Is there a surface on which a hexagon can have all right angles?
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27
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So I was watching a video that features astronomer and topologist Cliff Stoll talking about how figures that aren't quadrilaterals can have all their angles equal 90 degrees on different surfaces. For example, on a sphere, you can create a triangle that has all of its angles equal $90^circ$. On a pseudosphere, you can create a pentagon that has all of its angles equal $90^circ$. Now, here's my question.
Is there a surface where a hexagon with this property is possible?
geometry
edited Dec 2 at 10:57
amWhy
191k27223439
asked Nov 30 at 1:35
Xavier Stanton
315210
|
show 2 more comments
up vote
27
down vote
favorite
So I was watching a video that features astronomer and topologist Cliff Stoll talking about how figures that aren't quadrilaterals can have all their angles equal 90 degrees on different surfaces. For example, on a sphere, you can create a triangle that has all of its angles equal $90^circ$. On a pseudosphere, you can create a pentagon that has all of its angles equal $90^circ$. Now, here's my question.
Is there a surface where a hexagon with this property is possible?
geometry
edited Dec 2 at 10:57
amWhy
191k27223439
asked Nov 30 at 1:35
Xavier Stanton
315210
7
On hyerbolic surfaces you can have a triangle where the sum of the angles is 0. And hexagons with 6 right angles.
– Doug M
Nov 30 at 1:38
you can find many questions on this site about the hyperbolic plane, usually in one of the popular models: disc model, upper half-plane model. The bad news is that we cannot fit the whole thing into a pseudosphere...
– Will Jagy
Nov 30 at 1:39
Related: math.stackexchange.com/q/691894/90543
– jgon
Nov 30 at 1:39
@DougM I think it is worth to note that the existence of a triangle where sum of angles is 0 depends on the definition of the triangle -- it is not clear whether ideal triangles are allowed, since their vertices are in infinity. If we require all the vertices to be in the hyperbolic plane, then the angles have to be greater than 0, and can be as small as we want.
– Zeno Rogue
Nov 30 at 11:40
Searching "hyperbolic crochet" on Google Images shows lots of pictures of the waviness that is the hyperbolic plane (or, at least, small sections of it) embedded in three-space. (It should be noted that the areas of circles on the hyperbolic plane grow exponentially as you increase the radius—unlike the flat Euclidean plane where they grow quadratically—meaning that, assuming the yarn has nonzero thickness, you'll quickly run out of space as you try to include larger and larger sections of the hyperbolic plane.)
– Akiva Weinberger
Nov 30 at 12:05
|
show 2 more comments
up vote
27
down vote
favorite
up vote
27
down vote
favorite
So I was watching a video that features astronomer and topologist Cliff Stoll talking about how figures that aren't quadrilaterals can have all their angles equal 90 degrees on different surfaces. For example, on a sphere, you can create a triangle that has all of its angles equal $90^circ$. On a pseudosphere, you can create a pentagon that has all of its angles equal $90^circ$. Now, here's my question.
Is there a surface where a hexagon with this property is possible?
geometry
edited Dec 2 at 10:57
amWhy
191k27223439
asked Nov 30 at 1:35
Xavier Stanton
315210
So I was watching a video that features astronomer and topologist Cliff Stoll talking about how figures that aren't quadrilaterals can have all their angles equal 90 degrees on different surfaces. For example, on a sphere, you can create a triangle that has all of its angles equal $90^circ$. On a pseudosphere, you can create a pentagon that has all of its angles equal $90^circ$. Now, here's my question.
Is there a surface where a hexagon with this property is possible?
geometry
geometry
edited Dec 2 at 10:57
amWhy
191k27223439
asked Nov 30 at 1:35
Xavier Stanton
315210
edited Dec 2 at 10:57
amWhy
191k27223439
asked Nov 30 at 1:35
Xavier Stanton
315210
edited Dec 2 at 10:57
amWhy
191k27223439
edited Dec 2 at 10:57
amWhy
191k27223439
edited Dec 2 at 10:57
amWhy
191k27223439
191k27223439
asked Nov 30 at 1:35
Xavier Stanton
315210
asked Nov 30 at 1:35
Xavier Stanton
315210
asked Nov 30 at 1:35
Xavier Stanton
315210
315210
7
On hyerbolic surfaces you can have a triangle where the sum of the angles is 0. And hexagons with 6 right angles.
– Doug M
Nov 30 at 1:38
you can find many questions on this site about the hyperbolic plane, usually in one of the popular models: disc model, upper half-plane model. The bad news is that we cannot fit the whole thing into a pseudosphere...
– Will Jagy
Nov 30 at 1:39
Related: math.stackexchange.com/q/691894/90543
– jgon
Nov 30 at 1:39
@DougM I think it is worth to note that the existence of a triangle where sum of angles is 0 depends on the definition of the triangle -- it is not clear whether ideal triangles are allowed, since their vertices are in infinity. If we require all the vertices to be in the hyperbolic plane, then the angles have to be greater than 0, and can be as small as we want.
– Zeno Rogue
Nov 30 at 11:40
Searching "hyperbolic crochet" on Google Images shows lots of pictures of the waviness that is the hyperbolic plane (or, at least, small sections of it) embedded in three-space. (It should be noted that the areas of circles on the hyperbolic plane grow exponentially as you increase the radius—unlike the flat Euclidean plane where they grow quadratically—meaning that, assuming the yarn has nonzero thickness, you'll quickly run out of space as you try to include larger and larger sections of the hyperbolic plane.)
– Akiva Weinberger
Nov 30 at 12:05
|
show 2 more comments
7
On hyerbolic surfaces you can have a triangle where the sum of the angles is 0. And hexagons with 6 right angles.
– Doug M
Nov 30 at 1:38
you can find many questions on this site about the hyperbolic plane, usually in one of the popular models: disc model, upper half-plane model. The bad news is that we cannot fit the whole thing into a pseudosphere...
– Will Jagy
Nov 30 at 1:39
Related: math.stackexchange.com/q/691894/90543
– jgon
Nov 30 at 1:39
@DougM I think it is worth to note that the existence of a triangle where sum of angles is 0 depends on the definition of the triangle -- it is not clear whether ideal triangles are allowed, since their vertices are in infinity. If we require all the vertices to be in the hyperbolic plane, then the angles have to be greater than 0, and can be as small as we want.
– Zeno Rogue
Nov 30 at 11:40
Searching "hyperbolic crochet" on Google Images shows lots of pictures of the waviness that is the hyperbolic plane (or, at least, small sections of it) embedded in three-space. (It should be noted that the areas of circles on the hyperbolic plane grow exponentially as you increase the radius—unlike the flat Euclidean plane where they grow quadratically—meaning that, assuming the yarn has nonzero thickness, you'll quickly run out of space as you try to include larger and larger sections of the hyperbolic plane.)
– Akiva Weinberger
Nov 30 at 12:05
7
7
On hyerbolic surfaces you can have a triangle where the sum of the angles is 0. And hexagons with 6 right angles.
– Doug M
Nov 30 at 1:38
On hyerbolic surfaces you can have a triangle where the sum of the angles is 0. And hexagons with 6 right angles.
– Doug M
Nov 30 at 1:38
you can find many questions on this site about the hyperbolic plane, usually in one of the popular models: disc model, upper half-plane model. The bad news is that we cannot fit the whole thing into a pseudosphere...
– Will Jagy
Nov 30 at 1:39
you can find many questions on this site about the hyperbolic plane, usually in one of the popular models: disc model, upper half-plane model. The bad news is that we cannot fit the whole thing into a pseudosphere...
– Will Jagy
Nov 30 at 1:39
Related: math.stackexchange.com/q/691894/90543
– jgon
Nov 30 at 1:39
Related: math.stackexchange.com/q/691894/90543
– jgon
Nov 30 at 1:39
@DougM I think it is worth to note that the existence of a triangle where sum of angles is 0 depends on the definition of the triangle -- it is not clear whether ideal triangles are allowed, since their vertices are in infinity. If we require all the vertices to be in the hyperbolic plane, then the angles have to be greater than 0, and can be as small as we want.
– Zeno Rogue
Nov 30 at 11:40
@DougM I think it is worth to note that the existence of a triangle where sum of angles is 0 depends on the definition of the triangle -- it is not clear whether ideal triangles are allowed, since their vertices are in infinity. If we require all the vertices to be in the hyperbolic plane, then the angles have to be greater than 0, and can be as small as we want.
– Zeno Rogue
Nov 30 at 11:40
Searching "hyperbolic crochet" on Google Images shows lots of pictures of the waviness that is the hyperbolic plane (or, at least, small sections of it) embedded in three-space. (It should be noted that the areas of circles on the hyperbolic plane grow exponentially as you increase the radius—unlike the flat Euclidean plane where they grow quadratically—meaning that, assuming the yarn has nonzero thickness, you'll quickly run out of space as you try to include larger and larger sections of the hyperbolic plane.)
– Akiva Weinberger
Nov 30 at 12:05
Searching "hyperbolic crochet" on Google Images shows lots of pictures of the waviness that is the hyperbolic plane (or, at least, small sections of it) embedded in three-space. (It should be noted that the areas of circles on the hyperbolic plane grow exponentially as you increase the radius—unlike the flat Euclidean plane where they grow quadratically—meaning that, assuming the yarn has nonzero thickness, you'll quickly run out of space as you try to include larger and larger sections of the hyperbolic plane.)
– Akiva Weinberger
Nov 30 at 12:05
|
show 2 more comments
3 Answers
3
active
oldest
votes
up vote
56
down vote
You would need a surface of negative curvature.
It is best to use a hyperbolic plane for this, where you can easily fit any regular n-gon with given angles as long as the sum of its external angles is greater than 360 degrees. The problem is that the hyperbolic plane does not fit in Euclidean space.
The pseudosphere is a small fragment of the hyperbolic plane. You can draw a right-angled hexagon on the pseudosphere only if you allow it to wrap over itself. (Edit: actually I am not completely sure about this; see here, you get a pseudosphere by cutting the part covered with white dots; it appears that a hexagon is slightly larger than the area covered by the pseudosphere, but I am not sure. Should be possible to prove.)
You can also draw it on a Dini's surface -- that is basically an unrolled pseudosphere where you have several layers, and thus you avoid the intersection problem. But it would be hard to see anything because it is rolled very tightly. See here.
Less smooth, but probably the best way would be to use something similar to a hyperbolic crochet. See our computer simulation (arrow keys to rotate).
edited Nov 30 at 22:38
psmears
70749
answered Nov 30 at 3:34
Zeno Rogue
91649
Other answers provides other (possibly simpler) planes, but they don't have constant curvature.
– user202729
Nov 30 at 11:00
It's been such a long time since I last played your game! I can't check now because I'm on mobile, but did you ever end up adding the azimuthal equidistant projection?
– Akiva Weinberger
Nov 30 at 13:17
On a related note, I just rewatched this video. The ending is so shocking!
– Akiva Weinberger
Nov 30 at 13:21
Sure! I have added azimuthal equidistant projection and some other cool projections too. They are listed here.
– Zeno Rogue
Nov 30 at 13:23
Ah, nice! (Oh my god it looks so weird)
– Akiva Weinberger
Nov 30 at 13:33
|
show 4 more comments
up vote
28
down vote
I'm afraid the constraints in the question as stated might be lower than intended, because it isn't difficult to fit a surface to most of 6-segment right-angled closed lines in space in a way that all angles lie in that surface.
Taking in account that the OP hasn't even stated that all right angles must turn to the same side, there is an even simpler solution: a cylinder. We just need to draw a 6-segment zigzag with right angles across a piece of paper and fold the paper in a cylinder to close the line.
If the fact the cylinder has has a hole is a problem, we can close it with a semi-sphere - not included in the photos because I didn't have at hand a ball of a suitable size.
Addition of a solution with all angles to the same side (no zigzag):
I beg your pardon for my poor drawing skills.
Please take two contiguous faces of a cube, make a circuit using all edges except for the common one, and attach an small square to each of the six vertexes, in the plane of the circuit.
Now put one band along each edge the circuit, joining the squares. Put it in a way that all turns at the vertexes keep to the same side. In the next picture I added normal vectors to the squares, twists and turns for clarification:
And since there are no complete twists, you can close the hole in the band to form a disc:
answered Nov 30 at 10:07
Pere
44837
5
This has to be a hexagon, not a zigzag line with 6 segments. A hexagon has an outside and an inside, and it cannot have a hole in the inside. Also the angles have to be 90 degrees on the INSIDE (otherwise they are actually 270 degrees). It is easily possible to make a hexagon with 90+90+90+90+90+270 degrees even on a plane, that would be an L shape. You need a negatively curved surface to make it 90+90+90+90+90+90, it will not be possible on a cylinder or a semi-sphere.
– Zeno Rogue
Nov 30 at 11:09
6
@ZenoRogue - As I said in the answer, if you need an inside and an outside, just cover the hole of the cylinder with half a sphere. And an L-shaped hexagon is still a hexagon. Furthermore, there are other solutions with all angles to the same side: you just need to fix a twisted band to the edges of a cube. Of course, you would be probably right if the question asked for a regular hexagon in an constant curvature surface, but that's not in the question - even if the OP had it in mind, which is possible.
– Pere
Nov 30 at 11:20
3
@ZenoRogue - I added an explanation and some drawings. I hope it helps.
– Pere
Nov 30 at 12:29
2
Thanks! I understand now, and it works indeed -- it is easier to see when we flip the odd corner to the right. By Gauss-Bonnet theorem the curvature of this surface must be negative somewhere.
– Zeno Rogue
Nov 30 at 13:31
1
At least, the twisted bands have negative curvature in my rendition. Other parts have positive curvature or no curvature.
– Pere
Nov 30 at 19:50
|
show 3 more comments
up vote
15
down vote
Take a cube and orient it so you're looking down a body diagonal. The edges that appear to be at the boundary of this view actually form a nonplanar object with six congruent edges, right angles wherever two adjacent edges meet, and $D_{3d}$ point group symmetry.
answered Nov 30 at 2:59
Oscar Lanzi
11.9k11936
There's one in every crowd
– Oscar Lanzi
Dec 1 at 23:18
add a comment |
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
56
down vote
You would need a surface of negative curvature.
It is best to use a hyperbolic plane for this, where you can easily fit any regular n-gon with given angles as long as the sum of its external angles is greater than 360 degrees. The problem is that the hyperbolic plane does not fit in Euclidean space.
The pseudosphere is a small fragment of the hyperbolic plane. You can draw a right-angled hexagon on the pseudosphere only if you allow it to wrap over itself. (Edit: actually I am not completely sure about this; see here, you get a pseudosphere by cutting the part covered with white dots; it appears that a hexagon is slightly larger than the area covered by the pseudosphere, but I am not sure. Should be possible to prove.)
You can also draw it on a Dini's surface -- that is basically an unrolled pseudosphere where you have several layers, and thus you avoid the intersection problem. But it would be hard to see anything because it is rolled very tightly. See here.
Less smooth, but probably the best way would be to use something similar to a hyperbolic crochet. See our computer simulation (arrow keys to rotate).
edited Nov 30 at 22:38
psmears
70749
answered Nov 30 at 3:34
Zeno Rogue
91649
Other answers provides other (possibly simpler) planes, but they don't have constant curvature.
– user202729
Nov 30 at 11:00
It's been such a long time since I last played your game! I can't check now because I'm on mobile, but did you ever end up adding the azimuthal equidistant projection?
– Akiva Weinberger
Nov 30 at 13:17
On a related note, I just rewatched this video. The ending is so shocking!
– Akiva Weinberger
Nov 30 at 13:21
Sure! I have added azimuthal equidistant projection and some other cool projections too. They are listed here.
– Zeno Rogue
Nov 30 at 13:23
Ah, nice! (Oh my god it looks so weird)
– Akiva Weinberger
Nov 30 at 13:33
|
show 4 more comments
up vote
56
down vote
You would need a surface of negative curvature.
It is best to use a hyperbolic plane for this, where you can easily fit any regular n-gon with given angles as long as the sum of its external angles is greater than 360 degrees. The problem is that the hyperbolic plane does not fit in Euclidean space.
The pseudosphere is a small fragment of the hyperbolic plane. You can draw a right-angled hexagon on the pseudosphere only if you allow it to wrap over itself. (Edit: actually I am not completely sure about this; see here, you get a pseudosphere by cutting the part covered with white dots; it appears that a hexagon is slightly larger than the area covered by the pseudosphere, but I am not sure. Should be possible to prove.)
You can also draw it on a Dini's surface -- that is basically an unrolled pseudosphere where you have several layers, and thus you avoid the intersection problem. But it would be hard to see anything because it is rolled very tightly. See here.
Less smooth, but probably the best way would be to use something similar to a hyperbolic crochet. See our computer simulation (arrow keys to rotate).
edited Nov 30 at 22:38
psmears
70749
answered Nov 30 at 3:34
Zeno Rogue
91649
Other answers provides other (possibly simpler) planes, but they don't have constant curvature.
– user202729
Nov 30 at 11:00
It's been such a long time since I last played your game! I can't check now because I'm on mobile, but did you ever end up adding the azimuthal equidistant projection?
– Akiva Weinberger
Nov 30 at 13:17
On a related note, I just rewatched this video. The ending is so shocking!
– Akiva Weinberger
Nov 30 at 13:21
Sure! I have added azimuthal equidistant projection and some other cool projections too. They are listed here.
– Zeno Rogue
Nov 30 at 13:23
Ah, nice! (Oh my god it looks so weird)
– Akiva Weinberger
Nov 30 at 13:33
|
show 4 more comments
up vote
56
down vote
up vote
56
down vote
You would need a surface of negative curvature.
It is best to use a hyperbolic plane for this, where you can easily fit any regular n-gon with given angles as long as the sum of its external angles is greater than 360 degrees. The problem is that the hyperbolic plane does not fit in Euclidean space.
The pseudosphere is a small fragment of the hyperbolic plane. You can draw a right-angled hexagon on the pseudosphere only if you allow it to wrap over itself. (Edit: actually I am not completely sure about this; see here, you get a pseudosphere by cutting the part covered with white dots; it appears that a hexagon is slightly larger than the area covered by the pseudosphere, but I am not sure. Should be possible to prove.)
You can also draw it on a Dini's surface -- that is basically an unrolled pseudosphere where you have several layers, and thus you avoid the intersection problem. But it would be hard to see anything because it is rolled very tightly. See here.
Less smooth, but probably the best way would be to use something similar to a hyperbolic crochet. See our computer simulation (arrow keys to rotate).
edited Nov 30 at 22:38
psmears
70749
answered Nov 30 at 3:34
Zeno Rogue
91649
You would need a surface of negative curvature.
It is best to use a hyperbolic plane for this, where you can easily fit any regular n-gon with given angles as long as the sum of its external angles is greater than 360 degrees. The problem is that the hyperbolic plane does not fit in Euclidean space.
The pseudosphere is a small fragment of the hyperbolic plane. You can draw a right-angled hexagon on the pseudosphere only if you allow it to wrap over itself. (Edit: actually I am not completely sure about this; see here, you get a pseudosphere by cutting the part covered with white dots; it appears that a hexagon is slightly larger than the area covered by the pseudosphere, but I am not sure. Should be possible to prove.)
You can also draw it on a Dini's surface -- that is basically an unrolled pseudosphere where you have several layers, and thus you avoid the intersection problem. But it would be hard to see anything because it is rolled very tightly. See here.
Less smooth, but probably the best way would be to use something similar to a hyperbolic crochet. See our computer simulation (arrow keys to rotate).
edited Nov 30 at 22:38
psmears
70749
answered Nov 30 at 3:34
Zeno Rogue
91649
edited Nov 30 at 22:38
psmears
70749
edited Nov 30 at 22:38
psmears
70749
edited Nov 30 at 22:38
psmears
70749
70749
answered Nov 30 at 3:34
Zeno Rogue
91649
answered Nov 30 at 3:34
Zeno Rogue
91649
answered Nov 30 at 3:34
Zeno Rogue
91649
91649
Other answers provides other (possibly simpler) planes, but they don't have constant curvature.
– user202729
Nov 30 at 11:00
It's been such a long time since I last played your game! I can't check now because I'm on mobile, but did you ever end up adding the azimuthal equidistant projection?
– Akiva Weinberger
Nov 30 at 13:17
On a related note, I just rewatched this video. The ending is so shocking!
– Akiva Weinberger
Nov 30 at 13:21
Sure! I have added azimuthal equidistant projection and some other cool projections too. They are listed here.
– Zeno Rogue
Nov 30 at 13:23
Ah, nice! (Oh my god it looks so weird)
– Akiva Weinberger
Nov 30 at 13:33
|
show 4 more comments
Other answers provides other (possibly simpler) planes, but they don't have constant curvature.
– user202729
Nov 30 at 11:00
It's been such a long time since I last played your game! I can't check now because I'm on mobile, but did you ever end up adding the azimuthal equidistant projection?
– Akiva Weinberger
Nov 30 at 13:17
On a related note, I just rewatched this video. The ending is so shocking!
– Akiva Weinberger
Nov 30 at 13:21
Sure! I have added azimuthal equidistant projection and some other cool projections too. They are listed here.
– Zeno Rogue
Nov 30 at 13:23
Ah, nice! (Oh my god it looks so weird)
– Akiva Weinberger
Nov 30 at 13:33
Other answers provides other (possibly simpler) planes, but they don't have constant curvature.
– user202729
Nov 30 at 11:00
Other answers provides other (possibly simpler) planes, but they don't have constant curvature.
– user202729
Nov 30 at 11:00
It's been such a long time since I last played your game! I can't check now because I'm on mobile, but did you ever end up adding the azimuthal equidistant projection?
– Akiva Weinberger
Nov 30 at 13:17
It's been such a long time since I last played your game! I can't check now because I'm on mobile, but did you ever end up adding the azimuthal equidistant projection?
– Akiva Weinberger
Nov 30 at 13:17
On a related note, I just rewatched this video. The ending is so shocking!
– Akiva Weinberger
Nov 30 at 13:21
On a related note, I just rewatched this video. The ending is so shocking!
– Akiva Weinberger
Nov 30 at 13:21
Sure! I have added azimuthal equidistant projection and some other cool projections too. They are listed here.
– Zeno Rogue
Nov 30 at 13:23
Sure! I have added azimuthal equidistant projection and some other cool projections too. They are listed here.
– Zeno Rogue
Nov 30 at 13:23
Ah, nice! (Oh my god it looks so weird)
– Akiva Weinberger
Nov 30 at 13:33
Ah, nice! (Oh my god it looks so weird)
– Akiva Weinberger
Nov 30 at 13:33
|
show 4 more comments
up vote
28
down vote
I'm afraid the constraints in the question as stated might be lower than intended, because it isn't difficult to fit a surface to most of 6-segment right-angled closed lines in space in a way that all angles lie in that surface.
Taking in account that the OP hasn't even stated that all right angles must turn to the same side, there is an even simpler solution: a cylinder. We just need to draw a 6-segment zigzag with right angles across a piece of paper and fold the paper in a cylinder to close the line.
If the fact the cylinder has has a hole is a problem, we can close it with a semi-sphere - not included in the photos because I didn't have at hand a ball of a suitable size.
Addition of a solution with all angles to the same side (no zigzag):
I beg your pardon for my poor drawing skills.
Please take two contiguous faces of a cube, make a circuit using all edges except for the common one, and attach an small square to each of the six vertexes, in the plane of the circuit.
Now put one band along each edge the circuit, joining the squares. Put it in a way that all turns at the vertexes keep to the same side. In the next picture I added normal vectors to the squares, twists and turns for clarification:
And since there are no complete twists, you can close the hole in the band to form a disc:
answered Nov 30 at 10:07
Pere
44837
5
This has to be a hexagon, not a zigzag line with 6 segments. A hexagon has an outside and an inside, and it cannot have a hole in the inside. Also the angles have to be 90 degrees on the INSIDE (otherwise they are actually 270 degrees). It is easily possible to make a hexagon with 90+90+90+90+90+270 degrees even on a plane, that would be an L shape. You need a negatively curved surface to make it 90+90+90+90+90+90, it will not be possible on a cylinder or a semi-sphere.
– Zeno Rogue
Nov 30 at 11:09
6
@ZenoRogue - As I said in the answer, if you need an inside and an outside, just cover the hole of the cylinder with half a sphere. And an L-shaped hexagon is still a hexagon. Furthermore, there are other solutions with all angles to the same side: you just need to fix a twisted band to the edges of a cube. Of course, you would be probably right if the question asked for a regular hexagon in an constant curvature surface, but that's not in the question - even if the OP had it in mind, which is possible.
– Pere
Nov 30 at 11:20
3
@ZenoRogue - I added an explanation and some drawings. I hope it helps.
– Pere
Nov 30 at 12:29
2
Thanks! I understand now, and it works indeed -- it is easier to see when we flip the odd corner to the right. By Gauss-Bonnet theorem the curvature of this surface must be negative somewhere.
– Zeno Rogue
Nov 30 at 13:31
1
At least, the twisted bands have negative curvature in my rendition. Other parts have positive curvature or no curvature.
– Pere
Nov 30 at 19:50
|
show 3 more comments
up vote
28
down vote
I'm afraid the constraints in the question as stated might be lower than intended, because it isn't difficult to fit a surface to most of 6-segment right-angled closed lines in space in a way that all angles lie in that surface.
Taking in account that the OP hasn't even stated that all right angles must turn to the same side, there is an even simpler solution: a cylinder. We just need to draw a 6-segment zigzag with right angles across a piece of paper and fold the paper in a cylinder to close the line.
If the fact the cylinder has has a hole is a problem, we can close it with a semi-sphere - not included in the photos because I didn't have at hand a ball of a suitable size.
Addition of a solution with all angles to the same side (no zigzag):
I beg your pardon for my poor drawing skills.
Please take two contiguous faces of a cube, make a circuit using all edges except for the common one, and attach an small square to each of the six vertexes, in the plane of the circuit.
Now put one band along each edge the circuit, joining the squares. Put it in a way that all turns at the vertexes keep to the same side. In the next picture I added normal vectors to the squares, twists and turns for clarification:
And since there are no complete twists, you can close the hole in the band to form a disc:
answered Nov 30 at 10:07
Pere
44837
5
This has to be a hexagon, not a zigzag line with 6 segments. A hexagon has an outside and an inside, and it cannot have a hole in the inside. Also the angles have to be 90 degrees on the INSIDE (otherwise they are actually 270 degrees). It is easily possible to make a hexagon with 90+90+90+90+90+270 degrees even on a plane, that would be an L shape. You need a negatively curved surface to make it 90+90+90+90+90+90, it will not be possible on a cylinder or a semi-sphere.
– Zeno Rogue
Nov 30 at 11:09
6
@ZenoRogue - As I said in the answer, if you need an inside and an outside, just cover the hole of the cylinder with half a sphere. And an L-shaped hexagon is still a hexagon. Furthermore, there are other solutions with all angles to the same side: you just need to fix a twisted band to the edges of a cube. Of course, you would be probably right if the question asked for a regular hexagon in an constant curvature surface, but that's not in the question - even if the OP had it in mind, which is possible.
– Pere
Nov 30 at 11:20
3
@ZenoRogue - I added an explanation and some drawings. I hope it helps.
– Pere
Nov 30 at 12:29
2
Thanks! I understand now, and it works indeed -- it is easier to see when we flip the odd corner to the right. By Gauss-Bonnet theorem the curvature of this surface must be negative somewhere.
– Zeno Rogue
Nov 30 at 13:31
1
At least, the twisted bands have negative curvature in my rendition. Other parts have positive curvature or no curvature.
– Pere
Nov 30 at 19:50
|
show 3 more comments
up vote
28
down vote
up vote
28
down vote
I'm afraid the constraints in the question as stated might be lower than intended, because it isn't difficult to fit a surface to most of 6-segment right-angled closed lines in space in a way that all angles lie in that surface.
Taking in account that the OP hasn't even stated that all right angles must turn to the same side, there is an even simpler solution: a cylinder. We just need to draw a 6-segment zigzag with right angles across a piece of paper and fold the paper in a cylinder to close the line.
If the fact the cylinder has has a hole is a problem, we can close it with a semi-sphere - not included in the photos because I didn't have at hand a ball of a suitable size.
Addition of a solution with all angles to the same side (no zigzag):
I beg your pardon for my poor drawing skills.
Please take two contiguous faces of a cube, make a circuit using all edges except for the common one, and attach an small square to each of the six vertexes, in the plane of the circuit.
Now put one band along each edge the circuit, joining the squares. Put it in a way that all turns at the vertexes keep to the same side. In the next picture I added normal vectors to the squares, twists and turns for clarification:
And since there are no complete twists, you can close the hole in the band to form a disc:
answered Nov 30 at 10:07
Pere
44837
I'm afraid the constraints in the question as stated might be lower than intended, because it isn't difficult to fit a surface to most of 6-segment right-angled closed lines in space in a way that all angles lie in that surface.
Taking in account that the OP hasn't even stated that all right angles must turn to the same side, there is an even simpler solution: a cylinder. We just need to draw a 6-segment zigzag with right angles across a piece of paper and fold the paper in a cylinder to close the line.
If the fact the cylinder has has a hole is a problem, we can close it with a semi-sphere - not included in the photos because I didn't have at hand a ball of a suitable size.
Addition of a solution with all angles to the same side (no zigzag):
I beg your pardon for my poor drawing skills.
Please take two contiguous faces of a cube, make a circuit using all edges except for the common one, and attach an small square to each of the six vertexes, in the plane of the circuit.
Now put one band along each edge the circuit, joining the squares. Put it in a way that all turns at the vertexes keep to the same side. In the next picture I added normal vectors to the squares, twists and turns for clarification:
And since there are no complete twists, you can close the hole in the band to form a disc:
answered Nov 30 at 10:07
Pere
44837
edited Dec 1 at 13:11
answered Nov 30 at 10:07
Pere
44837
answered Nov 30 at 10:07
Pere
44837
answered Nov 30 at 10:07
Pere
44837
44837
5
This has to be a hexagon, not a zigzag line with 6 segments. A hexagon has an outside and an inside, and it cannot have a hole in the inside. Also the angles have to be 90 degrees on the INSIDE (otherwise they are actually 270 degrees). It is easily possible to make a hexagon with 90+90+90+90+90+270 degrees even on a plane, that would be an L shape. You need a negatively curved surface to make it 90+90+90+90+90+90, it will not be possible on a cylinder or a semi-sphere.
– Zeno Rogue
Nov 30 at 11:09
6
@ZenoRogue - As I said in the answer, if you need an inside and an outside, just cover the hole of the cylinder with half a sphere. And an L-shaped hexagon is still a hexagon. Furthermore, there are other solutions with all angles to the same side: you just need to fix a twisted band to the edges of a cube. Of course, you would be probably right if the question asked for a regular hexagon in an constant curvature surface, but that's not in the question - even if the OP had it in mind, which is possible.
– Pere
Nov 30 at 11:20
3
@ZenoRogue - I added an explanation and some drawings. I hope it helps.
– Pere
Nov 30 at 12:29
2
Thanks! I understand now, and it works indeed -- it is easier to see when we flip the odd corner to the right. By Gauss-Bonnet theorem the curvature of this surface must be negative somewhere.
– Zeno Rogue
Nov 30 at 13:31
1
At least, the twisted bands have negative curvature in my rendition. Other parts have positive curvature or no curvature.
– Pere
Nov 30 at 19:50
|
show 3 more comments
5
This has to be a hexagon, not a zigzag line with 6 segments. A hexagon has an outside and an inside, and it cannot have a hole in the inside. Also the angles have to be 90 degrees on the INSIDE (otherwise they are actually 270 degrees). It is easily possible to make a hexagon with 90+90+90+90+90+270 degrees even on a plane, that would be an L shape. You need a negatively curved surface to make it 90+90+90+90+90+90, it will not be possible on a cylinder or a semi-sphere.
– Zeno Rogue
Nov 30 at 11:09
6
@ZenoRogue - As I said in the answer, if you need an inside and an outside, just cover the hole of the cylinder with half a sphere. And an L-shaped hexagon is still a hexagon. Furthermore, there are other solutions with all angles to the same side: you just need to fix a twisted band to the edges of a cube. Of course, you would be probably right if the question asked for a regular hexagon in an constant curvature surface, but that's not in the question - even if the OP had it in mind, which is possible.
– Pere
Nov 30 at 11:20
3
@ZenoRogue - I added an explanation and some drawings. I hope it helps.
– Pere
Nov 30 at 12:29
2
Thanks! I understand now, and it works indeed -- it is easier to see when we flip the odd corner to the right. By Gauss-Bonnet theorem the curvature of this surface must be negative somewhere.
– Zeno Rogue
Nov 30 at 13:31
1
At least, the twisted bands have negative curvature in my rendition. Other parts have positive curvature or no curvature.
– Pere
Nov 30 at 19:50
5
5
This has to be a hexagon, not a zigzag line with 6 segments. A hexagon has an outside and an inside, and it cannot have a hole in the inside. Also the angles have to be 90 degrees on the INSIDE (otherwise they are actually 270 degrees). It is easily possible to make a hexagon with 90+90+90+90+90+270 degrees even on a plane, that would be an L shape. You need a negatively curved surface to make it 90+90+90+90+90+90, it will not be possible on a cylinder or a semi-sphere.
– Zeno Rogue
Nov 30 at 11:09
This has to be a hexagon, not a zigzag line with 6 segments. A hexagon has an outside and an inside, and it cannot have a hole in the inside. Also the angles have to be 90 degrees on the INSIDE (otherwise they are actually 270 degrees). It is easily possible to make a hexagon with 90+90+90+90+90+270 degrees even on a plane, that would be an L shape. You need a negatively curved surface to make it 90+90+90+90+90+90, it will not be possible on a cylinder or a semi-sphere.
– Zeno Rogue
Nov 30 at 11:09
6
6
@ZenoRogue - As I said in the answer, if you need an inside and an outside, just cover the hole of the cylinder with half a sphere. And an L-shaped hexagon is still a hexagon. Furthermore, there are other solutions with all angles to the same side: you just need to fix a twisted band to the edges of a cube. Of course, you would be probably right if the question asked for a regular hexagon in an constant curvature surface, but that's not in the question - even if the OP had it in mind, which is possible.
– Pere
Nov 30 at 11:20
@ZenoRogue - As I said in the answer, if you need an inside and an outside, just cover the hole of the cylinder with half a sphere. And an L-shaped hexagon is still a hexagon. Furthermore, there are other solutions with all angles to the same side: you just need to fix a twisted band to the edges of a cube. Of course, you would be probably right if the question asked for a regular hexagon in an constant curvature surface, but that's not in the question - even if the OP had it in mind, which is possible.
– Pere
Nov 30 at 11:20
3
3
@ZenoRogue - I added an explanation and some drawings. I hope it helps.
– Pere
Nov 30 at 12:29
@ZenoRogue - I added an explanation and some drawings. I hope it helps.
– Pere
Nov 30 at 12:29
2
2
Thanks! I understand now, and it works indeed -- it is easier to see when we flip the odd corner to the right. By Gauss-Bonnet theorem the curvature of this surface must be negative somewhere.
– Zeno Rogue
Nov 30 at 13:31
Thanks! I understand now, and it works indeed -- it is easier to see when we flip the odd corner to the right. By Gauss-Bonnet theorem the curvature of this surface must be negative somewhere.
– Zeno Rogue
Nov 30 at 13:31
1
1
At least, the twisted bands have negative curvature in my rendition. Other parts have positive curvature or no curvature.
– Pere
Nov 30 at 19:50
At least, the twisted bands have negative curvature in my rendition. Other parts have positive curvature or no curvature.
– Pere
Nov 30 at 19:50
|
show 3 more comments
up vote
15
down vote
Take a cube and orient it so you're looking down a body diagonal. The edges that appear to be at the boundary of this view actually form a nonplanar object with six congruent edges, right angles wherever two adjacent edges meet, and $D_{3d}$ point group symmetry.
answered Nov 30 at 2:59
Oscar Lanzi
11.9k11936
There's one in every crowd
– Oscar Lanzi
Dec 1 at 23:18
add a comment |
up vote
15
down vote
Take a cube and orient it so you're looking down a body diagonal. The edges that appear to be at the boundary of this view actually form a nonplanar object with six congruent edges, right angles wherever two adjacent edges meet, and $D_{3d}$ point group symmetry.
answered Nov 30 at 2:59
Oscar Lanzi
11.9k11936
There's one in every crowd
– Oscar Lanzi
Dec 1 at 23:18
add a comment |
up vote
15
down vote
up vote
15
down vote
Take a cube and orient it so you're looking down a body diagonal. The edges that appear to be at the boundary of this view actually form a nonplanar object with six congruent edges, right angles wherever two adjacent edges meet, and $D_{3d}$ point group symmetry.
answered Nov 30 at 2:59
Oscar Lanzi
11.9k11936
Take a cube and orient it so you're looking down a body diagonal. The edges that appear to be at the boundary of this view actually form a nonplanar object with six congruent edges, right angles wherever two adjacent edges meet, and $D_{3d}$ point group symmetry.
answered Nov 30 at 2:59
Oscar Lanzi
11.9k11936
answered Nov 30 at 2:59
Oscar Lanzi
11.9k11936
answered Nov 30 at 2:59
Oscar Lanzi
11.9k11936
answered Nov 30 at 2:59
Oscar Lanzi
11.9k11936
11.9k11936
There's one in every crowd
– Oscar Lanzi
Dec 1 at 23:18
add a comment |
There's one in every crowd
– Oscar Lanzi
Dec 1 at 23:18
There's one in every crowd
– Oscar Lanzi
Dec 1 at 23:18
There's one in every crowd
– Oscar Lanzi
Dec 1 at 23:18
add a comment |
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On hyerbolic surfaces you can have a triangle where the sum of the angles is 0. And hexagons with 6 right angles.
– Doug M
Nov 30 at 1:38
you can find many questions on this site about the hyperbolic plane, usually in one of the popular models: disc model, upper half-plane model. The bad news is that we cannot fit the whole thing into a pseudosphere...
– Will Jagy
Nov 30 at 1:39
Related: math.stackexchange.com/q/691894/90543
– jgon
Nov 30 at 1:39
@DougM I think it is worth to note that the existence of a triangle where sum of angles is 0 depends on the definition of the triangle -- it is not clear whether ideal triangles are allowed, since their vertices are in infinity. If we require all the vertices to be in the hyperbolic plane, then the angles have to be greater than 0, and can be as small as we want.
– Zeno Rogue
Nov 30 at 11:40
Searching "hyperbolic crochet" on Google Images shows lots of pictures of the waviness that is the hyperbolic plane (or, at least, small sections of it) embedded in three-space. (It should be noted that the areas of circles on the hyperbolic plane grow exponentially as you increase the radius—unlike the flat Euclidean plane where they grow quadratically—meaning that, assuming the yarn has nonzero thickness, you'll quickly run out of space as you try to include larger and larger sections of the hyperbolic plane.)
– Akiva Weinberger
Nov 30 at 12:05