+&.% is the operation for combining parallel electrical resistances,
another example that immediately demonstrates the associativity.
(Come to think of it, if you calculate the resistance using the method
of tracing squares, resistance and aspect ratio are entirely dual concepts)
Henry Rich
On 9/26/2012 6:59 AM, Boyko Bantchev wrote:
Here is a nice and very simple example of 'under' I don't recall
being mentioned in the docs or in the forums.
Place two rectangles one beside the other, stretching them as needed
but preserving their aspect ratios, so that the left side of the one
coincides with the right side of the other. If the aspect ratios
(w:h) of the given rectangles are a and b, that of the resulting
rectangle is a +&.% b .
The above operation being both commutative and associative, an
obvious generalization suggests itself as the verb +/&.:% .
(Note that &.: is needed in place of &. .)
Notes.
1. It is interesting to observe that the associativity of the
arithmetical operation on ratios is not immediately obvious, while
for some reason its geometric equivalent seems self-evident.
2. The said operation on rectangles underlies one of the operations
on pictures in the so called functional geometry, invented by
P.Henderson and implemented in, e.g., the GeomLab programming system.
3. jwiki/Essays/Under has no example of &.: so far, and does not
even mention it.
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