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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1731.0. "Crux Mathematicorum 1815" by RUSURE::EDP (Always mount a scratch monkey.) Wed Mar 17 1993 13:47

    Proposed by Stan Wagon, Macalester College, St. Paul, Minnesota.
    
    An old puzzle (see _Mathematical Puzzles and Diversions_, Martin
    Gardner, Simon & Schuster, New York, 1959, p. 114, or _Puzzlegrams_,
    Simon & Schuster/Fireside, New York, 1989, p. 171) asks that five
    congruent coins be placed in space so that each touches the other four. 
    The solution often given is as illustrated below:  one coin supports
    two others, which meet over the center of the bottom coin, with two
    tilted coins forming the sides of the tent-like figure.
    
    [I can't enter the illustration, but there's one coin flat at the
    bottom with two coins flat above that apparently meeting each other
    above the center of the bottom coin, and the final two coins are
    slanted, touching the bottom coin where it is exposed under the two
    two other coins and touching each of those two other coins and then
    rising to touch the other of the last two coins.  -- edp]
    
    Show that this solution is _invalid_ if the coins are nickels.  Assume
    (despite the picture) that the nickels are ordinary cylinders with
    diameter to height ratio of 11 to 1.
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1731.1HERON::BUCHANANThe was not found.Wed Mar 17 1993 14:1316
In the same section of the Gardner book, if I recall, there is listed:

	(a) a configuration of 6 matches, each touching each of the others
	(b) a configuration of 7 cigarettes, each touching each of the others

I wonder:

	(i) What is the maximum number of identical convex bounded objects, 
which can each touch each of the others, without interpenetrating?   
(Conjecture: answer = 7)

	(ii) What is the maximum number of identical (not necessarily convex)
bounded objects, which can each touch each of the others, without 
interpenetrating?   (Conjecture: answer is infinite)

Andrew	
1731.2(ii) answeredHERON::BUCHANANThe was not found.Wed Mar 17 1993 18:2016
>	(ii) What is the maximum number of identical (not necessarily convex)
>bounded objects, which can each touch each of the others, without 
>interpenetrating?   (Conjecture: answer is infinite)

	Yes it is infinite.   The basic object I used is a unit cube, with some
thin pipes coming out of the top, heading nearly East, and then turning down 
again.   Arrange an infinite number of cubes from the origin out towards the 
East.   The easy trick is to set up the pipes so that they don't collide with
one another.   For instance, all the pipes, P, that connect cube n with cube n+3
occupy the region 1/2 < y < 3/4.   P are all parallel to one another, and are
very thin, and head *just* North of West.   So there are no collisions between
them.

	(i) is a good question

Andrew.
1731.3VMSDEV::HALLYBFish have no concept of fire.Wed Mar 17 1993 18:296
>	Yes it is infinite.   The basic object I used is a unit cube, with some
> thin pipes coming out of the top, heading nearly East, and then turning down 
    
    You don't happen to have a ray-traced image of that, do you?
    
      John :-)
1731.4slapdash remarksHERON::BUCHANANThe was not found.Wed Mar 17 1993 19:3926
>	(i) What is the maximum number of identical convex bounded objects, 
>which can each touch each of the others, without interpenetrating?   
>(Conjecture: answer = 7)

	A simpler, related problem to tackle first.

	What's the maximum number of identical convex objects in 2 dimensions
which can touch each of the others, without interpenetrating?

(Conjecture: answer is 4.)

	Obviously, 5 is not possible, since the complete graph on 5 vertices
is not planar.   Obviously, 3 is possible, since we just stick 3 circles
together (being careful to define whether the circle boundaries are included
or not!)

	It's this boundary issue which I still don't grok for the case of
4 objects.   4 equilateral triangles arranged in the shape of a larger 
equilateral triangle nearly solve the problem.   But are the corner vertices
included in the object?   Oops.

	Using a flat isosceles triangle, it all becomes easier, but I still
have problems with the boundary.

Later,
Andrew.
1731.5HANNAH::OSMANsee HANNAH::IGLOO$:[OSMAN]ERIC.VT240Thu Mar 18 1993 13:1220
Well, with sloppy thinking, one could talk about 4 square floor tiles arranged
to form a large square, and claim that all four touch each other in the middle.

Looks like this:

	+---------------+
	|   a	|   b	|
	|	|	|
	+-------X-------+
	|   c	|  d	|
	|	|	|
	+-------+-------+

They suppsedly all touch each other.  But in 2 dimensions, if a is touching d,
it seems to me that their contact *prevents* b from touching c.

So, I don't know if you really can have 4 objects cotouching in 2 dimensions...

/Eric
1731.6clarificationHERON::BUCHANANThe was not found.Tue Mar 23 1993 11:2218
1731.7AUSSIE::GARSONTue Mar 23 1993 21:375
1731.8Touch is topologicalHERON::BLOMBERGTrapped inside the universeWed Mar 24 1993 10:564
    Maybe better with a topological definition of touching, something like
    
    Two sets A and B are said to touch, if there is a point p, belonging
    to A or B, such that every neighbourhood of p intersects both A and B.
1731.9RUSURE::EDPAlways mount a scratch monkey.Tue Feb 01 1994 13:3212
    The proof appears in the January _Crux_.  I won't attempt to reproduce
    it here.  The diameter to height ratio must be at least 2/(x*sqrt(3)),
    where x is the solution of 9x^3-3x^2+11x-1=0 around .0926, which is
    about 12.47.
    
    
    				-- edp
    
    
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