An extension of the back-propagation algorithm to three dimensions by vector product

A 3D vector version of the backpropagation algorithm is proposed for multilayered neural networks in which a vector product operation is performed, and whose weights, threshold values, input and output signals are all 3D real numbered vectors. This new algortihm can be used to learn patterns considered of 3D vectors in a natural way. A 3D example was used to successfully test the new formulation.