Constructed meshes, branch claude/constructive-classification
Every surface here is a mesh the construction actually output, refined to grid
cells and embedded in space. Drag any picture to turn it; scroll on it to zoom; double-click to
reset. The heavy black lines are the isolines u ∈ Z and v ∈ Z
of the parametrization, which are the edges of the original squares.
Not one of these signatures satisfies the hypothesis of the sufficient condition in the prior
work, which asks for gcd(m1, …, mn) = 1. For
image(ρ) equal to 0 or 2Z4 that is
automatic, since the generator divides every order; the rows with full holonomy were chosen with
a common factor on purpose. The classification answers all of them, in both directions.
A cone of valence v has exactly
v heavy lines running into it. Checked on every dart before drawing.
A simple pole: the surface closes on itself through an angle of π. Two isolines meet there, not four.
The sheets are translucent and the cone markers are sorted in with them, so a cone on the far side shows through dimmed rather than floating on top.
Only the shape. A flat cone metric like this has no isometric embedding in R³, so the surface is a distortion carrying an exact grid.
Signatures that no surface realizes. A correct algorithm has to fail on these, which is what makes them worth keeping.
| Parametrization | Genus | Cone orders | Valences | image(ρ) | Squares | gcd | Prior sufficient condition | Produced by |
|---|---|---|---|---|---|---|---|---|
|
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torus, umbilics of index +1/2 and -1/2 |
1 | -2, 2 | 2, 6 | Z4 | 2 minimal |
2 | not covered | search |
No surface with this signature exists, at any size. same torus, holonomy raised: no such surface |
1 | -2, 2 | — | 2Z4 | — | 2 | not covered | unrealizable |
|
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two umbilics of each sign: now it exists |
1 | -2, -2, 2, 2 | 2, 2, 6, 6 | 2Z4 | 4 minimal |
2 | not covered | exhaustive search |
|
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and with the full holonomy too |
1 | -2, -2, 2, 2 | 2, 2, 6, 6 | Z4 | 4 minimal |
2 | not covered | search |
|
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one cone of angle 6 pi, primitive quartic |
2 | 8 | 12 | Z4 | 3 minimal |
8 | not covered | construction |
|
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same cone, translation surface in H(2) |
2 | 8 | 12 | 0 | 3 minimal |
8 | not covered | closed form |
No surface with this signature exists, at any size. same cone, quadratic: Q(4) is empty |
2 | 8 | — | 2Z4 | — | 8 | not covered | unrealizable |
|
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angles 3 pi and 5 pi, primitive quartic |
2 | 2, 6 | 6, 10 | Z4 | 4 minimal |
2 | not covered | construction |
No surface with this signature exists, at any size. same angles, quadratic: Q(1,3) is empty |
2 | 2, 6 | — | 2Z4 | — | 2 | not covered | unrealizable |
|
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same curvature spread over four cones |
2 | 2, 2, 2, 2 | 6, 6, 6, 6 | Z4 | 6 minimal |
2 | not covered | construction |
|
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and the quadratic one exists here |
2 | 2, 2, 2, 2 | 6, 6, 6, 6 | 2Z4 | 6 minimal |
2 | not covered | exhaustive search |
|
drag to turn
two cones of angle 4 pi |
2 | 4, 4 | 8, 8 | Z4 | 4 minimal |
4 | not covered | construction |
|
drag to turn
|
2 | 4, 4 | 8, 8 | 2Z4 | 4 minimal |
4 | not covered | exhaustive search |
|
drag to turn
|
2 | 4, 4 | 8, 8 | 0 | 4 minimal |
4 | not covered | closed form |
|
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a pole and a cone of valence 14 |
2 | -2, 10 | 2, 14 | 2Z4 | 4 minimal |
2 | not covered | exhaustive search |
|
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one cone of angle 10 pi, primitive quartic |
3 | 16 | 20 | Z4 | 5 minimal |
16 | not covered | construction |
|
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the minimal quadratic stratum Q(8) |
3 | 16 | 20 | 2Z4 | 5 minimal |
16 | not covered | exhaustive search |
|
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the minimal abelian stratum H(4) |
3 | 16 | 20 | 0 | 5 minimal |
16 | not covered | closed form |
|
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four cones of angle 3 pi |
3 | 4, 4, 4, 4 | 8, 8, 8, 8 | Z4 | 8 minimal |
4 | not covered | construction |
|
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|
3 | 4, 4, 4, 4 | 8, 8, 8, 8 | 0 | 8 minimal |
4 | not covered | closed form |
|
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one cone of angle 14 pi |
4 | 24 | 28 | Z4 | 7 minimal |
24 | not covered | construction |
|
drag to turn
|
4 | 24 | 28 | 0 | 7 minimal |
24 | not covered | closed form |
|
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odd orders, so the holonomy is forced; gcd is 3 |
4 | 3, 21 | 7, 25 | Z4 | 8 minimal |
3 | not covered | construction |
|
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genus 5, four cones of angle 5 pi |
5 | 8, 8, 8, 8 | 12, 12, 12, 12 | 0 | 12 minimal |
8 | not covered | closed form |
|
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the pillowcase, four poles |
0 | -2, -2, -2, -2 | 2, 2, 2, 2 | 2Z4 | 2 minimal |
2 | not covered | exhaustive search |