Here is an illustration of the basic structure of spatial connectivity matrices for N-dimensional Euclidean spaces.

A general network has a non-dimensional topology however by placing particular constraints on the network we can create networks with dimensional topologies.

These matrices can then be used in various calculations, computations and simulations involving dimensional spaces and systems interacting within dimensional spaces.

This matrix perspective clearly reveals the underlying fractal pattern that repeats in order to define higher and higher dimensional spaces.

Note: I only illustrate small spaces for simplicity (e.g. length = 3 units). Just increase the sizes for larger spaces.

Note: each network 'edge' is represented by either '0' for absence of connectivity or '1' for presence of connectivity - this merely illustrates the shape. Just use complex numbers for a more realistic edge weighting scheme.

0 1 0 1 0 1 0 1 0

For clarity, here is another view that highlights the pattern of connected edges.

# # # #

This represents a 1D line of connected points, which could be illustrated as :

*--*--*

Consider a column and row that intersect on a diagonal element, for instance, the top row and the left column.

The column represents outgoing connections.

The row represents incoming connections.

The diagonal element represents self-interactions.

In this way we can represent all of the nodes and edges within a network.

Such as the 1D case shown above. Notice that each end point is only connected to the middle node whilst the middle node is connected to the node on either side.

To construct a 3x3 area, we take three 1D lines and connect them up, in parallel. This is the general process of creating the next higher dimensional connectivity matrix.

First we take the previous dimensional pattern

0 1 0 1 0 1 0 1 0

and place three of them diagonally

0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0

This is like placing the three 1D lines into the same space - but not connecting them up yet.

*--*--* *--*--* *--*--*

Now we connect them, point to point, in corresponding lines.

0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0

The blank areas are all zeroes.

This gives us the network:

*--*--* | | | *--*--* | | | *--*--*

For clarity, here is another view that highlights the pattern of connected edges.

# # # # # # # # # # # # # # # # # # # # # # # #

Notice that this is the same basic pattern as the 1D matrix, because it is three things connected in line, except that now each 'thing' is a 1D line rather than a 0D node.

In general, the previous (next lower-dimensional) matrix serves as a building block, e.g. above it was

0 1 0 1 0 1 0 1 0

and there is also another sub-matrix that serves as the glue that binds the building blocks together.

1 0 0 0 1 0 0 0 1

These combine (like bricks and mortar) to create a fractal pattern of connectivity between the points within a dimensional space.

To build the next level, lay out the previous matrix along the diagonal:

0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0

Create a 'glue' matrix of the right size:

1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1

And now we 'glue' the 'blocks' together - in this case we are connecting three areas into a volume.

0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0

For clarity, here is another view that highlights the pattern of connected edges.

# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # #

So above is the basic pattern of connectivity within a 3D Euclidean space.

This pattern can be repeated to model higher dimensional spaces up to any number of dimensions.

Furthermore, this is one way that dimensional spatial contexts can be represented in SMN (System Matrix Notation) and then used in various simulation scenarios. However this is not how they arise when SMN is used to model the actual metaphysical arising of dimensional spaces – then it gets a lot more complicated... for more, see the mathematical analysis.