Biological Dipoles and Neural Networks from Trees: the figure-eight as a basic element, two-dimensional planting geometry as network topology, and the biological Hopfield network

This paper is the sixth in the series on forest bioelectric systems and extends Paper 5 concepts to neural networks with learning. We propose the figure-eight — a cross-connection of upper and lower tree branches — as an element creating a symmetric bioelectric dipole inside the trunk. The dipole node inside the trunk is a zone of anomalous electric field influencing cambium division and wood growth. We analyse: connecting multiple dipole trees into an antenna array with a steerable bioelectric field direction in soil; planting geometric configurations (pentagram, circle, cross, complete graph); correspondence of each connection topology to a specific computational architecture. A complete graph of N trees with adaptive mycelial nodes at conductor crossing points is a biological implementation of the Hopfield network — associative memory restoring complete patterns from partial or noisy inputs. Memory capacity is ~0.14N patterns for N trees. The figure-eight applied to one tree also creates an internal dipole through the stem: currents from the two crossed branches redistribute internally creating opposing current flows that form a symmetric dipole. The first direct branch-to-root connection is proposed as a new class of bioelectric experiments producing a double-scale dipole between two trees. Swastika geometry is analysed as a vortex antenna creating rotating circularly polarised bioelectric field (addressed in the supplement). Priority date: April 2026.