Topology: curvature under refinement and stretching

Question

How do refinement and positive axis scaling affect vertex angle defects and their total?

Mathematical background

For a closed nondegenerate triangular surface, the sum of vertex angle defects is 2πχ. The Euler characteristic χ is calculated from the mesh’s vertices, edges and faces.

Individual angle defects are integrated curvature values at vertices. They can change with geometry even when the total is constrained by the topology.

\chi=V-E+F

Sources: [M1]

\sum_v\delta_v=2\pi\chi

Sources: [M1]

\sum_{v\in V_{\mathrm{int}}}\left(2\pi-\sum_{f\ni v}\theta_{f,v}\right)+\sum_{v\in V_{\partial}}\left(\pi-\sum_{f\ni v}\theta_{f,v}\right)=2\pi\chi

Sources: [M1]

Read the mathematical derivations.

Hypothesis or expected result

Theoretical expectations

The sphere cases are expected to have χ = 2 and total defect 4π; the torus cases are expected to have χ = 0 and total defect 0.

Nonuniform scaling may redistribute the local defects. Floating-point residuals are expected to be small within the documented, mesh-scaled tolerance; they are not assumed to equal zero.

Parameters and assumptions

All faces are nondegenerate, and every edge belongs to two faces.
Sphere levels 0–3 and torus levels 0–2 use the project’s identified vertices and periodic connections.
Each resolution uses the scales (1,1,1), (1.5,0.75,1) and (1,1.5,0.6). Triangulation is enabled.

Each case’s complete parameter object appears below and in its JSON file. Display rounding is used only in this article; the saved data retain full numeric precision.

Method and numerical settings

Run the shared mesh kernel for all 21 parameter sets. Calculate angles from each run’s coordinates, accumulate vertex defects, and independently count V − E + F.

Retain the actual total, predicted value, absolute residual, tolerance and complete per-vertex CSV. GPU buffers and image pixels are not used to calculate these quantities.

Methods reported by the executed kernel

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Software identifier: 0.1.0+0e6017e6a747.dirty.02665ba9098c
Git revision: 0e6017e6a747d3c610b956612b4125e232283001
Uncommitted worktree state: Present when the batch was generated
Source SHA-256: 02665ba9098c7b8fe2a4b728fbd2b3749b1724fe9999f830d8fa90cb5d8b209b
Runtime: v26.5.1

Results from an actual run

This saved batch contains 21 completed calculations. The table and downloads below refer to those actual runs.

The largest recorded absolute diagnostic residual was 5.47118e-13. 21 of 21 residual-bearing cases were within their own reported numerical tolerance.

Across 21 recorded readings, Euler characteristic χ ranged from 0 to 2.

Across 21 recorded readings, Total integrated curvature ranged from -5.47118e-13 to 12.56637.

Across 21 recorded readings, Theorem prediction 2πχ ranged from 0 to 12.56637.

Download summary CSV Download complete summary JSON

Actual case outputs. JSON retains every observable and the complete computed data.
Case	Selected measurements	Residual and tolerance	Data
1. Sphere · level 0 · scale (1, 1, 1)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 3.55271e-15 rad	3.55271e-15 Tolerance: 1.39287e-11	JSON · CSV
2. Sphere · level 0 · scale (1.5, 0.75, 1)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 0 rad	0 Tolerance: 1.39287e-11	JSON · CSV
3. Sphere · level 0 · scale (1, 1.5, 0.6)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 3.55271e-15 rad	3.55271e-15 Tolerance: 1.39287e-11	JSON · CSV
4. Sphere · level 1 · scale (1, 1, 1)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 8.88178e-15 rad	8.88178e-15 Tolerance: 2.46435e-11	JSON · CSV
5. Sphere · level 1 · scale (1.5, 0.75, 1)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 7.10543e-15 rad	7.10543e-15 Tolerance: 2.46435e-11	JSON · CSV
6. Sphere · level 1 · scale (1, 1.5, 0.6)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 8.88178e-15 rad	8.88178e-15 Tolerance: 2.46435e-11	JSON · CSV
7. Sphere · level 2 · scale (1, 1, 1)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 2.48690e-14 rad	2.48690e-14 Tolerance: 6.75024e-11	JSON · CSV
8. Sphere · level 2 · scale (1.5, 0.75, 1)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 5.50671e-14 rad	5.50671e-14 Tolerance: 6.75024e-11	JSON · CSV
9. Sphere · level 2 · scale (1, 1.5, 0.6)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 3.90799e-14 rad	3.90799e-14 Tolerance: 6.75024e-11	JSON · CSV
10. Sphere · level 3 · scale (1, 1, 1)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 1.43885e-13 rad	1.43885e-13 Tolerance: 2.38938e-10	JSON · CSV
11. Sphere · level 3 · scale (1.5, 0.75, 1)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 1.79412e-13 rad	1.79412e-13 Tolerance: 2.38938e-10	JSON · CSV
12. Sphere · level 3 · scale (1, 1.5, 0.6)	Euler characteristic χ: 2 Total integrated curvature: 12.56637 rad Theorem prediction 2πχ: 12.56637 rad Absolute residual: 2.34479e-13 rad	2.34479e-13 Tolerance: 2.38938e-10	JSON · CSV
13. Torus · level 0 · scale (1, 1, 1)	Euler characteristic χ: 0 Total integrated curvature: -3.37508e-14 rad Theorem prediction 2πχ: 0 rad Absolute residual: 3.37508e-14 rad	3.37508e-14 Tolerance: 4.42871e-11	JSON · CSV
14. Torus · level 0 · scale (1.5, 0.75, 1)	Euler characteristic χ: 0 Total integrated curvature: -3.01981e-14 rad Theorem prediction 2πχ: 0 rad Absolute residual: 3.01981e-14 rad	3.01981e-14 Tolerance: 4.42871e-11	JSON · CSV
15. Torus · level 0 · scale (1, 1.5, 0.6)	Euler characteristic χ: 0 Total integrated curvature: -2.57572e-14 rad Theorem prediction 2πχ: 0 rad Absolute residual: 2.57572e-14 rad	2.57572e-14 Tolerance: 4.42871e-11	JSON · CSV
16. Torus · level 1 · scale (1, 1, 1)	Euler characteristic χ: 0 Total integrated curvature: -1.39444e-13 rad Theorem prediction 2πχ: 0 rad Absolute residual: 1.39444e-13 rad	1.39444e-13 Tolerance: 1.47149e-10	JSON · CSV
17. Torus · level 1 · scale (1.5, 0.75, 1)	Euler characteristic χ: 0 Total integrated curvature: -1.44773e-13 rad Theorem prediction 2πχ: 0 rad Absolute residual: 1.44773e-13 rad	1.44773e-13 Tolerance: 1.47149e-10	JSON · CSV
18. Torus · level 1 · scale (1, 1.5, 0.6)	Euler characteristic χ: 0 Total integrated curvature: -1.32339e-13 rad Theorem prediction 2πχ: 0 rad Absolute residual: 1.32339e-13 rad	1.32339e-13 Tolerance: 1.47149e-10	JSON · CSV
19. Torus · level 2 · scale (1, 1, 1)	Euler characteristic χ: 0 Total integrated curvature: -5.28466e-13 rad Theorem prediction 2πχ: 0 rad Absolute residual: 5.28466e-13 rad	5.28466e-13 Tolerance: 5.58594e-10	JSON · CSV
20. Torus · level 2 · scale (1.5, 0.75, 1)	Euler characteristic χ: 0 Total integrated curvature: -5.10703e-13 rad Theorem prediction 2πχ: 0 rad Absolute residual: 5.10703e-13 rad	5.10703e-13 Tolerance: 5.58594e-10	JSON · CSV
21. Torus · level 2 · scale (1, 1.5, 0.6)	Euler characteristic χ: 0 Total integrated curvature: -5.47118e-13 rad Theorem prediction 2πχ: 0 rad Absolute residual: 5.47118e-13 rad	5.47118e-13 Tolerance: 5.58594e-10	JSON · CSV

Interpretation

At sphere level 1, the recorded vertex-defect range was [0.2738442, 0.3093413] radians for scale (1,1,1), and [0.1211419, 0.7997491] radians for scale (1.5,0.75,1). These ranges come from the saved per-vertex fields.

A small total-curvature residual is evidence about the implementation on these meshes. The closed-mesh derivation supplies the theorem; the experiments do not establish the complete classification of surfaces.

Compare local field ranges at a fixed model and resolution before interpreting the effect of stretching. Changing resolution changes the locations and number of vertices.

Limitations and possible errors

The study covers closed sphere and torus models. Disk and annulus boundary terms are available in the exhibit but are not included in this run matrix.
Only the specified positive axis scalings are tested. No pointwise smooth-curvature estimate or arbitrary vertex editing is used.

The kernel reported no warnings for this batch. This is a recorded software status, not an independent review.

Reproduce this study

Open Topology and enter the model, subdivision and three scale values shown for a case.
Import the case JSON to restore its complete inputs and recompute them in the current build. Compare the recomputed values with the downloaded record, checking model and software versions first.

Open topology

1. Sphere · level 0 · scale (1, 1, 1)

Recorded 17 September 2026 at 00:39 UTC. 12 exported data rows.

{
  "model": "sphere",
  "subdivision": 0,
  "scaleX": 1,
  "scaleY": 1,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 12
Edges E: 30
Faces F: 20
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 3.553e-15 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

2. Sphere · level 0 · scale (1.5, 0.75, 1)

Recorded 17 September 2026 at 00:39 UTC. 12 exported data rows.

{
  "model": "sphere",
  "subdivision": 0,
  "scaleX": 1.5,
  "scaleY": 0.75,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 12
Edges E: 30
Faces F: 20
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 0 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

3. Sphere · level 0 · scale (1, 1.5, 0.6)

Recorded 17 September 2026 at 00:39 UTC. 12 exported data rows.

{
  "model": "sphere",
  "subdivision": 0,
  "scaleX": 1,
  "scaleY": 1.5,
  "scaleZ": 0.6,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 12
Edges E: 30
Faces F: 20
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 3.553e-15 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

4. Sphere · level 1 · scale (1, 1, 1)

Recorded 17 September 2026 at 00:39 UTC. 42 exported data rows.

{
  "model": "sphere",
  "subdivision": 1,
  "scaleX": 1,
  "scaleY": 1,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 42
Edges E: 120
Faces F: 80
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 8.882e-15 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

5. Sphere · level 1 · scale (1.5, 0.75, 1)

Recorded 17 September 2026 at 00:39 UTC. 42 exported data rows.

{
  "model": "sphere",
  "subdivision": 1,
  "scaleX": 1.5,
  "scaleY": 0.75,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 42
Edges E: 120
Faces F: 80
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 7.105e-15 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

6. Sphere · level 1 · scale (1, 1.5, 0.6)

Recorded 17 September 2026 at 00:39 UTC. 42 exported data rows.

{
  "model": "sphere",
  "subdivision": 1,
  "scaleX": 1,
  "scaleY": 1.5,
  "scaleZ": 0.6,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 42
Edges E: 120
Faces F: 80
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 8.882e-15 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

7. Sphere · level 2 · scale (1, 1, 1)

Recorded 17 September 2026 at 00:39 UTC. 162 exported data rows.

{
  "model": "sphere",
  "subdivision": 2,
  "scaleX": 1,
  "scaleY": 1,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 162
Edges E: 480
Faces F: 320
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 2.487e-14 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

8. Sphere · level 2 · scale (1.5, 0.75, 1)

Recorded 17 September 2026 at 00:39 UTC. 162 exported data rows.

{
  "model": "sphere",
  "subdivision": 2,
  "scaleX": 1.5,
  "scaleY": 0.75,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 162
Edges E: 480
Faces F: 320
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 5.507e-14 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

9. Sphere · level 2 · scale (1, 1.5, 0.6)

Recorded 17 September 2026 at 00:39 UTC. 162 exported data rows.

{
  "model": "sphere",
  "subdivision": 2,
  "scaleX": 1,
  "scaleY": 1.5,
  "scaleZ": 0.6,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 162
Edges E: 480
Faces F: 320
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 3.908e-14 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

10. Sphere · level 3 · scale (1, 1, 1)

Recorded 17 September 2026 at 00:39 UTC. 642 exported data rows.

{
  "model": "sphere",
  "subdivision": 3,
  "scaleX": 1,
  "scaleY": 1,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 642
Edges E: 1920
Faces F: 1280
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 1.439e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

11. Sphere · level 3 · scale (1.5, 0.75, 1)

Recorded 17 September 2026 at 00:39 UTC. 642 exported data rows.

{
  "model": "sphere",
  "subdivision": 3,
  "scaleX": 1.5,
  "scaleY": 0.75,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 642
Edges E: 1920
Faces F: 1280
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 1.794e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

12. Sphere · level 3 · scale (1, 1.5, 0.6)

Recorded 17 September 2026 at 00:39 UTC. 642 exported data rows.

{
  "model": "sphere",
  "subdivision": 3,
  "scaleX": 1,
  "scaleY": 1.5,
  "scaleZ": 0.6,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 642
Edges E: 1920
Faces F: 1280
Euler characteristic χ: 2
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: 12.566371 rad
Boundary turning: 0 rad
Total integrated curvature: 12.566371 rad
Theorem prediction 2πχ: 12.566371 rad
Absolute residual: 2.345e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

13. Torus · level 0 · scale (1, 1, 1)

Recorded 17 September 2026 at 00:39 UTC. 96 exported data rows.

{
  "model": "torus",
  "subdivision": 0,
  "scaleX": 1,
  "scaleY": 1,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 96
Edges E: 288
Faces F: 192
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -3.375e-14 rad
Boundary turning: 0 rad
Total integrated curvature: -3.375e-14 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 3.375e-14 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

14. Torus · level 0 · scale (1.5, 0.75, 1)

Recorded 17 September 2026 at 00:39 UTC. 96 exported data rows.

{
  "model": "torus",
  "subdivision": 0,
  "scaleX": 1.5,
  "scaleY": 0.75,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 96
Edges E: 288
Faces F: 192
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -3.020e-14 rad
Boundary turning: 0 rad
Total integrated curvature: -3.020e-14 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 3.020e-14 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

15. Torus · level 0 · scale (1, 1.5, 0.6)

Recorded 17 September 2026 at 00:39 UTC. 96 exported data rows.

{
  "model": "torus",
  "subdivision": 0,
  "scaleX": 1,
  "scaleY": 1.5,
  "scaleZ": 0.6,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 96
Edges E: 288
Faces F: 192
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -2.576e-14 rad
Boundary turning: 0 rad
Total integrated curvature: -2.576e-14 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 2.576e-14 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

16. Torus · level 1 · scale (1, 1, 1)

Recorded 17 September 2026 at 00:39 UTC. 384 exported data rows.

{
  "model": "torus",
  "subdivision": 1,
  "scaleX": 1,
  "scaleY": 1,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 384
Edges E: 1152
Faces F: 768
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -1.394e-13 rad
Boundary turning: 0 rad
Total integrated curvature: -1.394e-13 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 1.394e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

17. Torus · level 1 · scale (1.5, 0.75, 1)

Recorded 17 September 2026 at 00:39 UTC. 384 exported data rows.

{
  "model": "torus",
  "subdivision": 1,
  "scaleX": 1.5,
  "scaleY": 0.75,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 384
Edges E: 1152
Faces F: 768
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -1.448e-13 rad
Boundary turning: 0 rad
Total integrated curvature: -1.448e-13 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 1.448e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

18. Torus · level 1 · scale (1, 1.5, 0.6)

Recorded 17 September 2026 at 00:39 UTC. 384 exported data rows.

{
  "model": "torus",
  "subdivision": 1,
  "scaleX": 1,
  "scaleY": 1.5,
  "scaleZ": 0.6,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 384
Edges E: 1152
Faces F: 768
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -1.323e-13 rad
Boundary turning: 0 rad
Total integrated curvature: -1.323e-13 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 1.323e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

19. Torus · level 2 · scale (1, 1, 1)

Recorded 17 September 2026 at 00:39 UTC. 1536 exported data rows.

{
  "model": "torus",
  "subdivision": 2,
  "scaleX": 1,
  "scaleY": 1,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 1536
Edges E: 4608
Faces F: 3072
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -5.285e-13 rad
Boundary turning: 0 rad
Total integrated curvature: -5.285e-13 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 5.285e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

Download complete case JSON · Download case CSV

20. Torus · level 2 · scale (1.5, 0.75, 1)

Recorded 17 September 2026 at 00:39 UTC. 1536 exported data rows.

{
  "model": "torus",
  "subdivision": 2,
  "scaleX": 1.5,
  "scaleY": 0.75,
  "scaleZ": 1,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 1536
Edges E: 4608
Faces F: 3072
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -5.107e-13 rad
Boundary turning: 0 rad
Total integrated curvature: -5.107e-13 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 5.107e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

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21. Torus · level 2 · scale (1, 1.5, 0.6)

Recorded 17 September 2026 at 00:39 UTC. 1536 exported data rows.

{
  "model": "torus",
  "subdivision": 2,
  "scaleX": 1,
  "scaleY": 1.5,
  "scaleZ": 0.6,
  "triangulate": true
}

Model version: topology-1.0.0

Unique polygonal cells; convex face triangulation; atan2 triangle angles; compensated summation; boundary turning included.

Vertices V: 1536
Edges E: 4608
Faces F: 3072
Euler characteristic χ: 0
Boundary components: 0
Counted cells: Triangular cells
Interior angle defect: -5.471e-13 rad
Boundary turning: 0 rad
Total integrated curvature: -5.471e-13 rad
Theorem prediction 2πχ: 0 rad
Absolute residual: 5.471e-13 radMeasured angle sum compared with 2π times the independently counted Euler characteristic.

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Each run’s download address is available in the summary JSON. The printed article preserves the methods, measurements and limitations.

References and review status

Project attribution: BIPH Math Society. Exposition and implementation were prepared with tool assistance.

Review status: Not reviewed. Independent teacher or member review has not yet been recorded.

[M1] Discrete Differential Geometry: An Applied Introduction — Keenan Crane
Version dated 29 January 2025. Chapter 2 and Sections 5.5–5.6 provide the background for triangular surfaces, angle defect and discrete Gauss–Bonnet. The exhibit’s explanations and implementation are original project work. Source checked 16 September 2026.