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Brunnian 5 links

Posted 9 years ago
POSTED BY: ali cherad
11 Replies
Posted 9 years ago
POSTED BY: Updating Name

I do not entirely get the question either. Do you mean 5 links or 5 intersections of a single link - because that is a thing too:

KnotData[{5, 1}, "Image"]

enter image description here

POSTED BY: Sam Carrettie
Posted 9 years ago

Hi Krause .Yes that is (Mandala) what I am using (I googled it) but my question is how to get Mathematica to draw the links so that I am able to change some parameters and alter the shape

POSTED BY: ali cherad

Do a redraw of a redraw of a redraw

(* Take 5 from http://katlas.org/wiki/File:6Loops-Brunnian-link.png *) 
Graphics3D[{{Yellow, Specularity[White, 70], 
   Tube[{{0, 12, 0}, {23, 12, 0}, {23, -12, 0}, {0, -12, 0}, {0, 12, 
      0}}, 1/3]},
  {Green, Specularity[White, 70], 
   Tube[{{1, 9, 0}, {22, 9, 0}, {22, -9, 0}, {1, -9, 0}, {1, 9, 0}}, 
    1/3]},
  {Blue, Specularity[White, 70], 
   Tube[{{2, 6, 0}, {21, 6, 0}, {21, -6, 0}, {2, -6, 0}, {2, 6, 0}}, 
    1/3]},
  {Cyan, Specularity[White, 70], 
   Tube[{{3, 3, 0}, {20, 3, 0}, {20, -3, 0}, {3, -3, 0}, {3, 3, 0}}, 
    1/3]},
  {Red, Specularity[White, 40], 
   Tube[BSplineCurve[{{23/2, 2, 0}, {19, 2, 0}, (* 
      über 1 *) {19, -3, 2},
      (* über 2, 3, 
      4 *) {19, -6, 2}, {19, -9, 2}, {19, -12, 2}, {19, -15, 0},
      (* unter 4, über 3 *) {18, -15, 0}, {18, -12, -2}, {18, -9, 2},
      (* unter 3, 
      unter 4 *) {18, -15/2, 0}, {17, -15/2, 
       0}, {17, -9, -2}, {17, -12, -2}, {17, -15, 0},
      (* über 4, unter 3, 
      über 2 *) {16, -15, 0}, {16, -12, 2}, {16, -9, -2}, {16, -6, 
       2}, {16, -9/2, 0}, {15, -9/2, 0},
      (* unter 2, unter 3, 
      über 4 *) {15, -6, -2}, {15, -9, -2}, {15, -12, 2}, {15, -15, 
       0}, {14, -15, 0},
      (* unter 4, 
      unter 3 *) {14, -12, -2}, {14, -9, -2}, {14, -15/2, 
       0}, {13, -15/2, 0},
      (* über 3, 
      unter 4 *) {13, -9, 2}, {13, -12, -2}, {13, -15, 0}, {12, -15, 
       0},
      (* über 4, über 3, unter 2, 
      über 1 *) {12, -12, 2}, {12, -9, 2}, {12, -6, -2}, {12, -3, 
       2}, {12, -3/2, 0}, {11, -3/2, 0},
      (* unter 1, unter 2, über 3, 
      über 4 *) {11, -3, -2}, {11, -6, -2}, {11, -9, 2}, {11, -12, 
       2}, {11, -15, 0}, {10, -15, 0},
      (* unter 4, 
      über 3 *){10, -12, -2}, {10, -9, 2}, {10, -15/2, 0}, {9, -15/2, 
       0},
      (* unter 3, 
      unter 4 *) {9, -9, -2}, {9, -12, -2}, {9, -15, 0}, {8, -15, 0},
      (* über 4, unter 3, 
      unter 2 *){8, -12, 2}, {8, -9, -2}, {8, -6, -2}, {8, -9/2, 
       0}, {7, -9/2, 0},
      (* über 2,unter 3, 
      über 4 *) {7, -6, 2}, {7, -9, -2}, {7, -12, 2}, {7, -15, 
       0}, {6, -15, 0},
      (* unter 4, 
      unter 3 *) {6, -12, -2}, {6, -9, -2}, {6, -15/2, 0}, {5, -15/2, 
       0},
      (* über 3, 
      unter 4 *) {5, -9, 2}, {5, -12, -2}, {5, -15, 0}, {4, -15, 0},
      (* über 4,über 3, über 2, 
      unter 1 *) {4, -12, 2}, {4, -9, 2}, {4, -6, 2}, {4, -3, -2}, {4,
        2, 2}, {23/2, 2, 0}
      }], 1/3]}
  }, ViewPoint -> Top, BoxRatios -> Automatic, Background -> Black, 
 Boxed -> False]

enter image description here

POSTED BY: Udo Krause
Posted 9 years ago

Hi Krause Thanks for the code. In your last graphics however the blues the green and yellow are not linked (only through he red). If I undo the red but only the red, all other loops will be trivial (loose). My situation is that every loop plays the same role, if I undo any loop (no only red) all will become undone!

POSTED BY: ali cherad
POSTED BY: Udo Krause
Posted 9 years ago

Hi Sam I meant 5 links (not 5 intersections)

POSTED BY: ali cherad
POSTED BY: Udo Krause
POSTED BY: Rick Mabry

Something like this

enter image description here

this one

... serves as an illustration of a symmetric Brunnian link of order 5.

So do you have a model of a Brunnian 5 link?

POSTED BY: Udo Krause
Posted 9 years ago

Hello Thanks for replying. Well yes I think i have a model. I am doing research in sociology and my model is to do with bureaucratic administration in the sense that there are 5 perspectives to be considered when a decision has to be made. However if one perspective is trivialized then the whole structure is functionally undone. So basically the model is not mathematically defined but I just want to use the graphics because they are very representative of my model.

POSTED BY: ali cherad
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