Why is a line segment called a digon?

[개구리]

This isn't really the etymology but it is the practical implication: polygon means "many 1-cells". All 1-cells are line segments. Therefore, a polygon is made of line segments. Why is a single line segment called a digon and not a monogon? Doesn't "digon" imply that there are two line segments connected at their two vertices? (This is degenerate in Euclidean geometry, they overlap, but in ellipsoidal geometry you can have either a lune or an "equatorial digon," as well as an "equatorial monogon.")

I've also seen "dyad" but I feel this alludes more to the idea of the 0-sphere, a disjoint point pair in 1D. "Ditelon" (polytelon=1D shape) seems like the most accurate name, but it requires you first qualify the idea of the telon, the 0-cell (point), as a constructing object, which is quite niche and limited in use. So what is a good and accurate name for a line segment that isn't the unwieldy "line segment," and why was it ever called a "digon" in the first place?

[wendy]

A digon is usually refered to as such in the symmetries of 2d geometries. In essence you can discuss it as a polygonal face.

A line is o------o. Two vertices, one edge. A digon is o======o. Two vertices, two edges. When you look at something like a cube, it is supposed that the square faces of the cube alternate with digons, rather than edges. This puts the regular figures in the same group as the icosadodecahedron ID or the cuboctahedron CO. The CO has faces 3,4; 2 sets at a vertex, The C has faces 4,2 ; 3 sets at a vertex.

In the Coxeter-Dynkin diagram, it amounts to the cube as @--4--o-----o If you cover a node (@ or o) with the finger, you get a polygon that forms in the remaining two mirrors (the node is a mirror, the branch between them is an angle between the mirrors). So covering up the first node gives one half-edge in the kaleidoscope (room of mirrors), which is repeated around the join --4-- four times normally and four times reversed. So four full edges.

The mirror marked @ puts the vertex right in the angle between the mirrors, so no marked nodes, no edges.

The middle node is surrounded by two mirrors with no marked branch. This is actually a '2' branch, the mirrors are at right-angles. Putting a vertex in between these would drop edges to both mirrors, and so the result would be four vertices, four edges, as a rectangle or square. You see these squares in the rhomboCO rhombotrunctated CO and the corresponding ID figures.

If the vertex is only on one mirror, as @ o, the resulting figure is an @2o, or digon.

Digonal symmetry is different to ordinary 'line-symmetry', in that it has extra mirrors. A line is o---|---o, the mirror reflects the vertices one onto another. The digonal symmetry has one that runs the length of the edge as well, as ---o======0----

digon, means 'two knees'.

[Klitzing]

The Greek word "polygon" does not mean "many sides / edges", it literally rather means "many corners / vertices".
This is why a single edge also is called a "digon": it still has 2 incident vertices, its ends.

--- rk

[개구리]

I think I understand now. A digon is distinct from a line segment? That doesn't seem great, why use it? Doesn't understanding the "-gon" suffix to mean vertices and not line segments conflict with the general pattern of polytope names, too? I suppose it usually isn't a problem because of vertices and edges typically being equal in count in two dimensions, but right here is the example of the distinction between the line segment and the degenerate digon. I don't know how comfortable I am with the idea that the faces of a polyhedron meet at digons. Do the cells of the pyrochoron meet at di-triangles? Do its edges meet at tri-gons? It feels silly.

[wendy]

The main use of a digon is in symmetries of 2d surfaces, including spheres.

A line segment only has one mirror, which reflects in the same space that the line segment is in. This reflects end-to-end.

A digon also has a second symmetry that reflects one side of the line to another. This is in keeping with even polygons, which have two sets of mirrors, vertex-to-vertex and edge middle-to-edge-middle. In the case of a digon, it's the first mirror that reflects what's on one side to what's on the other.

When one deals with symmetries in 3d, using the „wythoff“ system, the base symmetry are the schwarz-triangles like 2 3 5. You then put a mark | to divide which mirrors the vertex is on or off. So p | q r has a cycle of polygons q,r,q,r... for p times. The p q | r has a cycle p,2r,q,2r, and p q r | has a cycle 2p, 2q, 2r. Any of these can be '2', and the digon (when r=2) becomes an edge, and a truncated digon (2r=4) becomes a square.

We also speak of zero-height prisms in higher dimensions, which arise in much the same way. One of the polytopes in the prism-product is a zero-height prism. In general, a digon is x2o, which means a rectangle of sides 1 and 0.

[ndl]

This isn't really the etymology but it is the practical implication: polygon means "many 1-cells". All 1-cells are line segments. Therefore, a polygon is made of line segments. Why is a single line segment called a digon and not a monogon? Doesn't "digon" imply that there are two line segments connected at their two vertices? (This is degenerate in Euclidean geometry, they overlap, but in ellipsoidal geometry you can have either a lune or an "equatorial digon," as well as an "equatorial monogon.")

I've also seen "dyad" but I feel this alludes more to the idea of the 0-sphere, a disjoint point pair in 1D. "Ditelon" (polytelon=1D shape) seems like the most accurate name, but it requires you first qualify the idea of the telon, the 0-cell (point), as a constructing object, which is quite niche and limited in use. So what is a good and accurate name for a line segment that isn't the unwieldy "line segment," and why was it ever called a "digon" in the first place?

See the following article:

http://www.steelpillow.com/polyhedra/ditela.html