Osaka Journal of Mathematics

The configuration space of equilateral and equiangular hexagons

Jun O'Hara

Full-text: Open access


We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration space.

Article information

Osaka J. Math., Volume 50, Number 2 (2013), 477-489.

First available in Project Euclid: 21 June 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65D18: Computer graphics, image analysis, and computational geometry [See also 51N05, 68U05]
Secondary: 51E12: Generalized quadrangles, generalized polygons 55R80: Discriminantal varieties, configuration spaces


O'Hara, Jun. The configuration space of equilateral and equiangular hexagons. Osaka J. Math. 50 (2013), no. 2, 477--489.

Export citation


  • J.A. Calvo and A.L. Pile: A spectrum for $\alpha$-regular unknots. J. Knot Theory Ramifications (2012) DOI: 10.1142/S0218216512501210.
  • G.M. Crippen: Exploring the conformation space of cycloalkanes by linearized embedding, J. Comput. Chem. 13 (1992), 351–361.
  • E.D. Demaine and J. O'Rourke: Geometric Folding Algorithms, Cambridge Univ. Press, Cambridge, 2007.
  • C. Frayer and C. Schafhauser: Alpha-regular stick unknots, J. Knot Theory Ramifications 21 (2012), 140–150.
  • J.-C. Hausmann: Sur la topologie des bras articulés; in Algebraic Topology Poznań 1989, Lecture Notes in Math. 1474, Springer, Berlin, 1991, 146–159.
  • T.F. Havel: Some examples of the use of distances as coordinates for Euclidean geometry, J. Symbolic Comput. 11 (1991), 579–593.
  • Y. Kamiyama: Topology of equilateral polygon linkages in the Euclidean plane modulo isometry group, Osaka J. Math. 36 (1999), 731–745.
  • M. Kapovich and J.J. Millson: The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 (1996), 479–513.
  • A.L. Pile: The space of regular polygons with a small number of edges, Ph.D. Dissertation, North Dakota State Univ. (2008).
  • R. Randell: Conformation spaces of molecular rings; in MATH/CHEM/COMP 1987, Studies in Physical and Theoretical Chemistry 54, Elsevier, 1988, 141–156.
  • H. Sachse: Über die Konfigurationen der polymethylenringe, Z. Phys. Chem. 10 (1892), 202–241.