Message #3834

From: Eduard Baumann <ed.baumann@bluewin.ch>
Subject: Re: [MC4D] Yes, there is handedness in 4D, 5D, etc
Date: Fri, 24 Nov 2017 14:41:23 +0100

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]
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