Message #3836

From: Marc Ringuette <ringuette@solarmirror.com>
Subject: Re: [MC4D] Yes, there is handedness in 4D, 5D, etc
Date: Fri, 24 Nov 2017 10:16:45 -0800

(While I was finishing this message, Joel made a correct, and
math-laden, reply.  Hi Joel!  This is my own reply to Ed, taking a
different more intuitive tack.)

Hi, Ed,

> Handedness of 2^4 appears only after the 4D/3D projection

No, sorry, this is wrong.

This is a really interesting and worthwhile topic to me.   I think
you’re missing something important.  I’m not sure I’ve got it 100%
straight either – perhaps some other people on the MC4D list can help
me not to screw it up.

I will start by spending a lot of time talking about 2D faces embedded
in 3D.

==

> A flat L-shape is achiral in 3D.

No, not always.   Only if its top and bottom cannot be distinguished.  
If the L-shape is colored blue on top and green on the bottom, then it
is chiral in 3D also.   It is permanently a blue L, and the backwards L
is permanently green.

This is not a sneaky trick.  Sneaky would have been making the L shape
indistinguishable on the top and the bottom; we typically want to color
all of the sides of our puzzles differently, not keep pairs of them
indistinguishable.

Another way to tell the sides of the L apart would be to glue the bottom
side onto our puzzle.   The physical constraints of a puzzle can provide
the distinction, so we don’t need an extra color to tell the inside from
the outside.

==

I will briefly mention a potentially important detail, and then I will
try to ignore it again:  a completely flat 2D object can’t be
manipulated in 3D because it does not exist.  There are no atoms in it.
  It is just an idea.   An "image".  We can PRETEND that a thin sheet
of rubber, or a piece of paper, or a wood tile, are 2D objects, but they
are not really.   In fact, all three of these versions have different 3D
properties.   The rubber can be both bent and stretched; the paper can
be bent (folded) but not stretched; the tile can be neither bent nor
stretched.   Most of the time we’re happy to pretend that a thin piece
of paper has zero thickness and forget about it, but sometimes we might
need to think about it explicitly, in order to make sure we’re not
making weird assumptions without noticing them.   For instance, when
making claims like "in 3D, a 2D object can be distorted until any two
points touch each other."   Well, only if, when we distort it, we follow
the rules of a 2D surface painted onto a piece of paper or rubber, and
not the rules of a 2D surface painted onto a wood tile.

Now I’ll go back to ignoring the thickness of the paper.

==

Here is an extended example, in 3 dimensions.   (If you wish, you may
substitute "is chiral" whenever I say "has a handedness", and "is
achiral" for "has no handedness".  I’ll use them interchangeably,
treating handedness as a synonym for chirality, because I usually prefer
the more familiar plain-English words.)

Consider a solid stiff wood block painted like a Rubik’s Cube. Each of
the six 2D faces has a distinct color painted on it.   Each face has an
outside (that we can see) and an inside (facing into the wood), that are
different and cannot be exchanged.  It has a handedness (white-red-green
clockwise).

Consider six flat flexible square sheets of rubber, of six different
colors (same color on both sides) glued together along their edges into
a cube without any holes or slices in it.   Fill the cube with a little
bit of colored gas, to remind us that it has an inside and an outside.  
Each face has an outside and an inside, that are different and cannot be
exchanged.  It has a handedness.

Now take the 6-color rubber-sided cube with colored gas in it, and cut a
slice in one side.  The gas rushes out through the slice. Now, because
the rubber is flexible and stretchy, we can invert the cube through the
slice, turning it inside out.   Each face no longer has a fixed outside
and inside.   The object no longer has a handedness.

Now, make the rubber-sided cube 12-colored, so we can distinguish the
outside from the inside again.   Say, we put black polka-dots all over
the exterior of the cube.  Now, if we turn the cube inside-out, we can
tell the difference.   It has a handedness again:  if the cube shows
"white-red-green clockwise with dots" and "white-red-green anticlockwise
without dots", then it can never be transformed into its mirror image
that shows "white-red-green clockwise without dots".

==

So, is a 3D cube, made from 2D colored faces, chiral or achiral? Chiral,
if it has distinguishable sides and a permanent outside or inside,
achiral if not.   Chiral if it is stiff, or if it can’t be turned
inside-out, or is 12-colored so we can tell when it is inside-out;
achiral if it is flexible, invertible, and 6-colored with the inside and
outside of each face the same color.

So, is it a flaw that the Rubik’s Cube is chiral?  Would the achiral
Rubik’s Cube, with indistinguishable outside and inside, be a better
puzzle?   Perhaps we should make flexible rubber Rubik’s Cubes and see
if we like them better when we can turn them inside-out.   Or not.

==

Getting back to 4D, it seems to me that the MC4D hypercube puzzles, like
the real 3D Rubik’s Cube puzzles, keep the inside and outside
separate.   There is no mirror operation that flips orientation (in the
sense of https://en.wikipedia.org/wiki/Orientation_(vector_space) ); nor
are there puzzle moves that mirror-flip a slice of the puzzle.   This is
not a byproduct of how we view the puzzle in MC4D, but rather is a
property of the 4D puzzle itself.

Cheers
Marc