Message-ID: <Pine.NEB.4.58.0312131512420.24361@panix3.panix.com>
From: Alan Sondheim <sondheim@panix.com>
To: Cyb <cybermind@listserv.aol.com>,
"WRYTING-L : Writing and Theory across Disciplines" <WRYTING-L@LISTSERV.UTORONTO.CA>
Subject: Musings
Date: Sat, 13 Dec 2003 15:12:52 -0500 (EST)
Musings Take any chain of N unit vectors mutually orthogonal. The chain is defined such that at most two vectors meet at any vertex, and the endpoints are necessarily disparate. Then the chain defines a unique measure polytope of dimension N. The length of the chain = N, and if N is the maximum chain in any space, then N is the dimension of that space. Connect the two endpoints of the chain. Then the length of the connection is N^1/2, and the vector is a major diagonal of the measure polytope. Connect the endpoints of any two adjacent vectors; the length is 2^1/2. Connect the endpoints of any X adjacent vectors, and the length is X^1/2. If the dimension is N create a table such as: 1 2 3 4 5 6 6 5 4 3 2 1 . There are 6 vectors of length 1, 5 diagonals of length 2^1/2, down to 1 diagonal of length 6^1/2. In three dimensions 1 2 3 3 2 1 and the total defines the number of edges of a tetrahedron embedded in a cube (6 of course), such that for any dimension N, E = (N+1)(N)/2 for the number of edges. Note that these tetrahedrons are only regular when N = 1. The point of the exercise is that the awkwardness of the chain resolves quickly when the diagonals are added in; clearly the chain forms a tetrahedron of dimension N within a stable measure polytope; the diagonals create strength and stability, and the chain itself is embedded and comfortable with its new surroundings. _