Meaning of "rigid" in geometry
wasknijper
"Problem of the Month (April 2005)
Start with a horizontal line called the ground. We say a tent is a
collection of unit line segments that are only joined to the top half of
the ground and each other at their endpoints. We call a tent rigid if
its framework of segments is rigid in the plane as long as the
connections to the ground are fixed. We define the area of a tent to be
the area enclosed by the tent and ground.
It is well-known that the tent made of n segments with the largest area
is half a 2n-gon. This month's problem is to investigate the rigid tent
made from n segments with the largest area. What are the best results
for various n? Can you give lower or upper bounds for the largest
possible area?"
http://www.stetson.edu/%7Eefriedma/mathmagic/0405.html
I have a vague, intuitive grasp of what "rigid" may mean here, and I can
see that half a 2n-gon is not valid ... but what /exactly/ is "rigid"? I
don't understand the definition that is given. How can I tell whether a
tent is rigid or not?
Patrick Hamlyn
Immagine every connection is made using a bolt. A rod connected to the ground
can rotate about the connected end, but the connection can't slide along the
ground.
Similarly with two rods connected.
Imagine three rods connected in a triangle. Despite the fact that every
connection is 'loose', none of the rods can in fact move relative to the others,
this is called a 'rigid' structure. On the other hand four rods in a square do
not form a rigid structure because the whole thing can collapse down into a flat
shape, forming a parallelogram along the way.
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms...@egroups.com)
wasknijper
> Immagine every connection is made using a bolt. A rod connected to the ground
> can rotate about the connected end, but the connection can't slide along the
> ground.
>
> Similarly with two rods connected.
>
> Imagine three rods connected in a triangle. Despite the fact that every
> connection is 'loose', none of the rods can in fact move relative to the others,
> this is called a 'rigid' structure. On the other hand four rods in a square do
> not form a rigid structure because the whole thing can collapse down into a flat
> shape, forming a parallelogram along the way.
Now I understand. Thank you, Patrick.
Graham Jones
writes
What about things like:
____
/\ /\
where the 4 bottom points are held fixed by the ground? Such structures,
if allowed, would be handy for the puzzle in question.
--
Graham Jones
http://www.visiv.co.uk
Emails to gra...@visiv.co.uk may be deleted as spam
Please add a j just before the @ to ensure delivery
wasknijper
> What about things like:
> ____
> /\ /\
>
> where the 4 bottom points are held fixed by the ground? Such structures,
> if allowed, would be handy for the puzzle in question.
Yes. It appears to be the optimal solution for n = 7-11.
http://www.stetson.edu/%7Eefriedma/mathmagic/0405.html