User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

n-Dimensions
Let the universe of discourse be an n-dimensional space, $n \in \N, n > 1$.


 * $\forall a,b,c,p: \left[{\displaystyle \bigwedge_{1 \le i < j \le n} p_i \ne p_j \land \bigwedge_{i=2}^n ap_1 \equiv ap_i \land \bigwedge_{i=2}^n bp_1 \equiv bp_i \land \bigwedge_{i=2}^n cp_1 \equiv cp_i}\right]$


 * $\implies \left[{\mathsf{B}abc \lor \mathsf{B}bca \lor \mathsf{B}cab}\right]$

Intuition
Any three points equidistant from $n$ distinct points are collinear.

In other words, the set of all points equidistant from $n$ distinct points form a line.

This is a big one and I'm going to need some time to think about it and try to visualize it. Does this answer your question, LF?


 * Yes, that answers my question (in that I can see where it goes now). However, the name is still awkward, but I can imagine no better exists.
 * Hm, typing that I realize that the axioms give only an upper bound on the dimension (because of vacuous truth); how very interesting. However, be careful to distinguish the degenerate case that some of the points are collinear (there might be more cases) as then the set of points satisfying the axiom appears to be empty. --Lord_Farin 17:54, 23 January 2012 (EST)
 * If you think of a better name feel free to change it. By the way, I don't like putting up material on the main wiki if I don't quite understand it. But as you understand it more quickly than I do and you have access to my source (the link is on the sources for the other axioms) feel free to take over this axiom and I'll contemplate it at my leisure. --GFauxPas 18:03, 23 January 2012 (EST)
 * If you don't mind, I will put it on the short-but-not-today term. I am developing some notions on Hilbert space atm. --Lord_Farin 18:09, 23 January 2012 (EST)

Pictures - File:UpperDimensionalAxiom2D.png, File:UpperDimensionalAxiom3d.png --GFauxPas 18:29, 23 January 2012 (EST)