Fujii, Hokusai: theculturetrip.com
In this article, we'll discuss the entropy reduction of a sequence
by representing it as a binary tree which is multi-dimensional.
In computer programming there are standard methods of representing
the elements of a sequences with a binary tree whose elements are
the nodes. The multi-dimensional tree  is the progressive expansion
of sequences from a vector of elements, to a binary tree of nodes,
and into forest of multiple overlapping of binary trees.
In the context of building neural networks, a sequence is an ordered
list of elements. It's a time ordered, Markov process which is
embedded in a type of binary tree called a trie. You can visualize
a trie as a two dimensional lattice or trellis. A three dimensional
trie is an overlay of a pair of two dimensional tries.
Propagating or traversing the 3-D tree is equivalent to propagating
each of the pairs of binary trees. The order of looping on 2-D trees
is 2 whereas the order on a 3-D tree is 1. In practice, in computer
programs we only propagate over 2-D trees instead of multi-dimensional
The philosophy, or the vocabulary of the metaworld, for content and
form are, respectively, parts and rules. The nodes at the vertices
of trees are the parts of the forest, and the rules are the links in
the tree of a rewriting system.  With respect to the programming
language Lisp, the content and form are the variables and functions.
The neural network is a dynamic system in which the content and form
of the trees are represented by neurons and their connections. The
computer simulation of this network is a recurrent temporally ordered
Markov process. In using the Markov model for simulating neural nets
you would, by trail and error, compute the probabilities for the
state transitions. In the binary tree formulation you would use
the structure or form of the connections between the nodes. By
trail and error you would try to geometrically build the connections
between the neurons.
This is as simple as I can get it so far. If you have any ideas,
whether from a philosophical, mathematical or programming view, and
want to collaborate on this, please contact me at:
glenn @ neuralmachines . com.
 Multidimensional Trees, William Baldwin and George Strawn, 1991
 Pattern Recognition, Satosi Watanabe, 1985, p. 385