1. Joined
    04 May '05
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    2621
    22 Sep '10 18:25
    Hi,

    What does the pieces look like, if you want to divide the surface area of a sphere into equal shapes that score high on symmetry?

    For two pieces - you slice it down the middle.

    But what about three and more pieces?

    What is the procedure for doing this? Any good links to graphical representations?


    regards, Paul
  2. Standard memberPBE6
    Bananarama
    False berry
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    22 Sep '10 19:07
    Originally posted by pulern
    Hi,

    What does the pieces look like, if you want to divide the surface area of a sphere into equal shapes that score high on symmetry?

    For two pieces - you slice it down the middle.

    But what about three and more pieces?

    What is the procedure for doing this? Any good links to graphical representations?


    regards, Paul
    Couldn't you just slice it into pieces like an orange?
  3. Germany
    Joined
    27 Oct '08
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    3081
    22 Sep '10 19:28
    How do you define your "symmetry score"?
  4. Subscribersonhouse
    Fast and Curious
    slatington, pa, usa
    Joined
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    22 Sep '10 20:33
    Originally posted by pulern
    Hi,

    What does the pieces look like, if you want to divide the surface area of a sphere into equal shapes that score high on symmetry?

    For two pieces - you slice it down the middle.

    But what about three and more pieces?

    What is the procedure for doing this? Any good links to graphical representations?


    regards, Paul
    Just like slicing an orange, your first example was a cut bisecting the sphere, 180 degrees apart. Trisecting, 120 degree cuts, 4, 90 degree cuts, 5 cuts, 72 degrees apart, 6 cuts, 60 degrees apart, 7 cuts, about 51.4 degrees apart, etc. Just divide 360 degrees by the # of cuts. Simple arithmetic.
  5. Joined
    04 May '05
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    2621
    22 Sep '10 22:14
    Thanks for the response!

    yes, sorry - I was a bit unclear.

    If you exclude the orange solution (i.e. no piece can include a part of both poles), are there other solutions?

    by symmetry I just wanted to minimize the opportunity to derive several solutions to cutting the orange in half. - say a curved cut.

    I realize that you can still cut the orange in half "vertically", then slice "horizontally" and work your way to all possible even number pieces... but what about odd and prime number patches?

    hope that cleared it up a bit?
  6. Standard memberflexmore
    Quack Quack Quack !
    Chesstralia
    Joined
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    23 Sep '10 10:168 edits
    Originally posted by sonhouse
    Just like slicing an orange, your first example was a cut bisecting the sphere, 180 degrees apart. Trisecting, 120 degree cuts, 4, 90 degree cuts, 5 cuts, 72 degrees apart, 6 cuts, 60 degrees apart, 7 cuts, about 51.4 degrees apart, etc. Just divide 360 degrees by the # of cuts. Simple arithmetic.
    pulern, you seem to have a good instinct, i can feel it but not yet calculate it ... what is it exactly that you look for?
    odd and prime is not enough ... they can be very orangeable as sonhouse showed:

    2 pieces ... prime
    3 pieces ... odd and prime
    4 pieces
    5 pieces , odd and prime
    6 pieces
    7 pieces odd and prime

    . Just divide 360 degrees by the # of pieces. Simple arithmetic
  7. Joined
    04 May '05
    Moves
    2621
    23 Sep '10 11:07
    Originally posted by flexmore
    ... what is it exactly that you look for?
    Thanks again for trying to answer my not so clear question guys.

    What is that i'm looking for?

    Let's say you wanted to devide the surface area of a sphere in 41 pieces.

    It's easy to calculate the actual area once you have the radius.

    But if all of these pieces needed to be identical (but you are not allowed to just slice 41 orange-boats), what would the patch look like? Is there a solution?

    Looking for small objects, that if you had X number of such pieces, you could add them together to get a sphere. How would the shape of these objects change as X grows. I particularly want these pieces to have the highest Area to Circumference proportinality possible.

    Why? I just bought this globe, and you know how a globe is devided in longitudes and latitudes... well I just can't get it out of my mind, what solutions would be available if those resulting patches had to be identical.

    Sorry for waisting your time guys, but hope that some of you might now be able to read my mind, despite my lack of ability to formalize my queery properly.


    Regards, Paul
  8. Cape Town
    Joined
    14 Apr '05
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    52945
    23 Sep '10 11:28
    I know you want an solution for arbitrary numbers, but for a specific set of numbers the best solution is to put a regular polyhedron inside the sphere then project its vertices onto the surface (a light place in the center might help)

    I suspect that for any number not in the regular polyhedra set, you might find that the orange slice solution is the best one.
  9. Standard memberwolfgang59
    invigorated
    Dunedin
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    23 Sep '10 23:33
    The solutions for fullerenes (bucky balls) might push you in the right direction?????
  10. Subscriberjoe shmo
    Strange Egg
    podunk, PA
    Joined
    10 Dec '06
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    7733
    24 Sep '10 04:34
    Originally posted by pulern
    Thanks again for trying to answer my not so clear question guys.

    What is that i'm looking for?

    Let's say you wanted to devide the surface area of a sphere in 41 pieces.

    It's easy to calculate the actual area once you have the radius.

    But if all of these pieces needed to be identical (but you are not allowed to just slice 41 orange-boats), what wou ...[text shortened]... read my mind, despite my lack of ability to formalize my queery properly.


    Regards, Paul
    could it be your looking for geodisic domes?
  11. Standard memberPalynka
    Upward Spiral
    Halfway
    Joined
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    24 Sep '10 09:565 edits
    Originally posted by pulern
    Thanks for the response!

    yes, sorry - I was a bit unclear.

    If you exclude the orange solution (i.e. no piece can include a part of both poles), are there other solutions?

    by symmetry I just wanted to minimize the opportunity to derive several solutions to cutting the orange in half. - say a curved cut.

    I realize that you can still cut the orange i er pieces... but what about odd and prime number patches?

    hope that cleared it up a bit?
    For any number, just forget about the horizontal cuts and cut along meridians. Divide the equator in X equal parts and cut from North pole to South pole passing through each point (along the relevant) semisphere.

    I couldn't find any graphical representation, but simply look at this:
    http://www.gnu.org/software/3dldf/graphics/sphere01.png
    and remove the horizontal lines.#

    Edit - Oops, too late. I didn't read the other posts properly.
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