Message #3835
From: Joel Karlsson <joelkarlsson97@gmail.com>
Subject: Re: [MC4D] Yes, there is handedness in 4D, 5D, etc
Date: Fri, 24 Nov 2017 18:57:21 +0100
What Ed is saying about an L-shape being chiral in 2D but achiral in 3D is
definitely correct. However, what Marc is saying about orientation is also
correct; a rotation matrix always has determinant 1 (and an n-2 dimensional
eigenspace associated with the eigenvalue 1, what we rotate "around")
whereas a matrix describing mirroring always has determinant -1 (and an n-1
dimensional eigenspace associated with the eigenvalue 1, what we mirror
"in"). Note that the inverse is also true: all matrices with determinant 1
is a rotation matrix and all matrices with determinant -1 is a mirror
matrix.
An object in an n-dimensional space is achiral if it cannot be
distinguished from its mirror image (per definition). Therefore there exist
a rotation matrix and a mirror matrix that (when the corresponding
transformation is performed on the object) produce the same result.
Combining this mirror transformation with the inverse of the rotation
(which also is a rotation) we get a new transformation that doesn’t affect
the object and has determinant (-1)*1 and, thus, is a mirror
transformation. So, the object is achiral if and only if there exists a
mirror transformation that doesn’t affect it (meaning that the object is
symmetrical with respect to an n-1 dimensional subspace corresponding to
the eigenspace of the transformation).
So, regarding the handedness of an n-dimensional "magic cube", it comes
down to the question "is the object symmetrical with respect to an n-1
dimensional subspace?". A cube (without coloured faces) is definitely
achiral but I don’t believe that a coloured cube is (assuming that all
colours are distinct) since this breaks the mirror symmetry. If the
coloured cube is achiral there must exist a rotation that only swaps places
of two opposite colours. However, since all colours are distinct that means
that the rotation can affect no other colour. This implies that there is an
n-1 dimensional eigenspace associated with the eigenvalue 1 which
contradicts the assumption (since a rotation has an n-2 dimensional
eigenspace associated with the eigenvalue 1) and, hence, a cube coloured
with distinct colours is chiral.
tl;dr "magic cubes" are chiral and we can indeed talk about handedness.
Best regards,
Joel
Den 24 nov. 2017 2:41 em skrev "‘Eduard Baumann’ ed.baumann@bluewin.ch
[4D_Cubing]" <4D_Cubing@yahoogroups.com>:
Take the 3D/2D analogon.
A flat L-shape is *achiral* in 3D but *chiral* in 2D.
You can chose 2 different projections from 3D to 2D to produce the two
forms of the chiral L in 2D.
The flat 2D L-shape *can be turned in 3D* to become a mirrored example.
Same in 4D/3D
The *achiral* tesseract in 4D can be projected to 3D by two different
projections (operations) to produce a right handed or a left handed 3D
projection (result).
The righthanded 3D projection (result) of 2^4 *can be turned in 4D* to the
lefthanded 3D projection (result) of 2^4.
Handedness of 2^4 appears only after the 4D/3D projection and you have two
possible projections.
MCD4 has chosen one specific projection. The physical 2^4 should chose the
same projection for practical reasons.
Hmph.
Best regards
Ed
—– Original Message —–
*From:* Marc Ringuette ringuette@solarmirror.com [4D_Cubing]
<ringuette@solarmirror.com+[4D_Cubing]>
*To:* 4D_Cubing@yahoogroups.com
*Sent:* Tuesday, November 21, 2017 7:04 PM
*Subject:* [MC4D] Yes, there is handedness in 4D, 5D, etc
(I’m re-sending this after 24 hours of not seeing it show up on the list)
Don’t believe everything you read in a book.
I spent a long time yesterday trying to figure out how to reconcile the
claim that Ed quoted, that "handedness has no meaning in spaces with 4
dimensions or more", with the fact that I observe a handedness in MC4D
(we cannot create the left-right mirror image of the solved position via
any sequence of rotations; nor could I conceive of any non-stretching
rotations that MC4D could be lacking).
The resolution is simple: the quote is WRONG, completely wrong. There
is handedness in n-space for every n, called "orientation".
https://en.wikipedia.org/wiki/Orientation_(vector_space)
There are always two orientations, corresponding to a positive and
negative determinant of the unique linear transformation between a pair
of ordered bases. In every dimension n, if we put distinct colors on
all 2n sides of an n-dimensional hypercube, the object can never be
rotated into its mirror image.
Now I will try to expunge that wrong idea from my head.
Hmph.
Marc