This page continues the page Associahedron and associator identities.

F-magmas

by Markus Rost (Notes, January 2026, 14 pages)

The text formalizes basic aspects of magmas, in particular of free magmas.

Full text (February 10, 2026): [pdf]

Loday's realization of the associahedron

Here are affine projections of Loday's realization of the 3-dimensional associahedron.

The first image is simply the result of a first attempt.

The second image is close to the drawing at the end of Loday's paper [Loday, Jean-Louis. Realization of the Stasheff polytope. Arch. Math. 83, no. 3, 267-278, 2004. MR 2108555.]

A projection of Loday's associahedron Loday's projection of Loday's associahedron

The third image is the orthogonal projection to a particular face (the pentagon in front). The 2-dimensional sub-associahedron and its opposite are drawn in green. The homotopy between them is indicated (see the page The 3-dimensional associahedron unfolded and the blue arrows in the “homotopy” variant of Tamari's diagram).

An orthogonal projection of Loday's associahedron

Image 1: [high resolution image] · [with coordinates] · [Source: cube-3L.tex] (with the coordinates of the 4-dim. case)
Image 2: [high resolution image] · [with coordinates] · [Source: cube-3LL.tex]
Image 3: [high resolution image] · [with coordinates] · [Source: cube-3LO.tex] (with the orthogonal projections to faces)

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