Associahedron
(𝑛−2)-dimensional convex polytope in which each vertex corresponds to a parenthesization of a word of 𝑛 letters and each edge corresponds to an application of the associativity rule
In mathematics, an associahedron Kn is an (n − 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of n letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with n + 1 sides, and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari.
Examples
The one-dimensional associahedron K3 represents the two parenthesizations ((xy)z) and (x(yz)) of three symbols, or the two triangulations of a square. It is itself a line segment.
The two-dimensional associahedron K4 represents the five parenthesizations of four symbols, or the five triangulations of a regular pentagon. It is itself a pentagon and is related to the pentagon diagram of a monoidal category.
The three-dimensional associahedron K5 is an enneahedron with nine faces (three disjoint quadrilaterals and six pentagons) and fourteen vertices, and its dual is the triaugmented triangular prism.
Realization
Initially Jim Stasheff considered these objects as curvilinear polytopes. Subsequently, they were given coordinates as convex polytopes in several different ways; see the introduction of Ceballos, Santos & Ziegler (2015) for a survey.
This brief starts where responsible research should: with the source description of “Associahedron” as (𝑛−2)-dimensional convex polytope in which each vertex corresponds to a parenthesization of a word of 𝑛 letters and each edge corresponds to an application of the associativity rule. Everything that follows is an evidence route, not borrowed authority.
Why this record matters
The subject matters to the general reference register because the source frames it as (𝑛−2)-dimensional convex polytope in which each vertex corresponds to a parenthesization of a word of 𝑛 letters and each edge corresponds to an application of the associativity rule. Its deeper value depends on whether names, dates, institutions and citations support that framing.
Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated May 29, 2026. The linked authority identifier is Q4809117. None of the 0 selected statements returned an explicit reference. The first chronological checks are 2015.
Overview language is designed for orientation and should not be treated as a substitute for the evidence cited beneath it. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Associahedron”, its source revision and the description used here.
- Expand the search: follow Associahedron primary sources, Associahedron archive and Associahedron research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Associahedron”?
- Which institution is responsible for the underlying evidence?
- Which cited source is closest to the event, object or claim?
Search terms from this dossier
This entry incorporates text from “Associahedron” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.