What Would Life Be Like on a Hypothetical Cube-Shaped Earth?

[Nathew]

Hypothetically, if the Earth were a cube, would walking to the corners(vertices etc.) feel like you were going uphill or would it feel flat?

 

[Danger]

I don't know, but it's a sure bet that I'd trip and fall off of the edge...

 

[Drakkith]

I'd guess it would feel like uphill.

 

[Evo]

I'd guess it would feel like uphill.

Both ways?

 

[collinsmark]

Yes, uphill (on your way toward a corner/edge, and assuming you are already closer to that corner/edge than the corner/edge behind you.) Your idea of what "down" is (and feels like) would always be pointed toward the center of the huge cube.

[Edit: Evo: ]

 

[Drakkith]

Both ways?

Nonsense. Once you step over the edge you just slide down the rest of the way!

 

[Evo]

Nonsense. Once you step over the edge you just slide down the rest of the way!

 

[SteamKing]

If the Earth were a cube, everybody would be like square, man.

 

[Danger]

If the Earth were a cube, everybody would be like square, man.

Oh, please, no! We'd still be up to our asses in Beatniks.

 

[Drakkith]

I'm far too hip to be square.

 

[FreeMitya]

I'm far too hip to be square.

Don't worry about being square, it's the new circular.

Yes, that's the best I've got.

 

[SW VandeCarr]

The surfaces would always look flat but feel increasingly tilted as you walked toward a corner or an edge.

 

[Eohlas]

Good question! Now I'm going to be asking everyone I know this question.

 

[Danger]

I'm far too hip to be square.

I've seen your hips; you're more pear-shaped than square.

Don't worry about being square, it's the new circular.

Yes, that's the best I've got.

My condolences.

Good question! Now I'm going to be asking everyone I know this question.

And you've been a devotee of Dale Carnegie for how long?

 

[chiro]

I'm far too hip to be square.

 

[Garth]

I've always known "It's a Square World"!

It's a Square World

A zany TV comedy programme that had me in stitches whilst a teenager; ah those were the days my friend...

Garth

 

[FlexGunship]

I'm most interested in the idea of a vertex of ocean. The question leads a lot of the topographical details to the imagination, but I'm fairly confident that, of any eight equidistant Terran-verticies, at least one would be in water.

"We'll take the boat to the top of Ocean Peak, and we'll water ski down!"

 

[ImATrackMan]

That would also bring up the matter of "What is 'sea-level'"

 

[SW VandeCarr]

I'm most interested in the idea of a vertex of ocean. The question leads a lot of the topographical details to the imagination, but I'm fairly confident that, of any eight equidistant Terran-verticies, at least one would be in water.

"We'll take the boat to the top of Ocean Peak, and we'll water ski down!"

Which way are the gravity vectors pointing at the vertices? It seems the vertices are like giant mountains which might support glaciers if the atmosphere could extend that "high", but liquid water would run off toward the center of each face.

 

[SW VandeCarr]

That would also bring up the matter of "What is 'sea-level'"

That's a good question. On earth, the gravity vectors are everywhere perpendicular to the ocean surface, as you would expect. The only places where the gravity vectors on a cubic planet are perpendicular to the cube surface are at the center of each face. So any surface water is constantly trying to get to that point, creating a lot of turbulence. If surface water behaves as it does on earth, the gravity vectors are always perpendicular to the mean water surface. So what would the ocean surface look like? It would not be "flat"; that is, conforming to the surface of the cube face.

 

[ImATrackMan]

Also, what would it [the planet's actual surface, mind you] look like to the people inhabiting it? As far as we're concerned, because we're so small compared to the earth, and so close to its surface, everything looks flat to us. In the case of a planet (or any celestial body, really) actually being flat (topography notwithstanding), the gravity would likely be biased toward the center of each face, but would it also look as if everything is tilted very slightly upward from the perspective of a person standing at the center of one of the faces (Probably not)?



EDIT: And what would seasonal changes bring about, or even be like?

 

[Andre]

So any surface water is constantly trying to get to that point, creating a lot of turbulence.

Would it? Turbulence implies conversion of energy. So you would expect an energy source or it would look like a perpetume mobile.

If surface water behaves as it does on earth, the gravity vectors are always perpendicular to the mean water surface. So what would the ocean surface look like? It would not be "flat"; that is, conforming to the surface of the cube face.

That would be a challenge to calculate. Incidently on a much smaller scale but the same principle, the effect on gravity of ice sheets melting after the last ice age is well understood, http://www.ngu.no/glacipet/photos/internal/pdfs_of_articles/geoidal.pdf. Sea levels are slightly modified as the gravity anomaly of the melted ice sheets have their (minute) effect on the direction of the gravity vector. I believe Nils Axel Mörner is credited for that but I can't find the applicable ref right now. http://www.jstor.org/discover/10.2307/30065695?uid=3738736&uid=2129&uid=2&uid=70&uid=4&sid=21102203212307.

 

[collinsmark]

That's a good question. On earth, the gravity vectors are everywhere perpendicular to the ocean surface, as you would expect. The only places where the gravity vectors on a cubic planet are perpendicular to the cube surface are at the center of each face. So any surface water is constantly trying to get to that point, creating a lot of turbulence. If surface water behaves as it does on earth, the gravity vectors are always perpendicular to the mean water surface. So what would the ocean surface look like? It would not be "flat"; that is, conforming to the surface of the cube face.

Yes, if an ocean of water was added to this cube the water would pool at the particular face in sort of upside down saucer sort of shape. And, (assuming that the weather is calm at the time) the gravitational acceleration vector would still be perpendicular to the surface of the water, just as it is here.

Also, what would it [the planet's actual surface, mind you] look like to the people inhabiting it? As far as we're concerned, because we're so small compared to the earth, and so close to its surface, everything looks flat to us. In the case of a planet (or any celestial body, really) actually being flat (topography notwithstanding), the gravity would likely be biased toward the center of each face, but would it also look as if everything is tilted very slightly upward from the perspective of a person standing at the center of one of the faces (Probably not)?

If you were at the center of one of the faces, and assuming you are not under water (we'll say the face in question does not have an ocean), the horizons would stretch out very far. The the land would still look as flat as can be. The land itself would not appear to be tilted at all.

But if you were to take out your telescope and look at people and buildings closer to the edges you would notice that they seem to be tilted: they would all seem to leaning away from you.

 

[nsaspook]

I'm sure some of you remember the old comic book series:
http://asitecalledfred.com/comics101/images/2003/sep24/bizarroworld.jpg

 

[BobG]

I would not like a cubical Earth where there were no spherical cows.

And spherical cows could not exist on a cubical Earth, since they'd all roll downhill into the water.

 

[SW VandeCarr]

Would it? Turbulence implies conversion of energy. So you would expect an energy source or it would look like a perpetume mobile.

I'm assuming the energy source is a star, otherwise we don't have an issue with liquid water.

 

[SW VandeCarr]

That would be a challenge to calculate.

We can make simplifying assumptions. The cube has smooth level surfaces everywhere, has the same volume as the Earth and mean density equal to mean density of the earth. We can then calculate the equipotential gravitational spherical surface centered on the center of gravity of the cube. I calculated the radius of this sphere as 0.6204 L where L is the length of a cube edge. So the sphere surface would be 0.1204 L above the face surface at the surface center.

One is tempted to say the water surface could rise this high since the gravity vectors would all be perpendicular on the water surface. However, water is considerably less dense than than the mean planet density so I suspect the water surface would not have the same curvature as the equipotential sphere. This means gravity vectors would be angled to the water surface with the greatest angles on the periphery where the water is shallowest. This is why I think there would be turbulence in these areas.

 

[collinsmark]

One is tempted to say the water surface could rise this high since the gravity vectors would all be perpendicular on the water surface. However, water is considerably less dense than than the mean planet density so I suspect the water surface would not have the same curvature as the equipotential sphere. This means gravity vectors would be angled to the water surface with the greatest angles on the periphery where the water is shallowest. This is why I think there would be turbulence in these areas.

Tidal forces might have particularly dramatic effects due to the lack of beaches. When the tide came it, it would come a long way in.

But neglecting the tide the "ocean" wouldn't be turbulent. It would just sit calmly in a upside down saucer sort of shape (as in a shallow dome looking thing).

Turbulence needs a energy source, and there is no energy source working on the water once it reaches equilibrium in its upside down saucer sort of shape (neglecting the tidal forces).

All of this of course ignores any weather effects.

 

[collinsmark]

Speaking of weather effects, the atmosphere would do the same thing as the water: concentrate itself around the center of the faces (the same sort of shallow dome sort of shape).

Most people don't realize how thin the Earth's atmosphere really is. I've heard it described this way: consider a standard sized globe, about the size of a basketball or so, that represents the Earth. Suppose this globe has a coat of shellac on it. The depth of the breathable, survivable atmosphere on Earth is roughly to scale with this layer of shellac.

So if one were to attempt to travel to one of the cube's edges, one would soon find oneself with no air to breath.

 

[SW VandeCarr]

Tidal forces might have particularly dramatic effects due to the lack of beaches. When the tide came it, it would come a long way in.

But neglecting the tide the "ocean" wouldn't be turbulent. It would just sit calmly in a upside down saucer sort of shape (as in a shallow dome looking thing).

Turbulence needs a energy source, and there is no energy source working on the water once it reaches equilibrium in its upside down saucer sort of shape (neglecting the tidal forces).

All of this of course ignores any weather effects.

OK. I'm thinking if the ocean were very large and deep, it could change the assumptions regarding the direction of the gravitational vectors over the water surface since water is so much less dense than the planet mean density, but I suppose the ocean surface would just conform to those vectors, whatever they are.

 

[hankaaron]

Then E would = MC3

 

[Nathew]

Then E would = MC3

I loled

 

[collinsmark]

Your idea of what "down" is (and feels like) would always be pointed toward the center of the huge cube.

I need to correct myself. On the cube Earth, the direction of what "down" is (and feels like) is not necessarily pointed precisely at the center. It's pointed near the center, but not necessarily directly at it.

 

[DiracPool]

I thought flat-earth models weren't allowed in this forum

 

[micromass]

I thought flat-earth models weren't allowed in this forum

A cubic Earth is not a flat Earth

 

[DiracPool]

A cubic Earth is not a flat Earth

Yes it is. There's just more "flat" there. In fact, there's 6 TIMES as much flat on a cubic Earth than your "regular" flat Earth. I did the math

 

[ImATrackMan]

Yes it is. There's just more "flat" there. In fact, there's 6 TIMES as much flat on a cubic Earth than your "regular" flat Earth. I did the math

No, it's flat³, silly.

 

[krash661]

Good question! Now I'm going to be asking everyone I know this question.

I also thought it was, it's kind of a good thought experiment if you think about it.

 

[SW VandeCarr]

The gravity vector field for an ideal cube is pretty straightforward. How about an ideal torus? Where can you have an ocean on an Earth mass/volume torus? (only undergrads please).

 

[collinsmark]

The gravity vector field for an ideal cube is pretty straightforward.

Are you sure about that?

I tried to find the vector field representing the force on a given point an any particular face last night, but needles to say I was not successful. Setting up the triple, definite integral is easy enough. But evaluating it is a bear. Even Mathematica gave up when evaluating it directly. As Micromass said, "well, if Mathematica can't solve it..."

Maybe it's easier to work with gravitational potential first like these folks did:
http://possiblywrong.wordpress.com/2011/09/09/if-the-earth-were-a-cube/. (I haven't checked their math yet, btw. Instead I gave up and went to sleep. Maybe later.) Once the gravitational potential is calculated, the vector force field can be found by taking the gradient.

Here is something else that might come in useful:
http://arxiv.org/pdf/1206.3857.pdf

 

[SW VandeCarr]

Are you sure about that?

haven't checked their math yet, btw. Instead I gave up and went to sleep. Maybe later.) Once the gravitational potential is calculated, the vector force field can be found by taking the gradient.

Here is something else that might come in useful:
http://arxiv.org/pdf/1206.3857.pdf

Sorry. I was only thinking of the direction of the vector, not it's length (representing the acceleration on a test mass). In line with OP, I was only concerned the perception of "tilt'. Given an ideal cube with uniform density, I just assumed all vectors point toward the center of mass.

That's not true for a torus.

EDIT: Thanks for the link. It seems there is some directional distortion of the vectors due to the mass around the vertices. This does doesn't surprise me. I didn't consider it important wrt the OP's question.

 

[I_am_learning]

I wonder

Just like spherical Earth isn't perfectly smooth and that there are horizontal plains where people live even in the mountainous regions, In the cube earth, there will be cities built on plains that are perpendicular to the gravitational vector fields at that place.
But as you move from cities on the center to cities towards the edges, you will always need to travel steep highways.
But the gravity in the cities that lie on the edges will be much lower (or will it be?)
Basketball baskets will be placed higher there because people can jump higher.
Oceans will be only located at the centers, rivers will be very steep and enormous potential for hydro-power. The edges are like enormous mountains.

How would an artificial satellite fly?

 

[collinsmark]

I think the most profound characteristic of the cube Earth would be its atmosphere. The only natural, survivable atmosphere would not extend to the cube corners or even the edges. (The same is true with water.)

Neglecting microbes that might rarely traverse faces via meteor impacts (and if they survive that), all of life on a given cube face is completely isolated from all other faces of the cube planet.

The natural, biological evolution of all species on the planet would be completely isolated between cube faces (again, except perhaps for those rare microbes that might survive a meteor blast).

Serious technology would have to first be created by any intelligent species before attempting to traverse faces. Airline travel is obviously right out (there is no air at the edges, so there can be no airplanes at the edges). Space-suits would have to be invented. Also, one couldn't drive a unmodified, conventional car to the edges either, since internal combustion engines require air to operate. Traveling from one face to another would be something sort of akin to an Apollo mission.

Resulting life on one face could/would be absolutely different than on other faces. Traveling from one face to another would be like traveling to an alien world.

How would an artificial satellite fly?

That's a good question. Low-earth-orbit satellites are right out. On our Earth, the International Space Station is only about 230 miles (370 km) above the surface. On a typically sized globe, that's to scale with about the width of a finger or thumb. It goes without saying that that wouldn't work for a cube.

Perhaps satellites farther out, such as geosynchronous satellites might be possible/practical. But it would be tricky.