Let C be a 2d container with height h and width w from the non-negative reals. Unit circles (radius 1) can be placed inside the container, and they obey the usual rules of gravity, and do not intersect. What is the minumum number of unit circles needed to place a circle such that its highest point is of height >= h? If w<2 then f(h,w)=oo as we cannot place any unit circles into a container of this width. At w=2, f(h,w)=ceil(h/2) but for w>2 f becomes much less defined. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com
Jon Perry wrote:
Let C be a 2d container with height h and width w from the non-negative reals. Unit circles (radius 1) can be placed inside the container, and they obey the usual rules of gravity, and do not intersect.
What is the minumum number of unit circles needed to place a circle such that its highest point is of height >= h?
If w<2 then f(h,w)=oo as we cannot place any unit circles into a container of this width.
At w=2, f(h,w)=ceil(h/2)
but for w>2 f becomes much less defined.
Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com
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The usual rules of gravity? I'm not sure what that means. With a "steady" hand, one should be able to always obtain ceil(h/2), albeit metastably.
jon perry wrote:
Let C be a 2d container with height h and width w from the non-negative reals. Unit circles (radius 1) can be placed inside the container, and they obey the usual rules of gravity, and do not intersect.
What is the minumum number of unit circles needed to place a circle such that its highest point is of height >= h?
once a circle falls and rolls, does it stick, or can it be pushed horizontally by future circles? either way, 2<w<4 is easy. drop all the circles in the middle of the container and they will (aside from possibly the first one) form a zig zag to the top. by removing one of your variables and assuming w=oo and that circles stick once they stop rolling, you still get an interesting non-trivial problem. or similarly, you could assume h=w and only consider square containers. a related problem about least dense circle packings can be found here: http://www.stetson.edu/~efriedma/rigid/ erich
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Jon Perry