Conical-gyroscopic suggests one solution (I think) to the gravity problem, albeit only for two faces. Spin the sucker up fast enough to counter gravity’s pulling everything to the center-of-the-face. Note that there probably should be a second rotation vis-a-vis the star around which the planet orbits (to deal with the energy problem), Thus, Spin 1 would be parallel to the orbital plane, so Spin 2 could bring each of the two gravity-stabilized faces into sunshine by turns. In other words, the cube spins on its side, not polar as with the Big Blue Marble.
Your initial assumption was correct. As I said earlier (or tried to say, but I got my terms mixed up), gravity would point to the center at the center of a face, the center of an edge, and at a corner. So if you walked from the center of a face to the center of an edge, it would seem like you started out on flat land (pretending there were no oceans) and ended up on a 45 degree incline. The only catch is that the land in between wouldn’t be the degree of incline that you would expect from trigonometric calculations. I’m thinking it would only be off by 2 or 3 degrees at most though.
A different case would be if you were walking to the edge, but not at the center or a corner. Walking up it wouldn’t simply seem like walking up a hill; there would seem to be a bit of a traversal as well. In other words, it would be like walking up a mountain on our Earth at an angle instead of straight up. This would actually happen even if gravity was pulling you to the center the whole time, but because it’s not, the traversal angle will seem slightly different.
And in the case of walking towards a corner, it would be like walking towards the center of the edge, but a longer distance and possibly with more/less deviation from the expected angle along the way. It wouldn’t end in a 45 degree incline either, but I can’t remember my geometry well enough right now to say what angle it would be. (For some reason, 54 degrees is popping into my head.)
To expand upon what I said, imagine being in a crater here on Earth with 4 mountains at equally spaced angles. The feet of the mountain are against each other, so one begins where another ends. From in your crater, every direction is uphill, but not equally so. You can walk between the peaks of the mountains (like walking towards the center of an edge), which is the least steep path and is strictly uphill, no traversing. You can also go over a peak (corner), which is a steeper angle but still no traversing. Or you can go halfway between the two, and the steepness will be between that of the peak and the valley, but you’ll also be leaning as you walk.
Gravity.
The corners are basically stone pyramids fifteen hundred miles high. The faces are exposing mantle material that very much wants to squirt upwards a thousand miles or so. In between, you get a strong torsion to accommodate those forces. If you stop and think about it, lots of stuff even smaller than the earth is spherical, including the moon, and the largest asteroids. They are not even large enough to have atmospheres. Venus and Mars and Mercury are all spherical, and much smaller than earth.
Equilibrium is the thing here. Spheres have it, cubes don’t. I don’t know how large a carbon honeycomb can be, but if it could be as large as the Earth[sup]3[/sup] it would be much less massive than the Earth[sup]o[/sup], and have a much lower gravity. Earth is mostly iron, nickel, silicon, and stuff like that. A pure diamond that size, even if we were willing to settle for an Octagon, would have far less mass than the Earth[sup]3[/sup].
Make it out of Scrith, (the material Niven imagined for the Ringworld) and you have to make it hollow, and line the inside of it with gravity generators. If you do that, you can have gravity perpendicular to the surface, and walk to the edge without perceiving any upward slope. Of course, then you don’t get oceans, without engineering them as well. (By the way, scrith has a greater tensile strength than the strong nuclear force, which, in my opinion makes Ringworld a magical construct. Of course we don’t need that much tensile strength, but we need compressive strength in the same ballpark)
Oh, and what about meteor strikes?
Also, is anyone else going to try to figure out the dynamics of solar energy distribution for the cubic faces? I am out of my depth in the geometry so far. I know that all areas of any one face receive the same amount of sunlight per square meter as any other area. Near as I can tell that means very boring weather. With three fourths of the time in darkness, and only partial atmospheric coverage, it sounds like freeze up and stay frozen, to me, but I am not confident about it.
Tris
If you were in a boat (or on an island) in the middle of a face I don’t think the corners would look like mountains as they are lower than what you would perceive as a right angle to the force of gravity.
Man am I jealous. I asked this exact same question many moons ago and didn’t get this any where near this kind of a response.
http://boards.straightdope.com/sdmb/showthread.php?t=133211&highlight=cube+earth
was it a cubic moon?
Would a cubical Earth composed of completely-cold and rigid rock and iron (no internal heating, no magma) be rigid enough to retain a cube shape?
Neat pix, but it looks like they just mapped the existing Earth to the shapes without taking into consideration gravity and mass re-distribution effects in the ways we have been talking about here. No “lens” oceans, for example.
No. But, I don’t have numbers, or cites to back that up.
One of the ancient pyramids began to collapse when it got so tall that the weight of the many rocks above began crushing the rocks on the bottom. The compressive strength of brick and rock is substantial. That’s why it gets used for building really big stuff. But it is limited on a scale of really big pyramids. Really big pyramids are measure in meters. The corners of the Earth[sup]3[/sup] measure in millions of meters, and their weight is trillions of times greater.
So, you get cracks. As noted, everywhere is downhill toward the center of the face, so, stuff rolls, or slides to the center, as it falls off the corners. It begins to look round. In fact, it begins to be round. Iron is pretty strong, but it bends. Bending a thousand mile thick chunk of iron would take a lot of force. But piling a few thousand miles of rock on one end is a lot of force. Also, the iron is much heavier than rock, so it adds its own problems.
There is one easy way to think of it. The Earth weighs about 5,972,000,000,000,000,000,000 metric tons. The parts nearer to the center have that weight sitting on top of them, proportional to how far they are from the outside. Nothing is strong enough to hold that up. So, it collapses. Or, more precisely, it collapsed as it formed. But, if you reshaped it with magic, (or engineering indistinguishable from magic) it would collapse immediately.
Here is the thing; it would collapse as much as it could so that everything was as close as possible to the center. That shape happens to be a sphere. Look around the universe, and everything really big is a sphere, unless it rotates, in which case it gets oblate, perhaps even oblate enough to be a disk. If it had enough strength, some of the stuff on the surface would get poked up above the mathematic surface of the minimal sphere, perhaps up to ten or so miles, like Mt. Everest. But, all the stuff that fell off as it went up, would fall into the lowest places, and leave your new Mt. Everest only seven miles above the average surface.
Please don’t try this at home.
Tris
Isn’t that one of the new parameters of our definition of planet? that it be massive enough to crush itself into a sphere?
Musicat: True, but in fairness, it was an ad campaign. And intended to suggest the question, not answer it.
Tris: Also true. I guess we’ll just have to count on the Magrathians coming up with some clever material that’ll do the job.
Sapo: Other way 'round, I think. The parameter is that the cube be strong enough not to devolve into a sphere.
I meant the REAL definition of a planet. The one that kicked out Pluto (for different reasons). Those guys would say that our CubEarth was not a planet. What nerve!
My bad. And, so, you’re right that Tris’ point is so true that it has become a basis of taxonomy.
OTOH, once the Magrathians build Cubic Earth, perhaps the astronomers will have to adjust the taxonomy again. 
Or perhaps a bit higher, e.g. Olympus Mons, 27km above the mean surface level of Mars.
Ahhh, Watch it, you went to Mars, and failed to mention that the gravity is less! Tut tut.
Tris
Well, you did say “Don’t try this at home.”
Damn kids, always coming to my Mars and messing with my landscape.
Oh, get a lawn!
You’re looking at it the wrong way round: if the guys that defined these things lived on Earth[sup]3[/sup] they’d come up with a different definition!