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I am assuming that a polygon is defined as a 2-dimensional polytope.

In that case, a 1-dimensional polytope will be a connected union of line segments. In other words, it is a physical realization of a graph.

Is there a short professional term for this concept?

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  • $\begingroup$ A chain of segments and a $1$-complex come to mind, depending on your set-up. Why not use "an embedding of a connected graph"? $\endgroup$ Commented Dec 22, 2018 at 20:29
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    $\begingroup$ Guy Inchbald suggests Polytelon, Ditelon, Dion, or Dyad $\endgroup$
    – Henry
    Commented Dec 22, 2018 at 20:32
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    $\begingroup$ Polytelon sounds nice and proper. Who is Guy Inchbald? I think there is not a consensus for this concept yet? $\endgroup$
    – user
    Commented Dec 22, 2018 at 20:41
  • $\begingroup$ @elasolova amazon.co.uk/Books-Guy-Inchbald/…. $\endgroup$
    – Paul Frost
    Commented Dec 23, 2018 at 0:15

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I am Guy Inchbald. In modern polytope theory, a 1-dimensional polytope is just a single closed line segment, meaning the end points are included as its vertices. A full-blown graph is effectively a polyhedron, as it has the same combinatorial structure (Search for Grünbaum on "Polyhedra as graphs, graphs as polyhedra"). There is indeed no consensus yet on what to call the 1-polytope. Amazingly, the first time that any term was formally used was not until 2018, in Prof. Norman Johnson's "Geometries and Transformations" (Cambridge University Press). He used "dion". This was some years after a few people had used "dyad" informally, which I thought unsuitable because it was used for other things and also begged the question, two of what? So I had proposed "ditelon". Johnson contracted it "dion" for his book, which I also find unsuitable because it begs the same question. A "ditelon" it a two-ended thing, "telos" being the end of say a rope, in the same way that a polygon has "many corners" and a polyhedron has "many seats" or faces. You know what you are talking about. Who am I? Nobody very much, I have just published a few papers on polyhedra. You can find a fuller explanation of the issue on my web site at http://www.steelpillow.com/polyhedra/ditela.html

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  • $\begingroup$ But I always thought that a single closed line segment is the 1-dimensional simplex. A triangle being 2-dimensional simplex. A connected composition of triangles and lesser dimensional simplexes(i.e. line segments) being a 2-dimensional polytope. By the same analogy, can't we say a connected composition of closed line-segments is a 1-dimensional polytope? I think I need to do some more research. Thanks for caring to answer my question. $\endgroup$
    – user
    Commented Jan 12, 2019 at 19:10
  • $\begingroup$ The simplest polytope in n dimensions is called the n-simplex: point, line segment, triangle, tetrahedron and so on. But they are polytopes first and foremost. The 1-simplex is the only 1-polytope. $\endgroup$ Commented Nov 25, 2019 at 17:19

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