Musings

Alan Sondheim sondheim at panix.com
Sat Dec 13 21:13:19 CET 2003





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.


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