Conference rusure::math

    distance = 
    
               oo?

	cosec ( pi / n )

    re .1
    
    Infinity if the beetles travelled around the circumference. These
    beetles are hungry! and will travel directly towards their successor.
    
    The trick is to note that the beetles form a regular n-gon and that
    their initial direction of travel is along the sides each of which
    makes a constant angle with the radius and then to note that this
    situation is invariant.
    
    re .2
    
    Your previous solution was correct but I see you have replaced it by
    something equivalent and more concise which is the answer I was going
    to post.
    
    
    
    I like this problem because it has a simple solution easily accessable
    and avoids any thoughts of finding the equation of the tractrix
    (I think) and then calculating the path length.
    
    
    
    Follow up:
    
    It should be evident that the beetles spiral in to a big crunch.
    
    Calculate the angle turned through by a beetle before its demise.
    

>    I like this problem because it has a simple solution easily accessable
>    and avoids any thoughts of finding the equation of the tractrix
>    (I think) and then calculating the path length.
    
	Yeah, it's cute.   All you need to do is to consider one beetle,
say Alexander, and look at his pursuit of the next beetle along, Bucephalus
say.   Alexander is heading straight for Bucephalus, who is moving always
on a line orthogonal to AB.   So the distance that A travels is just the same
as if B remained stationary, ie: the length of one side of the n-gon.

	In fact, this problem for the case n=4 turned up a little while ago,
in the guise of 3 beetles stalking one another over the surface of a cube.
A fourth "virtual" beetle appeared when we tried to map the surface of the
cube into the plane.

>    Calculate the angle turned through by a beetle before its demise.
    
	Since in a constant angle rotation, the beetles separation decreases
by a fixed ratio, the beetles turn through an infinite angle (in a finite
time.)

Regards,
Andrew.

>	Yeah, it's cute.   All you need to do is to consider one beetle,
>say Alexander, and look at his pursuit of the next beetle along, Bucephalus
>say.   Alexander is heading straight for Bucephalus, who is moving always
>on a line orthogonal to AB.   So the distance that A travels is just the same
>as if B remained stationary, ie: the length of one side of the n-gon.

>	In fact, this problem for the case n=4 turned up a little while ago,
>in the guise of 3 beetles stalking one another over the surface of a cube.
>A fourth "virtual" beetle appeared when we tried to map the surface of the
>cube into the plane.
    
    I think your argument in paragraph 1 *only* applies to the case n=4
    where B is always moving at right angles to A. For n <> 4 the angle is
    not 90� and thus the distance travelled is not just the length of one
    side of the n-gon. This is evident from the fact that as n increases
    the length of the side decreases but the distance travelled increases.
    
    Thanks for the extra insight into the special case n=4 though.
    
>	Since in a constant angle rotation, the beetles separation decreases
>by a fixed ratio, the beetles turn through an infinite angle (in a finite
>time.)
    
    Yes, that's what I get but I am having great difficulty believing the
    answer. Does this mean that the situation is not physically realisable?

>>    Yes, that's what I get but I am having great difficulty believing the
>>    answer. Does this mean that the situation is not physically realisable?

	Yes, unless they are point-beetles and are very agile. :-)
	Real beetles with width and length and height will bump into
	each other near the center.

	Dan

re .6
    
>	Yes, unless they are point-beetles and are very agile. :-)
>	Real beetles with width and length and height will bump into
>	each other near the center.

    I was prepared to allow the beetles to be point masses. (There's
    reality and then there's reality. :-) )
    
    Are there any laws of physics being broken? For example, the initial
    linear momentum of the system is 0 and this is conserved. (My physics is
    very rusty!)

>>    Are there any laws of physics being broken?

	Hmmm.  It would seem that the force required to turn
	each beetle as it approaches the center must increase
	without bound.  Also, wouldn't they radiate an ever
	increasing amount of energy in the form of gravity
	waves?

	Dan

    re .8
    
>	Hmmm.  It would seem that the force required to turn
>	each beetle as it approaches the center must increase
>	without bound.  
    
    Yes, I guess that the centripetal force is inversely proportional to r
    and thus tends to infinity as r tends to 0. Notwithstanding the fact
    that we know nothing about the force between the beetles that attracts
    them, it is not without precedent that force varies inversely with the
    square of the distance (e.g. gravitational, eletric). These forces
    therefore increase faster than they need to in order to keep the
    beetles turning.
    
    Dredging through my head for some long forgotten formulae suggests that
    angular momentum may not be being conserved. Methinks it's decreasing.
    
    Gravity waves? I'll have to pass on that one.

    Whenever you consider bugs (or people, or whatever) walking (or
    driving, or rolling, or whatever) on a fixed platform, you will violate
    conservation of linear and angular momenta.  Alternately, you can take
    the platform to have infinite mass and therefore able to "count" for
    completely arbitrary momenta without any change in velocity.  Either
    way, you are throwing out the momentum conservation laws.

					Topher

>    I think your argument in paragraph 1 *only* applies to the case n=4
>    where B is always moving at right angles to A. For n <> 4 the angle is
>    not 90� and thus the distance travelled is not just the length of one
>    side of the n-gon. This is evident from the fact that as n increases
>    the length of the side decreases but the distance travelled increases.

	Gasp, but of course you're right.   I was multitasking while waiting
for a job to finish, and I allocated a duff brain cell to the problem.   The
corrected version of the argument gives the right value:  cosec(pi/n).
    
Regards,
Andrew.

    
    
    re .10
    
>    Whenever you consider bugs (or people, or whatever) walking (or
>    driving, or rolling, or whatever) on a fixed platform, you will violate
>    conservation of linear and angular momenta.
    
    True.
    
    However, I had in my thoughts dispensed with the platform. The universe
    consists of n beetles whose "initial" relative positions are as described
    in .0. Naturally you would wonder how they can move without a surface
    on which to crawl or something to eject.
    
    I am proposing that they are not in fact moving of their own free will
    but in reality under the influence of an attractive force like for example
    gravitation but with unusual properties that lead to each beetle moving
    "directly" towards its successor. A beetle does not appear to be attracted
    by any beetle other than its successor.
    
    To my knowledge this has no known precedent in "reality". I am just
    exploring the possibilities.

RE: .12

    A force which operates from A to B but not vice versa would certainly
    result in a violation of the momentum conservation laws.  The immobile
    platform at least is a reasonable approximation of a realistic
    situation -- a platform so massive, or rigidly attached to something so
    massive (e.g., the Earth), that its movement can be neglected.

					Topher

>    However, I had in my thoughts dispensed with the platform. The universe
>    consists of n beetles whose "initial" relative positions are as described
>    in .0. Naturally you would wonder how they can move without a surface
>    on which to crawl or something to eject.
                          ^^^^^^^^^^^^^^^^^^

	Hence we now the nature of the force...
        
>    To my knowledge this has no known precedent in "reality". I am just
>    exploring the possibilities.

Adrian

The ideal assumption of point volume to allow infinite rotation (in finite time)
causes other problems.  Namely, if these beetles have zero volume and positive 
mass, then they have infinite mass density and are therefore black holes!  Thus,
to avoid all kinds of other problems, you also have to assume zero mass. 

So what?  It's all for fun.  If you can get sensible results from one set of 
impossible assumptions, by all means, do it.

Avery