[CG] Re: Logical Graphs • First Impressions

Logical Graphs • First Impressions 4 • https://inquiryintoinquiry.com/2023/08/24/logical-graphs-first-impressions/ Duality : Logical and Topological (cont.) — Last time we took up the axiom or initial equation shown below. Logical Graph • Initial Equation I₂ • https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-arithm… We noted it could be written inline as “( ( ) ) = ” or set off in a text display: ( ( ) ) = When we turn to representing the corresponding expressions in computer memory, where they can be manipulated with the greatest of ease, we begin by transforming the planar graphs into their topological duals. The planar regions of the original graph correspond to nodes (or points) of the dual graph, and the boundaries between planar regions in the original graph correspond to edges (or lines) between the nodes of the dual graph. For example, overlaying the corresponding dual graphs on the plane-embedded graphs shown above, we get the following composite picture. Initial Equation I₂ • Planar Graphs Overlaid By Dual Trees • https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-i2-pla… Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the root node of the corresponding dual graph. In the present style of Figure the root nodes are marked by horizontal strike-throughs. Extracting the dual graphs from their composite matrix, we get the following equation. Initial Equation I₂ • Dual Trees • https://inquiryintoinquiry.files.wordpress.com/2023/08/logical-graph-i2-tre… Regards, Jon cc: https://www.academia.edu/community/ldYE85 cc: https://mathstodon.xyz/@Inquiry/110945139629369891