Saturday, December 29, 2007

Demi-orthogonality

I had strange theoretical dreaming as I was rousing from sleep.

Three distinctly different, and offbeat, concepts presented.
The first was on the use of tree structures as representations of the program paths and execution histories of running programs... simulacra of an executing program as it were. It is theoretically possible now to represent the entire history of a program, from its inception through development, instances of its executions, and archival, in a hypernetwork -- which hypernetwork can be exposed as tree structured markup. This was evident to me from my first exposure to SGML in 1992, but not something I felt was practical or useful to discuss with those around me. I'm not sure why it occurred to me again.

The second was the idea that time is... I'm grasping for it now as it fades... the virtual interface between spacial dimensions. In my dream I was making a presentation of some sort of simulation or model, and explaining that "we can choose to view space as having only one dimension", space-time being what results when a singularity is expanded. Time is the interface between the vectors as they span out from the center. (I don't know what that was supposed to really mean. I get the image of the universe as an grain of puffed wheat, foamy with slight tearing spread throughout, the distance between the edges of the tearing of which constitutes time. But as far as I know, that's just gobbledygook.)

The third thing that occurred to me was a formal definition of something called "demi-orthogonality", in conjunction with the "isometric vertexion" defined by Fuller. Fuller's construction involves 12 vectors radiating out from a center point, each in alignment with the edges of an octahedral/tetrahedral truss -- that is, the center-to-vertex vectors of an icosahedron. So in my dream ideation, demi-othogonality as it were, was defined as being a property of such a set of vectors, an analog of the orthogonality among the standard basis vectors in 3-space. I awoke as it was being defined... something about some subset of the vectors summing to the negation of one of the other vectors.

After I awoke, I thought about it for a minute, and considered that would mean demi-orthogonality couldn't be connected to orthogonality. So I thought about the definition as:
a set of vectors {a, b, ...n} in the M-space is demi-orthogonal to a vector q in the M-space if and only if there exists a vector r orthogonal to q such that the vector sum of q and r is the negation of the vector sum of {a, b, ...n}.

Don't take that definition as meaning anything important. It was only a dream. I haven't even tried to look up any other definition(s) of demi-orthogonal (but just now Google shows none.)