Message #3839

From: Marc Ringuette <ringuette@solarmirror.com>
Subject: Chirality, orientation (~= handedness, inside-outness)
Date: Fri, 24 Nov 2017 11:36:10 -0800

I wonder if my example utilizing inside-outness was perhaps an unwise
addition.  I think there are two concepts, handedness (= chirality) and
inside-outness (= vector space orientation), that I failed to
distinguish clearly.

I tried to write some of this down last week, but got tangled up and
didn’t post it then.  Since it came up, though, I’ll try again.

==

2D surface:  In Flatland, a solid 2D circle is either heads or tails,
permanently.   One coin will have headness, and another will have
tailness.   Orientation is fixed.   Objects can be chiral.

2D surface embedded in 3-space:   Take a solid 2D circle and give it a
small extent in the z-dimension (a coin).  In the 3rd dimension a coin
can be flipped.  A coin and can have headness and tailness
simultaneously, and alternately display one side or the other when
placed flat on a table.   If we define a 2D tile puzzle in 3-space, we
have the option of allowing flips or not, depending on the rules of the
puzzle.   A wooden tile in the shape of a human handprint, if unpainted,
is achiral if it is allowed to be flipped.  If the tile is painted white
on one side and yellow on the other side, however, it is permanently
chiral even in 3D with flips allowed.

3D solid:  In 3D, an object can be either left-handed or right-handed,
permanently.   Orientation is fixed.  Objects can be chiral.

3D solid embedded in 4-space:   if we create a 3-dimensional object and
then give it an arbitrarily small extent in the w-dimension, we can
choose to "slide it around on a 4-dimensional table" in an embedded
3-space, in which case it can have a fixed handedness (chirality), or we
can "pick it up off the table" and, without bending the solid 4D object,
"flip" it into an opposite-handed version and place it back on the
4-dimensional table.   Or we can construct our puzzle to disallow this,
either by constraining the available moves, or by coloring the +w and -w
sides of the "4-tile" differently.

==

Now, I’ll add inside-outness to the mix.

2D border of 3D solid:  In 3D, a solid cubical shell (say, a cardboard
box that is all "taped up") has a distinct inside and outside.  
However, if I make a few slices along the edges, I can turn the
cardboard box inside-out and re-tape it into the "same box", adjacency
wise, but inside-out.   Or, if the box were made of flexible rubber, it
could be inverted like a rubber glove if it has a hole or slice through
which it can be inverted.   The insides and outsides have been
exchanged.   It is achiral, if the inside and outside of each face are
indistinguishable.   Chiral, if they are given different colors.

3D border of 4D solid:  In 4-space, let’s take eight 3D cubes, give each
one some arbitrarily small extent in the w-dimension, and use 4D paint
to color each one the same color on both "sides".   Use 4D glue to
fasten these thin 4D objects together into a hypercube version of a
cardboard box.    It is chiral and has fixed orientation.   It can hold
some 4-gas inside it that will not escape to infinity.   If we are
allowed to slice the box with our 4D knife, turn it inside-out, and
re-glue it, we can flip its orientation (while unfortunately allowing
the 4-gas to escape).   Since it has only 8 colors, with both the +w and
-w sides of each cubical 4-tile the same color, these inside-out
operations make the 4-box achiral.   Or, we can make the inside and
outside surfaces distinguishable and it becomes chiral again.

==

One reason why I’m so interested in inside-outness is that I think
Melinda’s 2^4 puzzle uses inside-outness as one of the ways to emulate
the 2^4 hypercube in 3 dimensions.   The other element to the emulation
involves "multiplexing" two dimensions along one axis, left-right and
in-out.  I’m only part of the way along to fully understanding the
tricks involved and seeing to what extent they can be generalized to
more slices or more dimensions.

Cheers
Marc