Some concrete iso-Delaunay tessellations

[videlec]

(tentative answer to a question by Chris Judge at the SoS conference 2022)

Can we find two distinct translation surfaces with the same iso-Delaunay tesselation #163? plausible strategy : Pick two square-tiled surfaces with the minimum possible number of squares+1 (ie with a single vertex with angle 2pi). Here are some concrete candidates extracted from the origami database in surface_dynamics (index below means index of the veech group)

index=3
H_2(2)
(1)(2,3)
(1,2)(3)

H_3(4)
(1)(2,3)(4,5)
(1,2)(3,4)(5)

H_4(6)
(1)(2,3)(4,5)(6,7)
(1,2)(3,4)(5,6)(7)

H_6(10)
(1)(2,3)(4,5)(6,7)(8,9)(10,11)
(1,2)(3,4)(5,6)(7,8)(9,10)(11)

index=10
H_5(4^2)
(1)(2,3)(4)(5,6)(7,8)(9,10)
(1,2,4,5,7)(3,8,9,10,6)

H_5(4^2)
(1)(2,3)(4,5)(6,7)(8,9)(10)
(1,2,4,3,6)(5,8,7,9,10)

index=16
H_3(1^4)
(1)(2,3,4,5,6,7,8)
(1,2)(3,5)(4,8)(6,7)

H_4(3^2)
(1)(2,3,4)(5)(6,7,8)
(1,2,3,5,6,4,7)(8)

H_4(3^2)
(1)(2,3,4)(5)(6,7,8)
(1,2)(3,5)(4,6)(7,8)

index=21
H_5(6, 2)
(1)(2,3,4,5)(6)(7,8)(9,10)
(1,2)(3,6)(4,7,9)(5,10,8)

H_5(6, 2)
(1)(2,3,4,5)(6)(7,8)(9,10)
(1,2,5,10,8)(3,6,7,9,4)

index=32
H_3(3, 1)
(1)(2,3)(4)(5,6)
(1,2,4,3,5)(6)

H_4(1^6)
(1)(2,3,4,5,6)(7)(8,9,10,11,12)
(1,2,3,7,8)(4)(5,10,9,12,11)(6)

index=72
H_4(5, 1)
(1)(2,3)(4,5,6,7)(8)
(1,2,3,4,5,7)(6,8)

H_4(5, 1)
(1)(2,3)(4,5,6,7)(8)
(1,2,3,4,7,5)(6,8)

H_5(5, 2, 1)
(1)(2,3)(4)(5,6)(7,8)(9,10)(11)
(1,2,4,5,7)(3,6,9,10,11)(8)

H_5(5, 2, 1)
(1)(2,3)(4)(5,6)(7,8)(9,10)(11)
(1,2,4,5,7,9,8,11,3,6,10)

[videlec]

To build them

sage: from flatsurf import translation_surfaces
sage: S = SymmetricGroup(3)
sage: o = translation_surfaces.origami(S('(1)(2,3)'), S('(1,2)(3)'))

[saraedum]

There are two tasks here:

• The actual one is to decide whether two tessellations are the same up to an isometry.
• The harder one is to decide whether two hyperbolic surfaces are the same up to an isometry (all the above examples might have that property.) The easiest way to do this is to build Voronoi cells by growing horocycles around the cusps and then comparing the pictures.