# 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