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Date: Fri, 17 Dec 1993 14:22:34 EST
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: Re: Size of the Cube Group
In-Reply-To: Message of 12/17/93 at 11:21:51 from ,
BRYAN%WVNVM.BITNET@mitvma.mit.edu
On 12/17/93 at 11:21:51 Jerry Bryan said:
>However, the alert reader should have noticed a problem. Why did I
>not divide by 2 to take into account the fact that odd edge
>permutations can only occur with odd corner permutations and vice
>versa? Actually, I did, but the division by 2 cancelled. The reason
>it canceled is slightly tricky. Also, remember that we are talking
>about equivalence classes, not specific cube configurations. Any
>equivalence class has both even and odd members, depending on how
^^^^^^^^^^^^^^^^^^^^^^^^^
>the members are rotated. Hence, any corner equivalence class can be
>matched up with any edge equivalence class, assuming the rotations
>are compatible. But you still have to worry about "dividing by 2",
>as follows.
It is pretty bad when you have to followup with
corrections to your own posts. I hurried to complete the previous
post before lunch, and just didn't think clearly enough -- till I
had time to think *during* lunch. Let's try this again.
A qturn of the whole cube (a 90 degree rotation of the whole cube)
is odd. However, if you think of a qturn rotation of the whole cube
as disjoint between edges and corners, a qturn rotation of the
corners is even, and a qturn rotation of the edges is odd. Hence,
for any equivalence class of the corners under M, either the whole
equivalence class is even, or the whole equivalence class is odd.
For any equivalence class of the edges under M, half of the equivalence
class is even and half is odd. Thus, any equivalence class of the
corners can occur with any equivalence class of the edges, but with only
half the members of the edge equivalence class -- namely those with
the same parity.
I believe my calculations were correct, but a piece of the justification
was not. I hope I am not still missing something. You do have to
"divide by 2", and my calculations do indeed "divide by 2" as previously
described, but the parity of edges vs. the parity of corners was
incorrect in the previous post.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan) (304) 293-5192
Associate Director, WVNET (304) 293-5540 fax
837 Chestnut Ridge Road BRYAN@WVNVM
Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU
If you don't have time to do it right today, what makes you think you are
going to have time to do it over again tomorrow?