JAM - Choreograhies wiki

n=3

D'(3,2)

  • Figure 8
    • winding nos = (-, 0).
  • with extra twist
    • winding nos = (-, 0).


D(3,4)

  • New and gorgeous! A Celtic knot !
    • Still a bit wobbly even with FS=30 - should try matlab?
    • winding nos = (1; -3).


D(3,5) - wishful thinking!

n=4

D(4,1)

  • Off-centre figure 8.
    • ''winding nos =" -1; 0, -1
  • With extra twist
    • ''winding nos =" 1; 2, 1


D(4,2)

  • Gerver's solution
    • winding nos = -1; 1, -1
  • With extra twist
    • winding nos =


D(4,6)

  • Beautiful: 6 ovals around a central hexagon.


n=5

D'(5,2)

  • Bowtie example.
    • - also many examples in paper of Simo


D(5,1)

  • Skew Fig. 8.


D(5,4)

  • 5 on a 4-daisy.
    • winding nos: 1, -3, ??


D(5,8)

  • circular chain with 8 links
    • winding nos: 1; -7, 1


D(5,8/3)

  • 5 on an 8-flower


n=6

Numerics6

The D(6,∞,1) (circular choreography) has

  • winding nos = (1; 1, 1, 1), and
  • action = 84.5.

C(6,1)

  • no symmetry (Simo)
    • winding nos = (1; 1, 0, 1)



(c)

(b)

(a)

D(6,4) This has core of order 2.

  • (a) A square with loops at vertices.
    • Action=136
    • winding nos = (1; -3, 1, 1).
  • (b) A square with diagonals.
    • Action=227
    • winding nos = (1; -3, -3, 1).
  • (c) Doubling up D(3,4) ...
    • Action=220
    • winding nos = (1; 1, -3, 1).


D(6,5)

  • 6 on a daisy (pentagon with loops).


D(6,5/2)

  • (a) a pentagram?
    • winding no. (-3; -3, 2, -3).
  • (b) round a circle
    • winding no. (2; 2, 2, 7).


heptagons and heptagrams:

D(6,7)

  • Heptagon chain with 'inward links'
    • winding nos = (1; 8, 1, 1)


D(6,7/2)

  • (a) Simple heptagram
    • winding nos = (2; 2, 2, -5)
  • (b) on 7 petals
    • winding nos = (2; 2, -5, -5)


D(6,7/3)

  • on a {7/3} heptagram,
    • winding nos = (-4; -4, -4, 3).


n=7

C'(7,2)

  • Reflection but not time-reversing:
    • Thm: winding nos = 0.


C(7,1)

  • no symmetry
    • winding nos = 1; 1, 0, 0.


D(7,4)

  • seven on ...
    • winding nos = .


n=8


(b)

(a)

D(8,3)

  • (a) Moustache
    • Action= 264
    • winding nos = -2; 1, -2, -2, 1.
  • (b)
    • Action= 217
    • winding nos = 1 ; -2, -2, 1, 1.



(d)

(c)
  • (c)
    • Action= 304
    • winding nos = 4 ; 4, 1, 4, 1.
  • (d)
    • Action= 227
    • winding nos = -2; -2, -2, -2, 1.

For comparison, action for D(8,∞) is 150.


(b)

(a)

D(8,4) - order 4 core

  • (a) 4 balloons
    • Action = 281
    • winding nos = -3; -3, -3, 1, -3.
  • (b) Square with loops
    • Action =
    • winding nos = 1; ?.



(b)

(a)

D(8,7)

  • (a) 8 on 7-petalled flower
    • Action = 396 ?
    • winding nos = 1; -6, -6, 1, 1.
  • (b) 8 on 7-daisy
    • Action = 325
    • winding nos = 1; -6, 1, 1, 1.


D(8,9/2)

  • {9/2}-enneagram
    • winding nos = .


D(8,9/4)

  • {9/4}-enneagram
    • winding nos = .


n=9


(b)

(a)

D(9,4)

  • (a) From D(3,4) -
    • action = 426
    • winding nos = 1; 5, 1, -3, 1.
  • (b) From D(6,4) -
    • action = 483
    • winding nos = 1; 1, 1, -3, 1.
Compare: action for D(9,∞/1) is 189


n=10

Attach:D-10-5.gif Δ|(a) D(10,5)

  • (a) 5 balloons
    • action =
    • winding nos = -4; -4, -4, 1, -4, 1.
Compare: action for D(9,∞/1) is 189


  • D(10,5) (b) seems unlikely, with the motion almost a circle, and with increased number of terms I can't get it to converge! See figure

D(10,5/2)

  • (a) Dancing pentagons
    • action =
    • winding nos = 2; -3, 2, 2, 2, -3.
  • (b) balloons
    • action =
    • winding nos = -3; -3, -3, 2, -3, -3.
  • (c) almost circular
    • action =
    • winding nos = -3; -3, -3, -8, -3, -3.


D(10,8/3)

  • (a) 10 on an {8/3} octagram
    • action =
    • winding nos = ??.