Thread: "Chirality, orientation (~= handedness, inside-outness)"

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



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




From: Joel Karlsson <joelkarlsson97@gmail.com>
Date: Fri, 24 Nov 2017 21:04:51 +0100
Subject: Re: [MC4D] Chirality, orientation (~= handedness, inside-outness)



--001a114d1d6ccd6d17055ec0104b
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

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,
Joel

2017-11-24 20:36 GMT+01:00 Marc Ringuette ringuette@solarmirror.com
[4D_Cubing] <4D_Cubing@yahoogroups.com>:

>
>
> I wonder if my example utilizing inside-outness was perhaps an unwise
> addition. I think there are two concepts, handedness (=3D chirality) and
> inside-outness (=3D 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.
>
> =3D=3D
>
> 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.
>
> =3D=3D
>
> 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.
>
> =3D=3D
>
> 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
>
>=20
>

--001a114d1d6ccd6d17055ec0104b
Content-Type: text/html; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

I, again, agree with most of what you said but w=
ould like to point something out. "Inside-outness" is not the sam=
e thing as the orientation of the vector space. If an object is achiral it&=
#39;s either possible to turn it into its mirror image with rotations (whic=
h is connected to the symmetry I talked about earlier and closely related t=
o the orientation of different bases in a vector space) or via nonlinear tr=
ansformations (and one then has to define what nonlinear transformations ar=
e allowed), such as turning the object inside out.

Best regard=
s,
Joel

l_quote">2017-11-24 20:36 GMT+01:00 Marc Ringuette tte@solarmirror.com">ringuette@solarmirror.com [4D_Cubing] "ltr"><4D=
_Cubing@yahoogroups.com
>
:
" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">












=20

=C2=A0







=20=20=20=20=20=20
=20=20=20=20=20=20

I wonder if my example utilizing inside-outness was perhaps an unw=
ise

addition.=C2=A0 I think there are two concepts, handedness (=3D chirality) =
and

inside-outness (=3D 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.=C2=A0 Since it came up, though, I'll try again=
.



=3D=3D



2D surface:=C2=A0 In Flatland, a solid 2D circle is either heads or tails, =


permanently.=C2=A0=C2=A0 One coin will have headness, and another will have=


tailness.=C2=A0=C2=A0 Orientation is fixed.=C2=A0=C2=A0 Objects can be chir=
al.



2D surface embedded in 3-space:=C2=A0=C2=A0 Take a solid 2D circle and give=
it a

small extent in the z-dimension (a coin).=C2=A0 In the 3rd dimension a coin=


can be flipped.=C2=A0 A coin and can have headness and tailness

simultaneously, and alternately display one side or the other when

placed flat on a table.=C2=A0=C2=A0 If we define a 2D tile puzzle in 3-spac=
e, we

have the option of allowing flips or not, depending on the rules of the >
puzzle.=C2=A0=C2=A0 A wooden tile in the shape of a human handprint, if unp=
ainted,

is achiral if it is allowed to be flipped.=C2=A0 If the tile is painted whi=
te

on one side and yellow on the other side, however, it is permanently

chiral even in 3D with flips allowed.



3D solid:=C2=A0 In 3D, an object can be either left-handed or right-handed,=


permanently.=C2=A0=C2=A0 Orientation is fixed.=C2=A0 Objects can be chiral.=




3D solid embedded in 4-space:=C2=A0=C2=A0 if we create a 3-dimensional obje=
ct 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 embedd=
ed

3-space, in which case it can have a fixed handedness (chirality), or we r>
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 th=
e

4-dimensional table.=C2=A0=C2=A0 Or we can construct our puzzle to disallow=
this,

either by constraining the available moves, or by coloring the +w and -w r>
sides of the "4-tile" differently.



=3D=3D



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



2D border of 3D solid:=C2=A0 In 3D, a solid cubical shell (say, a cardboard=


box that is all "taped up") has a distinct inside and outside.=C2=
=A0=C2=A0

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", adja=
cency

wise, but inside-out. =C2=A0 Or, if the box were made of flexible rubber, i=
t

could be inverted like a rubber glove if it has a hole or slice through >
which it can be inverted.=C2=A0=C2=A0 The insides and outsides have been r>
exchanged.=C2=A0=C2=A0 It is achiral, if the inside and outside of each fac=
e are

indistinguishable. =C2=A0 Chiral, if they are given different colors.



3D border of 4D solid:=C2=A0 In 4-space, let's take eight 3D cubes, giv=
e each

one some arbitrarily small extent in the w-dimension, and use 4D paint

to color each one the same color on both "sides". =C2=A0 Use 4D g=
lue to

fasten these thin 4D objects together into a hypercube version of a

cardboard box.=C2=A0=C2=A0=C2=A0 It is chiral and has fixed orientation.=C2=
=A0=C2=A0 It can hold

some 4-gas inside it that will not escape to infinity.=C2=A0=C2=A0 If we ar=
e

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). =C2=A0 Since it has only 8 colors, with both the +w a=
nd

-w sides of each cubical 4-tile the same color, these inside-out

operations make the 4-box achiral.=C2=A0=C2=A0 Or, we can make the inside a=
nd

outside surfaces distinguishable and it becomes chiral again.



=3D=3D



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.=C2=A0=C2=A0 The other element to the emu=
lation

involves "multiplexing" two dimensions along one axis, left-right=
and

in-out.=C2=A0 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






=20=20=20=20=20

=20=20=20=20







=20=20









--001a114d1d6ccd6d17055ec0104b--




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



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.

3. 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




From: Marc Ringuette <ringuette@solarmirror.com>
Date: Sun, 26 Nov 2017 18:35:14 -0600
Subject: Re: [MC4D] Chirality, orientation (~= handedness, inside-outness)




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



--001a114d1d6c1e4a64055ef83e61
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

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:

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.

3. 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



I

--001a114d1d6c1e4a64055ef83e61
Content-Type: text/html; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

Hi Marc,

">1) That's a good definition (and is what is usually used in linear al=
gebra).

2)=C2=A0 This mi=
ght depend on the precise definition of chirality. We could say that an obj=
ect is chiral with respect to a set of transformations if it can be disting=
uished from its mirror image under those transformations combined (set unio=
n) with ordinary rotations. Usually, when talking about chirality, only rot=
ations are allowed (I think), meaning that we are talking about chirality w=
ith 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.=C2=A0
">
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)=C2=
=A0 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 ta=
kes the vectors of a basis. Consider an n-dimensional cube in an n-dimensio=
nal space. Let's (arbitrarily) choose a corner. For every face that the=
corner is a part of (the ones "touching" the corner) imagine a u=
nit vector pointing out of that face (origin in the center of the cube). Th=
ese vectors form an orthogonal normalized basis. Now, let's turn the cu=
be 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 tr=
ansformation if n is even. However, since the inside out turning is definit=
ely not just a rotation or a mirror transformation (these do not change whi=
ch side of a face is pointing out of the cube) it cannot be a linear transf=
ormation.=C2=A0

Best reg=
ards,=C2=A0
Joel

>
Den 24 nov. 2017 10:04 em skrev "Marc =
Ringuette ri=
nguette@solarmirror.com
[4D_Cubing]" <ng@yahoogroups.com" target=3D"_blank">4D_Cubing@yahoogroups.com>: type=3D"attribution">
le=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">












=20

=C2=A0







=20=20=20=20=20=20
=20=20=20=20=20=20

Hi, Joel,



Thanks, yeah, trying to equate orientation and inside-outness was a poor r>
idea on my part.=C2=A0=C2=A0 I'm trying to properly define the inside-o=
ut

operation. =C2=A0 How about this instead:



1.=C2=A0 I will define a mirror operation to be any linear transformation <=
br>
that flips the sign of the orientation.



2.=C2=A0 If even one mirror operation is permitted, then even asymmetric r>
objects become achiral.



3. 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.=C2=A0 The inside-out operation results in a mirror operation (a linear <=
br>
tranformation that flips orientation), but cannot be broken down into a >
sequence of infinitesimal linear transformations.=C2=A0 We could stretch th=
e

object and squeeze it through the puncture nonlinearly (think rubber

balloon).=C2=A0=C2=A0 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






=20=20=20=20=20

=20=20=20=20







=20=20






I=C2=A0


--001a114d1d6c1e4a64055ef83e61--





Return to MagicCube4D main page
Return to the Superliminal home page