On 11/14/94 at 13:54:31 email@example.com said:
Despite having read all of the archives, I still don't know what an
antipode is. I suspect I'd have to know more about group theory, but can
you briefly describe what one is (you may want to CC the cube-lovers list
as well, in case more don't understand the term).
I guess the most limited definition is two points on the opposite sides
of a sphere, at the ends of a diameter -- e.g., the north pole and
the south pole. However, the definition need not be limited to
three dimensions (points on the opposite ends of a diameter of a
circle are sometimes referred to as antipodes, I think) nor to
circles and spheres (I have seen opposite corners of a square
referred to as antipodes).
Generalizing further, antipodes are "opposite" or "maximally distant"
points of any sort of structure, depending on what "opposite" or
"maximally distant" mean in the context at hand. With respect to
Rubik's cube, antipodes of Start are states which are maximally
distant from Start, and it is a matter of great interest what that
maximal distance might be.
I have to admit to a certain discomforture with one aspect of the way
we tend to refer to antipodes in the Rubik's cube. Most Rubik
structures that have been investigated do not have a single point which
is maximally distant from Start; rather, they have several or many
maximally distant points, and all the maximally distant points are
called antipodes. I would be more comfortable using "antipode" only
when the maximally distant point is unique. One example where the
maximally distant point is unique is the subgroup consisting of
edges only (no corners or centers) where only Q-turns are allowed.
In this case, the maximally distant point has been called the
"unique antipode". The description "unique antipode" seems
redundant somehow -- "antipode" ought to imply "unique", but that
has not been the custom on Cube-Lovers.
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