| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1731.1 | | HERON::BUCHANAN | The was not found. | Wed Mar 17 1993 11:13 | 16 |
| 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) answered | HERON::BUCHANAN | The was not found. | Wed Mar 17 1993 15:20 | 16 |
| > (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.
|
| 1731.3 | | VMSDEV::HALLYB | Fish have no concept of fire. | Wed Mar 17 1993 15:29 | 6 |
| > 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 :-)
|
| 1731.4 | slapdash remarks | HERON::BUCHANAN | The was not found. | Wed Mar 17 1993 16:39 | 26 |
| > (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.
|
| 1731.5 | | HANNAH::OSMAN | see HANNAH::IGLOO$:[OSMAN]ERIC.VT240 | Thu Mar 18 1993 10:12 | 20 |
|
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
|
| 1731.6 | clarification | HERON::BUCHANAN | The was not found. | Tue Mar 23 1993 08:22 | 18 |
| Statement of the 2-d problem
============================
Can you draw 4 identical�, convex, disjoint shapes in the plane such
that each touches� each of the others?
� identical up to rotation and translation.
� two shapes A & B touch if no line, disjoint to A union B, can separate A from
B.
The careful definitions are necessary to avoid the example suggested
in -.1. The central point can belong to only one of the four shapes, and then
a line through this point can separate two of the other shapes from one
another.
I think I have an answer for this one, but I won't put it in yet.
Andrew.
|
| 1731.7 | | AUSSIE::GARSON | | Tue Mar 23 1993 18:37 | 5 |
| re .6
>� two shapes A & B touch if no line
What's a line? (I'm serious.)
|
| 1731.8 | Touch is topological | HERON::BLOMBERG | Trapped inside the universe | Wed Mar 24 1993 07:56 | 4 |
| 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.
|
| 1731.9 | | RUSURE::EDP | Always mount a scratch monkey. | Tue Feb 01 1994 10:32 | 12 |
| 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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