Planets in the Fourth Dimension

https://johncarlosbaez.wordpress.com/2015/03/17/planets_in_the_4th_dimension/

by diego898 • 11 years ago

252 53 11 years ago

johncarlosbaez.wordpress.com

tokai• 11 years ago

Awesome!

superobserver• 11 years 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/Pi...

claar2• 11 years 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.

fixermark2• 11 years 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• 11 years 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).

valarauca12• 11 years 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.

Lutin• 11 years 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-Princi... https://github.com/mockingbirdnest/Principia

Florin_Andrei2• 11 years 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.

gohrt3• 11 years 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.

TeMPOraL4• 11 years 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.

fixermark2• 11 years 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

troubled5• 11 years 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.

claar2• 11 years 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• 11 years ago

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

dashoffset• 11 years ago

probably

Xeoncross2• 11 years 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• 11 years ago

I think you missed the point. Planets orbit in ellipses in the regular, 3D world.

Xeoncross2• 11 years 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?

ebbv2• 11 years 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.

gohrt3• 11 years 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

subnaught2• 11 years ago

General Relativity is not needed to explain elliptical orbits. This is a very clever reformulation of classical mechanics, and certainly not horseshit.

ebbv2• 11 years 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.

pqomdv• 11 years ago

Ebbv might be referring to problems with orbital precession, which was explained completely by general relativity.

zamalek• 11 years ago

Also, General Relativity is 4 dimensional.

Chathamization• 11 years ago

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).

fsloth• 11 years ago

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.

impostervt• 11 years ago

I'm not adding a fourth dimension to my code.

http://www.aretheplanetsaligned.com

mattxxx• 11 years 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• 11 years 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• 11 years 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

mappu• 11 years 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.

Rumford2• 11 years 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• 11 years 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• 11 years 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/Googl...

mjcohen• 11 years 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• 11 years 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.

Planets in the Fourth Dimension

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