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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 (circles), edge overlaps (dark blue) and face collapses (black).
The projection heights (altitudes) of the vertices are shown in images in section Images and source code.
[click on the images for high resolution versions]
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° and the length ratios are 1:1:3:2:2, 2:1:4:3:2, 1:1:2:2:1, respectively.
Let us generally call any pentagon with angles 120°,60° a 120°/60°-pentagon. Furthermore, for such a pentagon, the corner is the vertex with the unique acute angle (of 60°), the base is the side opposite to the corner, the sides adjacent to the corner are called the long legs and the sides adjacent to the base the short legs. A short leg is adjacent to a long leg and the other long leg is parallel to it. (A long leg can be shorter than the adjacent short leg.)
Extending sides of a 120°/60°-pentagon erects outside equilateral triangles at the 120°-120° sides (the base and the short legs):
In particular, one gets an equilateral triangle and a parallelogram with angles 120°,60° as follows:
Conversely, the pentagon is obtained by cutting off accordingly equilateral triangles from an equilateral triangle resp. from a 120°/60°-parallelogram.
Clearly, a 120°/60°-pentagon is determined by the lengths of its base and its short legs. The extension to the parallelogram shows:
The length of a long leg is the sum of the lengths of the parallel short leg and the base.
This follows also from the extension to the equilateral triangle: its side length is long leg + adjacent short leg = base + short legs.
Besides the TeX/TikZ source files there are the images with
Here are related Python scripts (also included in [Source: assoc-L3A.tex]):
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