Animated Logical Graphs • 62

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (57)(58)(59)(60)(61)

Another way of looking at the dual interpretation of logical graphs from a group-theoretic point of view is provided by the following Table.  In this arrangement we have sorted the rows of the previous Table to bring together similar graphs \gamma belonging to the set X, the similarity being determined by the action of the group G = \{ 1, t \}.  Transformation group theorists refer to the corresponding similarity classes as orbits of the group action under consideration.  The orbits are defined by the group acting transitively on them, meaning elements of the same orbit can always be transformed into one another by some group operation while elements of different orbits cannot.

\text{Peirce Duality as Group Symmetry} \stackrel{_\bullet}{} \text{Orbit Order}

Peirce Duality as Group Symmetry • Orbit Order

Scanning the Table we observe the 16 points of X fall into 10 orbits total, divided into 4 orbits of 1 point each and 6 orbits of 2 points each.  The points in singleton orbits are called fixed points of the transformation group since they are not moved but mapped into themselves by all group actions.  The bottom row of the Table tabulates the total number of fixed points for the group operations 1 and t respectively.  The group identity 1 always fixes all points, so its total is 16.  The group action t fixes only the four points in singleton orbits, giving a total of 4.

I leave it as an exercise for the reader to investigate the relationship between the group order |G| = 2, the number of orbits 10, and the total number of fixed points 16 + 4 = 20.

Resources

cc: Cybernetics (1) (2)Laws of FormFB | Logical Graphs • Ontolog Forum (1) (2)
• Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) • Structural Modeling (1) (2)
• Systems Science (1) (2)

This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization and tagged , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

12 Responses to Animated Logical Graphs • 62

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