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

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

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

is it going to be some polar graph?

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

How can one move in two direction at once

they are moving in cycle.

A->B ->C->A

oh

Great riddle let us think, thanks

Well I think I know the beginning:

I guess there will always be an equilateral triangle between them

Something like this:|dw:1347902736343:dw|

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

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

d theta = sin theta * dx

\[\frac{ d \theta }{ dx } = \sin \theta\]

This is integrable easily

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

\[\theta(x) = \cos(x)\]

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

|dw:1347903100408:dw|

|dw:1347903134187:dw|

I think there is a possibility of that also

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

by the way my soln is a piece of spiral

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

ya did it that way @experimentX

http://www.wolframalpha.com/input/?i=polar+plot+r+%3D+theta

this is diverging spiral.

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

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

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

ok

Well I think I have got it - by geberal consideration

http://www.wolframalpha.com/input/?i=Archimedes+spiral

Sorry r Linear in theta

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

the problem is supposed to be calculus problem.

well ... sorry my bad :/

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

0.002 is simply to underscore the constancy of\[\Delta t\]

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

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

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

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

Have no clue on polar equation or parametric equation

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

http://www.wolframalpha.com/input/?i=polar+plot+r%28theta%29+%3D+1*exp%280.3*theta%29

Well guys I have to go now.

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

http://www.wolframalpha.com/input/?i=plot+r%28theta%29+%3D+theta

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

@mathslover please check my solution of exponential spiral

hold on ... I'm hunting for spirals

Hunting is good, but hunting for WRONG is just funny

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

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

all right ... i'll check.

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

This GIVES THE SHAPE OF THE PATHS FOR ANY REGULAR POLYGON

Square, Pentagon, Hexagon, Triangle whatever your n is

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

|dw:1347906117851:dw|

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