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geometry 4d

Klein bottle and Veronese surface embedded in 4D

Build the Klein bottle and the real projective plane (Veronese surface) as embedded surfaces of R4, prove non-orientability and Euler characteristics on their meshes, show their 3D shadows cross themselves (Steiner's Roman surface), and write both shadows coloured by the fourth coordinate.

Recorded render of Klein bottle and Veronese surface embedded in 4D

What this component contains

  • ATTRIBUTION.md
  • LICENSE
  • README.md
  • component.json
  • example.gltf
  • fourd.py
  • klein_bottle_veronese.py
  • preview.png
  • test_package.py
  • verification/native.json

How it works

Entry points
klein, klein_mesh, veronese, veronese_mesh, euler_characteristic, is_orientable, closest_far_pair, roman_residual and 5 more
Outputs
example.gltf
Coordinates
surfaces in x, y, z, w; the shadow drops w (Klein) or the fourth Veronese coordinate, Godot/glTF +Y up

Recorded checks

godot 4.7.2.stable.official.ed1daf0bf

  • cli_rebuilds_example
  • gltf_structure_example
  • cli_example_runs
  • cli_example_gltf_structure
  • godot_loads_example
  • godot_loads_cli_coarse_surfaces
  • preview_render

Limits to consider

Meshes are flat-shaded with duplicated vertices because a non-orientable surface has no consistent normal field; materials are double-sided. The embedding check measures the smallest distance between mesh vertices more than three edges apart, a sampled test rather than a proof. The colour scale of the dropped coordinate is per surface.