Message #3840

From: Joel Karlsson <>
Subject: Re: [MC4D] Chirality, orientation (~= handedness, inside-outness)
Date: Fri, 24 Nov 2017 21:04:51 +0100

I, again, agree with most of what you said but would like to point
something out. "Inside-outness" is not the same thing as the orientation of
the vector space. If an object is achiral it’s either possible to turn it
into its mirror image with rotations (which is connected to the symmetry I
talked about earlier and closely related to the orientation of different
bases in a vector space) or via nonlinear transformations (and one then has
to define what nonlinear transformations are allowed), such as turning the
object inside out.

Best regards,

2017-11-24 20:36 GMT+01:00 Marc Ringuette
[4D_Cubing] <>:

> 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