Nets (Unfoldings) of Platonic Solids

[Evgeny.Makarov]

I was asked to help with a problem of constructing nets (unfoldings) of Platonic solids. This is supposed to be a high school research project.

In my opinion, the problem is not well-defined. There are only five Platonic solids, and their nets are well-known. So I am trying first to come up with some problem statement. One possible question is to come up with one's own net since, for example, there are 43380 nets for dodecahedron. Another is to devise an algorithm that, given a picture that looks like a net, to determine whether it is indeed a net of a Platonic solid. There may be something to prove about the connection between, say, the spanning tree of a graph of a polyhedron or its dual graph with a net of that polyhedron.

Has anyone encountered a high-school level problem related to nets of Platonic solids that is more mathematical in nature than cutting the net from a piece of paper?

 

[Evgeny.Makarov]

Coming up with a new net, not found on the Internet or in a book, is not completely trivial. Suppose we want to build a regular icosahedron (a polyhedron with 20 triangular faces). Once we prove its existence, it is clear that its net consists of triangles connected by edges. Therefore such net is determined by a graph, actually, a tree, whose vertices represent triangular faces and edges connect (vertices representing) triangles that have a common edge. This is the definition of a dual graph. Thus, to build a net we must take the skeleton of the icosahedron (its vertices and edges), take its dual graph and find its spanning tree.

It turns out that the dual graph of an icosahedron is a dodecahedron. For a plane embedding of the dodecahedron graph I took its Schlegel diagram, i.e., a perspective projection viewed from a point outside of the polyhedron, above the center of a facet. Then I used depth-first search to find a spanning tree. The Schlegel diagram of a dodecahedron and its spanning tree are shown below.

View attachment 4186 View attachment 4187
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The tree can also be represented as follows.

View attachment 4188
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The tree leads to the following net.

View attachment 4189
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However, to draw this net the spanning three must be an ordered tree, i.e., it should be known whether every child is a left or a a right one. Similarly, when constructing a dodecahedron (each face is a regular pentagon), whose dual graph is an icosahedron where the degree of each vertex is 5, each child must have a number from 1 to 4. In a dodecahedron's net, we can assign 0 to the edge that faces the root of the spanning tree, and we can enumerate edges clockwise. Then each face connected with the given face must have a number saying which edge is common between the two faces.

In the net above, the neighbors of face 1 are 2, 3 and 4, in clockwise order. This is also the case in the spanning tree and Schlegel_diagram. If in the spanning we put vertex 4 above 1 and between 2 and 3 and construct the net accordingly, it won't fold into an icosahedron. So how can I be sure that the Schlegel diagram correctly lists the order of child vertices? Is it possible to take a different plane embedding of the dual graph where the order is wrong?

Also I would be interested in an algorithm that, given an ordered tree, says whether it corresponds to a net of a given polyhedron.