How many axis of symmetry of the cube are there?












2












$begingroup$


In my final mathematics test, I have a bonus question: How many axis of symmetry of the cube are there?



The teacher gives me the definition:




Definition: If we rotate a 3-dimension object around the line d for 180 degrees and it result in an exactly same shape in an exactly same position, line d is a axis of symmetry of that object.




I have found 9 axis of symmetry, 3 of which pass through the centers of 2 opposite faces, the other 6 pass through the midpoints of the 2 opposite edges. But my teacher told that 9 is wrong and said that the correct answer is not what we're going to expect.



So, what is the correct answer? And how can we prove it?






geometry rubiks-cube







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Octahedral_symmetry
    $endgroup$
    – vadim123
    May 10 '17 at 15:24










  • $begingroup$
    Your teacher must have misled you. The correct answer is 13.
    $endgroup$
    – Parcly Taxel
    May 10 '17 at 15:26






  • 1




    $begingroup$
    Are you sure only $180$ degree rotations are allowed? A cube has $3$-fold rotational symmetry about an axis which passes through two opposite corners.
    $endgroup$
    – kccu
    May 10 '17 at 15:28










  • $begingroup$
    @vadim123: Acually the definition of my teacher is not the same as usual. It's is related to rotation, and 'real' symmetry probably won't work.
    $endgroup$
    – Lê Đức Minh
    May 10 '17 at 15:28






  • 3




    $begingroup$
    There is something interesting. A cube has $12$ edges and $13 $($=12+1$) axes of symmetry. $6$ of them are of order $2$ (and $6times 2=12$), $4$ of them are of order $3$ (and $4times 3=12$) and $3$ of them are of order $4$ (and $3times 4=12$). A regular dodecahedron has $30$ edges and $31$ axes of symmetry. $15$ of them are of order $2$, $10$ of them are of order $3$ and $6$ of them are of order $5$. And we have $15times2=10times3=6times5=30$.
    $endgroup$
    – CY Aries
    May 10 '17 at 16:00
















2












$begingroup$


In my final mathematics test, I have a bonus question: How many axis of symmetry of the cube are there?



The teacher gives me the definition:




Definition: If we rotate a 3-dimension object around the line d for 180 degrees and it result in an exactly same shape in an exactly same position, line d is a axis of symmetry of that object.




I have found 9 axis of symmetry, 3 of which pass through the centers of 2 opposite faces, the other 6 pass through the midpoints of the 2 opposite edges. But my teacher told that 9 is wrong and said that the correct answer is not what we're going to expect.



So, what is the correct answer? And how can we prove it?






geometry rubiks-cube







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Octahedral_symmetry
    $endgroup$
    – vadim123
    May 10 '17 at 15:24










  • $begingroup$
    Your teacher must have misled you. The correct answer is 13.
    $endgroup$
    – Parcly Taxel
    May 10 '17 at 15:26






  • 1




    $begingroup$
    Are you sure only $180$ degree rotations are allowed? A cube has $3$-fold rotational symmetry about an axis which passes through two opposite corners.
    $endgroup$
    – kccu
    May 10 '17 at 15:28










  • $begingroup$
    @vadim123: Acually the definition of my teacher is not the same as usual. It's is related to rotation, and 'real' symmetry probably won't work.
    $endgroup$
    – Lê Đức Minh
    May 10 '17 at 15:28






  • 3




    $begingroup$
    There is something interesting. A cube has $12$ edges and $13 $($=12+1$) axes of symmetry. $6$ of them are of order $2$ (and $6times 2=12$), $4$ of them are of order $3$ (and $4times 3=12$) and $3$ of them are of order $4$ (and $3times 4=12$). A regular dodecahedron has $30$ edges and $31$ axes of symmetry. $15$ of them are of order $2$, $10$ of them are of order $3$ and $6$ of them are of order $5$. And we have $15times2=10times3=6times5=30$.
    $endgroup$
    – CY Aries
    May 10 '17 at 16:00














2












2








2





$begingroup$


In my final mathematics test, I have a bonus question: How many axis of symmetry of the cube are there?



The teacher gives me the definition:




Definition: If we rotate a 3-dimension object around the line d for 180 degrees and it result in an exactly same shape in an exactly same position, line d is a axis of symmetry of that object.




I have found 9 axis of symmetry, 3 of which pass through the centers of 2 opposite faces, the other 6 pass through the midpoints of the 2 opposite edges. But my teacher told that 9 is wrong and said that the correct answer is not what we're going to expect.



So, what is the correct answer? And how can we prove it?






geometry rubiks-cube







share|cite|improve this question











$endgroup$




In my final mathematics test, I have a bonus question: How many axis of symmetry of the cube are there?



The teacher gives me the definition:




Definition: If we rotate a 3-dimension object around the line d for 180 degrees and it result in an exactly same shape in an exactly same position, line d is a axis of symmetry of that object.




I have found 9 axis of symmetry, 3 of which pass through the centers of 2 opposite faces, the other 6 pass through the midpoints of the 2 opposite edges. But my teacher told that 9 is wrong and said that the correct answer is not what we're going to expect.



So, what is the correct answer? And how can we prove it?






geometry rubiks-cube




geometry rubiks-cube






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited May 10 '17 at 15:37







Lê Đức Minh

















asked May 10 '17 at 15:22









Lê Đức MinhLê Đức Minh

197117




197117








  • 1




    $begingroup$
    en.wikipedia.org/wiki/Octahedral_symmetry
    $endgroup$
    – vadim123
    May 10 '17 at 15:24










  • $begingroup$
    Your teacher must have misled you. The correct answer is 13.
    $endgroup$
    – Parcly Taxel
    May 10 '17 at 15:26






  • 1




    $begingroup$
    Are you sure only $180$ degree rotations are allowed? A cube has $3$-fold rotational symmetry about an axis which passes through two opposite corners.
    $endgroup$
    – kccu
    May 10 '17 at 15:28










  • $begingroup$
    @vadim123: Acually the definition of my teacher is not the same as usual. It's is related to rotation, and 'real' symmetry probably won't work.
    $endgroup$
    – Lê Đức Minh
    May 10 '17 at 15:28






  • 3




    $begingroup$
    There is something interesting. A cube has $12$ edges and $13 $($=12+1$) axes of symmetry. $6$ of them are of order $2$ (and $6times 2=12$), $4$ of them are of order $3$ (and $4times 3=12$) and $3$ of them are of order $4$ (and $3times 4=12$). A regular dodecahedron has $30$ edges and $31$ axes of symmetry. $15$ of them are of order $2$, $10$ of them are of order $3$ and $6$ of them are of order $5$. And we have $15times2=10times3=6times5=30$.
    $endgroup$
    – CY Aries
    May 10 '17 at 16:00














  • 1




    $begingroup$
    en.wikipedia.org/wiki/Octahedral_symmetry
    $endgroup$
    – vadim123
    May 10 '17 at 15:24










  • $begingroup$
    Your teacher must have misled you. The correct answer is 13.
    $endgroup$
    – Parcly Taxel
    May 10 '17 at 15:26






  • 1




    $begingroup$
    Are you sure only $180$ degree rotations are allowed? A cube has $3$-fold rotational symmetry about an axis which passes through two opposite corners.
    $endgroup$
    – kccu
    May 10 '17 at 15:28










  • $begingroup$
    @vadim123: Acually the definition of my teacher is not the same as usual. It's is related to rotation, and 'real' symmetry probably won't work.
    $endgroup$
    – Lê Đức Minh
    May 10 '17 at 15:28






  • 3




    $begingroup$
    There is something interesting. A cube has $12$ edges and $13 $($=12+1$) axes of symmetry. $6$ of them are of order $2$ (and $6times 2=12$), $4$ of them are of order $3$ (and $4times 3=12$) and $3$ of them are of order $4$ (and $3times 4=12$). A regular dodecahedron has $30$ edges and $31$ axes of symmetry. $15$ of them are of order $2$, $10$ of them are of order $3$ and $6$ of them are of order $5$. And we have $15times2=10times3=6times5=30$.
    $endgroup$
    – CY Aries
    May 10 '17 at 16:00








1




1




$begingroup$
en.wikipedia.org/wiki/Octahedral_symmetry
$endgroup$
– vadim123
May 10 '17 at 15:24




$begingroup$
en.wikipedia.org/wiki/Octahedral_symmetry
$endgroup$
– vadim123
May 10 '17 at 15:24












$begingroup$
Your teacher must have misled you. The correct answer is 13.
$endgroup$
– Parcly Taxel
May 10 '17 at 15:26




$begingroup$
Your teacher must have misled you. The correct answer is 13.
$endgroup$
– Parcly Taxel
May 10 '17 at 15:26




1




1




$begingroup$
Are you sure only $180$ degree rotations are allowed? A cube has $3$-fold rotational symmetry about an axis which passes through two opposite corners.
$endgroup$
– kccu
May 10 '17 at 15:28




$begingroup$
Are you sure only $180$ degree rotations are allowed? A cube has $3$-fold rotational symmetry about an axis which passes through two opposite corners.
$endgroup$
– kccu
May 10 '17 at 15:28












$begingroup$
@vadim123: Acually the definition of my teacher is not the same as usual. It's is related to rotation, and 'real' symmetry probably won't work.
$endgroup$
– Lê Đức Minh
May 10 '17 at 15:28




$begingroup$
@vadim123: Acually the definition of my teacher is not the same as usual. It's is related to rotation, and 'real' symmetry probably won't work.
$endgroup$
– Lê Đức Minh
May 10 '17 at 15:28




3




3




$begingroup$
There is something interesting. A cube has $12$ edges and $13 $($=12+1$) axes of symmetry. $6$ of them are of order $2$ (and $6times 2=12$), $4$ of them are of order $3$ (and $4times 3=12$) and $3$ of them are of order $4$ (and $3times 4=12$). A regular dodecahedron has $30$ edges and $31$ axes of symmetry. $15$ of them are of order $2$, $10$ of them are of order $3$ and $6$ of them are of order $5$. And we have $15times2=10times3=6times5=30$.
$endgroup$
– CY Aries
May 10 '17 at 16:00




$begingroup$
There is something interesting. A cube has $12$ edges and $13 $($=12+1$) axes of symmetry. $6$ of them are of order $2$ (and $6times 2=12$), $4$ of them are of order $3$ (and $4times 3=12$) and $3$ of them are of order $4$ (and $3times 4=12$). A regular dodecahedron has $30$ edges and $31$ axes of symmetry. $15$ of them are of order $2$, $10$ of them are of order $3$ and $6$ of them are of order $5$. And we have $15times2=10times3=6times5=30$.
$endgroup$
– CY Aries
May 10 '17 at 16:00










1 Answer
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$begingroup$

An axis of symmetry can only passes through



(1) mid-points of two opposite edges. (As a cube has 12 edges, there are $12div2=6$ axes of this type.)



(2) two opposite vertices. (As a cube has 8 vertices, there are $8div2=4$ axes of this type.)



(3) the centres of two opposite faces. (As a cube has 6 faces, there are $6div2=3$ axes of this type.)



So it has $13$ axes of symmetry.



Note: The number of symmetry is equal to



$$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$






share|cite|improve this answer









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    active

    oldest

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    active

    oldest

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    2












    $begingroup$

    An axis of symmetry can only passes through



    (1) mid-points of two opposite edges. (As a cube has 12 edges, there are $12div2=6$ axes of this type.)



    (2) two opposite vertices. (As a cube has 8 vertices, there are $8div2=4$ axes of this type.)



    (3) the centres of two opposite faces. (As a cube has 6 faces, there are $6div2=3$ axes of this type.)



    So it has $13$ axes of symmetry.



    Note: The number of symmetry is equal to



    $$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      An axis of symmetry can only passes through



      (1) mid-points of two opposite edges. (As a cube has 12 edges, there are $12div2=6$ axes of this type.)



      (2) two opposite vertices. (As a cube has 8 vertices, there are $8div2=4$ axes of this type.)



      (3) the centres of two opposite faces. (As a cube has 6 faces, there are $6div2=3$ axes of this type.)



      So it has $13$ axes of symmetry.



      Note: The number of symmetry is equal to



      $$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        An axis of symmetry can only passes through



        (1) mid-points of two opposite edges. (As a cube has 12 edges, there are $12div2=6$ axes of this type.)



        (2) two opposite vertices. (As a cube has 8 vertices, there are $8div2=4$ axes of this type.)



        (3) the centres of two opposite faces. (As a cube has 6 faces, there are $6div2=3$ axes of this type.)



        So it has $13$ axes of symmetry.



        Note: The number of symmetry is equal to



        $$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$






        share|cite|improve this answer









        $endgroup$



        An axis of symmetry can only passes through



        (1) mid-points of two opposite edges. (As a cube has 12 edges, there are $12div2=6$ axes of this type.)



        (2) two opposite vertices. (As a cube has 8 vertices, there are $8div2=4$ axes of this type.)



        (3) the centres of two opposite faces. (As a cube has 6 faces, there are $6div2=3$ axes of this type.)



        So it has $13$ axes of symmetry.



        Note: The number of symmetry is equal to



        $$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered May 10 '17 at 16:06









        CY AriesCY Aries

        17k11743




        17k11743






























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