## experimentX 3 years ago Three snails at the point of equilateral triangle start to move at each other. What is the shape of curve?

1. psi9epsilon

assuming they start from vertices, to meet at centre then there are different possibilites they can meet at incentres also or at centroid too

2. psi9epsilon

in either case it would be along a line or a ray

3. experimentX

it can't be a ray ... they aren't moving to center. they are moving to each other ( the other snail) |dw:1347897411584:dw|

4. FweaBoyLive

they could meet by the vertices and be moving clockwise to each other

5. experimentX

nop ... they all in the direction of another snail.

6. Agent47

is it going to be some polar graph?

7. experimentX

don't know ..haven't solved it yet. i tried to do this couple of months ago ... couldn't do it.

8. sauravshakya

How can one move in two direction at once

9. experimentX

they are moving in cycle.

10. experimentX

A->B ->C->A

11. sauravshakya

oh

12. Mikael

Great riddle let us think, thanks

13. Mikael

Well I think I know the beginning:

14. sauravshakya

I guess there will always be an equilateral triangle between them

15. sauravshakya

Something like this:|dw:1347902736343:dw|

16. across

The equilateral triangle shrinks and rotates at a speed proportional to the snails' pace.

17. sauravshakya

Yep.. @across

18. Mikael

|dw:1347902612412:dw| One writes the differential equation Remembering v*dt = dx

19. Mikael

d theta = sin theta * dx

20. Mikael

$\frac{ d \theta }{ dx } = \sin \theta$

21. Mikael

This is integrable easily

22. experimentX

sorry ... had been away ... carry on

23. Mikael

$\theta(x) = \cos(x)$

24. Mikael

This is 1-st try one needs to improve the geometry

25. sauravshakya

|dw:1347903100408:dw|

26. experimentX

|dw:1347903134187:dw|

27. sauravshakya

I think there is a possibility of that also

28. experimentX

might be ... let's assume unit velocity for now ... we will generalize it later.

29. Mikael

by the way my soln is a piece of spiral

30. experimentX

|dw:1347903217601:dw| yep .. it should be spiral ...

31. sauravshakya

ya did it that way @experimentX

32. sauravshakya

@Mikael do u know the equation of a spiral........ Its beyond my knowledge

33. experimentX
34. experimentX

this is diverging spiral.

35. experimentX

also note that I don't the answer of this Q.

36. sauravshakya

This may be funny but this will depend upon the length of the snail

37. sauravshakya

Because when they meet there has to be an equilateral triangle with length of the side equal to the length of the snail

38. experimentX

|dw:1347903644328:dw| kinda seems this way.

39. experimentX

@sauravshakya you have a knack for making problems difficult ... well i agree that this gives insight into things ... but for this case. try to take the size of snail that would simplify the problem. take it as a point size snail.

40. sauravshakya

ok

41. Mikael

Well I think I have got it - by geberal consideration

42. Mikael

Here 1) After constant time Delat t ( constant) the pictire undergoes similarity transformation by A) rotating, B) Shrinking by CONSTANT factor Z(DElta t) The only spiral that is self similar is Archimedian Spiral where r is a LInear function of t

43. Mikael
44. Mikael

Sorry r Linear in theta

45. experimentX

well ... i didn't know that. let's say we know the answer. any method arising from calculus?

46. psi9epsilon

@experimentX , the question can be solved in numerous ways, and is open to interpretation you can also use graph theory to find the shortest path which infact will define the curve other solutions like rotation of triangle are justified if sufficient and necessary conditions are provided

47. experimentX

the problem is supposed to be calculus problem.

48. psi9epsilon

well that should have been indicated well in advance before requesting help, makes life easy than scratching heads : )

49. Mikael

Very high-brow @psi9epsilon yet I have offered a definite solution (right or wrong) but precise. Either disprove it or show my argument deficient.

50. experimentX

well ... sorry my bad :/

51. psi9epsilon

@Mikael, do not approve or disapprove your solution as it MAY NOT be the ONLY solution

52. Mikael

I CLAIM that the only spiral that has in 0.002 time-units A) constant rotation B) Constant zoom-down factor Z Is the above

53. experimentX

it's a spiral ... at least we know that.

54. Mikael

0.002 is simply to underscore the constancy of$\Delta t$

55. sauravshakya

Can any one show me how the equation of a spiral looks

56. Mikael

R(t) = exp(Kt) theta(t) = kt

57. experimentX

let's solve this Q for particular case. assume side is 5 and speed is 1

58. experimentX

@sauravshakya there are numerous way in which you can have a spiral ... try to look for polar equation or parametric equation ... this is same a circle except the radius changes.

59. Mikael

$R(\theta) = R_0e^{k*\theta}$

60. sauravshakya

Have no clue on polar equation or parametric equation

61. experimentX

well .. that's one hell of one spiral ..

62. Mikael
63. sauravshakya

Well guys I have to go now.

64. Mikael

AAND I HAVE PROOF ! Here it is$R(t + \Delta T) = R(t) * e^{k \Delta T} ---- constant factor$

65. experimentX
66. Mikael

$\theta(t + \Delta T) = \theta(t) + k*\Delta T ----> constant-rotation$

67. Mikael

So @experimentX @across @siddhantsharan and whoever want - check this solution !

68. Mikael

Please CHECK this solution @estudier @mukushla @hartnn

69. experimentX
70. Mikael

@experimentX Please accept the proof - it is rock solid. Do not evade - this is THE only Only solution

71. Mikael

@mathslover please check my solution of exponential spiral

72. experimentX

hold on ... I'm hunting for spirals

73. Mikael

By the way THIS IS A GENERAL SOLUTION FORM FOR ANY EQUILATERAL POLYGON WITH N-SNAILS IN EACH VERTEX !!!!

74. Mikael

Hunting is good, but hunting for WRONG is just funny

75. Mikael

THIS IS A GENERAL SOLUTION FORM FOR ANY EQUILATERAL POLYGON WITH N-SNAILS IN EACH VERTEX !!!!

76. experimentX

no ... just for fun. man fun is good.

77. Mikael

If I am right - PLEASE say so. Check - this is justified because you asked the. And I solved it (or not - but check) question

78. experimentX

i thought this would work out .. lol http://www.wolframalpha.com/input/?i=parametric+plot+x%28t%29+%3D+e^t+cos%28t%29%2C+y%28t%29+%3D+e^t+sin%28t%29+from+t%3D0+to+t%3D2+pi

79. experimentX

all right ... i'll check.

80. Mikael

$R( \Theta) = R_0 e^{k \Theta}$

81. Mikael

This GIVES THE SHAPE OF THE PATHS FOR ANY REGULAR POLYGON

82. Mikael

Square, Pentagon, Hexagon, Triangle whatever your n is

83. experimentX

|dw:1347905702311:dw| yeah this is definitely a r = k e^(theta)

84. experimentX

|dw:1347906117851:dw|

85. experimentX

let's try to solve this problem using calculus method some other day.