Log in

No account? Create an account

January 25th, 2005



Take this ball and run with it, if you Will. I start out with a premise about which I know – or take myself to know – something about; that unacknowledged Energy variously described as chi, prana, bindu, od, orgone et al. It heals, it enlightens, it continuously either invigorates or, bottled up, destroys. The Taoists and Reich rightly associated it with sexual energy, but that’s a long story. But, consider this. It may be the key to that question which most people who fancy themselves magicians seem to have little curiosity about. Right after the legitimate “does it work?”comes “how does it work?”.

So, I’m stumbling around in the universe of graph theory, and I’m struck by the vision of the ever-brilliant “Flatland” once again. That’s the now century-old book showing what it’s like to be a two-dimensional conscious being and encountering, variously, other two dimensional beings, a one dimensional being (or as it would have it, THE one-dimensional being), and then a three dimensional – Something – viewed from a two dimensional being’s perspective.

I postulate that magick is the art – some day to become science – of channeling chi-prana-od-orgone-whatsit into and through one’s Self by act of focus or Will into an instrument (such as a wand) or token (such as a talisman) which, like a sort of device or machine of an electrical nature, or something of the type, transforms said energy into something more familiar, like matter.

Enter graph theory, stage left.

"One major problem that has plagued graph theory since its inception is the consistent lack of consistency in terminology..." Wikipedia, the free encyclopedia

Now, that’s promising. Very Deconstructionist, void, darkness upon the face of the Deep, hot for the anvil.

"But there is a fundamental result in graph theory that shows that if a network is not planar, then it must always be possible to identify in it a specific part that can be reduced to one of the two forms...
"So this implies that one can in fact meaningfully associate a definite structure with non-planarity..." Stephen Wolfram, A New Kind of Science

"The word graph may refer to the familiar curves of analytic geometry and function theory, or it may refer to simple geometric figures consisting of points and lines connecting some of these points; the latter are sometimes called linear graphs, although there is little confusion within a given context. Such graphs have long been associated with puzzles..." Encyclopedia Britannica

OK, we are in fact dealing with a puzzle of sorts here. The Great Work is a puzzle. Sort of like the little box in “Hellraiser” perhaps, but a puzzle nevertheless.

Every nonplanar graph is a supergraph of an expansion of the utility graph (i.e., the complete bipartite graph on two sets of three vertices) or the complete graph . This theorem was also proven earlier by Pontryagin (1927-1928), and later by Frink and Smith (1930). Kennedy et al. (1985) give a detailed history of the theorem, and there exists a generalization known as the Robertson-Seymour theorem.

"graph theory-
The mathematical study of the properties of the formal mathematical structures called graphs.

"Kazimierz Kuratowski

Born: 2 Feb 1896 in Warsaw, Russian Empire (now Poland)
Died: 18 June 1980 in Warsaw, Poland"

"Kuratowski's theorem-
A graph is nonplanar if and only if it contains a subgraph homeomorphic to K3, 3 or K5 "

"Kuratowski's Theorem). A graph is planar if and only if it does not contains a subdivision of K5 or K3,3.

Take the proof in discrete steps:

If G is nonplanar, then every subdivision of G is nonplanar.
If G is planar, then every subgraph of G is planar.
If a graph G has a 2 vertex cutset {u, v}, then G-{u,v} is disconnected. Let H1 be the subgraph induced on one of the resulting connected pieces along with vertices u, v, and an edge connecting u and v. Let H2 be a subgraph induced on the rest of the connected pieces along with vertices u, v, and an edge connecting u and v. Then if G is nonplanar, then at least one of H1 or H2 is also nonplanar.
Let G be a nonplanar connected graph that contains no subdivision of K5 or K3,3, and has a few edges as possible. Then G is simple and 3-connected.
Now you are ready to finish off Kuratowski's Theorem. (See Bondy and Murty for a good source.)"
"(Wagner's Theorem) Every planar graph has a plane drawing where each edge is a straight line. Hint: Apply induction on the order of maximal planar graphs by omitting a suitable vertex."


"A graph is called planar if it can be drawn in the plane without any two edges crossing. A famous theorem of Kuratowski says that a graph is planar if and only if it does not "contain" any copies of the following two graphs — K5 (the complete graph on five vertices, in which each vertex is connected to every other vertex) and K3, 3 (the complete bipartite graph with three vertices in each of the two sets, in which an edge is drawn between each pair of vertices that lie in different sets):

The Pentagram – K5

The Unicursal Hexagram – K3,3

You can fairly quickly convince yourself that K5 is nonplanar by trying to draw it without edge crossings. Trying to draw K3, 3 without edge crossings will result in similar difficulties, but in this case it is less obvious that there is no way of drawing it without crossings."

Thank you to the U.S. SGC for the certificate regarding my years at the OTO Prison Ministry. I am truly honored.

Thanks to D/S Oasis for honoring my long time deal friends Dan and Shannon on occasion of the former's Lesser Feast. A most interesting party after the packed and jammed Mass; a most interesting discussion regarding parties and what's happened to trust with Alex and James, and, later, with Alyssa and John.

We can change the world. We can. We must.

Latest Month

November 2019


Powered by LiveJournal.com
Designed by Tiffany Chow