Determining the number of ways to color the faces of a triangular prism with 4 colours using the counting theorem.
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Consider the symmetry group of rotations in $3$ dimensions of the triangular prism, acting on colourings of the faces of a triangular prism with $4$ colours (red, blue, yellow and green). Note that the triangular faces are equilateral triangles.
triangular prism
Use the counting theorem to determine the number of colourings of the faces of the prism with $4$ colours, where two colourings are considered the same if one can be obtained from the other by applying an element from the symmetry group of rotations of the prism.
For the counting theorem I know that the size of my $|G| = 12$ since there are $12$ symmetries for my triangular prism and the fix (Identity) will be $3^4$. I'm having trouble finding out what my other fix functions should be.
combinatorics symmetric-groups
edited Aug 28 at 14:56
N. F. Taussig
39.1k93153
asked Aug 28 at 12:21
Andrew Chung
171
add a comment |Â
up vote
0
down vote
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Consider the symmetry group of rotations in $3$ dimensions of the triangular prism, acting on colourings of the faces of a triangular prism with $4$ colours (red, blue, yellow and green). Note that the triangular faces are equilateral triangles.
triangular prism
Use the counting theorem to determine the number of colourings of the faces of the prism with $4$ colours, where two colourings are considered the same if one can be obtained from the other by applying an element from the symmetry group of rotations of the prism.
For the counting theorem I know that the size of my $|G| = 12$ since there are $12$ symmetries for my triangular prism and the fix (Identity) will be $3^4$. I'm having trouble finding out what my other fix functions should be.
combinatorics symmetric-groups
edited Aug 28 at 14:56
N. F. Taussig
39.1k93153
asked Aug 28 at 12:21
Andrew Chung
171
How do you define the triangular prism? I just googled it (I am not native), and it seems to have only three orientation preserving symmetries (you wrote rotations).
– A. Pongrácz
Aug 28 at 12:31
2
Is your prism really a tetrahedron? (That would make it $12$ symmetries rather than the $6$ you have for what is conventionally called a triangular prism.) And by "the counting theorem", do you mean Burnside's lemma?
– Arthur
Aug 28 at 12:36
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the symmetry group of rotations in $3$ dimensions of the triangular prism, acting on colourings of the faces of a triangular prism with $4$ colours (red, blue, yellow and green). Note that the triangular faces are equilateral triangles.
triangular prism
Use the counting theorem to determine the number of colourings of the faces of the prism with $4$ colours, where two colourings are considered the same if one can be obtained from the other by applying an element from the symmetry group of rotations of the prism.
For the counting theorem I know that the size of my $|G| = 12$ since there are $12$ symmetries for my triangular prism and the fix (Identity) will be $3^4$. I'm having trouble finding out what my other fix functions should be.
combinatorics symmetric-groups
edited Aug 28 at 14:56
N. F. Taussig
39.1k93153
asked Aug 28 at 12:21
Andrew Chung
171
Consider the symmetry group of rotations in $3$ dimensions of the triangular prism, acting on colourings of the faces of a triangular prism with $4$ colours (red, blue, yellow and green). Note that the triangular faces are equilateral triangles.
triangular prism
Use the counting theorem to determine the number of colourings of the faces of the prism with $4$ colours, where two colourings are considered the same if one can be obtained from the other by applying an element from the symmetry group of rotations of the prism.
For the counting theorem I know that the size of my $|G| = 12$ since there are $12$ symmetries for my triangular prism and the fix (Identity) will be $3^4$. I'm having trouble finding out what my other fix functions should be.
combinatorics symmetric-groups
edited Aug 28 at 14:56
N. F. Taussig
39.1k93153
asked Aug 28 at 12:21
Andrew Chung
171
edited Aug 28 at 14:56
N. F. Taussig
39.1k93153
edited Aug 28 at 14:56
N. F. Taussig
39.1k93153
edited Aug 28 at 14:56
N. F. Taussig
39.1k93153
39.1k93153
asked Aug 28 at 12:21
Andrew Chung
171
asked Aug 28 at 12:21
Andrew Chung
171
asked Aug 28 at 12:21
Andrew Chung
171
171
How do you define the triangular prism? I just googled it (I am not native), and it seems to have only three orientation preserving symmetries (you wrote rotations).
– A. Pongrácz
Aug 28 at 12:31
2
Is your prism really a tetrahedron? (That would make it $12$ symmetries rather than the $6$ you have for what is conventionally called a triangular prism.) And by "the counting theorem", do you mean Burnside's lemma?
– Arthur
Aug 28 at 12:36
add a comment |Â
How do you define the triangular prism? I just googled it (I am not native), and it seems to have only three orientation preserving symmetries (you wrote rotations).
– A. Pongrácz
Aug 28 at 12:31
2
Is your prism really a tetrahedron? (That would make it $12$ symmetries rather than the $6$ you have for what is conventionally called a triangular prism.) And by "the counting theorem", do you mean Burnside's lemma?
– Arthur
Aug 28 at 12:36
How do you define the triangular prism? I just googled it (I am not native), and it seems to have only three orientation preserving symmetries (you wrote rotations).
– A. Pongrácz
Aug 28 at 12:31
How do you define the triangular prism? I just googled it (I am not native), and it seems to have only three orientation preserving symmetries (you wrote rotations).
– A. Pongrácz
Aug 28 at 12:31
2
2
Is your prism really a tetrahedron? (That would make it $12$ symmetries rather than the $6$ you have for what is conventionally called a triangular prism.) And by "the counting theorem", do you mean Burnside's lemma?
– Arthur
Aug 28 at 12:36
Is your prism really a tetrahedron? (That would make it $12$ symmetries rather than the $6$ you have for what is conventionally called a triangular prism.) And by "the counting theorem", do you mean Burnside's lemma?
– Arthur
Aug 28 at 12:36
add a comment |Â
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How do you define the triangular prism? I just googled it (I am not native), and it seems to have only three orientation preserving symmetries (you wrote rotations).
– A. Pongrácz
Aug 28 at 12:31
2
Is your prism really a tetrahedron? (That would make it $12$ symmetries rather than the $6$ you have for what is conventionally called a triangular prism.) And by "the counting theorem", do you mean Burnside's lemma?
– Arthur
Aug 28 at 12:36