Quick update: I suspect many/most problems where thinking in terms of symmetry can be more helpfwly reframed in terms of isthmuses[1]. Here’s the chain-of-thought I was writing which caused me to think this:
(Background: I was trying to explain the general relevance of symmetry when finding integrals.)
In the context of finding integrals for geometric objects¹, look for simple subregions² for which manipulating a single variable³ lets you continuously expand to the whole object.⁴
The general feature to learn to notice as you search through subregions here is: shared symmetriesfor the object and its subregion. hmmmmm
Actually, “symmetry” is a distracting concept here. It’s the “isthmus” between subregions you should be looking for.
WHEN: Trying to find an integral
THEN: Search for a single isthmus-variable connecting subregions which together fill the whole area
FINALLY: Integrate over that variable between those regions.
or said differently… THEN: Look for simple subregions which transform into the whole area via a single variable, then integrate over that variable.
Hm. This btw is in general how you find generalizations. Start from one concept, find a cheap action which transforms it into a different concept, then define the second in terms of the first plus its distance along that action.
That action is then the isthmus that connects the concepts.
If previously from a given context (assuming partial memory-addresses A and B), fetching A* and B* each cost you 1000 search-points separately, now you can be more efficient by storing B as the delta between them, such that fetching B only costs 1000+[cost of delta].
Or you can do a similar (but more traditional) analysis where “storing” memories has a cost in bits of memory capacity.
This example is from a 3B1B vid, where he says “this should seem promising because it respects the symmetry of the circle”. While true (eg, rotational symmetry is preserved in the carve-up), I don’t feel like the sentence captures the essence of what makes this a good step to take, at least not on my semantics.
Quick update: I suspect many/most problems where thinking in terms of symmetry can be more helpfwly reframed in terms of isthmuses[1]. Here’s the chain-of-thought I was writing which caused me to think this:
(Background: I was trying to explain the general relevance of symmetry when finding integrals.)
“An isthmus is a narrow piece of land connecting two larger areas across an expanse of water by which they are otherwise separated.”
This example is from a 3B1B vid, where he says “this should seem promising because it respects the symmetry of the circle”. While true (eg, rotational symmetry is preserved in the carve-up), I don’t feel like the sentence captures the essence of what makes this a good step to take, at least not on my semantics.