On Thu, 3 Mar 2022, Elijah Stone wrote:
Is there a well-defined notion of distance defined in higher-order
'space'? If so, is it still scalar?
A thought:
In ordinary cartesian space, the components of a given coordinate are
completely independent of one another. When measuring the distance
between points A and B, we care if A's X-coordinate is similar to B's
X-coordinate, and we care if A's Y-coordinate is similar to B's
Y-coordinate, but we do not care at all if A's X-coordinate is similar to
B's Y-coordinate.
Now consider a 'space' of order 4 whose 'rank' is 2 2. Number its 'axes'
from 0 below 4, as in i.2 2:
0 1
2 3
Now, 'axis' 0 seems 'close' to 'axes' 1 and 2, and 'far' from axis 3.
Perhaps that means that there is an interesting distance metric for
'coordinates' A and B in this 'space' such that we care if A's 0-component
is similar to B's 1-component, and we care less if A's 0-component is
similar to B's 3-component.
I am reaching, but it seems plausible. A distance metric defined
according to this principle should probably work on ordinary vector
coordinates, and give the same results as the traditional distance metric.
-E
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
