Planets in the Fourth Dimension(johncarlosbaez.wordpress.com)
johncarlosbaez.wordpress.com
Planets in the Fourth Dimension
https://johncarlosbaez.wordpress.com/2015/03/17/planets_in_the_4th_dimension/
54 comments
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.
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.
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.
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.
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.
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.
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. :)
Then I remembered that it's still accurate enough to be awesome and stopped worrying about it. :)
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.
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.
Thankfully, we only used KSP math, and not KSP engineering.
> 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.
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.
>I've seen very detailed plots of the free return trajectories that the Apollo missions followed, too.
Free return trajectories can be calculated with patched conics. 3 Body problem was solved, but actually doing all the math involved dynamically was far to complex for mission computers.
You forget that at the time NASA was using IBM System/360's which were struggling to maintain 5 megaFLOPs
Free return trajectories can be calculated with patched conics. 3 Body problem was solved, but actually doing all the math involved dynamically was far to complex for mission computers.
You forget that at the time NASA was using IBM System/360's which were struggling to maintain 5 megaFLOPs
I get that the idea that we "went to the moon with a slide rule" and similar ideas are appealing, but I'm not convinced that your statements are resting on a basis of fact.
For one thing, I'm not convinced that your mental models of what a "mission computer" is and what a "mission computer" would be doing, or when, make sense from an engineering perspective. I'm also not convinced that you know what an orbital perturbation is, nor how they would be used to plan and fly a mission.
To take an example from aviation, we didn't have practical simulations of general viscous fluid flow in the 1960s, but it would be meaningless to say that we "used Bernoulli's principle" to fly across the Pacific.
The Apollo vehicles had an on-board IMU. A trajectory could thus be pre-planned and flown to using feedback control. That trajectory would certainly have been calculated ahead of time. Thus, while there would be no need to have "[done] the math involved dynamically," that by no means implies that patched conics were used while in flight, nor does it imply that they were in any meaningful way used for the final design of the missions' flight paths.
Further, on any deviation from the planned flight path, numerical integration methods would yield results that would be good enough for later correction, again, by using feedback control.
If you still need to believe that "KSP math" is how we got to the Moon, go ahead. You've certainly done nothing to convince anyone else, though.
For one thing, I'm not convinced that your mental models of what a "mission computer" is and what a "mission computer" would be doing, or when, make sense from an engineering perspective. I'm also not convinced that you know what an orbital perturbation is, nor how they would be used to plan and fly a mission.
To take an example from aviation, we didn't have practical simulations of general viscous fluid flow in the 1960s, but it would be meaningless to say that we "used Bernoulli's principle" to fly across the Pacific.
The Apollo vehicles had an on-board IMU. A trajectory could thus be pre-planned and flown to using feedback control. That trajectory would certainly have been calculated ahead of time. Thus, while there would be no need to have "[done] the math involved dynamically," that by no means implies that patched conics were used while in flight, nor does it imply that they were in any meaningful way used for the final design of the missions' flight paths.
Further, on any deviation from the planned flight path, numerical integration methods would yield results that would be good enough for later correction, again, by using feedback control.
If you still need to believe that "KSP math" is how we got to the Moon, go ahead. You've certainly done nothing to convince anyone else, though.
Aren't Lagrange points present in Newton's model? Why would they be missing from the game?
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.
Well if you're still interested in following the ongoing work to change that check out Principia :)
http://forum.kerbalspaceprogram.com/threads/68502-WIP-Princi... https://github.com/mockingbirdnest/Principia
http://forum.kerbalspaceprogram.com/threads/68502-WIP-Princi... https://github.com/mockingbirdnest/Principia
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
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).
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
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/NbodyChoreoGamm...
http://tuvalu.santafe.edu/~moore/crisscross123.loop.gif
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/NbodyChoreoGamm...
http://tuvalu.santafe.edu/~moore/crisscross123.loop.gif
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.
Seriously, it's amazing how this game can let you understand space travel and make you want to know the math behind it.
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
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
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.
After that point no maneuver was daunting in KSP, just something that was a matter of time and making sure I quick save occasionally.
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.
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.
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.
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.
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.
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
>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.
e.g. gimbal lock solved by quaternian rotation.
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/Pi...
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/Pi...
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.
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?
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?
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.
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.
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/Googl...
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/Googl...
Google is sometimes doing the same to me, but the picture is of a younger, better looking guy, so I'm not complaining.
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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.
I'm not adding a fourth dimension to my code.
http://www.aretheplanetsaligned.com
http://www.aretheplanetsaligned.com
Awesome!
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.
Well, obfuscation with a hefty dose of "gee whiz!" at least. Goransson's paper that he links too at least states that it's about a "nonstandard parameterisation," as opposed to talking about how it's another form of time (hey Baez, I suppose you could explain MPH to people as being another form of time if your goal was to confuse rather than enlighten).
Anyway, people are better off looking at the paper itself (which is rather sober) than reading Baez's summary, as the two are about the same length. I've noticed there's an unfortunate tendency among too many physicists/mathematicians to focus more on trying to amaze their audience rather than to educate them (in it's worst manifestations, you find them spouting outright falsehoods).
Anyway, people are better off looking at the paper itself (which is rather sober) than reading Baez's summary, as the two are about the same length. I've noticed there's an unfortunate tendency among too many physicists/mathematicians to focus more on trying to amaze their audience rather than to educate them (in it's worst manifestations, you find them spouting outright falsehoods).
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
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
General Relativity is not needed to explain elliptical orbits. This is a very clever reformulation of classical mechanics, and certainly not horseshit.
Also, General Relativity is 4 dimensional.
Ebbv might be referring to problems with orbital precession, which was explained completely by general relativity.
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.
This article about a "weird fourth dimension" that's "like time but not time" certainly seems like horse shit to me.
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
In fact, coordinate transformations are at the heart of the modern conception of classical mechanics: http://en.wikipedia.org/wiki/Lagrangian_mechanics
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.
As stated in the top portion of the thread, the original paper is completely not horsesh!t. It is a reformulation of a 3 dimensional problem in 4 dimensions that significantly simplifies the specific calculations/relationships of interest.
The author could instead have used only abstract mathematical language (like "a configuration space with a basis of cardinality 4" and "parameter s defined by s = ..."), but this would be accessible to fewer readers. If you prefer dryer language, read the second half of the article, or the other resources it links to.
Using a configuration space that is a good fit for a specific system is not unusual in math or physics. It's the same thing that astrophysicists do when they work in orbital elements, or that engineers and programmers do when they make calculations in Fourier space. It's worth understanding these concepts even if you're not a physicist.
Configuration spaces are so fundamental and useful that they are introduced in the very first lecture on classical mechanics in Leornard Susskind's excellent series of Stanford physics courses: http://theoreticalminimum.com/courses
(Neal Stephenson's novel Anathem also has a simple introduction to configuration spaces in an appendix, because that's the sort of thing that ends up in a Neal Stephenson novel.)
Using a configuration space that is a good fit for a specific system is not unusual in math or physics. It's the same thing that astrophysicists do when they work in orbital elements, or that engineers and programmers do when they make calculations in Fourier space. It's worth understanding these concepts even if you're not a physicist.
Configuration spaces are so fundamental and useful that they are introduced in the very first lecture on classical mechanics in Leornard Susskind's excellent series of Stanford physics courses: http://theoreticalminimum.com/courses
(Neal Stephenson's novel Anathem also has a simple introduction to configuration spaces in an appendix, because that's the sort of thing that ends up in a Neal Stephenson novel.)
May I suggest Leonard Susskinds "Theoretical minimum" as an accessible introduction to classical mechanics which attempts to show among other things why reparametrizations are cool.
> 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.
Um, no. They are moving in 3 dimensions and when they are projected down to 2-dimensional space they become ellipses.
I think you missed the point. Planets orbit in ellipses in the regular, 3D world.
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?
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.