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This is a work in progress - all rights reserved.
Copyright © 2006-2007 Tony Giovia

 

15. The Rules of Power Relationships

 

15.1 - Geometric Outlines (GOs) can be constructed of one dimension, or they can be constructed of more than one dimension. (Definition)

15.2 - When a GO is constructed of one dimension, it is defined by that dimension. (Definition)

15.3 - When a GO is constructed of more than one dimension, all the dimensions used in its construction are related by a rule. That rule is a GO defined by its dimensions. (Definition)

15.4 - GOs can enter relationships with other GOs by sharing dimensions with those GOs.

15.5 – GOs can break existing relationships with other GOs by disassociating themselves from the dimensions shared with the other GOs. This disassociation changes the definition of all participating GOs.

15.6 - When a new shared dimension is added to a GO, the original rule that related the original dimensions undergoes a change in definition to include the additional dimension. (Definition)

15.7- Addition is a rule that joins two or more GOs not in a current relationship by identifying a dimension included in the construction of each GO, effectively creating a shared dimension between the GOs and therefore a relationship between the GOs. This shared dimension creates a new uniquely defined GO that includes part or all of the rules that defined the original unrelated GOs. Addition is symbolized by the “+” sign. (Definition)

a) Valence is a method of addition in which GOs directly share one or more dimensions in their respective constructions. Each new shared dimension changes the relationship between the GOs and therefore the definitions of the participating GOs. (Definition)

The rules creating these “joinings” are uniquely identifiable as GOs in their own right. Because rules define levels, these “joinings” may define the Dominant Rule (DR) of the context, or simply define a Recessive Rule (RR) within a larger defined context.

b) Mixing is a method of addition that applies a rule to GOs not currently in a relationship. That rule is not an existing Dominant or Recessive rule in any of the GOs to which it is applied. That rule identifies at least one shared dimension in the construction of those GOs and creates a relationship between the GOs via that rule, and no other rule. Because contexts are simply groups of GOs, and because rules define levels, this rule always becomes the Dominant Rule of the context when using mixed addition. (Definition)

15.8 - Multiplication is an expression of addition using multiple “+” signs, or the “x” sign indicating multiple additions of a particular dimension or GO.

15.9 - Subtraction is a rule that disassociates one or more shared dimensions from a context, creating a new uniquely defined context. Subtraction is symbolized by the “-” sign. (Definition)

a) Subtraction, like addition, moves dimensions out of one context and into another context.

15.10 - Division is an expression of subtraction using multiple “-” signs, or the “÷” or “√” signs indicating multiple subtractions.

15.11 - The Vectors of addition and multiplication (VA) are shared dimensions and connectivity. (Definition)

15.12 - The vectors of subtraction and division (VS) are non-shared dimensions and discontinuity. (Definition)

15.13- Identity Rule: For any uniquely defined context A, A = A (Definition)

15.14 - Inequality: A ≠ B (Definition)

15.15 - Irrationality: √2 (Definition)

15.16 - Imaginary: i (Definition)

15.17 - Impossible: A / 0 (Definition)

 

We are using logic as our tool for understanding and developing Dimensional Thinking. Rules and laws are equivalent concepts; however, in the case of multiple rules within a context, it is sometimes convenient to recognize a hierarchy of rules. When this happens, I will use “Dominant Recessive Rule” (DRR) to mean an inclusive shell rule that contains other related recessive rules.

The operative laws of logic and mathematics are the same – there is no logical law that cannot be converted to a mathematical law, and vice versa. This means that Dimensional Thinking, to be consistent, must completely obey all known and unknown mathematical and logical laws as they pertain to geometric objects.

Addition is a rule that joins the uniquely defined contextual dimensions that comprise Geometric Outlines (GOs), creating a new uniquely defined GO. There is one important difference between common mathematical notation and GO notation. Mathematically, A + 0 = A is correct notation. However, in terms of GOs, A + 0 = 0 is incorrect, unequal or inexact – however your POV chooses to define it. Remember that GOs are physical designs. For an identity to work contextually, the correct expression must be A + 0 = A + 0.

Valence is a method of addition that uses shared dimensions to join GOs. These “joinings” are bonds uniquely identifiable as GOs in their own right. The key point here is that contexts do not necessarily have to share all their dimensions to be added together. You may like 5 policy choices made by a political representative, but dislike 2 other choices. Here you are joined in 5 places and separated in two other places.

Because rules define levels, these bonds may define the Dominant Rule (DR) of the context, or simply define a Recessive Rule (RR) within a context defined by a DR. It may help you visualize this if you think of a complex context as a collection of dimensions organized in patterns controlled by rules. The stability of the context depends on the relationships of the rules that compose it – contexts are weakened and may break apart when internal RRs conflict with each other and/or with the DR.

Mixing is a method of addition that applies a rule to GOs not currently in a relationship. That rule is not an existing Dominant or Recessive rule in any of the GOs to which it is applied. That rule identifies at least one shared dimension in the construction of those GOs and creates a relationship between the GOs via that rule, and no other rule. “List everything in this room.” is an example of GOs connected by a DR that directly includes all the identical dimensions and indirectly all the non-identical dimensions of its component contexts.

Subtraction is a rule that removes one or more shared dimensions from a context, creating a new uniquely defined context. Essentially, subtraction and addition move dimensions from one rule to another rule.

Beyond the simple laws that are the foundations of math and logic are the laws that require more than one rule to build – these laws require more than one context for their uniqueness.

Inequality: A ≠ B (Identity, Not)

Irrationality: √2 (Square root, Infinite, Inexact)

Imaginary: I (Number, Inexact, Not on the number line, Not a real number)

Impossible: A / 0 (Division Law, Not on the number line, Inexact)

 

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