@phdthesis{Lummerzheim2022, type = {Master Thesis}, author = {Jonas Lummerzheim}, title = {On 3D fixed-angle chains that are locked, equilateral, equiangular, and obtuse}, doi = {10.57683/EPUB-1952}, url = {https://nbn-resolving.org/urn:nbn:de:hbz:832-epub4-19524}, pages = {64}, year = {2022}, abstract = {For most classes of chains, it is known if these contain locks, but especially for fixed-angle equilateral equiangular obtuse open polygonal chains in 3D, which can be used to model protein backbones, this is unknown. Fixed-angle equilateral equiangular obtuse closed and open polygonal chains can be used to model polymers. For these, it is clear, that locks based on knots exist, but not which chains are generally locked. We therefore examine both open and closed fixed-angle equilateral equiangular obtuse chains. For this purpose, those chains are divided into various subgroups and, depending on the subgroup, other aspects are investigated to show locks. Techniques from knot theory, graph theory, and specifically robot arm reachability and motion planning are combined. Algorithms are developed to create chains in desired configurations and to study them. It is shown why all fixed-angle equilateral equiangular obtuse closed chains are expected to be locked or in rare cases rigid and non-locked, but never non-locked and non-rigid. For fixed-angle equilateral equiangular obtuse open chains it is shown why it is expected that there are open chains that are locked and that the smallest locked open chain has 𝑛=7.}, language = {en} }