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Here are affine projections to the plane of the Loday realization of the 3-dimensional associahedron as a polytope.
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, Zbl 1059.52017].
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).
The 9 faces (6 pentagons and 3 rectangles) yield 4 essentially different orthogonal projections.
[click on the images for more detailed high resolution versions]
The realization lies in the hyperplane
x+y+z+t=10
and has the symmetry
T: (x,y,z,t) ↔ (t,z,y,x)
The quadrilateral faces are the parallelograms
Quadrilateral Q0 is a square invariant under T. Quadrilaterals Q1, Q2 are 1x3 rectangles interchanged by T and lying in parallel planes.
The lines orthogonal to some face are
Counting faces: (1+1)+(2+2)+1+2=9.
In the following, "Projection 1,2,3,4" refers to the projection along L1, L2, L3, L4, respectively. L1', L2' are implied via T.
Projection 1 makes vertex 4123 incident to edge 1612-4312. It is otherwise generic and appears to be the least degenerate among the 4 projections.
Projection 2 identifies vertices 3214, 4321. The projections of the adjacent pairs of parallel edges 3214-3124, 4321-4141 (direction [0,-1,1,0], length ratio 1:2) and 3214-1414, 4321-1621 (direction [-1,1,0,0], length ratio 2:3) overlap accordingly.
Projection 3 collapses Q1, Q2 like this:
Projection 4 collapses Q0 along its diagonal 3124-4213 and lets Q1, Q2 overlap, identifying corners 2134, 4312. It identifies edges 3124-2134 and 4213-4312.
The images below with thick blue edges indicate the vertex incidences, edge overlaps (darker blue) and face collapses (black).
The pentagon orthogonal to L1 and the 2 pentagons orthogonal to L2 are of the forms
The 3rd pentagon is the Loday realization in the 2-dimensional case.
The pentagons have angles 4x120°+60°. For a pentagon with such angles, extending edges like so
adds equilateral triangles and yields an equilateral triangle resp. a parallelogram with angles 60°,120°. Conversely, it follows that the pentagon is obtained by cutting accordingly equilateral triangles from an equilateral triangle resp. from a 60°/120°-parallelogram.
(One wonders about specific names for pentagons and parallelograms with such angles.)
The extension to the parallelogram shows that the length of a side at the 60° angle equals length of the parallel side + length of the base (the side opposite to the 60° angle). The length ratios of the 3 pentagons are 1:1:3:2:2, 2:1:4:3:2, 1:1:2:2:1, respectively.
[click on the images for high resolution versions]
Here are again the images with Loday coordinates. Further with the internal vertex names used in the source files (the names are the same as on cube-4-84-names-large.png from the page The 4-dimensional cubical associahedron).
Go to: Associahedron and associator identities · Publications and Preprints · Markus Rost's Web Page
