Necessary and Sufficient Criteria for the Realizability of Kaleidocycles Composed of Six Congruent Isosceles Tetrahedra
Abstract
A classical kaleidocycle is a closed cyclic chain of congruent, edge-hinged tetrahedra with successive hinge axes at right angles, capable of continuous inside-out motion. Kaleidocycles composed of six isosceles tetrahedra (disphenoids) form a natural generalization in which consecutive hinge axes need not be perpendicular. Their kinematics are governed by the closure equation for threefold-symmetric Bricard 6R linkages. We derive a new compact closure equation directly from threefold symmetry using the composition of rotations, and show its equivalence to the classical formulation. Furthermore, we establish two equivalent necessary and sufficient criteria for a kaleidocycle to admit continuous, unblocked motion in terms of the geometric parameters of the constituent disphenoid. These criteria yield sharp bounds on the admissible geometric parameters of realizable kaleidocycles. Finally, we study assemblies of eight such kaleidocycles joined face-to-face. We identify a configuration exhibiting full octahedral symmetry, which forms a polyhedral structure closely related to the disdyakis dodecahedron.
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BibTeX
@misc{kociemba2026necessary,
title = {{Necessary and Sufficient Criteria for the Realizability of Kaleidocycles Composed of Six Congruent Isosceles Tetrahedra}},
author = {Herbert Kociemba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13961},
eprint = {2610.13961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13961},
}
RIS
TY - UNPB
TI - Necessary and Sufficient Criteria for the Realizability of Kaleidocycles Composed of Six Congruent Isosceles Tetrahedra
AU - Herbert Kociemba
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.13961
UR - https://arxiv.org/abs/2610.13961
AB - A classical kaleidocycle is a closed cyclic chain of congruent, edge-hinged tetrahedra with successive hinge axes at right angles, capable of continuous inside-out motion. Kaleidocycles composed of six isosceles tetrahedra (disphenoids) form a natural generalization in which consecutive hinge axes need not be perpendicular. Their kinematics are governed by the closure equation for threefold-symmetric Bricard 6R linkages. We derive a new compact closure equation directly from threefold symmetry using the composition of rotations, and show its equivalence to the classical formulation. Furthermore, we establish two equivalent necessary and sufficient criteria for a kaleidocycle to admit continuous, unblocked motion in terms of the geometric parameters of the constituent disphenoid. These criteria yield sharp bounds on the admissible geometric parameters of realizable kaleidocycles. Finally, we study assemblies of eight such kaleidocycles joined face-to-face. We identify a configuration exhibiting full octahedral symmetry, which forms a polyhedral structure closely related to the disdyakis dodecahedron.
ER -