Re: Continued training with octahedron.
- From: "mina_world" <mina_world@xxxxxxxxxxx>
- Date: Fri, 4 Jan 2008 19:11:37 +0900
"David Bernier" <david250@xxxxxxxxxxxx> wrote in message
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mina_world wrote:
"mina_world" <mina_world@xxxxxxxxxxx> wrote in message
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"mina_world" <mina_world@xxxxxxxxxxx> wrote in message
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Hello sir~Sorry. modify : octahdron ==> icosahedron.
I showed icosahedral graph by octahedron.
http://mathworld.wolfram.com/IcosahedralGraph.html
There is Hamiltonian cycle count 2560 in the below table of link.
Maybe, this is equal to edge's cycle in octahedron.
(Namely, One ant is on a vertex of octahedron.
This ant moves on edge of octahedron.
This ant visits each vertex exactly once and also returns to the
starting vertex.
This count is 2560.)
Anyway, my question is...How can you calculate it(2560) ?
For reference,
Tetrahedral graph ==> Hamiltonian cycle count 6
Here I think this means there are 6 Hamiltonian
cycles, where cycles have a direction, which
can be shown by arrows on the edges that are visited.
For example, Wikipedia has a figure of a directed cycle
here:
< http://en.wikipedia.org/wiki/Path_%28graph_theory%29 >
Also, I think in graph theory for the graphs
of the icosahedron and so on, "rotating in
space" is not allowed and rotating in the
plane is not allowed when counting
Hamiltonian cycles. It's a convention
of graph theory.
Cubical graph ==> Hamiltonian cycle count 12
Octahedral graph ==> Hamiltonian cycle count 32
Dodecahedral graph ==> Hamiltonian cycle count 60
Isosahedral graph ==> Hamiltonian cycle count 2560
Maybe, It looks like direct counting.
http://board-2.blueweb.co.kr/user/math565/data/math/hem001.jpg
and It seems that these diagrams are equivalence in solid figure.
so,
One ant is on a vertex of icosahedron.
This ant moves on edge of icosahedron.
This ant visits each vertex exactly once and also returns to the starting
vertex.
I think...
If I igonore the rotation permutation of solid,
this ant's pass(cycle) is unique.
so, maybe...
answer is...(with rotation group)
I don't think rotations count. The graph
sits in the plane in a fixed position. An
ant walks on the edges and returns, having visited
every vertex exactly once. How many ways, if clockwise
and anticlockwise are considered different?
That's my impression ...
tetrahedron ==> 12
cube ==> 24
octahedron ==> 24
dodecahedron ==> 60
icosahedron ==> 60
Yes, it's false. You're right.
Start with labeled Platonic solid.(Labeled vertices.)
This ant's circuit of icosahedron is equivalent to Hamiltonian circuit of
icosahedral graph.
.
- References:
- Continued training with octahedron.
- From: mina_world
- Re: Continued training with octahedron.
- From: mina_world
- Re: Continued training with octahedron.
- From: mina_world
- Re: Continued training with octahedron.
- From: David Bernier
- Continued training with octahedron.
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