Originally posted by FabianFnas The answer is that it can't be done. Unless you have a torus like geometry or a paper with a hole in it or otherwise, then it can be done with a trick. But on a flat surface, like a paper, it can't be done.
My calculations say it could be done in a Black Hole if approaching the speed of light.
Originally posted by FabianFnas A paper is flat, a torus is not.
Yes I am aware of this but your putting a hole in the middle of the page and then drawing on it as if its a torus.
To post said solution is hard but here goes: the o's are your dots, then rip a hole around where the brackets are and use that to get to your last unconnected line by going around the back of the page.....
O O O
p ( )
O O O
Had to put the P in to make it center justify the brackets....
upper dots must connect with all bottom dots and either way, with a line..
cross line is illegal...
easy punch two holes inthe paper and have the line on the back going through both holes and connecting the dots this is the last line of course all other work out or go allthe way around the paper
Originally posted by Mexico Yes I am aware of this but your putting a hole in the middle of the page and then drawing on it as if its a torus.
To post said solution is hard but here goes: the o's are your dots, then rip a hole around where the brackets are and use that to get to your last unconnected line by going around the back of the page.....
O O O
p ...[text shortened]... )
O O O
Had to put the P in to make it center justify the brackets....
...and so we are led to ask the next natural question (well, I am anyway):
On a sphere any graph with K[5] or K[3,3] is non-planar - it cannot be drawn without the lines crossing. This is obviously not true on a Torus, and for other shapes with added handle-things. So, what are the graphs that force other graphs to be non-planar on a Torus? What are the graphs with "minimum vertexes" that are non-planar on a Torus?
Originally posted by Swlabr No - but they are connected. The 4-colour theorem applies to any planar graph (this is a graph - a collection of nodes (points) and vertexes (lines)). A planar graph is any graph that can be drawn on a sphere such that no lines cross.
It can be shown that a graph is not planar if and only if it contains the graph described above (called k[3,3]) or t ...[text shortened]... lem.
I know this reads quite awfully - http://tinyurl.com/33zkwr
[wiki] describes it better.
nice summation of planarity with graph theoretic ideas. your mention of drawing graphs on a sphere gets at what Fabian was saying about paper being flat but not a torus...
a plane (such as the flat piece of paper) is homeomorphic to the surface of a sphere (as long as you discount the "north pole", as it represents moving infinitely off in some direction in the plane), and similarly a torus can be thought of as any "almost planar" structure with a single "escape route" or hole... that is, thinking of punching a hole in the piece of paper indeed would make it a toroidal structure.
we classically view a torus as a "donut" but with the use of continuous deformation, it doesn't necessarily have to have this classic shape to maintain the qualities that make it uniquely toroidal.
25 Feb '08 18:38
puzzle
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