Proving the count of symmetric configurations of pentagon
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In a 3 × 3 dot grid, there are 5 configurations of symmetric pentagons. I am confused about how to prove that it is really just 5. Can anyone enlighten me?
geometry combinatorics
edited Feb 2 at 7:31
Omega Krypton
4,4051440
asked Feb 2 at 2:48
Sierra SorongonSierra Sorongon
465
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add a comment |
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In a 3 × 3 dot grid, there are 5 configurations of symmetric pentagons. I am confused about how to prove that it is really just 5. Can anyone enlighten me?
geometry combinatorics
edited Feb 2 at 7:31
Omega Krypton
4,4051440
asked Feb 2 at 2:48
Sierra SorongonSierra Sorongon
465
$endgroup$
$begingroup$
Hint: each pentagon has either a straight or diagonal line of symmetry.
$endgroup$
– Hugh
Feb 2 at 3:27
add a comment |
$begingroup$
In a 3 × 3 dot grid, there are 5 configurations of symmetric pentagons. I am confused about how to prove that it is really just 5. Can anyone enlighten me?
geometry combinatorics
edited Feb 2 at 7:31
Omega Krypton
4,4051440
asked Feb 2 at 2:48
Sierra SorongonSierra Sorongon
465
$endgroup$
In a 3 × 3 dot grid, there are 5 configurations of symmetric pentagons. I am confused about how to prove that it is really just 5. Can anyone enlighten me?
geometry combinatorics
geometry combinatorics
edited Feb 2 at 7:31
Omega Krypton
4,4051440
asked Feb 2 at 2:48
Sierra SorongonSierra Sorongon
465
edited Feb 2 at 7:31
Omega Krypton
4,4051440
asked Feb 2 at 2:48
Sierra SorongonSierra Sorongon
465
edited Feb 2 at 7:31
Omega Krypton
4,4051440
edited Feb 2 at 7:31
Omega Krypton
4,4051440
edited Feb 2 at 7:31
Omega Krypton
4,4051440
4,4051440
asked Feb 2 at 2:48
Sierra SorongonSierra Sorongon
465
asked Feb 2 at 2:48
Sierra SorongonSierra Sorongon
465
asked Feb 2 at 2:48
Sierra SorongonSierra Sorongon
465
465
$begingroup$
Hint: each pentagon has either a straight or diagonal line of symmetry.
$endgroup$
– Hugh
Feb 2 at 3:27
add a comment |
$begingroup$
Hint: each pentagon has either a straight or diagonal line of symmetry.
$endgroup$
– Hugh
Feb 2 at 3:27
$begingroup$
Hint: each pentagon has either a straight or diagonal line of symmetry.
$endgroup$
– Hugh
Feb 2 at 3:27
$begingroup$
Hint: each pentagon has either a straight or diagonal line of symmetry.
$endgroup$
– Hugh
Feb 2 at 3:27
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Here are 5 symmetric pentagons on a $3times3$ grid:
A proof is to notice that there are only two lines of mirror symmetry on the grid, vertical (or horizontal by rotation) and diagonal. A line of symmetry can only contain 1 vertex, because the other vertices are mirrored, and 3 in a row is impossible (a vertex would have 3 edges). There are only 3 variations for each of the two possible LoS vertices, giving only 12 cases to test, and 12 ways to draw each one (12345 is the same as 23451 is the same as 54321), so there are only 144 possibilities to check.
answered Feb 2 at 3:23
JonMark PerryJonMark Perry
19.2k63991
$endgroup$
$begingroup$
I have a proof, but I'm on mobile. I think I'll just bail and give this one to you. Good luck ! 😀
$endgroup$
– Hugh
Feb 2 at 3:25
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So this is an example of proof by exhaustive search.
$endgroup$
– Dr Xorile
Feb 2 at 4:01
add a comment |
$begingroup$
@JonMarkPerry got it and indicated that he'd looked through all the possibilities. But to outline the proof, you can note that:
- The axis of symmetry must go through one of the 5 vertices (call it $A$)
- The other 4 vertices must be symmetric to each other about the axis of symmetry.
Now note that there are only 3 vertices to choose from for vertex $A$: The center, the edge, and the corner.
The center can have an orthogonal axis of symmetry or a diagonal one.
The edge and corner will be symmetric about the line through that vertex and the center vertex.
Putting this together, there are only four cases which leads to the 5 cases already identified:
- Orthogonal axis of symmetry through the center vertex: 1 possibility.
- Diagonal axis of symmetry through the center vertex: 1 possibility
- Orthogonal axis of symmetry through the edge vertex: 1 possibility
- Diagonal axis of symmetry through the corner vertex: 2 possibilities
answered Feb 2 at 4:20
Dr XorileDr Xorile
12.6k22569
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add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here are 5 symmetric pentagons on a $3times3$ grid:
A proof is to notice that there are only two lines of mirror symmetry on the grid, vertical (or horizontal by rotation) and diagonal. A line of symmetry can only contain 1 vertex, because the other vertices are mirrored, and 3 in a row is impossible (a vertex would have 3 edges). There are only 3 variations for each of the two possible LoS vertices, giving only 12 cases to test, and 12 ways to draw each one (12345 is the same as 23451 is the same as 54321), so there are only 144 possibilities to check.
answered Feb 2 at 3:23
JonMark PerryJonMark Perry
19.2k63991
$endgroup$
$begingroup$
I have a proof, but I'm on mobile. I think I'll just bail and give this one to you. Good luck ! 😀
$endgroup$
– Hugh
Feb 2 at 3:25
$begingroup$
So this is an example of proof by exhaustive search.
$endgroup$
– Dr Xorile
Feb 2 at 4:01
add a comment |
$begingroup$
Here are 5 symmetric pentagons on a $3times3$ grid:
A proof is to notice that there are only two lines of mirror symmetry on the grid, vertical (or horizontal by rotation) and diagonal. A line of symmetry can only contain 1 vertex, because the other vertices are mirrored, and 3 in a row is impossible (a vertex would have 3 edges). There are only 3 variations for each of the two possible LoS vertices, giving only 12 cases to test, and 12 ways to draw each one (12345 is the same as 23451 is the same as 54321), so there are only 144 possibilities to check.
answered Feb 2 at 3:23
JonMark PerryJonMark Perry
19.2k63991
$endgroup$
$begingroup$
I have a proof, but I'm on mobile. I think I'll just bail and give this one to you. Good luck ! 😀
$endgroup$
– Hugh
Feb 2 at 3:25
$begingroup$
So this is an example of proof by exhaustive search.
$endgroup$
– Dr Xorile
Feb 2 at 4:01
add a comment |
$begingroup$
Here are 5 symmetric pentagons on a $3times3$ grid:
A proof is to notice that there are only two lines of mirror symmetry on the grid, vertical (or horizontal by rotation) and diagonal. A line of symmetry can only contain 1 vertex, because the other vertices are mirrored, and 3 in a row is impossible (a vertex would have 3 edges). There are only 3 variations for each of the two possible LoS vertices, giving only 12 cases to test, and 12 ways to draw each one (12345 is the same as 23451 is the same as 54321), so there are only 144 possibilities to check.
answered Feb 2 at 3:23
JonMark PerryJonMark Perry
19.2k63991
$endgroup$
Here are 5 symmetric pentagons on a $3times3$ grid:
A proof is to notice that there are only two lines of mirror symmetry on the grid, vertical (or horizontal by rotation) and diagonal. A line of symmetry can only contain 1 vertex, because the other vertices are mirrored, and 3 in a row is impossible (a vertex would have 3 edges). There are only 3 variations for each of the two possible LoS vertices, giving only 12 cases to test, and 12 ways to draw each one (12345 is the same as 23451 is the same as 54321), so there are only 144 possibilities to check.
answered Feb 2 at 3:23
JonMark PerryJonMark Perry
19.2k63991
edited Feb 2 at 11:04
answered Feb 2 at 3:23
JonMark PerryJonMark Perry
19.2k63991
answered Feb 2 at 3:23
JonMark PerryJonMark Perry
19.2k63991
answered Feb 2 at 3:23
JonMark PerryJonMark Perry
19.2k63991
19.2k63991
$begingroup$
I have a proof, but I'm on mobile. I think I'll just bail and give this one to you. Good luck ! 😀
$endgroup$
– Hugh
Feb 2 at 3:25
$begingroup$
So this is an example of proof by exhaustive search.
$endgroup$
– Dr Xorile
Feb 2 at 4:01
add a comment |
$begingroup$
I have a proof, but I'm on mobile. I think I'll just bail and give this one to you. Good luck ! 😀
$endgroup$
– Hugh
Feb 2 at 3:25
$begingroup$
So this is an example of proof by exhaustive search.
$endgroup$
– Dr Xorile
Feb 2 at 4:01
$begingroup$
I have a proof, but I'm on mobile. I think I'll just bail and give this one to you. Good luck ! 😀
$endgroup$
– Hugh
Feb 2 at 3:25
$begingroup$
I have a proof, but I'm on mobile. I think I'll just bail and give this one to you. Good luck ! 😀
$endgroup$
– Hugh
Feb 2 at 3:25
$begingroup$
So this is an example of proof by exhaustive search.
$endgroup$
– Dr Xorile
Feb 2 at 4:01
$begingroup$
So this is an example of proof by exhaustive search.
$endgroup$
– Dr Xorile
Feb 2 at 4:01
add a comment |
$begingroup$
@JonMarkPerry got it and indicated that he'd looked through all the possibilities. But to outline the proof, you can note that:
- The axis of symmetry must go through one of the 5 vertices (call it $A$)
- The other 4 vertices must be symmetric to each other about the axis of symmetry.
Now note that there are only 3 vertices to choose from for vertex $A$: The center, the edge, and the corner.
The center can have an orthogonal axis of symmetry or a diagonal one.
The edge and corner will be symmetric about the line through that vertex and the center vertex.
Putting this together, there are only four cases which leads to the 5 cases already identified:
- Orthogonal axis of symmetry through the center vertex: 1 possibility.
- Diagonal axis of symmetry through the center vertex: 1 possibility
- Orthogonal axis of symmetry through the edge vertex: 1 possibility
- Diagonal axis of symmetry through the corner vertex: 2 possibilities
answered Feb 2 at 4:20
Dr XorileDr Xorile
12.6k22569
$endgroup$
add a comment |
$begingroup$
@JonMarkPerry got it and indicated that he'd looked through all the possibilities. But to outline the proof, you can note that:
- The axis of symmetry must go through one of the 5 vertices (call it $A$)
- The other 4 vertices must be symmetric to each other about the axis of symmetry.
Now note that there are only 3 vertices to choose from for vertex $A$: The center, the edge, and the corner.
The center can have an orthogonal axis of symmetry or a diagonal one.
The edge and corner will be symmetric about the line through that vertex and the center vertex.
Putting this together, there are only four cases which leads to the 5 cases already identified:
- Orthogonal axis of symmetry through the center vertex: 1 possibility.
- Diagonal axis of symmetry through the center vertex: 1 possibility
- Orthogonal axis of symmetry through the edge vertex: 1 possibility
- Diagonal axis of symmetry through the corner vertex: 2 possibilities
answered Feb 2 at 4:20
Dr XorileDr Xorile
12.6k22569
$endgroup$
add a comment |
$begingroup$
@JonMarkPerry got it and indicated that he'd looked through all the possibilities. But to outline the proof, you can note that:
- The axis of symmetry must go through one of the 5 vertices (call it $A$)
- The other 4 vertices must be symmetric to each other about the axis of symmetry.
Now note that there are only 3 vertices to choose from for vertex $A$: The center, the edge, and the corner.
The center can have an orthogonal axis of symmetry or a diagonal one.
The edge and corner will be symmetric about the line through that vertex and the center vertex.
Putting this together, there are only four cases which leads to the 5 cases already identified:
- Orthogonal axis of symmetry through the center vertex: 1 possibility.
- Diagonal axis of symmetry through the center vertex: 1 possibility
- Orthogonal axis of symmetry through the edge vertex: 1 possibility
- Diagonal axis of symmetry through the corner vertex: 2 possibilities
answered Feb 2 at 4:20
Dr XorileDr Xorile
12.6k22569
$endgroup$
@JonMarkPerry got it and indicated that he'd looked through all the possibilities. But to outline the proof, you can note that:
- The axis of symmetry must go through one of the 5 vertices (call it $A$)
- The other 4 vertices must be symmetric to each other about the axis of symmetry.
Now note that there are only 3 vertices to choose from for vertex $A$: The center, the edge, and the corner.
The center can have an orthogonal axis of symmetry or a diagonal one.
The edge and corner will be symmetric about the line through that vertex and the center vertex.
Putting this together, there are only four cases which leads to the 5 cases already identified:
- Orthogonal axis of symmetry through the center vertex: 1 possibility.
- Diagonal axis of symmetry through the center vertex: 1 possibility
- Orthogonal axis of symmetry through the edge vertex: 1 possibility
- Diagonal axis of symmetry through the corner vertex: 2 possibilities
answered Feb 2 at 4:20
Dr XorileDr Xorile
12.6k22569
edited Feb 2 at 5:04
answered Feb 2 at 4:20
Dr XorileDr Xorile
12.6k22569
answered Feb 2 at 4:20
Dr XorileDr Xorile
12.6k22569
answered Feb 2 at 4:20
Dr XorileDr Xorile
12.6k22569
12.6k22569
add a comment |
add a comment |
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$begingroup$
Hint: each pentagon has either a straight or diagonal line of symmetry.
$endgroup$
– Hugh
Feb 2 at 3:27