Message #3844

From: Joel Karlsson <joelkarlsson97@gmail.com>
Subject: Re: [MC4D] Chirality, orientation (~= handedness, inside-outness)
Date: Mon, 27 Nov 2017 16:06:09 +0100

Hi Marc,

1) That’s a good definition (and is what is usually used in linear algebra).

2) This might depend on the precise definition of chirality. We could say
that an object is chiral with respect to a set of transformations if it can
be distinguished from its mirror image under those transformations combined
(set union) with ordinary rotations. Usually, when talking about chirality,
only rotations are allowed (I think), meaning that we are talking about
chirality with respect to the empty set with my previous definition. So if
the set is not specified we can assume the empty set. What I believe you
ment becomes "all objects are achiral with respect to a set containing at
least one mirror transformation", which is correct.

3 and 4) What you are describing is indeed a way to mirror an object but it
is not a mirror transform (in the sense of 1). To clarify, the
transformation you are describing mirrors the object in the sense that it
looks like its mirror image (after the transformation) but is not a mirror
transformation (using the definition in 1). Let me explain why. A linear
transformation can be defined in terms of where it takes the vectors of a
basis. Consider an n-dimensional cube in an n-dimensional space. Let’s
(arbitrarily) choose a corner. For every face that the corner is a part of
(the ones "touching" the corner) imagine a unit vector pointing out of that
face (origin in the center of the cube). These vectors form an orthogonal
normalized basis. Now, let’s turn the cube inside out. One of the basis
vectors will remain unchanged (x |-> x) and the others will change
direction (x |-> -x). This defines (precisely) one linear transformation
which is a rotation if n is odd and a mirror transformation if n is even.
However, since the inside out turning is definitely not just a rotation or
a mirror transformation (these do not change which side of a face is
pointing out of the cube) it cannot be a linear transformation.

Best regards,
Joel

Den 24 nov. 2017 10:04 em skrev "Marc Ringuette ringuette@solarmirror.com
[4D_Cubing]" <4D_Cubing@yahoogroups.com>:

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:

I will define a mirror operation to be any linear transformation
that flips the sign of the orientation.
If even one mirror operation is permitted, then even asymmetric
objects become achiral.
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).
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