ABSTRACT (original article): A pseudo-edge graph of a convex polyhedron
K
is a 3-connected embedded graph in
K
whose vertices coincide with those of
K
, whose edges are distance minimizing geodesics, and whose faces are convex. We construct a convex polyhedron
K
in Euclidean 3-space with a pseudo-edge graph with respect to which K is not unfoldable. The proof is based on a result of Pogorelov on convex caps with prescribed curvature, and an unfoldability obstruction for almost flat convex caps due to Tarasov. Our example, which has 340 vertices, significantly simplifies an earlier construction by Tarasov, and confirms that Dürer’s conjecture does not hold for pseudo-edge unfoldings. CITATION (original article): Barvinok, N., Ghomi, M. Pseudo-Edge Unfoldings of Convex Polyhedra. Discrete Comput Geom 64, 671–689 (2020). https://doi.org/10.1007/s00454-019-00082-1
Here we give the coordinates for the convex subdivision of the equilateral triangle discussed in Section 5 of our paper, and verify some basic properties of it.

The Coordinates


Figures

Basic figures

edges=Table[edge[i],{i,1,144}];​​​​Graphics[PrependTo[edges,RGBColor[0.35,0.5,0.78]],AspectRatioAutomatic]
edges=Table[edge[i],{i,1,135}];​​​​Graphics[PrependTo[edges,RGBColor[0.35,0.5,0.78]],AspectRatioAutomatic]

Labels

Computations (Proofs of Lemmas 5.2, 5.3)

Basic Formulas

Angles
angle[x_,y_,z_]:=ArcCos
(p[x]-p[y]).(p[z]-p[y])
Norm[p[x]-p[y]]Norm[p[z]-p[y]]
*
180
π
;
Obtuseness
obtuse[x_,y_,z_]=angle[x,y,z]-90;

Proof of Lemma 5.2

Using the “obtuse” function defined above we check that for each vertex
v
i
of the proposed subpath,
v
i+1
is the only adjacent vertex of
v
i
such that the angle spanned by
v
i+1
,
v
i
, and
p
1
is greater than or equal to π/2.
Note that
p
2
or p[2] has only two adjacent vertices: p[3] and p[18]. So we compute that:
{obtuse[1,2,3],obtuse[1,2,18]}
{2.38594,-2.77307}
Thus the only possible choice for the successor of p[2] into our path is p[3]. Next we check vertices that are adjacent to p[3], and continue this procedure until we reach p[-41]. Then the last computation shows that the path must leave the square.
{obtuse[1,3,4],obtuse[1,3,19]}
{13.3136,-3.32091}
{obtuse[1,4,5],obtuse[1,4,20]}
{6.79788,-3.74739}
{obtuse[1,5,6],obtuse[1,5,21]}
{7.98332,-4.20512}

Rest of vertices


Proof of Lemma 5.3

We follows the same general procedure described in the proof of Lemma 5.2.
p[o]={0,0};
{obtuse[o,24,25],obtuse[o,24,-22]}
{78.7933,-57.7029}
{obtuse[o,25,27],obtuse[o,25,26]}
{4.74045,-3.30413}
{obtuse[o,27,9],obtuse[o,27,28]}
{7.76125,-1.61334}
{obtuse[o,9,10],obtuse[o,9,27],obtuse[o,9,8]}
{63.3187,-59.3735,-76.0431}
{obtuse[o,10,11],obtuse[o,10,29]}
{53.5484,-48.359}
{obtuse[o,11,12],obtuse[o,11,31]}
{46.9183,-38.4005}
{obtuse[o,12,13],obtuse[o,12,32]}
{31.9493,-29.2577}
{obtuse[o,13,14],obtuse[o,13,33]}
{25.71,-20.6315}
{obtuse[o,14,15],obtuse[o,14,34]}
{17.793,-12.6559}

CITE THIS NOTEBOOK

Pseudo-edge unfoldings of convex polyhedra​
by Nicholas Barvinok and Mohammad Ghomi​
Wolfram Community, STAFF PICKS, January 29, 2025
​https://community.wolfram.com/groups/-/m/t/3367059