Discrete connections (part VI)

The main block of the euclidean fractal
For any four complex numbers u,v,w and z shaping a trapezoid let us consider the entity defined by the following elements:

the two base endpoints

[; u \; and \; z;]

the two top endpoints

[;v \; and \; w;]

the top line

[;\delta(v,w);]

the parallelism master line

[;\delta(u,v);]

the top value

[;\beta \in \sigma(v,w) \; such \; that \; \delta(\alpha,\beta) \| \delta(u,v);]

where:

[;\delta(z_1,z_2) \; - \; the \; complex \; line \; defined \; by \; z_1 \; and \; z_2;]

[;\sigma(z_1,z_2) \; - \; the \; open \; complex \; line \; segment \; defined \; by \; z_1 \; and \; z_2;]

[;\alpha \; -\; the \; limit \; of \; the \; top \; value \; base \; sequence ;]

The self similarity block of rank k of the euclidean fractal

For any four complex numbers u,v,w and z shaping a trapezoid let us consider the entity defined by the following elements:

the two base endpoints

[;\tilde{\alpha}_{k+1} \; and \; \tilde{\beta}_{k+2};]

the two top endpoints

[;\tilde{\beta}_{k+1} \; and \; 2\cdot \tilde{\beta}_{k+2} - \tilde{\alpha}_{k+2};]

the top line

[;\delta(\tilde{\beta}_{k+1}, 2 \cdot \tilde{\beta}_{k+2} - \tilde{\alpha}_{k+2});]

the parallelism master line

[;\delta(\tilde{\alpha}_{k+1}, \tilde{\beta}_{k+1});]

the top value

[;\beta^{(k)} = \overline{\sigma}(\tilde{\alpha}_0,\beta) \cap \sigma(\tilde{\beta}_{k+1}, 2 \cdot \tilde{\beta}_{k+2} - \tilde{\alpha}_{k+2});]

where:

[;\beta \; - \; defined \; as \; above;]

[;\overline{\sigma}(z_1,z_2) \; the \; closed \; complex \; line \; segment \; defined \; by \; z_1 \; and \; z_2;]

Moreover, the self similarity block of rank k is by itself an euclidean fractal main block.
Now for its link to the euclidean summation, consider the euclidean fractal main block defined by u=0,v=i,w=1+i and z=1. (suggestive figure pending)
Next: Fibonacci decomposition of n