Message #3841

From: Marc Ringuette <ringuette@solarmirror.com>
Subject: Re: [MC4D] Chirality, orientation (~= handedness, inside-outness)
Date: Fri, 24 Nov 2017 13:05:11 -0800

Hi, Joel,

Thanks, yeah, trying to equate orientation and inside-outness was a poor
idea on my part.   I’m trying to properly define the inside-out
operation.   How about this instead:

1.  I will define a mirror operation to be any linear transformation
that flips the sign of the orientation.

2.  If even one mirror operation is permitted, then even asymmetric
objects become achiral.

1. An n-1 dimensional shell around an n dimensional object can be
   mirrored by puncturing or slicing it and turning it inside-out (if the
   inside and outside colors are the same at every point).

4.  The inside-out operation results in a mirror operation (a linear
tranformation that flips orientation), but cannot be broken down into a
sequence of infinitesimal linear transformations.  We could stretch the
object and squeeze it through the puncture nonlinearly (think rubber
balloon).   Or, if the object is composed of n-1 dimensional flat faces,
we may cut a number of n-2 dimensional seams along the face
intersections, allowing some previously adjacent points to become
non-adjacent for the duration of the inside-out operation, and then
perform a distinct series of infinitesimal linear transformations for
each face (think cardboard box).

Does that hold together better?

Cheers
Marc