A counterexample for Steiner triangulation

I’ve been hesitating in writing up a blog post about my latest preprint, “Minimum-weight Steiner triangulation of convex polygons requires interior Steiner points” (with my student Zahra Hadizadeh, arXiv:2606.25302, to appear at CCCG) because, despite being a completely concrete two-dimensional construction, its exponential scale makes it difficult to visualize. In the paper we included an illustration using logarithmic coordinates but I didn’t find that entirely satisfactory so I’m trying again in a different way here. The new paper is about a problem from a much older paper of mine, “Approximating the minimum weight Steiner triangulation” (SODA 1992 and Discrete Comput. Geom. 1994), on “Steiner triangulation”, adding points to a geometric input to reduce the total edge length of a triangulation of the input. The added points are called Steiner points. The old paper proved that one could get within a constant factor of the minimum possible total edge length, for various kinds of…