um. First is based on the setup of Voronoi diagram processor (Sect. \[voronoi\]). Previously, we wrote that when propagating fronts of Physarum meet they stop. This is true. But what happens after they stop is equally interesting. Physarum forms a bridge — a single protoplasmic tube —- connecting the stationary Physarum fronts. Such protoplasmic tubes typically span geographically neighbouring sites of inoculation and they cross sites of first contacts of growing wave fronts. These tubes represent edges of the Delaunay triangulation. This is why we proposed in [@shirakawa2009simultaneous] that the Voronoi diagram and the Delaunay triangulation are constructed simultaneously by Physarum growing on a nutrient agar.

Second method of approximating the Delaunay triangulation is implemented on a non-nutrient agar. We represent planar data points by inoculation sites. The inoculants propagate and form a planar proximity graph spanning all inoculation sites. At the beginning of such development a Gabriel graph [@gabriel1969new] is formed. Then additional protoplasmic tubes emerge and the Gabriel graph is transformed to the Delaunay triangulation [@adamatzky2009developing].

Concave hull {#concavehull}
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The $\alpha$-shape of a planar set $\mathbf P$ is an intersection of the complement of all closed discs of radius $1/\alpha$ that includes no points of $\mathbf P$ [@edelsbrunner1983shape]. A concave hull is a connected $\alpha$-shape without holes. This is a non-convex polygon representing area occupied by $\mathbf P$. Given planar set $\mathbf P$ represented by physical objects Physarum must approximate concave hull of $\mathbf P$ by its thickest protoplasmic tube. We represent data points by somniferous pills placed directly on a non-nutrient agar. The pills emits attractants to ‘pull’ Physarum towards $\mathbf P$ but they also emit repellents preventing Physarum from growing inside $\mathbf P$ [@adamatzky2012slime]. The combination of long-distance attracting forces and short-distance (‘sho