Thread: "Testing for end of major Yahoo Groups outage"

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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 form=
s 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 project=
ions (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 le=
fthanded 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 -----=20
From: Marc Ringuette ringuette@solarmirror.com [4D_Cubing]=20
To: 4D_Cubing@yahoogroups.com=20
Sent: Tuesday, November 21, 2017 7:04 PM
Subject: [MC4D] Yes, there is handedness in 4D, 5D, etc


=20=20=20=20
(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=20
claim that Ed quoted, that "handedness has no meaning in spaces with 4=20
dimensions or more", with the fact that I observe a handedness in MC4D=20
(we cannot create the left-right mirror image of the solved position via=
=20
any sequence of rotations; nor could I conceive of any non-stretching=20
rotations that MC4D could be lacking).

The resolution is simple: the quote is WRONG, completely wrong. There=20
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=20
negative determinant of the unique linear transformation between a pair=20
of ordered bases. In every dimension n, if we put distinct colors on=20
all 2n sides of an n-dimensional hypercube, the object can never be=20
rotated into its mirror image.

Now I will try to expunge that wrong idea from my head.

Hmph.

Marc



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