Tootfinder

Cutting along a symmetric quadrilateral to construct an embedded flexible dodecahedron
Elvar Atlason
https://arxiv.org/abs/2510.06897 https://arxiv.org/pdf…

Cutting along a symmetric quadrilateral to construct an embedded flexible dodecahedron
Until recently, the simplest known flexible polyhedron was Steffen's polyhedron on nine vertices. However, in 2024, an embedded flexible polyhedron on eight vertices was announced. It attains the known lower bound for the number of vertices, showing that the simplest embedded flexible polyhedron has eight vertices. We introduce a method for making new flexible polyhedral surfaces from old ones. This general method applies to the above minimal example, giving another proof of its flexibility. We…

Approximating value functions via corner Benders' cuts
Matheus Jun Ota, Ricardo Fukasawa, Aleksandr M. Kazachkov
https://arxiv.org/abs/2509.21758 https://

Approximating value functions via corner Benders' cuts
We introduce a novel technique to generate Benders' cuts from a conic relaxation ("corner") derived from a basis of a higher-dimensional polyhedron that we aim to outer approximate in a lower-dimensional space. To generate facet-defining inequalities for the epigraph associated to this corner, we develop a computationally-efficient algorithm based on a compact reverse polar formulation and a row generation scheme that handles the redundant inequalities. Via a known connection between arc-flow a…

B-facets in dimension 4
Fedor Selyanin
https://arxiv.org/abs/2509.19118 https://arxiv.org/pdf/2509.19118

Constructing flexible polyhedra by twinning
Elvar Atlason, Simon Guest
https://arxiv.org/abs/2510.05280 https://arxiv.org/pdf/2510.05280

Constructing flexible polyhedra by twinning
Polyhedra are generically rigid, but can be made to flex under certain symmetry conditions. We generalise Raoul Bricard's 1897 method for making flexible octahedra to construct an infinite family of flexible polyhedra with self-intersections. Removing an edge from any of these models gives a crinkle, and these can be used to create flexible polyhedra without self-interesection. We show this in a particular example, giving a flexible embedded polyhedron with a large range of motion. We also disc…

Equilibrium Points and Surface Dynamics About Comet 67P/Churyumov-Gerasimenko
Leonardo Braga, Andre Amarante, Alessandra Ferreira, Caio Gomes, Luis Ceranto
https://arxiv.org/abs/2509.11613

Equilibrium Points and Surface Dynamics About Comet 67P/Churyumov-Gerasimenko
Small bodies in our Solar System are considered remnants of their early formation. Studying their physical and dynamic properties can provide insights into their evolution, stability, and origin. ESA's Rosetta mission successfully landed and studied comet Churyumov-Gerasimenko (67P) for approximately two years. In this work, the aim is to analyze the surface and orbital dynamics of comet 67P in detail, using a suitable 3-D polyhedral shape model. We applied the polyhedron method to calculate dy…