I am trying to grasp an article about the kissing problem in three dimensions (Das Problem der dreizehn Kugeln, by K. Schütte and B.L. van der Waerden). The article deals with irreducible graphs on a sphere and polygons in the graph, but I cannot find any definitions.
It seems like the edges all have to have the same length. Could anybody confirm (if so, why is it possible to have such a graph) or give definitions.
Also, it seems like edges of the polygons that are considered are equal. (If the above is true, this would follow.) Does there exists a special(?) definition of polygons on spheres.
Thanks in advance
The graph in question is the touching graph of the sphere arrangement. Two vertices in the graph share an edge if and only if the corresponding spheres touch, or equivalently if the distance between their centers is exactly $1$. This is because all spheres are considered to have a radius of $\frac12$. So the distance between the centers of any two touching spheres will be the same.
The above may me more easily understandable if you consider the previous work of these authors, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand Eins Platz? There they went to greater length to explain some of their vocabulary, like the meaning of the graph (§4), reductions and irreducibility (§5) and polygons (§6).
Regarding a definition of polygons: from §6 of that eralier work, it seems as if they were actually considering curved polygons on the surface of the sphere, as opposed to planar polygons with only their corners on the sphere. So the edges of these polygons are geodesics, i.e. great circle arcs. The length ratio between the lengths of these arcs and the length of the corresponding secants (which is $1$ as explained above) plays a major role in their texts.