Geometric Realizations of the 3D Associahedron (Multimedia Exposition)

Authors Satyan L. Devadoss, Daniel D. Johnson, Justin Lee, Jackson Warley

Thumbnail PDF


  • Filesize: 0.82 MB
  • 4 pages

Document Identifiers

Author Details

Satyan L. Devadoss
Daniel D. Johnson
Justin Lee
Jackson Warley

Cite AsGet BibTex

Satyan L. Devadoss, Daniel D. Johnson, Justin Lee, and Jackson Warley. Geometric Realizations of the 3D Associahedron (Multimedia Exposition). In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 75:1-75:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


The associahedron is a convex polytope whose 1-skeleton is isomorphic to the flip graph of a convex polygon. There exists an elegant geometric realization of the associahedron, using the remarkable theory of secondary polytopes, based on the geometry of the underlying polygon. We present an interactive application that visualizes this correspondence in the 3D case.
  • associahedron
  • secondary polytope
  • realization


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Louis J Billera, Susan P Holmes, and Karen Vogtmann. Geometry of the space of phylogenetic trees. Advances in Applied Mathematics, 27(4):733-767, 2001. Google Scholar
  2. Michael Bostock, Vadim Ogievetsky, and Jeffrey Heer. D³ data-driven documents. IEEE transactions on visualization and computer graphics, 17(12):2301-2309, 2011. Google Scholar
  3. Ricardo Cabello et al. Three.js, 2010. URL:
  4. Cesar Ceballos, Francisco Santos, and Günter M Ziegler. Many non-equivalent realizations of the associahedron. Combinatorica, 35:513-551, 2015. Google Scholar
  5. Frederic Chapoton, Sergey Fomin, and Andrei Zelevinsky. Polytopal realizations of generalized associahedra. Bulletin Canadien de Mathématiques, 45:537-566, 2002. Google Scholar
  6. Satyan L Devadoss. Tessellations of moduli spaces and the mosaic operad. Contemporary Mathematics, 239:91-114, 1999. Google Scholar
  7. Satyan L Devadoss and Joseph O'Rourke. Discrete and Computational Geometry. Princeton University Press, 2011. Google Scholar
  8. Satyan L Devadoss, Rahul Shah, Xuancheng Shao, and Ezra Winston. Deformations of associahedra and visibility graphs. Contributions to Discrete Mathematics, 7:68-81, 2012. Google Scholar
  9. Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Lagrangian intersection Floer theory: Anomaly and obstruction. American Mathematical Society, 2010. Google Scholar
  10. Israel M Gelfand, Mikhail Kapranov, and Andrei Zelevinsky. Discriminants, Resultants, and Multidimensional Determinants. Springer, 2008. Google Scholar
  11. Christophe Hohlweg and Carsten Lange. Realizations of the associahedron and cyclohedron. Discrete &Computational Geometry, 37(4):517-543, 2007. Google Scholar
  12. Ivan Kuckir. PolyK.js. URL:
  13. Carl W Lee. The associahedron and triangulations of the n-gon. European Journal of Combinatorics, 10:551-560, 1989. Google Scholar
  14. Chiu-Chu Melissa Liu. Moduli of J-holomorphic curves with Lagrangian boundary conditions and open Gromov-Witten invariants for an S¹-equivariant pair. arXiv math/0210257, 2002. Google Scholar
  15. Sébastien Loisel. Numeric javascript. URL:
  16. Alexander Postnikov. Permutohedra, associahedra, and beyond. International Mathematics Research Notices, 2009:1026-1106, 2009. Google Scholar
  17. Jim Stasheff. From operads to ‘physically’ inspired theories. In Operads: Proceedings of Renaissance Conferences, volume 202, page 53. American Mathematical Society, 1997. Google Scholar
  18. M Yoshida. Hypergeometric functions, my love. Vieweg, 1997. Google Scholar