15 ms·

Planets in the Fourth Dimension

252 points by diego898 12y ago

tokai 12y ago
Awesome!
superobserver 12y ago
Cool find. Added spatial dimensions always intrigue when presented visually (rather than logically or symbolically). Edit: this makes me wonder what other eliptical orbits might look like. Would the 4-sphere have to rotate about the barycenter to give the effect of this: http://www.polaris.iastate.edu/EveningStar/Unit4/Graphics/PicES4_8.jpg http://www.polaris.iastate.edu/EveningStar/Unit4/Graphics/Pi...
claar 12y ago
I've been playing the Kerbal Space Program (KSP) game lately; it's funny how when you actively use orbital mechanics, thinking about them is easier. This fourth dimension observation is cool and likely very useful mathematically, but it's also kinda obvious (at least after playing KSP) that if you subtract the time/gravity element from an orbit, it becomes circular. It's equivalent to saying that if you subtract out the gravity effects of the planet you're orbiting, your orbital speed is constant, which just makes sense intuitively. Calling this observation a fourth dimension is useful, but perhaps unnecessarily complicated for the simple concept.
- fixermark 12y ago
  I was a little sad to learn that because KSP's orbital model uses conic sections, it's not physically accurate enough to model things like Lagrange points. Then I remembered that it's still accurate enough to be awesome and stopped worrying about it. :)
  - Coincoin 12y ago
    As a matter of fact, I think that its simplicity is what makes KSP so fun to play. If it was real n-body orbital mechanics with real drag the game would quickly become much too complicated and tedious for casual play (and too slow).
    - TeMPOraL 12y ago
      Yup. One of the Fun things that are less known that comes from n-body orbital mechanics is the instability of orbits. You can't just leave your satellite out there and fast-forward two years because you have an Eve mission going - your satellite needs to make correction burns as other bodies perturb its orbit. http://en.wikipedia.org/wiki/Orbital_station-keeping http://en.wikipedia.org/wiki/Orbital_station-keeping But anyway, I still hope that Principia mod (the n-body simulation addon mentioned elsewhere in this thread) will get released. This will really bring KSP into Dwarf Fortress level of enjoyment. Also, you'd get to make some really weird-ass stuff: http://images.scholarpedia.org/w/images/f/f6/NbodyChoreoGamma1Var.gif http://images.scholarpedia.org/w/images/f/f6/NbodyChoreoGamm... http://tuvalu.santafe.edu/~moore/crisscross123.loop.gif http://tuvalu.santafe.edu/~moore/crisscross123.loop.gif
    - gohrt 12y ago
      For broader accessibility: a system like Civ's, where the system can play itself / do computations for you, but lets you dial up the amount of player-control.
      - TeMPOraL 12y ago
        You have MechJeb mod for that.
  - valarauca1 12y ago
    Actually Newtonian Conic Sections is how we (humans) calculated orbital, and interplanetary trajectories up until the 1980's. There weren't powerful enough computers to model full Eisenstein N-Bodies Systems until actually very recently (also its complete over kill for inner-solar system travel). Relativity is really complex math, and even modern computer clusters struggle to model very complex systems. Yes we sent astronauts to the moon using KSP math.
    - TeMPOraL 12y ago
      Thankfully, we only used KSP math, and not KSP engineering.
    - titanomachy 12y ago
      Aren't Lagrange points present in Newton's model? Why would they be missing from the game?
      - Rumford 12y ago
        They're missing from KSP not merely because it uses Newtonian physics, but because it only uses two-body gravitational physics at any given time. The planets and moons are on rails and you are in only one gravity well at a time. Lagrange points exist in the intersections between two gravity wells.
    - theothermkn 12y ago
      > Yes we sent astronauts to the moon using KSP math. I'm almost positive that we did not. While patched conics would have been used very early on for rough mission analysis and design, we had a very good understanding of perturbation theory at the time. Wikipedia tells me that the restricted 3-body problem was also essentially solved in 1917. I've seen very detailed plots of the free return trajectories that the Apollo missions followed, too. Using patched conics for the earth moon system would have resulted in an error on the scale of lunar escape velocity upon entering the sphere of influence of the Moon.
      - valarauca1 12y ago
  - Lutin 12y ago
    Well if you're still interested in following the ongoing work to change that check out Principia :) http://forum.kerbalspaceprogram.com/threads/68502-WIP-Principia-N-Body-Gravitation-and-Better-Integrators-for-Kerbal-Space-Program http://forum.kerbalspaceprogram.com/threads/68502-WIP-Princi... https://github.com/mockingbirdnest/Principia https://github.com/mockingbirdnest/Principia
    - titanomachy 12y ago
      I thought for a second you were facetiously referring to Philosophiae Naturalis Principia Mathematica... "hey, check out this cool new work on orbital mechanics" :P
- Florin_Andrei 12y ago
  Well, it's a different parameter space, that's all. We tend to operate in the usual 6-dimensional parameter space for everything we do (3 position vectors, 3 speed vectors), but there are equivalent parameter spaces that do the job just as well. In some cases, the alternate spaces are more useful - polar coordinates are the most common example. The paper quoted in the article found a parameter space where very interesting transformations and symmetries take place.
  - gohrt 12y ago
    This is exacly right, and it's a shame that article pushes the "woo, magic" perspective, instead of using the opportunity to highlight the little-known fact (among the lay public and students) of theoretical phyisics: it's a about model-building, "dimensions" aren't real except in the sense that a certain mathematical model (with N dimensions) is (a) consistent with experiment, and (b) has a lot of symmetry and simplicity relative to its explanatory/predictive power.
    - Florin_Andrei 12y ago
      TBH, when you stumble upon a really useful parameter space, it does seem like magic. Also, these days I'm reading Max Tegmark's "Our Mathematical Universe" and, being immersed in that argumentation, the whole "real vs unreal" dichotomy seems a bit transparent. Anyway, having formerly played a bit with computational physics, I'm primarily after what's useful.
      - sonoffett 12y ago
        General semantics helped me be aware of how easy it is to confuse the map for the territory, as well as adopt models based on their usefulness. As a great philosopher once said, reality is what you can get away with.
- TeMPOraL 12y ago
  With barely 200 hours of flight under my belt, I think I haven't played the game enough, because I haven't reached that intuition yet to see this article's insight as obvious. What this means, of course, is that I need to go back playing more KSP :). Seriously, it's amazing how this game can let you understand space travel and make you want to know the math behind it.
  - fixermark 12y ago
    Here's the shortest, sweetest "a-ha" I've yet learned in KSP. So you're piloting ship A and you want to rendezvous with ship B in a stable orbit. Let's assume that A and B are on identical, basically circular orbits, and A is behind B, so your distances from each other aren't really changing much. The maneuver you're looking for is.... Point retrograde (i.e. "away from the direction that will take you to where B is now") and burn. In the short run, you increase your distance from B (of course), but you also dropped your orbital altitude and therefore the area of the arc swept between your ship and your center of orbit over time... and by Kepler's second law, you are now orbiting faster (more full revolutions per unit of time). So you'll swing under B and eventually come up "in front" of it; time your burns right, and you come up epsilon distance in front of B; rendezvous successful. It's totally counter-intuitive to accelerate away from something to approach it, except you're basically on a sphere. :-p
    - hobs 12y ago
      Agreed, and I had played over 200 hours of KSP before I really got a handle on rendezvous, and had many failed attempts before intuitively getting it, and it becoming easy. After that point no maneuver was daunting in KSP, just something that was a matter of time and making sure I quick save occasionally.
- troubled5 12y ago
  How can you subtract gravity from an orbit and end up with a circle? If you have no gravity the particle will simply move in a straight line, and that's not a motion I would call an orbit.
  - claar 12y ago
    You're absolutely right, of course. I was talking about the "time-gravity" component they subtract in the article to circularize the orbit, not all of gravity. During an orbit, gravity is of course stronger when you're near the planet, and weaker when you're away from the planet. So they equalize the gravity by subtracting the relative time/gravity differential and calling it a separate dimension. This leaves a constant gravity force and velocity, making certain calculations and transformations more natural in this model.
jwcacces 12y ago
It's true, I do that planets go around the sun in elliptical orbits...
- dashoffset 12y ago
  probably
Xeoncross 12y ago
> In fact, they’re moving in circles in 4 dimensions. But when these circles are projected down to 3-dimensional space, they become ellipses! Um, no. They are moving in 3 dimensions and when they are projected down to 2-dimensional space they become ellipses.
- drcube 12y ago
  I think you missed the point. Planets orbit in ellipses in the regular, 3D world.
  - Xeoncross 12y ago
    Are we talking about "orbiting in elliptical paths" or "planets are ellipses"? In regular Height-Width-Depth (3D) space (excluding time) the plants are circling in ellipses. In only Height-Width (2D) space plants are flat circles and when projected onto a plane produce an elliptical path. So why the downvote?
deleted 12y ago
[deleted]
ebbv 12y ago
I'm no physics major but I'm pretty sure this is a bunch of horse shit and the elliptical orbits are explained perfectly by General Relativity.
- gohrt 12y ago
  This article is a visualization / coordinatization of Newtonian physics. GR has nowhere near enough effect to make a simple planetry orbit elliptical. Baez has earned more credibility than to be written off with a vulgar thoughtless one-liner. He is a Physics major. http://en.wikipedia.org/wiki/John_C._Baez http://en.wikipedia.org/wiki/John_C._Baez
- subnaught 12y ago
  General Relativity is not needed to explain elliptical orbits. This is a very clever reformulation of classical mechanics, and certainly not horseshit.
  - ebbv 12y ago
    I could "very clever"-ly reformulate Newtonian physics as a series of invisible frogs who push things around by jumping into them. It's still horse shit. This article about a "weird fourth dimension" that's "like time but not time" certainly seems like horse shit to me.
    - subnaught 12y ago
      So it's horseshit because it "seems like horseshit" to you? In fact, coordinate transformations are at the heart of the modern conception of classical mechanics: http://en.wikipedia.org/wiki/Lagrangian_mechanics http://en.wikipedia.org/wiki/Lagrangian_mechanics
    - ozankabak 12y ago
      Your "invisible frogs" would not be a reformulation of Newtonian Mechanics (NM). A reformulation needs to make the exact same predictions as the original theory. There are many reformulations of both NM and GR, and they are quite useful (and interesting) in various contexts. This is a solid, interesting work that does not deserve being dubbed "horse shit" by a person who does not even have a formal training in the field.
impostervt 12y ago
I'm not adding a fourth dimension to my code. http://www.aretheplanetsaligned.com http://www.aretheplanetsaligned.com
mattxxx 12y ago
Cool. Physics is about modeling. If you create a model, and it's consistent with the real world, then it's inscrutable. Cool. The only time you can contradict a model in Physics, is if the model conflicts with something observable. Isn't that why we have so many coexisting cosmological theories?
peter303 12y ago
I've seen a similar trick in numerical analysis. You add an fourth dimension to a computation grid and it reduces numerical problems with the 3D computation. I dont know if there is a name for this trick.
- DarkUranium 12y ago
  There is also a trick (read: algorithm) in computational geometry that computes (2D) Delaunay triangulation by mapping the points onto a (3D) paraboloid, and then computing the convex hull of that. E.g.: http://i.stack.imgur.com/OuWmZ.png http://i.stack.imgur.com/OuWmZ.png
- mappu 12y ago
  >You add an fourth dimension to a computation grid and it reduces numerical problems with the 3D computation e.g. gimbal lock solved by quaternian rotation.
Rumford 12y ago
You could also think of this time like dimension as acceleration. When above the plane, the planet is accelerating, and it is decelerating when below the plane.
stevebmark 12y ago
This is fundamentally wrong and very poorly described. All this unusual article amounts to is "you can rotate things." A planet is not a 3d projection of a 4d object. Space itself contains at least four spatial dimensions but it is not shaped like a hypersphere around every star that a planet revolves around.
danneu 12y ago
I noticed sci-fi author Greg Egan in the comments. Looks like he's still battling against Google using the wrong image for him when you google his name: http://gregegan.customer.netspace.net.au/ESSAYS/GOOGLE/Google.html http://gregegan.customer.netspace.net.au/ESSAYS/GOOGLE/Googl...
- mjcohen 12y ago
  Google is sometimes doing the same to me, but the picture is of a younger, better looking guy, so I'm not complaining.
madaxe_again 12y ago
This is a topic I like to muse on day-to-day - I find visualising hyperspace interesting. Extending the same notion of there being a time dimension that is less dense the further it is from a gravity well - well, it changes certain aspects of how one might look at the universe, such as, say, why distant galaxies are redshifted, why galaxies rotate evenly throughout their volumes, and why the speed of light is a thing.

252 points by diego898 12y ago

tokai 12y ago

Awesome!

Cool find. Added spatial dimensions always intrigue when presented visually (rather than logically or symbolically). Edit: this makes me wonder what other eliptical orbits might look like. Would the 4-sphere have to rotate about the barycenter to give the effect of this: http://www.polaris.iastate.edu/EveningStar/Unit4/Graphics/PicES4_8.jpg http://www.polaris.iastate.edu/EveningStar/Unit4/Graphics/Pi...

claar 12y ago

I've been playing the Kerbal Space Program (KSP) game lately; it's funny how when you actively use orbital mechanics, thinking about them is easier. This fourth dimension observation is cool and likely very useful mathematically, but it's also kinda obvious (at least after playing KSP) that if you subtract the time/gravity element from an orbit, it becomes circular. It's equivalent to saying that if you subtract out the gravity effects of the planet you're orbiting, your orbital speed is constant, which just makes sense intuitively. Calling this observation a fourth dimension is useful, but perhaps unnecessarily complicated for the simple concept.

fixermark 12y ago
I was a little sad to learn that because KSP's orbital model uses conic sections, it's not physically accurate enough to model things like Lagrange points. Then I remembered that it's still accurate enough to be awesome and stopped worrying about it. :)
- Coincoin 12y ago
  As a matter of fact, I think that its simplicity is what makes KSP so fun to play. If it was real n-body orbital mechanics with real drag the game would quickly become much too complicated and tedious for casual play (and too slow).
  - TeMPOraL 12y ago
    Yup. One of the Fun things that are less known that comes from n-body orbital mechanics is the instability of orbits. You can't just leave your satellite out there and fast-forward two years because you have an Eve mission going - your satellite needs to make correction burns as other bodies perturb its orbit. http://en.wikipedia.org/wiki/Orbital_station-keeping http://en.wikipedia.org/wiki/Orbital_station-keeping But anyway, I still hope that Principia mod (the n-body simulation addon mentioned elsewhere in this thread) will get released. This will really bring KSP into Dwarf Fortress level of enjoyment. Also, you'd get to make some really weird-ass stuff: http://images.scholarpedia.org/w/images/f/f6/NbodyChoreoGamma1Var.gif http://images.scholarpedia.org/w/images/f/f6/NbodyChoreoGamm... http://tuvalu.santafe.edu/~moore/crisscross123.loop.gif http://tuvalu.santafe.edu/~moore/crisscross123.loop.gif
  - gohrt 12y ago
    For broader accessibility: a system like Civ's, where the system can play itself / do computations for you, but lets you dial up the amount of player-control.
    - TeMPOraL 12y ago
      You have MechJeb mod for that.
- valarauca1 12y ago
  Actually Newtonian Conic Sections is how we (humans) calculated orbital, and interplanetary trajectories up until the 1980's. There weren't powerful enough computers to model full Eisenstein N-Bodies Systems until actually very recently (also its complete over kill for inner-solar system travel). Relativity is really complex math, and even modern computer clusters struggle to model very complex systems. Yes we sent astronauts to the moon using KSP math.
  - TeMPOraL 12y ago
    Thankfully, we only used KSP math, and not KSP engineering.
  - titanomachy 12y ago
    Aren't Lagrange points present in Newton's model? Why would they be missing from the game?
    - Rumford 12y ago
      They're missing from KSP not merely because it uses Newtonian physics, but because it only uses two-body gravitational physics at any given time. The planets and moons are on rails and you are in only one gravity well at a time. Lagrange points exist in the intersections between two gravity wells.
  - theothermkn 12y ago
    > Yes we sent astronauts to the moon using KSP math. I'm almost positive that we did not. While patched conics would have been used very early on for rough mission analysis and design, we had a very good understanding of perturbation theory at the time. Wikipedia tells me that the restricted 3-body problem was also essentially solved in 1917. I've seen very detailed plots of the free return trajectories that the Apollo missions followed, too. Using patched conics for the earth moon system would have resulted in an error on the scale of lunar escape velocity upon entering the sphere of influence of the Moon.
    - valarauca1 12y ago
- Lutin 12y ago
  Well if you're still interested in following the ongoing work to change that check out Principia :) http://forum.kerbalspaceprogram.com/threads/68502-WIP-Principia-N-Body-Gravitation-and-Better-Integrators-for-Kerbal-Space-Program http://forum.kerbalspaceprogram.com/threads/68502-WIP-Princi... https://github.com/mockingbirdnest/Principia https://github.com/mockingbirdnest/Principia
  - titanomachy 12y ago
    I thought for a second you were facetiously referring to Philosophiae Naturalis Principia Mathematica... "hey, check out this cool new work on orbital mechanics" :P
Florin_Andrei 12y ago
Well, it's a different parameter space, that's all. We tend to operate in the usual 6-dimensional parameter space for everything we do (3 position vectors, 3 speed vectors), but there are equivalent parameter spaces that do the job just as well. In some cases, the alternate spaces are more useful - polar coordinates are the most common example. The paper quoted in the article found a parameter space where very interesting transformations and symmetries take place.
- gohrt 12y ago
  This is exacly right, and it's a shame that article pushes the "woo, magic" perspective, instead of using the opportunity to highlight the little-known fact (among the lay public and students) of theoretical phyisics: it's a about model-building, "dimensions" aren't real except in the sense that a certain mathematical model (with N dimensions) is (a) consistent with experiment, and (b) has a lot of symmetry and simplicity relative to its explanatory/predictive power.
  - Florin_Andrei 12y ago
    TBH, when you stumble upon a really useful parameter space, it does seem like magic. Also, these days I'm reading Max Tegmark's "Our Mathematical Universe" and, being immersed in that argumentation, the whole "real vs unreal" dichotomy seems a bit transparent. Anyway, having formerly played a bit with computational physics, I'm primarily after what's useful.
    - sonoffett 12y ago
      General semantics helped me be aware of how easy it is to confuse the map for the territory, as well as adopt models based on their usefulness. As a great philosopher once said, reality is what you can get away with.
TeMPOraL 12y ago
With barely 200 hours of flight under my belt, I think I haven't played the game enough, because I haven't reached that intuition yet to see this article's insight as obvious. What this means, of course, is that I need to go back playing more KSP :). Seriously, it's amazing how this game can let you understand space travel and make you want to know the math behind it.
- fixermark 12y ago
  Here's the shortest, sweetest "a-ha" I've yet learned in KSP. So you're piloting ship A and you want to rendezvous with ship B in a stable orbit. Let's assume that A and B are on identical, basically circular orbits, and A is behind B, so your distances from each other aren't really changing much. The maneuver you're looking for is.... Point retrograde (i.e. "away from the direction that will take you to where B is now") and burn. In the short run, you increase your distance from B (of course), but you also dropped your orbital altitude and therefore the area of the arc swept between your ship and your center of orbit over time... and by Kepler's second law, you are now orbiting faster (more full revolutions per unit of time). So you'll swing under B and eventually come up "in front" of it; time your burns right, and you come up epsilon distance in front of B; rendezvous successful. It's totally counter-intuitive to accelerate away from something to approach it, except you're basically on a sphere. :-p
  - hobs 12y ago
    Agreed, and I had played over 200 hours of KSP before I really got a handle on rendezvous, and had many failed attempts before intuitively getting it, and it becoming easy. After that point no maneuver was daunting in KSP, just something that was a matter of time and making sure I quick save occasionally.
troubled5 12y ago
How can you subtract gravity from an orbit and end up with a circle? If you have no gravity the particle will simply move in a straight line, and that's not a motion I would call an orbit.
- claar 12y ago
  You're absolutely right, of course. I was talking about the "time-gravity" component they subtract in the article to circularize the orbit, not all of gravity. During an orbit, gravity is of course stronger when you're near the planet, and weaker when you're away from the planet. So they equalize the gravity by subtracting the relative time/gravity differential and calling it a separate dimension. This leaves a constant gravity force and velocity, making certain calculations and transformations more natural in this model.

jwcacces 12y ago

It's true, I do that planets go around the sun in elliptical orbits...

dashoffset 12y ago
probably

Xeoncross 12y ago

> In fact, they’re moving in circles in 4 dimensions. But when these circles are projected down to 3-dimensional space, they become ellipses! Um, no. They are moving in 3 dimensions and when they are projected down to 2-dimensional space they become ellipses.

drcube 12y ago
I think you missed the point. Planets orbit in ellipses in the regular, 3D world.
- Xeoncross 12y ago
  Are we talking about "orbiting in elliptical paths" or "planets are ellipses"? In regular Height-Width-Depth (3D) space (excluding time) the plants are circling in ellipses. In only Height-Width (2D) space plants are flat circles and when projected onto a plane produce an elliptical path. So why the downvote?

deleted 12y ago

[deleted]

ebbv 12y ago

I'm no physics major but I'm pretty sure this is a bunch of horse shit and the elliptical orbits are explained perfectly by General Relativity.

gohrt 12y ago
This article is a visualization / coordinatization of Newtonian physics. GR has nowhere near enough effect to make a simple planetry orbit elliptical. Baez has earned more credibility than to be written off with a vulgar thoughtless one-liner. He is a Physics major. http://en.wikipedia.org/wiki/John_C._Baez http://en.wikipedia.org/wiki/John_C._Baez
subnaught 12y ago
General Relativity is not needed to explain elliptical orbits. This is a very clever reformulation of classical mechanics, and certainly not horseshit.
- ebbv 12y ago
  I could "very clever"-ly reformulate Newtonian physics as a series of invisible frogs who push things around by jumping into them. It's still horse shit. This article about a "weird fourth dimension" that's "like time but not time" certainly seems like horse shit to me.
  - subnaught 12y ago
    So it's horseshit because it "seems like horseshit" to you? In fact, coordinate transformations are at the heart of the modern conception of classical mechanics: http://en.wikipedia.org/wiki/Lagrangian_mechanics http://en.wikipedia.org/wiki/Lagrangian_mechanics
  - ozankabak 12y ago
    Your "invisible frogs" would not be a reformulation of Newtonian Mechanics (NM). A reformulation needs to make the exact same predictions as the original theory. There are many reformulations of both NM and GR, and they are quite useful (and interesting) in various contexts. This is a solid, interesting work that does not deserve being dubbed "horse shit" by a person who does not even have a formal training in the field.

impostervt 12y ago

I'm not adding a fourth dimension to my code. http://www.aretheplanetsaligned.com http://www.aretheplanetsaligned.com

mattxxx 12y ago

Cool. Physics is about modeling. If you create a model, and it's consistent with the real world, then it's inscrutable. Cool. The only time you can contradict a model in Physics, is if the model conflicts with something observable. Isn't that why we have so many coexisting cosmological theories?

peter303 12y ago

I've seen a similar trick in numerical analysis. You add an fourth dimension to a computation grid and it reduces numerical problems with the 3D computation. I dont know if there is a name for this trick.

DarkUranium 12y ago
There is also a trick (read: algorithm) in computational geometry that computes (2D) Delaunay triangulation by mapping the points onto a (3D) paraboloid, and then computing the convex hull of that. E.g.: http://i.stack.imgur.com/OuWmZ.png http://i.stack.imgur.com/OuWmZ.png
mappu 12y ago
>You add an fourth dimension to a computation grid and it reduces numerical problems with the 3D computation e.g. gimbal lock solved by quaternian rotation.

Rumford 12y ago

You could also think of this time like dimension as acceleration. When above the plane, the planet is accelerating, and it is decelerating when below the plane.

stevebmark 12y ago

This is fundamentally wrong and very poorly described. All this unusual article amounts to is "you can rotate things." A planet is not a 3d projection of a 4d object. Space itself contains at least four spatial dimensions but it is not shaped like a hypersphere around every star that a planet revolves around.

danneu 12y ago

I noticed sci-fi author Greg Egan in the comments. Looks like he's still battling against Google using the wrong image for him when you google his name: http://gregegan.customer.netspace.net.au/ESSAYS/GOOGLE/Google.html http://gregegan.customer.netspace.net.au/ESSAYS/GOOGLE/Googl...

mjcohen 12y ago
Google is sometimes doing the same to me, but the picture is of a younger, better looking guy, so I'm not complaining.

madaxe_again 12y ago

This is a topic I like to muse on day-to-day - I find visualising hyperspace interesting. Extending the same notion of there being a time dimension that is less dense the further it is from a gravity well - well, it changes certain aspects of how one might look at the universe, such as, say, why distant galaxies are redshifted, why galaxies rotate evenly throughout their volumes, and why the speed of light is a thing.