2025/11/11 by Fabian Parsch, Parsch, Fabian, Hanrui Zhang +1
Computer Science · Materials Science · Mathematics · #52C45 (Primary) 53A70 #53C22 #58E10 (Secondary) #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2511.07810
openalex publication_date 2025/11/11 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28
We construct a geodesic net in the plane with four boundary (unbalanced) vertices that has 25 balanced vertices and that is irreducible, i.e. it does not contain nontrivial subnets. This net is novel and remarkable for several reasons: (1) It increases the previously known maximum for balanced vertices of nets of this kind from 16 to 25. (2) It is, to our knowledge, the first such net that includes balanced vertices whose incident edges are not exhibiting symmetries of any kind. (3) The approach taken in the construction is quite promising as it might have the potential for generalization. This would allow to construct a series of irreducible geodesic nets with four boundary vertices and an arbitrary number of balanced vertices, answering a conjecture that the number of balanced vertices is in fact unbounded for nets with four boundary vertices. This would stand in stark contrast to the previously proven theorem that for three boundary vertices, there can be at most one single balanced vertex.