How many axis of symmetry of the cube are there?
$begingroup$
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
asked May 10 '17 at 15:22
Lê Đức MinhLê Đức Minh
197117
$endgroup$
|
show 9 more comments
$begingroup$
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
asked May 10 '17 at 15:22
Lê Đức MinhLê Đức Minh
197117
$endgroup$
1
$begingroup$
en.wikipedia.org/wiki/Octahedral_symmetry
$endgroup$
– vadim123
May 10 '17 at 15:24
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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
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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
|
show 9 more comments
$begingroup$
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
asked May 10 '17 at 15:22
Lê Đức MinhLê Đức Minh
197117
$endgroup$
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
asked May 10 '17 at 15:22
Lê Đức MinhLê Đức Minh
197117
asked May 10 '17 at 15:22
Lê Đức MinhLê Đức Minh
197117
edited May 10 '17 at 15:37
Lê Đức Minh
asked May 10 '17 at 15:22
Lê Đức MinhLê Đức Minh
197117
asked May 10 '17 at 15:22
Lê Đức MinhLê Đức Minh
197117
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
|
show 9 more comments
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
|
show 9 more comments
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
$$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$
answered May 10 '17 at 16:06
CY AriesCY Aries
17k11743
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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
$$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$
answered May 10 '17 at 16:06
CY AriesCY Aries
17k11743
$endgroup$
add a comment |
$begingroup$
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
$$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$
answered May 10 '17 at 16:06
CY AriesCY Aries
17k11743
$endgroup$
add a comment |
$begingroup$
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
$$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$
answered May 10 '17 at 16:06
CY AriesCY Aries
17k11743
$endgroup$
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
$$frac{E}{2}+frac{V}{2}+frac{F}{2}=frac{E}{2}+frac{E+2}{2}=E+1$$
answered May 10 '17 at 16:06
CY AriesCY Aries
17k11743
answered May 10 '17 at 16:06
CY AriesCY Aries
17k11743
answered May 10 '17 at 16:06
CY AriesCY Aries
17k11743
answered May 10 '17 at 16:06
CY AriesCY Aries
17k11743
17k11743
add a comment |
add a comment |
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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