"angle\":0.7549149699122814},{\"triangleId\":6,\"faceId\":6,\"angle\":1.279834615464023},{\"triangleId\":15,\"faceId\":15,\"angle\":1.279834615464023}]",5.525365785983109,0.7578195211964776,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,6,0,-0.3942983340893502,-0.85065080835204,0,"[{\"triangleId\":8,\"faceId\":8,\"angle\":1.279834615464023},{\"triangleId\":12,\"faceId\":12,\"angle\":1.4558666152305002},{\"triangleId\":13,\"faceId\":13,\"angle\":0.7549149699122814},{\"triangleId\":17,\"faceId\":17,\"angle\":0.7549149699122814},{\"triangleId\":18,\"faceId\":18,\"angle\":1.279834615464023}]",5.525365785983109,0.7578195211964776,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,7,0,0.3942983340893502,-0.85065080835204,0,"[{\"triangleId\":2,\"faceId\":2,\"angle\":1.4558666152305002},{\"triangleId\":3,\"faceId\":3,\"angle\":0.7549149699122814},{\"triangleId\":8,\"faceId\":8,\"angle\":1.279834615464023},{\"triangleId\":9,\"faceId\":9,\"angle\":0.7549149699122814},{\"triangleId\":18,\"faceId\":18,\"angle\":1.279834615464023}]",5.525365785983109,0.7578195211964776,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,8,1.27597621252806,0,-0.5257311121191336,0,"[{\"triangleId\":9,\"faceId\":9,\"angle\":1.0090315813430704},{\"triangleId\":13,\"faceId\":13,\"angle\":1.0090315813430704},{\"triangleId\":14,\"faceId\":14,\"angle\":0.9910322577384362},{\"triangleId\":18,\"faceId\":18,\"angle\":0.5819234226617471},{\"triangleId\":19,\"faceId\":19,\"angle\":0.9910322577384362}]",4.58205110082476,1.7011342063548263,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,9,1.27597621252806,0,0.5257311121191336,0,"[{\"triangleId\":5,\"faceId\":5,\"angle\":1.0090315813430704},{\"triangleId\":10,\"faceId\":10,\"angle\":1.0090315813430704},{\"triangleId\":14,\"faceId\":14,\"angle\":0.9910322577384362},{\"triangleId\":15,\"faceId\":15,\"angle\":0.5819234226617471},{\"triangleId\":19,\"faceId\":19,\"angle\":0.9910322577384362}]",4.58205110082476,1.7011342063548263,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,10,-1.27597621252806,0,-0.5257311121191336,0,"[{\"triangleId\":3,\"faceId\":3,\"angle\":1.0090315813430704},{\"triangleId\":4,\"faceId\":4,\"angle\":0.9910322577384362},{\"triangleId\":7,\"faceId\":7,\"angle\":0.9910322577384362},{\"triangleId\":8,\"faceId\":8,\"angle\":0.5819234226617471},{\"triangleId\":17,\"faceId\":17,\"angle\":1.0090315813430704}]",4.58205110082476,1.7011342063548263,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0],["sphere",0,"triangles",1.5,0.75,1,11,-1.27597621252806,0,0.5257311121191336,0,"[{\"triangleId\":0,\"faceId\":0,\"angle\":1.0090315813430704},{\"triangleId\":4,\"faceId\":4,\"angle\":0.9910322577384362},{\"triangleId\":6,\"faceId\":6,\"angle\":0.5819234226617471},{\"triangleId\":7,\"faceId\":7,\"angle\":0.9910322577384362},{\"triangleId\":16,\"faceId\":16,\"angle\":1.0090315813430704}]",4.58205110082476,1.7011342063548263,12,30,20,2,0,12.566370614359172,0,12.566370614359172,12.566370614359172,0]]}},"build":{"packageVersion":"0.1.0","softwareVersion":"0.1.0+0e6017e6a747.dirty.02665ba9098c","gitRevision":"0e6017e6a747d3c610b956612b4125e232283001","gitDirty":true,"sourceDigest":"02665ba9098c7b8fe2a4b728fbd2b3749b1724fe9999f830d8fa90cb5d8b209b","runtime":"v26.5.1"}}
