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The Idea of Holonomy by Robert Bryant

from the MAA:

“Can I roll the ball from any point to any other point and have it wind up in a given orientation that we want?” Bryant asked.

If I draw a dot with this marker, can you eventually roll the ball enough times so that the dot would touch down anywhere on the table, anywhere at all? Or is the logic of the situation constrained, so that certain spots on the ball pair with certain spots on the table? The answer, he said, has consequences for fields from robotics to control theory.

To me, the idea of constrained motion sounds more like the fundamental economic dilemma.

  • You can’t live in as nice of a house as you want and work as little as you want and have all the other stuff you want.
  • Even if you had $100,000,000, you still couldn’t spend the weekend fishing in Chile and attend the Davos seminar and go to your son’s art exhibition.
  • There’s a direct tradeoff between how long you work on building the perfect product (say, a game console) and how soon it will be released. You might be able to achieve a little more of both by investing more money into the project … but that comes at the expense of something else.

The “optimal path" — if such a thing even exists — will never be feasible, because the choice space is fundamentally characterised by tradeoffs.

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