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The Loday realization of the associahedron (projections)

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).

• Image 1: [high resolution image] · [with Loday coordinates] · [Source: cube-3L.tex] (with the coordinates of the 4-dim. case)
• Image 2: [high resolution image] · [with Loday coordinates] · [Source: cube-3LL.tex]
• Image 3: [high resolution image] · [with Loday coordinates] · [Source: cube-3LO.tex] (with the orthogonal projections to faces)
• Image 1: [with parenthetical expressions] (and a color change)
• Image 2: [with parenthetical expressions] (ditto)
• Image 3: [with parenthetical expressions]

Orthogonal projections to faces

The 9 faces (6 pentagons and 3 rectangles) yield 4 essentially different orthogonal projections.

[click on the images for more detailed high resolution versions]

Some specifics

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

• Q₀: (3,2,1,4) + (1,0,0,-1) + (0,-1,1,0) : 3214, 4213, 3124, 4123
• Q₁: (1,2,3,4) + (1,-1,0,0) + (0,0,3,-3) : 1234, 2134, 1261, 2161
• Q₂=T(Q₁): (4,3,2,1) + (0,0,-1,1) + (-3,3,0,0) : 4321, 4312, 1621, 1612

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

• L₁ = [1,-3,1,1] : Orthogonal to a single face, a pentagon. The projection along L₁ is shown above (Image 3).
• L₁' = T(L₁) = [1,1,-3,1]: Like L₁ orthogonal to a single face, another pentagon.
• L₂ = [1,1,1,-3] : Orthogonal to the 2-dimensional sub-associahedron NNN4 and the parallel face (a further pentagon).
• L₂' = T(L₂) = [-3,1,1,1] : Orthogonal to the 2-dimensional sub-associahedron 4NNN and the parallel face (pentagon no. 6).
• L₃ = [1,-1,-1,1] : Orthogonal to a single face, the square Q₀.
• L₄ = [1,1,-1,-1] : Orthogonal to Q₁, Q₂.

Counting faces: (1+1)+(2+2)+1+2=9.

See also [Source: cube-3LO.tex].

[Projection 1] [Projection 2] [Projection 3] [Projection 4] are flat high resolution images for the projections along the L_k (also linked from the small condensed versions above).

Projection 1 makes vertex 4123 incident to edge 1612-4312 but is otherwise generic. It appears to be the least degenerate among the 4 projections.

Projection 2 identifies vertices 3214, 4321.

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.

Pentagon shapes

The pentagon orthogonal to L₁ and the 2 pentagons orthogonal to L₂ have the shapes

The 3rd pentagon is the Loday realization in the 2-dimensional case.

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