The associahedron and triangulations of the \(n\)-gon. (English) Zbl 0682.52004

For a convex \(n\)-gon \(P_n\), \(n\geq 3\), in the plane let \(\Sigma_n\) denote the collection of all sets of mutually noncrossing diagonals of \(P_n\). The author proves that \(\Sigma_n\) is isomorphic to the boundary complex of some \((n-3)\)-dimensional simplicial convex polytope, and that this polytope can be geometrically realized with the dihedral group \(D_n\) as its symmetry group.


52B99 Polytopes and polyhedra
Full Text: DOI


[1] Brøndsted, A., An introduction to convex polytopes, (1983), Springer-Verlag New York · Zbl 0509.52001
[2] Cayley, A., On the partitions of a polygon, Proc. lond. math. soc, 22, 1, 237-262, (1890-1891) · JFM 23.0541.01
[3] Dantzig, G.B.; Hoffman, A.J.; Hu, T.C., Triangulations (tilings) and certain block triangular matrices, Math. programm, 31, 1-14, (1985) · Zbl 0571.90098
[4] Dörrie, H., Euler’s problem of polygon division, (), 21-27
[5] Gardner, M., (), 253-266
[6] Grünbaum, B., Convex polytopes, (1967), Interscience London · Zbl 0163.16603
[7] M. Haiman, Constructing the associahedron, manuscript.
[8] Huguet, D.; Tamari, D., La structure polyedrale des complexes de parenthésages, J. comb., inf. & syst. sci., 3, 69-81, (1978) · Zbl 0441.68080
[9] Kirkman, T.P., On the k-partitions of the r-gon and r-ace, Phil. trans. R. soc. lond., 147, 217-272, (1857)
[10] Lee, C.W., (), 173-181
[11] McMullen, P.; Shephard, G.C., Convex polytopes and the upper bound conjecture, (1971), Cambridge University Press · Zbl 0217.46702
[12] Perles, M.A., List of problems, conference on convexity, oberwolfach, (July 1984), Federal Republic of Germany
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.