Can a curve intersect 0 to inf without crossing inself?
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Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?
Wouldn't accomplishing this feat mean infinitely approaching every point on the numberline? Can that be done without a straight horizontal line at y=0?
If so what would stop you from creating a function that does the same, not to just a numberline but a 2 dimensional plane?
If such a line was attempted, isn't there an infinite about of space between all points that a line should never need to cross itself to get out of a spiral it's created?
In order to do this would you need an infinitely complex function? If so is it possible to prove that the function to do this could in principle exist, but we able to prove that since it must be infinitely complex the function can't be known?
In short, is it possible to intersect every infinite point on a plane once and only once with a continuous line?
Am I missing something fundamental about lines and infinities?
Has this kind of question been explored before?
intersection-theory
asked Aug 31 at 4:45
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1216
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show 3 more comments
up vote
1
down vote
favorite
Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?
Wouldn't accomplishing this feat mean infinitely approaching every point on the numberline? Can that be done without a straight horizontal line at y=0?
If so what would stop you from creating a function that does the same, not to just a numberline but a 2 dimensional plane?
If such a line was attempted, isn't there an infinite about of space between all points that a line should never need to cross itself to get out of a spiral it's created?
In order to do this would you need an infinitely complex function? If so is it possible to prove that the function to do this could in principle exist, but we able to prove that since it must be infinitely complex the function can't be known?
In short, is it possible to intersect every infinite point on a plane once and only once with a continuous line?
Am I missing something fundamental about lines and infinities?
Has this kind of question been explored before?
intersection-theory
asked Aug 31 at 4:45
Legit Stack
1216
1
The answer to the question in your first paragraph, "Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?", is just the line along the $x$-axis. For the rest of your question, you may enjoy reading about space-filling curves.
– Rahul
Aug 31 at 5:14
1
It is not clear to me that the $x$-axis is the only such curve even if we ask for something as strong as continuity.
– Fimpellizieri
Aug 31 at 6:05
1
I can imagine the function $f(x) = sin(x/C)$, where $C$ is a constant. Now I take the limit of $C$ to $+0$. You end up with an "infinitely fast" oscillating function that intersects the x-axis "everywhere". Is that the kind of answer you were looking for ?
– M. Wind
Aug 31 at 6:06
1
Do I understand correctly that you are looking for a continuous bijection $mathbbR to mathbbR^2$?
– Paul Frost
Aug 31 at 7:43
1
A continuous space filling curve that is also injective would be a continuous bijection $f:mathbb Rlongrightarrow mathbb R^2$. This is impossible; see this answer.
– Fimpellizieri
Sep 1 at 6:49
 |Â
show 3 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?
Wouldn't accomplishing this feat mean infinitely approaching every point on the numberline? Can that be done without a straight horizontal line at y=0?
If so what would stop you from creating a function that does the same, not to just a numberline but a 2 dimensional plane?
If such a line was attempted, isn't there an infinite about of space between all points that a line should never need to cross itself to get out of a spiral it's created?
In order to do this would you need an infinitely complex function? If so is it possible to prove that the function to do this could in principle exist, but we able to prove that since it must be infinitely complex the function can't be known?
In short, is it possible to intersect every infinite point on a plane once and only once with a continuous line?
Am I missing something fundamental about lines and infinities?
Has this kind of question been explored before?
intersection-theory
asked Aug 31 at 4:45
Legit Stack
1216
Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?
Wouldn't accomplishing this feat mean infinitely approaching every point on the numberline? Can that be done without a straight horizontal line at y=0?
If so what would stop you from creating a function that does the same, not to just a numberline but a 2 dimensional plane?
If such a line was attempted, isn't there an infinite about of space between all points that a line should never need to cross itself to get out of a spiral it's created?
In order to do this would you need an infinitely complex function? If so is it possible to prove that the function to do this could in principle exist, but we able to prove that since it must be infinitely complex the function can't be known?
In short, is it possible to intersect every infinite point on a plane once and only once with a continuous line?
Am I missing something fundamental about lines and infinities?
Has this kind of question been explored before?
intersection-theory
intersection-theory
asked Aug 31 at 4:45
Legit Stack
1216
asked Aug 31 at 4:45
Legit Stack
1216
edited Aug 31 at 5:08
asked Aug 31 at 4:45
Legit Stack
1216
asked Aug 31 at 4:45
Legit Stack
1216
asked Aug 31 at 4:45
Legit Stack
1216
1216
1
The answer to the question in your first paragraph, "Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?", is just the line along the $x$-axis. For the rest of your question, you may enjoy reading about space-filling curves.
– Rahul
Aug 31 at 5:14
1
It is not clear to me that the $x$-axis is the only such curve even if we ask for something as strong as continuity.
– Fimpellizieri
Aug 31 at 6:05
1
I can imagine the function $f(x) = sin(x/C)$, where $C$ is a constant. Now I take the limit of $C$ to $+0$. You end up with an "infinitely fast" oscillating function that intersects the x-axis "everywhere". Is that the kind of answer you were looking for ?
– M. Wind
Aug 31 at 6:06
1
Do I understand correctly that you are looking for a continuous bijection $mathbbR to mathbbR^2$?
– Paul Frost
Aug 31 at 7:43
1
A continuous space filling curve that is also injective would be a continuous bijection $f:mathbb Rlongrightarrow mathbb R^2$. This is impossible; see this answer.
– Fimpellizieri
Sep 1 at 6:49
 |Â
show 3 more comments
1
The answer to the question in your first paragraph, "Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?", is just the line along the $x$-axis. For the rest of your question, you may enjoy reading about space-filling curves.
– Rahul
Aug 31 at 5:14
1
It is not clear to me that the $x$-axis is the only such curve even if we ask for something as strong as continuity.
– Fimpellizieri
Aug 31 at 6:05
1
I can imagine the function $f(x) = sin(x/C)$, where $C$ is a constant. Now I take the limit of $C$ to $+0$. You end up with an "infinitely fast" oscillating function that intersects the x-axis "everywhere". Is that the kind of answer you were looking for ?
– M. Wind
Aug 31 at 6:06
1
Do I understand correctly that you are looking for a continuous bijection $mathbbR to mathbbR^2$?
– Paul Frost
Aug 31 at 7:43
1
A continuous space filling curve that is also injective would be a continuous bijection $f:mathbb Rlongrightarrow mathbb R^2$. This is impossible; see this answer.
– Fimpellizieri
Sep 1 at 6:49
1
1
The answer to the question in your first paragraph, "Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?", is just the line along the $x$-axis. For the rest of your question, you may enjoy reading about space-filling curves.
– Rahul
Aug 31 at 5:14
The answer to the question in your first paragraph, "Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?", is just the line along the $x$-axis. For the rest of your question, you may enjoy reading about space-filling curves.
– Rahul
Aug 31 at 5:14
1
1
It is not clear to me that the $x$-axis is the only such curve even if we ask for something as strong as continuity.
– Fimpellizieri
Aug 31 at 6:05
It is not clear to me that the $x$-axis is the only such curve even if we ask for something as strong as continuity.
– Fimpellizieri
Aug 31 at 6:05
1
1
I can imagine the function $f(x) = sin(x/C)$, where $C$ is a constant. Now I take the limit of $C$ to $+0$. You end up with an "infinitely fast" oscillating function that intersects the x-axis "everywhere". Is that the kind of answer you were looking for ?
– M. Wind
Aug 31 at 6:06
I can imagine the function $f(x) = sin(x/C)$, where $C$ is a constant. Now I take the limit of $C$ to $+0$. You end up with an "infinitely fast" oscillating function that intersects the x-axis "everywhere". Is that the kind of answer you were looking for ?
– M. Wind
Aug 31 at 6:06
1
1
Do I understand correctly that you are looking for a continuous bijection $mathbbR to mathbbR^2$?
– Paul Frost
Aug 31 at 7:43
Do I understand correctly that you are looking for a continuous bijection $mathbbR to mathbbR^2$?
– Paul Frost
Aug 31 at 7:43
1
1
A continuous space filling curve that is also injective would be a continuous bijection $f:mathbb Rlongrightarrow mathbb R^2$. This is impossible; see this answer.
– Fimpellizieri
Sep 1 at 6:49
A continuous space filling curve that is also injective would be a continuous bijection $f:mathbb Rlongrightarrow mathbb R^2$. This is impossible; see this answer.
– Fimpellizieri
Sep 1 at 6:49
 |Â
show 3 more comments
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1
The answer to the question in your first paragraph, "Can you draw a curve such that it intersects every x at y=0 without ever crossing over itself (that is to say, without hitting the same value twice)?", is just the line along the $x$-axis. For the rest of your question, you may enjoy reading about space-filling curves.
– Rahul
Aug 31 at 5:14
1
It is not clear to me that the $x$-axis is the only such curve even if we ask for something as strong as continuity.
– Fimpellizieri
Aug 31 at 6:05
1
I can imagine the function $f(x) = sin(x/C)$, where $C$ is a constant. Now I take the limit of $C$ to $+0$. You end up with an "infinitely fast" oscillating function that intersects the x-axis "everywhere". Is that the kind of answer you were looking for ?
– M. Wind
Aug 31 at 6:06
1
Do I understand correctly that you are looking for a continuous bijection $mathbbR to mathbbR^2$?
– Paul Frost
Aug 31 at 7:43
1
A continuous space filling curve that is also injective would be a continuous bijection $f:mathbb Rlongrightarrow mathbb R^2$. This is impossible; see this answer.
– Fimpellizieri
Sep 1 at 6:49