If the Earth was a cube

Factual Questions
81

Okay, aside from all the technical stuff, what would life be like on earth-cubed?

Recreation: I mentioned hang-gliding. How about skiing? you could have the mother of all ski hills.

What would navigation be like? I’m trying to imagine how strange it would be trying to fly an airplane out from the center. If you flew straight out, you’d have to tilt the nose up more and more as the gravity vector tilted behind you. It would feel like you’re in a constant climb. At some distance, you couldn’t maintain the climb any more and would stall out. So airplanes would probably have to fly in arcs like flying across the face of a slope. It would be very difficult. Event at modest distances from the center, there would be enough tilt to the gravity vector that you’d be flying ‘uphill’ in one direction and ‘downhill’ in the other.

Religion: What kind of religious belief would form on a world which looked perfectly flat, but which applied a progressively stronger force against attempts to explore outwards? The fist of God pushing you back from the forbidden zone?

Weather: How would the weather propagate? With a lens-shaped atmosphere, I wonder if the center would look like the eye of a storm, with progressively stronger winds as you moved out to the edges? Would the whole thing rotate like the Great Spot on Jupiter?

Rain would be interesting. A huge percentage of the surface would probably be the driest desert you ever saw, as the combination of lower air pressure and higher energy requirements to get there means that very little water vapor would ever make it up there.

Come to think of it, the edges could be pretty hellish: Bone dry, windy, increasingly steep, and getting harder and harder to breathe. Wth that in all directions, maybe the people’s religion would be that they were essentially in heaven, surrounded by an increasingly inhospitable barren wasteland.

Would the center eventually be the world’s biggest collection of floating junk as every bit of detritus that hit the water began flowing down to the center?

82

I believe a cube-world modeled closely to young earth would present a fascinating study in biological evolution.

Let’s assume that each of the six surfaces on our cube-world had a zone favorable for the genesis of life, similar to that on earth ~4 billion years ago. I imagine this zone would extend as a circle (or shoreline band) of “X” miles radius from the center of each surface, the periphery being far from all edges and vertices. Abiogenesis and the resulting biological evolution would take place separately and uniquely on each surface. Unless someone argues convincingly to the contrary, I’ll assume that the vertices and edges will remain sterile (maybe some viruses or spores of some type could withstand the wasteland) for billions of years until the first technologically advanced species crosses over into adjacent surface bio-zones.
Comparative analysis: the lifeforms found on different cube surfaces should be much more unique than comparing any lifeforms on our round earth, due primarily to lack of common ancestry. However, they should not be as unique as those compared to lifeforms on alien planets, since the starting conditions and parameters on each of the 6 cube-world surfaces were virtually identical.
We know that life (in toto, as the biosphere) changed, and continues to change, planet earth (i.e.anaerobic to aerobic atmosphere, climatic changes etc.). Let’s imagine 5 surfaces with greatly different evolved bio-zones and one surface where abiogenesis never took place. Could the resulting varying morphological changes in terrain lead to devastating global effects (e.g. planetary wobble or akimbo tilt), perhaps jeopardizing life on all surfaces? I believe some degree of instability would be likely.
Lets imagine the “diaspora” of a species similar to modern man into another bio-zone propagated by Neanderthal-type beings. Would first contact result in severe mortality due to immunological systems overwhelmed by wholly un-recognized microbial flora, or perhaps (again, due to non-common ancestry), the alien microbes would be so different as to be non-pathological.
Apparently, the Cro-Magnon v. Neanderthal showdown here on earth resulted in the more intellectually advanced species driving the other to extinction. Would this necessarily be the case on our cube-world? The mingling of dominant species would most likely take place at a later stage of species evolution on the cube (i.e. a Cro-Magnon type man would not have the wherewithal to cross a planetary edge, IMO). Presumably, the dominant species in both bio-zones would have more time to evolve and dominate their respective environments before meeting for the first time. How technologically advanced would a species need to be in order to mount an expedition to other bio-zones? My guess is one comparative to late 20th century man. IMO, extinction of the invaded species would be a less likely scenario than assimilation, or perhaps non-lethal subjugation.

83

donut planet report by Chronos
donut planet thread
I swear there was a previous discussion on cube planets as well, but damned if I can find it. Anyway, I think that the surfaces describing the air and water on a cube planet would not be perfect spheres due to the gravity of the corners. Rather, the surfaces would be pulled slightly in the direction of the corners. This has been observed on our regular old oblate sphereoid Earth using satellite mapping techniques.

84

First of all, let me just say that, now that I am at home again and have done some of the calculations, I apologize for earlier skepticism. I think I seriously underestimated (by some substantial factors of ten!) the difference between the oceanic/atmospheric cover and the “height” of the corners of the world. :smack:

For those interested: The volume of oceanic water on the Earth is estimated at 1340 x 10[sup]6[/sup] cubic kilometers of water. Thus, each face would have about 223.3 million cubic km. of water in its lens. Data on water from Britannica Concise.

I haven’t bothered with the data on atmosphere because it is clear that the atmosphere wouldn’t reach the edges, even at the center, to say nothing of the Vertices. Note, too, that the amount of surface variation on the Earth is so slight that, even if Cubearth has the same variation, it is negligible in comparison to the effects of the cube shape itself. A mountain like Everest, for example, even if placed at a vertex, would barely register as a pimple added to it.

Gravity of this world is interesting. I was thinking of trying to restructure Cubearth so that it had equal gravity all over it surface, but then you don’t get lens oceans, and I find those more interesting. But assume for the nonce that g[sub]Cubearth[/sub] is equal to g[sub]Earth[/sub] at all points on Cubearth. Assume the axis of rotation is through two vertices. There would still be some really wierd atmospheric effects! That alone might be worth considering, instead of the relatively boring lens of atmosphere on each face if the density of Cubearth is relatively constant through its volume.

Imagine the storms, as they go over an edge!!!

85

I’m not saying it would be more or less near a corner, although I’m pretty sure it would be less since most of the matter would be farther away. I am saying it would be pointing a bit towards that corner, but not by nearly enough to cancel the uphill sensation. It would just seem like less of a hill in places than if the gravity was from the center of the planet. The middle of the face would still seem flat and the vertice would still seem like a 45 degree incline, but you would expect halfway between the center of the face and the middle of a vertice to seem like a 26.57 degree incline (tan^-1(1/2)) whereas it might actually seem like 26.5 degrees, or 26 degrees, or even less. Without making a computer simulation, there’s no way to even estimate by how much it would be less, but it would be less.

I’m going to see if I can come up with such a simulation. I think I can do it if I learn the formulas for converting between cartesian and polar coordinates in a 3-D space.

86

Hard to imagine if the atmosphere doesn’t reach it.

We’ve got time. Dum te dum, dum…

87

Okay, I designed a program that assumes a planet goes from -1 to 1 along each dimension of a three-dimensional cartesian plane and uses vector addition to calculate the sum of gravity of divisions of the planet 0.01 units wide. It’s also supposed to calculate the angular offset of the pull of gravity from the center of the planet, but that doesn’t seem to be working. It took a long time to debug the other calculations, so I’m not messing with that right now, but I’ll include the source code if anyone else wants to look at it. The gravity isn’t in any real units like m/s^2, but can be used to calculate ratios from one point to another. (I could easily adapt it to actual units, but I need a break.) I believe the gravity calculations are correct because the gravity in the center of a face is somewhat more than at a corner as I predicted, and the gravity at 50 units away from the planet is roughly 4 times what it is at 100 units away. At the center of the planet, it’s a very small but non-zero number but I don’t expect it to be 0 because of the way I estimated. That estimation should cause very little inprecision outside the center of the planet.

Here are some examples of what I came up with. At the center of a face (1,0,0), gravity is 5166947. At the center of a vertice (1,1,0), gravity is 4296988, or 83% of at the face. At a corner (1,1,1), it’s 3261420, or 63% of the face. Halfway between the center of a face and the center of a vertice (1,0.5,0), it’s 5025222. Halfway between the center of a face and a corner (1,0.5,0.5), it’s 4867273. In orbit above a face 5 units from the center of the planet (5,0,0), it’s 318782, while 5 units from the center above a corner (2.887,2.887,2.887), it’s 319258. So you don’t have to be orbiting all that high before the preturbance becomes really small.

Here’s the source code. You guys can help me debug the calculations.



#include <math.h>

int main(int argc, char *argv[])
{
	double loc_x;
	double loc_y;
	double loc_z;
	double pos_x;
	double pos_y;
	double pos_z;
	double ang1;
	double ang2;
	double rad;
	double grav_x;
	double grav_y;
	double grav_z;
	double grav_ang1;
	double grav_ang2;
	double grav_weight;
	printf("X location: ");
	scanf("%lf", &loc_x);
	printf("Y location: ");
	scanf("%lf", &loc_y);
	printf("Z location: ");
	scanf("%lf", &loc_z);
	grav_weight = 0;
	for (pos_x = -1; pos_x <= 1; pos_x += 0.01)
	{
		for (pos_y = -1; pos_y <= 1; pos_y += 0.01)
		{
			for (pos_z = -1; pos_z <= 1; pos_z += 0.01)
			{
				grav_x = grav_weight * sin(grav_ang1) * cos(grav_ang2);
				grav_y = grav_weight * sin(grav_ang1) * sin(grav_ang2);
				grav_z = grav_weight * cos(grav_ang1);
				rad = sqrt((loc_x - pos_x) * (loc_x - pos_x) + \
				  (loc_y - pos_y) * (loc_y - pos_y) + (loc_z - \
				  pos_z) * (loc_z - pos_z));
				if (rad < 0.0025)
					rad = 0.0025;
				grav_x += (loc_x - pos_x) / rad / rad / rad;
				grav_y += (loc_y - pos_y) / rad / rad / rad;
				grav_z += (loc_z - pos_z) / rad / rad / rad;
				grav_weight = sqrt(grav_x * grav_x + grav_y * \
				  grav_y + grav_z * grav_z);
				grav_ang1 = acos(grav_z / grav_weight);
				grav_ang2 = atan(grav_y / grav_x);
			}
		}
	}
	ang1 = acos((loc_z + pos_z) - sqrt((loc_x - pos_x) * (loc_x - pos_x) + (loc_y - \
	  pos_y) * (loc_y - pos_y) + (loc_z - pos_z) * (loc_z - pos_z)));
	ang2 = atan((loc_y - pos_y) / (loc_x - pos_x));
	printf("Gravity is %lf at %lf radians from center.
", grav_weight, sqrt((grav_ang1 - \
	  ang1) * (grav_ang1 - ang1) + (grav_ang2 - ang2) * (grav_ang2 - ang2)));
	return 0;
}
88

You shouldn’t <snip> relevant things I say, silly. I was postulating a world with even gravity, in which case the atmosphere and the oceans can reach the whole planet. :wink:

89

So nothing terribly different from the previous estimate.

Very interesting to see that having a satellite won’t be so much of a problem. It would suck not being able to have a phone call with aunt Edna at a different face of the planet.

According to someone’s previous estimate of an average depth of 20 Kms for the oceans and the cited volume of the water on earth, we get oceans of 1000-1500 Kms in diameter (by mental calculations with tons of rounding). I am inclined to believe that it should me more than that since the lenses should flatten. Either way, it is turning out that the oceans would not be too too small. If anyone has the tools to get a decent estimate, please feel welcome.

90

With respect, what is the difference between a “corner” and a “vertice” (properly called a vertex, btw. :wink: ).

91

Awesome calculations. The numbers sound about right.

I do think there would be biological transfer between the sides. Meteor impacts would blast stuff into space, and the detritus that didn’t reach escape velocity would come down all over the place. Some of it would have living microbes on it. So at least at the microbial level, there would be a transfer of biological material from face to face.

We’ve found bacterial life in the strangest places on the earth. Volcanic vents, deep underground and deep under iced over lakes. We find it almost everywhere we look.

As the societies became more technically advanced, it would be interesting to see how they travelled from face to face. Would they eventually build train-like transports to haul them up to one edge, and then down the other? The energy costs wouldn’t be any greater for a train on earth, as you’d go ‘uphill’ in one direction, then ‘downhill’ one you crossed over to the other plane. You could reclaim the energy going down the other side that it cost to go up the first, minus friction.

92

Point taken. I apologize.

They are the same thing, no?

93

But if you were on the satellite, and it had a close-to-earth orbit, I imagine it would be a bumpy ride even if it didn’t crash. Lots of mascons to contend with.

94

[nitpick]
If the Earth were a cube…
[/nitpick]

95

Nope, was. Were, too, of course. Were to be as well, come to think of it (I fervently hope, anyway!).

We can talk about it in the past as well as the present. :slight_smile:

96

I’ve got a handy little Earth[sup]3[/sup] model going. I’m using actual Earth numbers for the masses and volumes of the planet, atmosphere, and oceans. Here are some first data showing surface gravity and the extent of the oceans. The oceans are 125 km deep in the middle!

oceans and gravity

97

Here’s a slice through the center of the cube showing the gravitational potential energy and its gradient (which shows the direction of the pull of gravity.) I show the interior for anyone interested in making a tunnel.

potential and gradient

98

Pasta, do those figures take into account the affect upon gravitational pull resulting from the planet not being spherical? Also, how did you come up with the lens on the oceans being so thick? What was your algorithm for figuring that out?

I just noticed that Cubearth has about 122 million km[sup]2[/sup] extra surface area than Earth does. This means that, even if the gravity on Cubearth was uniform at all points on the surface, the oceans would be a bit less deep, assuming that the local variation in surface elevation was distributed in a fashion similar to Earth’s.

99

Yes. My code has a cube with the Earth’s mass and volume. I calculate everything by numerically integrating the contributions from all over the cube.

One thing I can calculate is the gravitation potential at any point. The surface of the ocean will follow an equipotential. I step through equipotentials, each time calculating the total volume of points outside the cube with potential greater than my current equipotential. When this volume (all six sides counted!) equals the volume of the oceans, I have the ocean’s surface potential V[sub]0[/sub]. (I do some quadratic interpolating to improve accuracy here.) Any point outside the cube with potential greater than V[sub]0[/sub] is under water. The depth D of the ocean in the middle is given by Potential(0,0,D)=V[sub]0[/sub], or by just reading the height off the first image.

I checked all this by fitting a curve to the blue/white boundary on that image and calculating the volume of revolution under the curve to verify that I got one-sixth the ocean volume. I did, to within the accuracy of my test curve.

100

Also, note that the vertical scale is not the same at the horizontal scale in the top panel in the first image. That may be causing confusion. The vertical scale is exaggerated 7.5x. The ocean lens isn’t very thick relative to the whole planet, as shown in the to-scale botton left panel of the same image. (The oceans are one pixel thick there because they actually are one pixel thick at this resolution.)