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This might be a fairly trivial question, but since I'm not familiar with the mathematical jargon, I'm asking it here.

Given a polyhedron $p$, what is the proper mathematical terminology for its boundary? For instance, suppose I have a 3D cube $c$, how do I describe the structure formed by just the 12 edges of the cube? It seems to me that perhaps I should refer to this wire-frame as the 1-skeleton of the cube, but I'm not quite sure. More specifically, is there an operation $\partial$ such that $\partial c$ describes the skeletal structure formed by the edges of the cube?

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  • $\begingroup$ I think $1$-skeleton is right. It's common and unambiguous. If you need a named operator for that then we need to see more context to make a reasonable suggestion. $\endgroup$ Commented Mar 21, 2018 at 18:20
  • $\begingroup$ Topologically, the boundary of a $3D$ polhyhedron is two dimensional: edges plus faces. To just talk about the edges you'd use the term $1$-skeleton. This is often denoted $c^{(1)}$. ($c^{(k)}$ being the $k$-skeleton of $c$.) $\endgroup$ Commented Mar 21, 2018 at 18:21

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Just based on terminology from simplicial and cellular complexes, I would call the graph induced by the edges the polyhedron's $1$-skeleton (Wikipedia). If it were the case the $1$-skeleton was a polyhedron, one might consider $1$-stratum instead, but it's not so we won't.

It's worth noting that it might matter how you think about the polyhedron $p$.

  • Is $p$ the combinatorial data of vertices, edges, and faces? Then the $1$-skeleton $p^{(1)}$ is the union of the vertices and edges. (Note that it's not exactly an operation. This is the name for this particular collection of parts.)
  • Is $p$ the combinatorial data of vertices, edges, and faces embedded in $\mathbb{R}^3$ in a particular way? Then just make sure when you use $p^{(1)}$ that you specify whether you mean the abstract graph or the actual embedding of the $1$-skeleton in $\mathbb{R}^3$.
  • Is $p$ given as an intersection of half-spaces? To use $p^{(1)}$, you are supposed to specify the combinatorial data of $p$. You could define $p^{(1)}$ to be the collection of points in $p$ such that in a small neighborhood of the point, $p$ does not look like a half-space. Where this definition might diverge from the others is if you chose combinatorial data for $p$ that happened to have adjacent faces that are coplanar.

I guess a takeaway is that it depends what you mean by a polyhedron. An amusing quote of Branko Grünbaum is that "the Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others [...] at each stage [...] the writers failed to define what are the polyhedra."

Depending on your audience, it might be worth just calling it the wireframe, and denote it by $Wp$. Or you could demonstrate your classical education by denoting it $\sigma p$, where $\sigma$ is the first letter of either "wire" or "skeleton" in Greek.

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