My brain keeps interpreting the pseudosphere as opening towards the camera and it borks up my mind when your fingers suddenly overlap (what my brain interprets as) the foreground
Beautiful model. There is another way to define the tractrix. Imagine a line of dominoes standing upright which are set to topple, but the last domino in the line is fixed upright. All the other dominoes end up leaning against it, but as you get further from the end of the line they lie down flatter and flatter. In the limit of infinitely many infinitely thin dominoes the top of each domino lies on a tractrix.
My brain completely lost perspective when I looked away for a second. It looked like it was bending in impossible ways at some points before it snapped back to reality near the end of the video.
Without _hyperbola_ EYE looked at US debt and lost all visionary perspective too - so many digits - March 2020 my auditory sense had ringing after working for Suzuki Porsche JCB Nissan Audi terminated with pneumonia with an octet of days in ICU (International Capitalist Usury) luckily my bones did not *snap back* in 2010 when I tried giving blood they said I was anaemic what they didn't follow up was I had Gluten intolerance a decade later [slightly] thinning bones which is naught compared to #LongCovid
I can only see this shape as part of the inside half of a torus. It's half of the donut hole. I'd been trying to figure out what shape you'd get if you flipped a sphere inside out, and I only got as far as the torus. Thank you for the demonstration- this is incredibly useful information.
Another way to define a pseudosphere is to take a sector of a horocycle (or a horodisk?) and roll it up to join the cut edges together (and embedding in Euclidean space). This is like how you can take a sector of a Euclidean disc and join the cut edges together to make a cone. In both cases the local curvature of the resulting shape is the same as curvature of the space it was cut out of.
When you showed how the curvature is defined with that illustration it reminded me of how a pistons up and down motion gets translated into rotational motion
It reminds me of those simplified models kf "infinite timespace curvature". Is it supposed to be the same geometric shape or perhaps they look the same in some ways but they are fundamentally different? (Like how hyperbola and parabola may seem similar to someone who hasn't seen or worked with them)
this is EXACTLY what I've been looking for. I was wondering what a sphere would look like with negative curvature but I couldn't figure out how phrase it to find it, and then I got this video in my recommended by pure coincidence.
This is a great video! A question I actually have is, and despite the representation on the model, would that mean that there is no end point of the pseudosphere?
I would say the volume is indefinetly and the surface is limited. (I think of a blackhole in the universe) - im a noob and Im curious to get an answer as well!
Another interesting fact about spheres and pseudospheres: On spheres u can have a Triangel with all ihrer angels being 90° and in a pseudospheres u can have a Pentagon with all die angels being 90°.
@@SuperSmashDolls Yep. Both are defined by constant negative curvature, thus it seems like more stuff fits in a smaller area. This can be seen because it is bent out into a larger space when projected in euclidean space whereas the surface of a sphere and the spherical plane are bent inwards closer to itself making things automatically feel closer.
I have some knowledge of mathematics involving these things, but it is not enough to fully comprehend it. So I kinda get what you said but itis hard to visualize XD
time to ruin the math class with the fact that this is a representation of a black hole, or at least the curvature of space time around and inside of it
What about the rotated tractrix makes the pseudosphere pseodospherical? I.e. non sphere spheroids also have constant positive curvature, what about this shape makes it pseodo*spherical*?
...this thing is trippy. on video i can't tell where the front and back is and if the hand is behind or inside it untill it covers a part of it. well done, my brain hurts.
He is right! The pseudosphere is the shape you get when 2D people wrongly imagine the 3D dimension and this could be the mistake we making when imagining the 4D dimension. The mistake is not shrinking the circles enough as you move in the x-direction!
The Wolfram Mathworld article has an explanation: "The pseudosphere therefore has the same volume as the sphere while having constant negative Gaussian curvature (rather than the constant positive curvature of the sphere), leading to the name "pseudo-sphere.""
Henry Segerman I don't understand the volume idea....to me the "skin" surface is not closed....so there is no enclosed volume to calculate...am I wrong?
Lau Ni There is some ambiguity in whether "pseudosphere" means the thing in the video, or the thing you get from two of these, glued together along the circular end. It is this thing that has an enclosed volume.
in that it is rotationally symmetric and smooth and extends off to infinity at the center, yes. the details of how far down at what distance from the center are quite different though. With those representations of a black hole, the whole thing like, gradually flattens out as you get further away, while with this shape, if you extend it at the not-pointy end, keeping the "the curvature is -1", eventually it ends up like, looping around and intersecting itself (if embedded in 3d euclidean space. I believe you can consider it intrinsically, not embedded in any larger space, and have it just continue without any problems.) That being said, those depiction of a black hole should have a negative curvature everywhere. This is not to say that the curvature of space around a black hole is negative. I don't understand general relativity enough to make that assertion. Maybe it is, but I don't know. I'm just saying that those particular visual representations of one should have negative curvature everywhere. Just not a constant negative curvature. The pseudosphere is specified by having specifically constant negative curvature everywhere. The surface of a normal chicken egg has positive curvature everywhere, but unlike a sphere, its curvature isn't constant over the shape. Analogously, these depictions of a black hole have negative curvature everywhere, but unlike a pseudosphere, its curvature isn't constant over the shape. That's not specific to a chicken egg btw. That was just an example of a shape with positive curvature everywhere but not constant curvature. Anything that looks largely like a sphere but like, not quite, and is convex, should fit the analogy as well.
To see inside the hole, look at the oval. To see how it actually is, look at the hand and move your eyes towards the finger. Recommend pausing to flip the perspective, in your brain.
@@redpepper74 the big trip to the big rip. expands so violently the atoms in your body are ripped apart. right up to it starts tearing quarks apart. the energy required to tear apart a quark is greater than the mass energy of a quark, so when you tear apart a quark, you just get two quarks, not half a quark in each hand. basically we end up with a quark soup. which some think may drive an inflation type epoch, leading to another universe, and ~10 odd billion years later, some beings looking back at a CMB and thinking about a big bang.
My brain keeps interpreting the pseudosphere as opening towards the camera and it borks up my mind when your fingers suddenly overlap (what my brain interprets as) the foreground
It’s such a cool effect to experience (and I don’t care that you commented this 8 months ago, you get to watch it happen again)
It also happened to me at the beginning, thought my eyes where broken
my mind too!
I Know Right!?
Yeah same
Looks similar to a paradoxical shape in mathematics known as the "Gabriel's Horn" which has infinite Surface Area but a finite Volume
It might be the same one, but I'm not sure
@@SlashCrash_Studios It looks similar, but they are mathematically very different
So we can paint it by filling it but not with a brush
But Gabriel's Horn is bounded unlike this one.
It looks like a truncated version of that
Beautiful model. There is another way to define the tractrix. Imagine a line of dominoes standing upright which are set to topple, but the last domino in the line is fixed upright. All the other dominoes end up leaning against it, but as you get further from the end of the line they lie down flatter and flatter. In the limit of infinitely many infinitely thin dominoes the top of each domino lies on a tractrix.
This is really cool! Thanks for sharing.
My brain completely lost perspective when I looked away for a second. It looked like it was bending in impossible ways at some points before it snapped back to reality near the end of the video.
Without _hyperbola_ EYE looked at US debt and lost all visionary perspective too - so many digits - March 2020 my auditory sense had ringing after working for Suzuki Porsche JCB Nissan Audi terminated with pneumonia with an octet of days in ICU (International Capitalist Usury) luckily my bones did not *snap back* in 2010 when I tried giving blood they said I was anaemic what they didn't follow up was I had Gluten intolerance a decade later [slightly] thinning bones which is naught compared to #LongCovid
@@OghamTheBold what kind of crack have you been smoking and where can i get some
@@ArcYT he on that tegridy shit
Ah, so you noticed that too. For a second I thought my eyes were tripping
@@OghamTheBold
Don't you have a brain?
I was wondering if you ever use it
Or it may just be there for decoration 😘💕
It is the perfect bell of a brass musical instrument. Like a trumpet or a tuba. The harmony of it all...
A model of the pseudosphere.
The demonstration with the weighted string is fantastic. I really really love those kinds of clear presentations of a concept.
Thanks Greg Burton!
Are you British because you really sound like the Birmingham area
do you have de 3D planes? tanks a lot
I can only see this shape as part of the inside half of a torus. It's half of the donut hole. I'd been trying to figure out what shape you'd get if you flipped a sphere inside out, and I only got as far as the torus. Thank you for the demonstration- this is incredibly useful information.
There's a famous video about turning spheres inside out here ua-cam.com/video/sKqt6e7EcCs/v-deo.html
Another way to define a pseudosphere is to take a sector of a horocycle (or a horodisk?) and roll it up to join the cut edges together (and embedding in Euclidean space). This is like how you can take a sector of a Euclidean disc and join the cut edges together to make a cone.
In both cases the local curvature of the resulting shape is the same as curvature of the space it was cut out of.
i love your videos. i'm doing math research right now, and this is just fascinating to me.
That reminds me of the model everyone uses for black holes
When you showed how the curvature is defined with that illustration it reminded me of how a pistons up and down motion gets translated into rotational motion
This whole video was trippy to watch and I love it
When you see the model that represents bending spacetime, a negative spherical shape is formed underneath a sphere (black hole/planet/star).
I wasn't looking closely at the thumbnail so I clicked this thinking it was going to be an animated 3D model and then a real hand showed up
It would've been cool if you just never allowed the pointy end to fall into frame as if the object in your hand might extend infinitely
It reminds me of those simplified models kf "infinite timespace curvature". Is it supposed to be the same geometric shape or perhaps they look the same in some ways but they are fundamentally different?
(Like how hyperbola and parabola may seem similar to someone who hasn't seen or worked with them)
Wouldn't the inside of a sphere also qualify?
My brain is interpreting all kind of cursed and forbidden shapes when he rotates and manipulate the pseudoephere
This really needs to be continued untilts a 3D cardioid rotated on node axis. And that shape is perfect model of manifold.
Is soo fantastic, is like a black hole, saludos desde Mexico!
8lack hole would be in 8lack plastic - the White hole is 39 times 10 to the power 13 - in Mexico is 13 lucky?
God damn this is a trip to look at I couldn't tell which way was up
Thanks for presenting a tactrix shape black hole . vow !!
this is EXACTLY what I've been looking for. I was wondering what a sphere would look like with negative curvature but I couldn't figure out how phrase it to find it, and then I got this video in my recommended by pure coincidence.
Concise and elegant.. Thank you.
I'd love it if he took a picture of him holding the pseudosphere with the pointed end on his wrist and he can become pseudospider-man 🤟
This is a great video! A question I actually have is, and despite the representation on the model, would that mean that there is no end point of the pseudosphere?
Correct, the sharp end goes on forever.
Does this have the same or similar properties as a sphere like volume and surface area?
I would say the volume is indefinetly and the surface is limited. (I think of a blackhole in the universe) - im a noob and Im curious to get an answer as well!
It looks like the visualization of a black hole's gravity well.
Another interesting fact about spheres and pseudospheres: On spheres u can have a Triangel with all ihrer angels being 90° and in a pseudospheres u can have a Pentagon with all die angels being 90°.
...wait that's just a hyperbolic plane
@@SuperSmashDolls Yep. Both are defined by constant negative curvature, thus it seems like more stuff fits in a smaller area. This can be seen because it is bent out into a larger space when projected in euclidean space whereas the surface of a sphere and the spherical plane are bent inwards closer to itself making things automatically feel closer.
I have some knowledge of mathematics involving these things, but it is not enough to fully comprehend it. So I kinda get what you said but itis hard to visualize XD
"Spider-Man, Spider-Man. Does whatever a spider can"
there is such a awesome supercool geek why few people noticed....
this concept is very interesting when applied to space-time and black holes. think about it..
Fascinating
time to ruin the math class with the fact that this is a representation of a black hole, or at least the curvature of space time around and inside of it
I psuedo-love it! Our brain makes it bend and curve!
this could have a perfect loop
some reason i keep sing the circle opening on the wrong direction, like the spinning silhouette thing
What about the rotated tractrix makes the pseudosphere pseodospherical? I.e. non sphere spheroids also have constant positive curvature, what about this shape makes it pseodo*spherical*?
...this thing is trippy.
on video i can't tell where the front and back is and if the hand is behind or inside it untill it covers a part of it.
well done, my brain hurts.
I thought this is gonna be a video about a Spider-Man toy
Man that was trippy
What is the mathematical eqn of that curve? Is it y=1/x
I had no idea this was a thing before watching this video, so this might be a stupid question, but would a pseudosphere not be infinite then?
The pseudosphere is supposed to be infinitely long, yes. The spiky end should go on forever.
@@henryseg Thanks for the reply. :) glad I watched the video!
i was expecting that a pseudosphere would be a shape with the same volume and surface area for a given radius, or something
The 1st thought I got from looking at this was Wormhole travel...
What does it look like if you view the negative sphere through the positive one?
Compound Pivot Hand Action
finally i think i understand surface curvature. 🤯🥳
Pierre Bézier would enjoy your shapely tractrix contrivance, however Dizzie Gillespie would not.
Spider web with a spherical end
i thought this was going to be something about a gravity well
Just handle this thing on Fool Us and take home the trophy. This video is an amazing optical illusion
Cool!!!!
He is right! The pseudosphere is the shape you get when 2D people wrongly imagine the 3D dimension and this could be the mistake we making when imagining the 4D dimension.
The mistake is not shrinking the circles enough as you move in the x-direction!
This is the shape of the curvature of spacetime of a black hole
You can make a 5 sided square on it I think
But why is it called pseudo sphere?
The Wolfram Mathworld article has an explanation: "The pseudosphere therefore has the same volume as the sphere while having constant negative Gaussian curvature (rather than the constant positive curvature of the sphere), leading to the name "pseudo-sphere.""
interesting
Henry Segerman I don't understand the volume idea....to me the "skin" surface is not closed....so there is no enclosed volume to calculate...am I wrong?
Lau Ni There is some ambiguity in whether "pseudosphere" means the thing in the video, or the thing you get from two of these, glued together along the circular end. It is this thing that has an enclosed volume.
Henry Segerman OK, thanks Henry
Looks like a fly ran into a spider web at mock 3
Se puede hacer lo mismo con el cubo?
0:15 beautiful crystal ball
Very good
But does it go in the square hole?
As a Yugioh player, all i hear is "Traptrix"
Она создаёт иллюзию изгибов :))
Thanks!
black hole gravity:
My head hurts
where can i buy one?
It’s interesting the tractrix is constant, as I would think it would change to due to coefficient of friction or something
This is messing with my brain omg
Pseudosphere looks like the interpretation of big bang
stop hurting my brain
looking like a spider-man web
now add another dimension to it.
my eyes tryna adjust like an cameleon watching this
❤
Hyperbolic!
Ya telling me a fkin dropper is related to a circle?
Gabriel's Horn
Someone took a Vuvuzela and rendered it in wireframe mode
I think you mean a pseudo-pseudo sphere, that thing doesn't extend to infinity :(
Ok this video is gonna blow up
I call it.
~50k vews rn
it looks like it bends
My sphere
Lineal sphere?
Isn't that similar to how black holes are often depicted?
in that it is rotationally symmetric and smooth and extends off to infinity at the center, yes.
the details of how far down at what distance from the center are quite different though.
With those representations of a black hole, the whole thing like, gradually flattens out as you get further away,
while with this shape, if you extend it at the not-pointy end, keeping the "the curvature is -1", eventually it ends up like, looping around and intersecting itself (if embedded in 3d euclidean space. I believe you can consider it intrinsically, not embedded in any larger space, and have it just continue without any problems.)
That being said,
those depiction of a black hole should have a negative curvature everywhere. This is not to say that the curvature of space around a black hole is negative. I don't understand general relativity enough to make that assertion. Maybe it is, but I don't know. I'm just saying that those particular visual representations of one should have negative curvature everywhere.
Just not a constant negative curvature.
The pseudosphere is specified by having specifically constant negative curvature everywhere.
The surface of a normal chicken egg has positive curvature everywhere, but unlike a sphere, its curvature isn't constant over the shape.
Analogously, these depictions of a black hole have negative curvature everywhere, but unlike a pseudosphere, its curvature isn't constant over the shape.
That's not specific to a chicken egg btw. That was just an example of a shape with positive curvature everywhere but not constant curvature. Anything that looks largely like a sphere but like, not quite, and is convex, should fit the analogy as well.
@@drdca8263 Thank you for the clarification (honestly you kinda lost me in the first part, but that analogy saved me xD)
It resembles the structure of a blackhole
0:50 - 1:00
The hole
I thought his thumb,
was inside
but then turn out to be outside.
To see inside the hole,
look at the oval.
To see how it actually is,
look at the hand and move your eyes towards the finger.
Recommend pausing to flip the perspective,
in your brain.
Apparently this is the shape of our universe.
So, wormholes?
In time direction.
@@lonestarr1490 huh… so it slowly expands for a long time until suddenly it expands faster and faster until it just _ends?_
@@redpepper74 the big trip to the big rip. expands so violently the atoms in your body are ripped apart. right up to it starts tearing quarks apart. the energy required to tear apart a quark is greater than the mass energy of a quark, so when you tear apart a quark, you just get two quarks, not half a quark in each hand. basically we end up with a quark soup. which some think may drive an inflation type epoch, leading to another universe, and ~10 odd billion years later, some beings looking back at a CMB and thinking about a big bang.
reminds me of Gabriels horn
p r i n g l e
Does this have anything mathematically in common with a tautochrone or is that just wishful thinking
Don’t you mean Spider-Man web?
i still don't understand why it's called a pseudosphere...
Gravity belike
My man that's a black hole
you can do this on a ballon
Vuvuzela
We live in a white hole at the opposite end of a Black hole
Doesn't pseudo mean "like"? Isn't this more like an opposite-sphere?