# What would a Steiner tree look like for the vertices of a heptagon?

As it happens, I am currently frantically writing loads and loads of words for NaNoWriMo. One of the chapters I will be writing tonight essentially has the characters approximate a Steiner tree on seven points with rope. In various places, I can find graphics of Steiner trees on three points, four points at the corners of a rectangle, six points (not as the vertices of a regular hexagon), and a couple others, but there seems to be no graphic online for seven points. As I will be writing all day, I don't have time to code up a program to generate these graphics.

Hence, I was hoping that someone else could 1) generate such a graphic so that I know what it looks like or 2) direct me to a program that I can use to generate such a graphic. What would the Steiner tree for seven points look like?

Unfortunately I don't have access to the paper, but the abstract claims that they show that for $7\leq n\leq 12$, a Steiner tree for $n$ points forming the corners of a regular $n$-gon is obtained by deleting one side of the $n$-gon.
Finding a Steiner tree for all other $n$ was already solved way back in 1934. So while I don't know why the range $7\leq n\leq 12$ is so special, proving that the above is a Steiner tree for $n=7$ is evidently nontrivial somehow.
• No problem. It's also unclear if the Steiner tree in the $n=7$ case is unique. Maybe there is a more interesting one! – Casteels Nov 29 '14 at 21:56