# Formal Formulations

The Form and Substance
of
Recursion and Iteration

Context and Motivation

The most striking example of the Form of Recursion and Iteration is the Form of the Initials of the Primary Arithmetic, itself.

A source for this lovely example is found in a discussion of a system to support the preservation of historic artifacts. The discussion leads to a further discussion of The Form and Substance of Sensors. In order to visualize the data that has been sensed, it is useful to clarify the Canonical Forms of Visualization. This presents a wonderful opportunity to directly apply the Laws of Form.

Initial Transitions

The Initial of the Primary Arithmetic are presented as:

 ```( ) ( ) = ( ), and, ( ( ) ) = . ```

As ( ), the actual Form may range in shape from a square through a circle. The formal Transition of State Marking, from a single Mark, may be either:

 ```( ) ===> ( ) ( ), or, ( ) ===> ( ( ) ). ```

As visual Forms, these Transitions have the Form of Iteration and Recursion, respectively.

The Form of Iteration

The first Transition is the Form of Iteration, and may be considered as iterating the Form as:

 ```( ) ===> ( ) ( ) ===> ( ) ( ) ( ) ===> ( ) ( ) ( ) ( ) ===> ... ```

Because our consciousness experiences Form in a spatial sense, the Form may exist in either an Open or Closed volume of space, and, the Iterated Form is one of either Extension or Intension, respectively.

The Form of Recursion

The second Transition is the Form of Recursion, and may be considered as recursing the Form as:

 ```( ) ===> ( ( ) ) ===> ( ( ( ) ) ) ===> ( ( ( ( ) ) ) ) ===> ... ```

Because our consciousness experiences Form in a spatial sense, the Form may exist in either an Open or Closed volume of space, and, the Recursed Form is one of either Explosion or Implosion, respectively.

Canonical Forms of Actual Visualization

For the actual visual display of a visual form, the display is limited to a small area of a graphic image, and, the above Forms must conform to the Forms of a Closed space.

As such, the Intensional Form of Iteration corresponds to an increase in the density of Forms within a Closed space.

This first Form suggests a uniform density, permeating the space, as with a gas.

The Explosive and Implosive Forms of Recursion may each be visualized in a Closed space, and also correspond to an increase in the density of Forms within a Closed space.

These second and third Forms suggest a non-uniform density, filling or emptying the space, as with a liquid.