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TimSweeneyEpic's profile
Tim Sweeney
Tim Sweeney
Tim Sweeney
@TimSweeneyEpic

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Tim Sweeney

@TimSweeneyEpic

Epic Games founder & CEO

epicgames.com
Joined August 2013

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    Tim Sweeney‏ @TimSweeneyEpic 3 Nov 2018
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    Can one integrate over the power set of the continuum? Is there a way to answer questions like: What is the average perimeter of all closed shapes with continuous boundaries and unit area?

    11:05 AM - 3 Nov 2018
    • 4 Retweets
    • 34 Likes
    • The alternator its me danilo Major League Polo Happy Frog Games Merritt Sergei aqwx Little sis
    17 replies 4 retweets 34 likes
      1. John Carlos Baez‏ @johncarlosbaez 3 Nov 2018
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        Replying to @TimSweeneyEpic

        The set of closed shapes you mention only has the cardinality of the continuum, not the power set of the continuum. It takes some work to prove this, but for starters note a continuous function defined on the reals is determined by its values on the rationals.

        0 replies 0 retweets 16 likes
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      2. D ␣ a ␣ n ␣ P ␣ i ␣ p ␣ o ␣ n ␣ i‏ @sigfpe 3 Nov 2018
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        Replying to @TimSweeneyEpic

        This is very close to the kind of path integrals that come up in physics. Eg. In string theory you might integrate over all 2d surfaces with a prescribed boundary. A few complications do come up :)

        1 reply 0 retweets 6 likes
      3. Jacques Carette‏ @jjcarett2 3 Nov 2018
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        Replying to @sigfpe @TimSweeneyEpic

        Or ray tracing. That would be the Veach-Guibas "Metropolis Light Transport" algorithm.

        0 replies 0 retweets 1 like
      4. End of conversation
      1. Four‏ @0diosis 3 Nov 2018
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        Replying to @TimSweeneyEpic

        I dont think I understand, but if your set allows for infinite possibilities, you could have a unit-area rectangle with w=∞ and h=1/∞. Wouldn't that obscure your perimeter average (or any question)?

        0 replies 0 retweets 3 likes
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      1. Merritt‏ @merritth_ 3 Nov 2018
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        Replying to @TimSweeneyEpic

        @kaylacostalot

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      1. Bob Burrough‏Verified account @bob_burrough 3 Nov 2018
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        Replying to @TimSweeneyEpic

        Doesn’t it go to infinity? e.g. rectangle with unit area where one axis goes to zero.

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      1. Dylan Kelly‏ @SnyggLich 3 Nov 2018
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        Replying to @TimSweeneyEpic

        I believe that is the definition of one "Absolute Unit"

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      2. Won Chun‏ @won3d 3 Nov 2018
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        Replying to @TimSweeneyEpic

        Lesbesgue measure?

        1 reply 0 retweets 0 likes
      3. Won Chun‏ @won3d 3 Nov 2018
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        Replying to @won3d @TimSweeneyEpic

        *Lebesgue

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      2. Michael Hale‏ @wakebrdkid 3 Nov 2018
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        Replying to @TimSweeneyEpic

        I know that the "calculus of variations" is used to prove circle has max area for a perimeter, shape of curve of fastest descent, etc. But I've never solved a problem myself with those techniques myself or know if they are developed enough to handle what you want.

        1 reply 0 retweets 0 likes
      3. Michael Hale‏ @wakebrdkid 3 Nov 2018
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        Replying to @wakebrdkid @TimSweeneyEpic

        But in that case they take the familiar gradient, generalizing to infinite dimensions (each point on the curve) to define an "uphill" direction to any curve's area. Then usual solve the derivative for 0 to get the max shape without having a distribution to enumerate all curves.

        0 replies 0 retweets 0 likes
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