Why does the real projective plane / Boy surface look like this?
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In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901
My question is, you can see that the Boy surface is made up of three identical parts. But how does the number $3$ come up? I cannot see it in the definition of $mathbb{R}P^2$.
general-topology soft-question projective-geometry
asked Jan 12 at 9:54
JiuJiu
496113
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add a comment |
$begingroup$
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901
My question is, you can see that the Boy surface is made up of three identical parts. But how does the number $3$ come up? I cannot see it in the definition of $mathbb{R}P^2$.
general-topology soft-question projective-geometry
asked Jan 12 at 9:54
JiuJiu
496113
$endgroup$
add a comment |
$begingroup$
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901
My question is, you can see that the Boy surface is made up of three identical parts. But how does the number $3$ come up? I cannot see it in the definition of $mathbb{R}P^2$.
general-topology soft-question projective-geometry
asked Jan 12 at 9:54
JiuJiu
496113
$endgroup$
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901
My question is, you can see that the Boy surface is made up of three identical parts. But how does the number $3$ come up? I cannot see it in the definition of $mathbb{R}P^2$.
general-topology soft-question projective-geometry
general-topology soft-question projective-geometry
asked Jan 12 at 9:54
JiuJiu
496113
asked Jan 12 at 9:54
JiuJiu
496113
asked Jan 12 at 9:54
JiuJiu
496113
asked Jan 12 at 9:54
JiuJiu
496113
asked Jan 12 at 9:54
JiuJiu
496113
496113
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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There are probably many ways to answer this question, and I'm totally unqualified to do so but here is one thing: it is possible to construct versions of Boy's surface with 5-fold (and larger odd numbered) symmetry, (there are some lovely illustrations in Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces
- Francois Apery, you may have access to this at https://link.springer.com/content/pdf/bbm%3A978-3-322-89569-1%2F1.pdf)
So the question could instead be why does the rotational symmetry of a Boy's type immersion have to be odd, it's probably something to do with the non-orientability of the surface.
answered Jan 12 at 11:53
Alex J BestAlex J Best
2,12211225
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$begingroup$
Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like.
$endgroup$
– Jiu
Jan 12 at 12:23
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first link is partly missing, looking forward to seeing it! (see my profile image)
$endgroup$
– uhoh
Jan 12 at 14:30
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@uhuh thanks, fixed!
$endgroup$
– Alex J Best
Jan 12 at 16:34
add a comment |
$begingroup$
3 occurs in the usual definition of $RP^2$ as the set of lines in $R^3$. That is, the quotient space of $R^3-0$ that identifies $xsim cx$ for all nonzero $xin R^3$ and nonzero real $c$. The homeomorphism $(x_1,x_2,x_3)to(x_2,x_3,x_1)$
for example induces a threefold symmetry of $RP^2$.
answered Jan 12 at 12:27
Bob TerrellBob Terrell
1,705710
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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
$begingroup$
There are probably many ways to answer this question, and I'm totally unqualified to do so but here is one thing: it is possible to construct versions of Boy's surface with 5-fold (and larger odd numbered) symmetry, (there are some lovely illustrations in Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces
- Francois Apery, you may have access to this at https://link.springer.com/content/pdf/bbm%3A978-3-322-89569-1%2F1.pdf)
So the question could instead be why does the rotational symmetry of a Boy's type immersion have to be odd, it's probably something to do with the non-orientability of the surface.
answered Jan 12 at 11:53
Alex J BestAlex J Best
2,12211225
$endgroup$
$begingroup$
Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like.
$endgroup$
– Jiu
Jan 12 at 12:23
$begingroup$
first link is partly missing, looking forward to seeing it! (see my profile image)
$endgroup$
– uhoh
Jan 12 at 14:30
$begingroup$
@uhuh thanks, fixed!
$endgroup$
– Alex J Best
Jan 12 at 16:34
add a comment |
$begingroup$
There are probably many ways to answer this question, and I'm totally unqualified to do so but here is one thing: it is possible to construct versions of Boy's surface with 5-fold (and larger odd numbered) symmetry, (there are some lovely illustrations in Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces
- Francois Apery, you may have access to this at https://link.springer.com/content/pdf/bbm%3A978-3-322-89569-1%2F1.pdf)
So the question could instead be why does the rotational symmetry of a Boy's type immersion have to be odd, it's probably something to do with the non-orientability of the surface.
answered Jan 12 at 11:53
Alex J BestAlex J Best
2,12211225
$endgroup$
$begingroup$
Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like.
$endgroup$
– Jiu
Jan 12 at 12:23
$begingroup$
first link is partly missing, looking forward to seeing it! (see my profile image)
$endgroup$
– uhoh
Jan 12 at 14:30
$begingroup$
@uhuh thanks, fixed!
$endgroup$
– Alex J Best
Jan 12 at 16:34
add a comment |
$begingroup$
There are probably many ways to answer this question, and I'm totally unqualified to do so but here is one thing: it is possible to construct versions of Boy's surface with 5-fold (and larger odd numbered) symmetry, (there are some lovely illustrations in Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces
- Francois Apery, you may have access to this at https://link.springer.com/content/pdf/bbm%3A978-3-322-89569-1%2F1.pdf)
So the question could instead be why does the rotational symmetry of a Boy's type immersion have to be odd, it's probably something to do with the non-orientability of the surface.
answered Jan 12 at 11:53
Alex J BestAlex J Best
2,12211225
$endgroup$
There are probably many ways to answer this question, and I'm totally unqualified to do so but here is one thing: it is possible to construct versions of Boy's surface with 5-fold (and larger odd numbered) symmetry, (there are some lovely illustrations in Models of the Real Projective Plane: Computer Graphics of Steiner and Boy Surfaces
- Francois Apery, you may have access to this at https://link.springer.com/content/pdf/bbm%3A978-3-322-89569-1%2F1.pdf)
So the question could instead be why does the rotational symmetry of a Boy's type immersion have to be odd, it's probably something to do with the non-orientability of the surface.
answered Jan 12 at 11:53
Alex J BestAlex J Best
2,12211225
edited Jan 12 at 16:33
answered Jan 12 at 11:53
Alex J BestAlex J Best
2,12211225
answered Jan 12 at 11:53
Alex J BestAlex J Best
2,12211225
answered Jan 12 at 11:53
Alex J BestAlex J Best
2,12211225
2,12211225
$begingroup$
Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like.
$endgroup$
– Jiu
Jan 12 at 12:23
$begingroup$
first link is partly missing, looking forward to seeing it! (see my profile image)
$endgroup$
– uhoh
Jan 12 at 14:30
$begingroup$
@uhuh thanks, fixed!
$endgroup$
– Alex J Best
Jan 12 at 16:34
add a comment |
$begingroup$
Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like.
$endgroup$
– Jiu
Jan 12 at 12:23
$begingroup$
first link is partly missing, looking forward to seeing it! (see my profile image)
$endgroup$
– uhoh
Jan 12 at 14:30
$begingroup$
@uhuh thanks, fixed!
$endgroup$
– Alex J Best
Jan 12 at 16:34
$begingroup$
Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like.
$endgroup$
– Jiu
Jan 12 at 12:23
$begingroup$
Thanks! Many interesting pictures in the link. I have a hard time imagining what a mobius strip with circular boundary looks like.
$endgroup$
– Jiu
Jan 12 at 12:23
$begingroup$
first link is partly missing, looking forward to seeing it! (see my profile image)
$endgroup$
– uhoh
Jan 12 at 14:30
$begingroup$
first link is partly missing, looking forward to seeing it! (see my profile image)
$endgroup$
– uhoh
Jan 12 at 14:30
$begingroup$
@uhuh thanks, fixed!
$endgroup$
– Alex J Best
Jan 12 at 16:34
$begingroup$
@uhuh thanks, fixed!
$endgroup$
– Alex J Best
Jan 12 at 16:34
add a comment |
$begingroup$
3 occurs in the usual definition of $RP^2$ as the set of lines in $R^3$. That is, the quotient space of $R^3-0$ that identifies $xsim cx$ for all nonzero $xin R^3$ and nonzero real $c$. The homeomorphism $(x_1,x_2,x_3)to(x_2,x_3,x_1)$
for example induces a threefold symmetry of $RP^2$.
answered Jan 12 at 12:27
Bob TerrellBob Terrell
1,705710
$endgroup$
add a comment |
$begingroup$
3 occurs in the usual definition of $RP^2$ as the set of lines in $R^3$. That is, the quotient space of $R^3-0$ that identifies $xsim cx$ for all nonzero $xin R^3$ and nonzero real $c$. The homeomorphism $(x_1,x_2,x_3)to(x_2,x_3,x_1)$
for example induces a threefold symmetry of $RP^2$.
answered Jan 12 at 12:27
Bob TerrellBob Terrell
1,705710
$endgroup$
add a comment |
$begingroup$
3 occurs in the usual definition of $RP^2$ as the set of lines in $R^3$. That is, the quotient space of $R^3-0$ that identifies $xsim cx$ for all nonzero $xin R^3$ and nonzero real $c$. The homeomorphism $(x_1,x_2,x_3)to(x_2,x_3,x_1)$
for example induces a threefold symmetry of $RP^2$.
answered Jan 12 at 12:27
Bob TerrellBob Terrell
1,705710
$endgroup$
3 occurs in the usual definition of $RP^2$ as the set of lines in $R^3$. That is, the quotient space of $R^3-0$ that identifies $xsim cx$ for all nonzero $xin R^3$ and nonzero real $c$. The homeomorphism $(x_1,x_2,x_3)to(x_2,x_3,x_1)$
for example induces a threefold symmetry of $RP^2$.
answered Jan 12 at 12:27
Bob TerrellBob Terrell
1,705710
answered Jan 12 at 12:27
Bob TerrellBob Terrell
1,705710
answered Jan 12 at 12:27
Bob TerrellBob Terrell
1,705710
answered Jan 12 at 12:27
Bob TerrellBob Terrell
1,705710
1,705710
add a comment |
add a comment |
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