Den fredagen den 18:e juli 2014 kl. 13:57:17 UTC+2 skrev Ben Bacarisse:
> jonas.thornvall@gmail.com writes:
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> > I was a bit tired tonight thinking about this, and very diffuse in my
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> > problem statement. I try be a bit more coherent below.
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> >
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> > But I think i need some help to formulate the problem in a more coherent manner.
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> > It is about how many corner/node names needed to create a collsion
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> > free network. Warning i am not that good formulate the actual problem.
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> >
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> > So i may need some help formulate the problem in a more coherent manner.
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> >
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> > I will start using the easiest case a square.
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> >
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> > A square is a unique individual that use different name for each corner.
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> > The arrangement of the corners is free, a square and its corner can
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> > never be revisited.
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> >
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> > Now i want to build a oneway network out from a cental starting
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> > square. I push squares together, creating outward nodes from a central
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> > square that will be collision free. ***You only push together corners
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> > holding same name*** thus a interconnected corner/node is named
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> > 1,2,3,4,5... and so on.
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> Personally, I'd call them colours. There's a long tradition of problems
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> involving colouring graphs (and therefore maps).
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> > To be collision free means that a corner can not point to two corners
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> > holding same name, but it can itself hold the same name as a corner it
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> > pointing to because it is a oneway path network.
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> Are you talking about a tessellation? I.e. must the shapes fill the
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> plain? If not, I think there needs to be much more said about the
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> constraints, but since the question you ask seems to be for a single
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> number it looks like you do mean to refer to infinite tessellations.
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> But then again you mention the tetrahedron, and space can't be filled
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> using regular tetrahedrons. Maybe the shapes do not have to be all the
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> same?
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> > How many corner names needed to create a collision free network.
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> Presumably you mean the minimum number.
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> I'd turn the description round: given a pattern of touching shapes (here
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> you can say an infinite tessellation if that is what you mean, or
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> specify other constraints on the pattern), what is the minimum number of
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> colours needed to colour the nodes so that the corners of every shape
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> have distinct colours.
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> (This raises another question -- how many distinct shapes are needed
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> when you take the coloured corners into account? To me, that seems like
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> the more interesting problem, but that's just a gut feeling.)
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> > 1. Triangle
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> > 2. Tetrahedron
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> > 3. Square
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> > 4. Cube
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> > And other polygons and platonic solids, is this group theory,
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> > computational complexity?
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> (I note in passing that the platonic solids are regular tessellations of
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> the sphere, so the problem can be asked *of* them as well as *about*
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> them if you relax the constraint that the pattern must be on the plain.)
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> It's most closely related to graph theory, I'd say, but group theory and
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> ordinary geometry are involved. Computational complexity will come up
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> if you ask about the computational aspects.
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> Anyway, one thing is for sure, it's not javascript! I've answered here
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> without setting followup-to because the obvious place is sci.maths, but
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> that become a cesspit of nonsense though there is the occasional bit if
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> real math that gets done. If you can face it, post there.
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> --
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> Ben.
I do mean connecting just one shape Ben in such a way all outgoing nodes have individual names. There is no problem if the current corner connect to a corner using the same name, but all the outgoing paths should lead to unique named corner/crosspoint.