I expect I am way out of my intellectual depth here, but I am reminded of another UA-cam video about a way of creasing the surface of a torus, invented by John Nash, which allows it to have the same length and area on any part of its surface as a Euclidean square.
The torus is supposed to have flat, euclidean geometry, but the usual "doughnut" embedding is not flat - it has curvature. Here's one way to fix the problem.
I'm now wondering about the properties of this vs. the sort of fractally corrugated flat torus embedding I've seen elsewhere. Do they have different degrees of (non)-differentiability?
If you have an arbitrary shape that covers the euclidian plane purely by translation in two directions, can it autmatically cover a torus with just one instance of this shape?
This can also be a really awesome physical demo to explain the idea of UV mapping when dealing with 3d computer graphics! Specifically primitives such as a cube, torus, cone etc.. I normally try to explain UV's like making a sewing pattern, but this is a nice tactile demonstration of the same principle!
the problem with using this particular video for that particular explanation is that this geometry does produce a shape that looks like a torus, but it is not a pure geometric torus. There are faces that overlap other faces, it's not convex, and I don't think it's even enclosed. Making matters even worse, the ideal UV map for a torus WOULD be a square, because computers don't actually have to physically wrap a rigid sheet. If you made the UV map for your torus look anything like this, your textures would be extremely distorted. With a square map, you only need some basic transformations to make your UV map look natural. A better physical demo to explain UV mapping would be nearly any other unwrapped net of a geometric 3d object, such as a cube or pyramid or dodecahedron. Those CAN be unwrapped, and you can easily see how it goes back together. If you want a physical activity for students, you could even have them do papercraft (often called pepakura), where you cut out the unfolded net of a 3d object and fold it together with glue or tape to hold it together. Cheap materials! You could have students draw on the template before they cut it out and fold it; this would literally be a UV map! Have them try to draw things in the correct locations or with textures that would line up along the seams, and have a laugh when it ends up all jumbled.
Oh my God! That's amazing! If I could figure out how to recreate this with wires I could fold a toroidal winding in seconds instead of painstakingly by hand over hours. Dude! Where can I find more information on this?
To explain even tho this does in fact fold flat nicer than other models perhaps, that doesn't make it a 'nice' fold, it has a lot of breaks that would be hard to mend on a several winding toroid, and impossible on a many winding toroid. Besides this the shape is pretty wonky and would impede efficiency and make calculations impractical.
For winding something onto a torus, you want something very different: 1) get yourself a wheel, with a slot cut into it's outer circumference- the wire/string/whatever that gets wound around the torus will get stored there; 2) cut out a segment of the wheel that's large enough for your torus to fit through when fully wound, and connect it back into place with a hinge & latch, so that you can easily swing it out of the way when needed, and latch it into place otherwise; 3) cut a hole at the spot where the two pieces meet but _don't_ have a hinge, and also place some sort of clamp nearby to hold the end of whatever you're winding onto the torus (note that the clamp is on the ring, _not_ the torus!); 4) get yourself a bit of machinery to turn the wheel at the same time as the torus (though probably at a different speed). You use this device as follows: 1) use the "swing segment" to put the ring through the center of the torus, then latch it closed, and mount the ring & torus onto the machine- they should now be interlocked; 2) take the end of whatever you want to wind onto the torus, run it through the hole in the ring (from outside to inside), and restrain to the _ring_ (not to the torus!) with the clamp; 3) run the machine so that some specific length of the material you're winding onto the torus is now wound onto the ring, then turn off the machine and cut the material between the ring and wherever it was being pulled from; 4) unclamp the material that you're winding onto the torus from the ring, in some way clamp it to the torus (tape or something, whatever works), and run the machine again. The reason that this works is that initially the ring is pulling the material into a giant loop that passes through the center of the torus, but after the wire or whatever is cut, and the clamping point is moved to the torus, that loop of material is transfered to the torus directly, with the hole through the ring serving to apply the tension that is used to initially pull the winding tight. I know of this device because I've seen a video of one being used to wind transformers.
I'd start with it being not really a torus since the crossections wil have polygon area instead of circular area.... it is just a genus 1 object with a hole, but since it's not circular base it doesn't need to pose any trouble unfolding it to 2d if You were to think of it; the problem with torus tiling is the circular nature of it You can tile ed cube with squares no problem, You can not tile a torus due to it's circular nature though as You'd have to use shapes, that are not flat I mean...unless You'd want to count tiling with infinitely small tiles that are too small to bend,,... but I'd argue, tthat the curvature is non-0 regardless of scale due to it only asymptotically aproaching 0; hence You should never be able to unfold torus to 2d Since this is not torus (as it's not having circular crossection) it does not pose such problem as You just have multi-polygon approximation of a torus which can be tiled as it's only polygon grid.
Why should a torus be circular ? It isn't a required constraint most of the time and certainly not if you consider going from a flat plane to a flat torus.
OK Henry, so what are your thoughts on the Duocylinder? upload.wikimedia.org/wikipedia/commons/7/7e/Duocylinder_ridge_animated.gif That's a really great example of a flat torus, only that it requires a fourth dimension of freedom to connect up all of the edges that way.. ;3
I expect I am way out of my intellectual depth here, but I am reminded of another UA-cam video about a way of creasing the surface of a torus, invented by John Nash, which allows it to have the same length and area on any part of its surface as a Euclidean square.
The torus is supposed to have flat, euclidean geometry, but the usual "doughnut" embedding is not flat - it has curvature. Here's one way to fix the problem.
Nifty. Who discovered that solution?
I'm not sure. I found out about this shape from www.mathcurve.com/polyedres/toreplat/toreplat.shtml
That's nice. No need to a fourth dimension to have a flat torus !
I'm now wondering about the properties of this vs. the sort of fractally corrugated flat torus embedding I've seen elsewhere. Do they have different degrees of (non)-differentiability?
The fractally corrugated one is C^1 (has first derivatives), this is only C^0 (continuous but not necessarily differentiable at all).
You make me feel very smart and very dumb at the same time, and fior this, I offer my eternal gratitude.
If you have an arbitrary shape that covers the euclidian plane purely by translation in two directions, can it autmatically cover a torus with just one instance of this shape?
Yes.
If you only use one shape, it will need to be really big! I recommend using multiple copies of the same shape 😉
This can also be a really awesome physical demo to explain the idea of UV mapping when dealing with 3d computer graphics! Specifically primitives such as a cube, torus, cone etc.. I normally try to explain UV's like making a sewing pattern, but this is a nice tactile demonstration of the same principle!
the problem with using this particular video for that particular explanation is that this geometry does produce a shape that looks like a torus, but it is not a pure geometric torus. There are faces that overlap other faces, it's not convex, and I don't think it's even enclosed.
Making matters even worse, the ideal UV map for a torus WOULD be a square, because computers don't actually have to physically wrap a rigid sheet. If you made the UV map for your torus look anything like this, your textures would be extremely distorted. With a square map, you only need some basic transformations to make your UV map look natural.
A better physical demo to explain UV mapping would be nearly any other unwrapped net of a geometric 3d object, such as a cube or pyramid or dodecahedron. Those CAN be unwrapped, and you can easily see how it goes back together. If you want a physical activity for students, you could even have them do papercraft (often called pepakura), where you cut out the unfolded net of a 3d object and fold it together with glue or tape to hold it together. Cheap materials! You could have students draw on the template before they cut it out and fold it; this would literally be a UV map! Have them try to draw things in the correct locations or with textures that would line up along the seams, and have a laugh when it ends up all jumbled.
an uniform tiling which is uniform for both R^2 and flat-T^2. Beautiful!
Origami masters must've known that it unfolds to a flat shape right from the thumbnail
Oh my God! That's amazing! If I could figure out how to recreate this with wires I could fold a toroidal winding in seconds instead of painstakingly by hand over hours. Dude! Where can I find more information on this?
ur 6 years late
Unfortunately it wouldn't work very well for that
To explain even tho this does in fact fold flat nicer than other models perhaps, that doesn't make it a 'nice' fold, it has a lot of breaks that would be hard to mend on a several winding toroid, and impossible on a many winding toroid. Besides this the shape is pretty wonky and would impede efficiency and make calculations impractical.
For winding something onto a torus, you want something very different:
1) get yourself a wheel, with a slot cut into it's outer circumference- the wire/string/whatever that gets wound around the torus will get stored there;
2) cut out a segment of the wheel that's large enough for your torus to fit through when fully wound, and connect it back into place with a hinge & latch, so that you can easily swing it out of the way when needed, and latch it into place otherwise;
3) cut a hole at the spot where the two pieces meet but _don't_ have a hinge, and also place some sort of clamp nearby to hold the end of whatever you're winding onto the torus (note that the clamp is on the ring, _not_ the torus!);
4) get yourself a bit of machinery to turn the wheel at the same time as the torus (though probably at a different speed).
You use this device as follows:
1) use the "swing segment" to put the ring through the center of the torus, then latch it closed, and mount the ring & torus onto the machine- they should now be interlocked;
2) take the end of whatever you want to wind onto the torus, run it through the hole in the ring (from outside to inside), and restrain to the _ring_ (not to the torus!) with the clamp;
3) run the machine so that some specific length of the material you're winding onto the torus is now wound onto the ring, then turn off the machine and cut the material between the ring and wherever it was being pulled from;
4) unclamp the material that you're winding onto the torus from the ring, in some way clamp it to the torus (tape or something, whatever works), and run the machine again.
The reason that this works is that initially the ring is pulling the material into a giant loop that passes through the center of the torus, but after the wire or whatever is cut, and the clamping point is moved to the torus, that loop of material is transfered to the torus directly, with the hole through the ring serving to apply the tension that is used to initially pull the winding tight. I know of this device because I've seen a video of one being used to wind transformers.
Does the fact that the inside of it is grooved not change anything?
It's still topologically a torus!
That's a cool mug
Where are you seeing a a mug? All I see are sewing needles
It's not a mug, there's no hole for the handle. It's a folded sheet, it's just not obvious.
0:56 WHOA IS THAT A 10X10?
*woah is that a 9 cube??????*
The ukulele plane... awesome lol
That's rollers, right? Then it's a hyperboloid bearing.
Seems that this shape could be folded in paper as origami?
Yes, absolutely.
That's great! That'll make it easier for me to practice!
very nice video. Thank you
That one may unhinge Lovecraft
0:14 Donut shape? What are you talking about? That's a coffee cup!
That moment when it was unfolded blew my mind
I legitimately thought this was legos until towards the end when you had a close up.
I'd start with it being not really a torus since the crossections wil have polygon area instead of circular area.... it is just a genus 1 object with a hole, but since it's not circular base it doesn't need to pose any trouble unfolding it to 2d
if You were to think of it; the problem with torus tiling is the circular nature of it
You can tile ed cube with squares no problem,
You can not tile a torus due to it's circular nature though as You'd have to use shapes, that are not flat
I mean...unless You'd want to count tiling with infinitely small tiles that are too small to bend,,... but I'd argue, tthat the curvature is non-0 regardless of scale due to it only asymptotically aproaching 0; hence You should never be able to unfold torus to 2d
Since this is not torus (as it's not having circular crossection) it does not pose such problem as You just have multi-polygon approximation of a torus which can be tiled as it's only polygon grid.
Why should a torus be circular ? It isn't a required constraint most of the time and certainly not if you consider going from a flat plane to a flat torus.
New word in topology~fidley
OK Henry, so what are your thoughts on the Duocylinder? upload.wikimedia.org/wikipedia/commons/7/7e/Duocylinder_ridge_animated.gif
That's a really great example of a flat torus, only that it requires a fourth dimension of freedom to connect up all of the edges that way.. ;3
If I understand this correctly, the boundary of the Duocylinder is a square flat torus, in fact the Clifford torus. Yes, it's a great example.
Can the shape, if extended to add extra faces and hinges, fold into larger toruses?
Reminds me of the camera iris shape.
Nice
Map projections hate him
Aperture Science
thats fucking awesome
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Who the fuck disliked this
Nobody knows anymore