The single celled organism Physarum polycephalum efficiently constructs and minimises dynamical nutrient transport networks resembling proximity graphs. We present a model multi-agent population which collectively approximates the network behaviours of Physarum. We demonstrate spontaneous transport network formation and evolution and show that the collective population also exhibits quasi-physical emergent properties, allowing the collective population to be considered as a virtual computing material - a synthetic Plasmodium. This material is used as an unconventional method to approximate spatially represented geometry problems. We demonstrate three different methods for the construction, evolution and minimisation of Physarum-like transport networks which approximate Steiner trees, relative neighbourhood graphs, convex hulls and concave hulls. The results span the Toussaint hierarchy of proximity graphs, suggesting that the foraging and minimising behaviours of Physarum reflect interplay between maximising foraging area and minimising transport distance. The properties and behaviours of the synthetic virtual Plasmodium may be useful in future physical instances of unconventional computing devices, and may also provide clues to the generation of emergent computation behaviours by Physarum.
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