Mixing
What shapes do these quotients represent? Do they have a name?
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I am unable to visualize complex shapes in space. I can draw simpler shapes and explain which points correspond to which but can't for the life of me determine the name of the shape. I would like somebody to help me identify those. I will present my thought process through a picture since it is very difficult to explain with words.
I am given the quotients $mathbb{R}/mathbb{Z} times[0,1]/sim$ where $sim$ denotes $(t,0) sim (-t,1)$ and $mathbb{R}/mathbb{Z} times[0,1]/sim'$ where $sim'$ denotes $(t,0) sim (t,1)$.
I think the first quotient represents a cylinder whose opposite edges at extremities are "identified", i.e. it looks kind of a Möbius strip but I don't think it is the Möbius strip because of instead of having sides as extremities there are circles. I have made a drawing to see things better but I still can't identify the correct shape. Maybe some kind of 8-looking strip that's not Möbius?
The second one seems easier and I guess it's just the torus... Am I right?
For the first one:
if I try to "identify" the circles in my head I just get some kind of strip that intersects itself like an "8" strip. Does this shape have a name? I have low spatial IQ
geometry geometric-topology
asked Jan 30 at 17:50
EvaristeEvariste
1,1862618
$endgroup$
add a comment |
$begingroup$
I am unable to visualize complex shapes in space. I can draw simpler shapes and explain which points correspond to which but can't for the life of me determine the name of the shape. I would like somebody to help me identify those. I will present my thought process through a picture since it is very difficult to explain with words.
I am given the quotients $mathbb{R}/mathbb{Z} times[0,1]/sim$ where $sim$ denotes $(t,0) sim (-t,1)$ and $mathbb{R}/mathbb{Z} times[0,1]/sim'$ where $sim'$ denotes $(t,0) sim (t,1)$.
I think the first quotient represents a cylinder whose opposite edges at extremities are "identified", i.e. it looks kind of a Möbius strip but I don't think it is the Möbius strip because of instead of having sides as extremities there are circles. I have made a drawing to see things better but I still can't identify the correct shape. Maybe some kind of 8-looking strip that's not Möbius?
The second one seems easier and I guess it's just the torus... Am I right?
For the first one:
if I try to "identify" the circles in my head I just get some kind of strip that intersects itself like an "8" strip. Does this shape have a name? I have low spatial IQ
geometry geometric-topology
asked Jan 30 at 17:50
EvaristeEvariste
1,1862618
$endgroup$
5
$begingroup$
This is better known as the Klein bottle. It has no boundary.
$endgroup$
– user98602
Jan 30 at 18:01
$begingroup$
@MikeMiller I did look at the Klein bottle before posting, but still couldn't see how circles were oppositely-identified. Now that I know the answer, I see it. (I didn't pay enough attention to the fact that the "bend" essentially makes this cross-identification I couldn't put my finger on...) Well thank you, I guess this qualifies as an answer, however short...
$endgroup$
– Evariste
Jan 30 at 18:06
1
$begingroup$
I would have drawn pictures if I had the capability :) Maybe you will be able to do so and post an answer. I would upvote.
$endgroup$
– user98602
Jan 30 at 18:11
$begingroup$
@MikeMiller A Klein bottle on paint will look like no Klein bottle, might as well quote or link en.wikipedia.org/wiki/Klein_bottle, still I can't answer my own question if somebody else just gave me the answer
$endgroup$
– Evariste
Jan 30 at 18:17
add a comment |
$begingroup$
I am unable to visualize complex shapes in space. I can draw simpler shapes and explain which points correspond to which but can't for the life of me determine the name of the shape. I would like somebody to help me identify those. I will present my thought process through a picture since it is very difficult to explain with words.
I am given the quotients $mathbb{R}/mathbb{Z} times[0,1]/sim$ where $sim$ denotes $(t,0) sim (-t,1)$ and $mathbb{R}/mathbb{Z} times[0,1]/sim'$ where $sim'$ denotes $(t,0) sim (t,1)$.
I think the first quotient represents a cylinder whose opposite edges at extremities are "identified", i.e. it looks kind of a Möbius strip but I don't think it is the Möbius strip because of instead of having sides as extremities there are circles. I have made a drawing to see things better but I still can't identify the correct shape. Maybe some kind of 8-looking strip that's not Möbius?
The second one seems easier and I guess it's just the torus... Am I right?
For the first one:
if I try to "identify" the circles in my head I just get some kind of strip that intersects itself like an "8" strip. Does this shape have a name? I have low spatial IQ
geometry geometric-topology
asked Jan 30 at 17:50
EvaristeEvariste
1,1862618
$endgroup$
I am unable to visualize complex shapes in space. I can draw simpler shapes and explain which points correspond to which but can't for the life of me determine the name of the shape. I would like somebody to help me identify those. I will present my thought process through a picture since it is very difficult to explain with words.
I am given the quotients $mathbb{R}/mathbb{Z} times[0,1]/sim$ where $sim$ denotes $(t,0) sim (-t,1)$ and $mathbb{R}/mathbb{Z} times[0,1]/sim'$ where $sim'$ denotes $(t,0) sim (t,1)$.
I think the first quotient represents a cylinder whose opposite edges at extremities are "identified", i.e. it looks kind of a Möbius strip but I don't think it is the Möbius strip because of instead of having sides as extremities there are circles. I have made a drawing to see things better but I still can't identify the correct shape. Maybe some kind of 8-looking strip that's not Möbius?
The second one seems easier and I guess it's just the torus... Am I right?
For the first one:
if I try to "identify" the circles in my head I just get some kind of strip that intersects itself like an "8" strip. Does this shape have a name? I have low spatial IQ
geometry geometric-topology
geometry geometric-topology
asked Jan 30 at 17:50
EvaristeEvariste
1,1862618
asked Jan 30 at 17:50
EvaristeEvariste
1,1862618
edited Jan 30 at 17:57
Evariste
asked Jan 30 at 17:50
EvaristeEvariste
1,1862618
asked Jan 30 at 17:50
EvaristeEvariste
1,1862618
asked Jan 30 at 17:50
EvaristeEvariste
1,1862618
1,1862618
5
$begingroup$
This is better known as the Klein bottle. It has no boundary.
$endgroup$
– user98602
Jan 30 at 18:01
$begingroup$
@MikeMiller I did look at the Klein bottle before posting, but still couldn't see how circles were oppositely-identified. Now that I know the answer, I see it. (I didn't pay enough attention to the fact that the "bend" essentially makes this cross-identification I couldn't put my finger on...) Well thank you, I guess this qualifies as an answer, however short...
$endgroup$
– Evariste
Jan 30 at 18:06
1
$begingroup$
I would have drawn pictures if I had the capability :) Maybe you will be able to do so and post an answer. I would upvote.
$endgroup$
– user98602
Jan 30 at 18:11
$begingroup$
@MikeMiller A Klein bottle on paint will look like no Klein bottle, might as well quote or link en.wikipedia.org/wiki/Klein_bottle, still I can't answer my own question if somebody else just gave me the answer
$endgroup$
– Evariste
Jan 30 at 18:17
add a comment |
5
$begingroup$
This is better known as the Klein bottle. It has no boundary.
$endgroup$
– user98602
Jan 30 at 18:01
$begingroup$
@MikeMiller I did look at the Klein bottle before posting, but still couldn't see how circles were oppositely-identified. Now that I know the answer, I see it. (I didn't pay enough attention to the fact that the "bend" essentially makes this cross-identification I couldn't put my finger on...) Well thank you, I guess this qualifies as an answer, however short...
$endgroup$
– Evariste
Jan 30 at 18:06
1
$begingroup$
I would have drawn pictures if I had the capability :) Maybe you will be able to do so and post an answer. I would upvote.
$endgroup$
– user98602
Jan 30 at 18:11
$begingroup$
@MikeMiller A Klein bottle on paint will look like no Klein bottle, might as well quote or link en.wikipedia.org/wiki/Klein_bottle, still I can't answer my own question if somebody else just gave me the answer
$endgroup$
– Evariste
Jan 30 at 18:17
5
5
$begingroup$
This is better known as the Klein bottle. It has no boundary.
$endgroup$
– user98602
Jan 30 at 18:01
$begingroup$
This is better known as the Klein bottle. It has no boundary.
$endgroup$
– user98602
Jan 30 at 18:01
$begingroup$
@MikeMiller I did look at the Klein bottle before posting, but still couldn't see how circles were oppositely-identified. Now that I know the answer, I see it. (I didn't pay enough attention to the fact that the "bend" essentially makes this cross-identification I couldn't put my finger on...) Well thank you, I guess this qualifies as an answer, however short...
$endgroup$
– Evariste
Jan 30 at 18:06
$begingroup$
@MikeMiller I did look at the Klein bottle before posting, but still couldn't see how circles were oppositely-identified. Now that I know the answer, I see it. (I didn't pay enough attention to the fact that the "bend" essentially makes this cross-identification I couldn't put my finger on...) Well thank you, I guess this qualifies as an answer, however short...
$endgroup$
– Evariste
Jan 30 at 18:06
1
1
$begingroup$
I would have drawn pictures if I had the capability :) Maybe you will be able to do so and post an answer. I would upvote.
$endgroup$
– user98602
Jan 30 at 18:11
$begingroup$
I would have drawn pictures if I had the capability :) Maybe you will be able to do so and post an answer. I would upvote.
$endgroup$
– user98602
Jan 30 at 18:11
$begingroup$
@MikeMiller A Klein bottle on paint will look like no Klein bottle, might as well quote or link en.wikipedia.org/wiki/Klein_bottle, still I can't answer my own question if somebody else just gave me the answer
$endgroup$
– Evariste
Jan 30 at 18:17
$begingroup$
@MikeMiller A Klein bottle on paint will look like no Klein bottle, might as well quote or link en.wikipedia.org/wiki/Klein_bottle, still I can't answer my own question if somebody else just gave me the answer
$endgroup$
– Evariste
Jan 30 at 18:17
add a comment |
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5
$begingroup$
This is better known as the Klein bottle. It has no boundary.
$endgroup$
– user98602
Jan 30 at 18:01
$begingroup$
@MikeMiller I did look at the Klein bottle before posting, but still couldn't see how circles were oppositely-identified. Now that I know the answer, I see it. (I didn't pay enough attention to the fact that the "bend" essentially makes this cross-identification I couldn't put my finger on...) Well thank you, I guess this qualifies as an answer, however short...
$endgroup$
– Evariste
Jan 30 at 18:06
1
$begingroup$
I would have drawn pictures if I had the capability :) Maybe you will be able to do so and post an answer. I would upvote.
$endgroup$
– user98602
Jan 30 at 18:11
$begingroup$
@MikeMiller A Klein bottle on paint will look like no Klein bottle, might as well quote or link en.wikipedia.org/wiki/Klein_bottle, still I can't answer my own question if somebody else just gave me the answer
$endgroup$
– Evariste
Jan 30 at 18:17