It took me several seconds to get the joke. Brilliant! 
Postulate additional physical laws that impose stable orbits, perhaps ?
So when I eat a 3 dimensional donut, I am denying my 2 dimensional self some donut goodness?
(I’ve been told that my one dimensional self doesn’t care about anything but sex)
You might appreciate the Robert Heinlein short story “—And He Built A Crooked House—”. (Weird-ass formatting on that link, but the text is all there.)
Doesn’t the gravitic curvature of space demonstrate a fourth dimension?
How could three dimension “bend” if not through a fourth?
Does light “bending” via gravity directly show the existence of the fourth dimension even though we can’t lay our hands on it directly?
Can I use any more question marks?
I’m in yer offis break room, eatin yer universes.
No.
Space-time is a four-dimensional construct that warps in the presence of mass. It’s not bending in the sense that a piece of paper bends.
No. Light follows the geodesic , the shortest distance, in a gravity well. There’s no need to postulate a fourth spatial dimension to explain that. A shortest distance route around the surface of a globe marks out a geodesic but stays in the same dimension.
Sure, but my guess is that the answers will be “no” to all of those as well. 
Non-scientist here but:
I thought that scientists were seriously considering that there are extra dimensions in the popular sense of the term, because it could explain why gravity is so weak: that it radiates out into all dimensions whereas the other forces are restricted to just 3 dimensions for whatever reason.
Also, understanding the extra dimensions of string theory as just like the regular dimensions of space is the only way I’ve managed to get my head around the concept…
I thought that when they talked about dimensions being “curled up small” they mean that the lengths of the extra dimensions are very small, and that they are geometrically closed. If that’s not what they mean…then I don’t know what they mean… 
Not really following this.
Yes the shortest distance around a globe marks out a geodesic and that line remains in the same dimension.
But the globe needs three dimensional space to curve even if it is not apparent to the line on the globe. The globe is now defining length, width and height. If we flatten it to 2d then it is no longer a globe but a flat plane.
Yes, but the surface of a globe can be considered in its entirety as a two-dimensional object. There’s no need to look at a third dimension to examine any aspect of it. Spherical geometry and trigonometry work on the surface of a globe without reference to its being embedded in a higher dimension.
That’s the same way we look at space-time. It can be completely explored without resorting to embedding it in a fourth spatial dimension. It’s mathematically analogous. Non-mathematicians (which includes me) have trouble seeing that because we live in three dimensions and our entire lifetime of experience prevents us from seeing anything else.
The unstable orbits aren’t directly caused by the number of dimensions. Rather, the unstable orbits are caused by the force laws, and in the simplest extra-dimension models, the force laws are different from what we’re familiar with. But there are slightly more complicated extra-dimensio models which leave the force laws intact, or mostly so: For instance, the electromagnetic force law can be kept at 1/r[sup]2[/sup] by confining the photon to a thin membrane (or “brane”) embedded in the higher-dimensional bulk, and the gravitational force law can be kept at 1/r[sup]2[/sup] (at least, at large distances) by putting periodic boundary conditions on the extra dimensions at a scale smaller than any at which we have measured gravity.
By the way, in our normal, 3-dimensional world, shadows are already three-dimensional. It’s just the intersection of the shadow with some projection surface that’s 2-d.
Not to be obtuse here but seems to me your 2d surface is curved in 3d space.
Yes we can explore it just from a 2d perspective. Say us Flatlanders draw out a huge equilateral triangle. Unless my geometry fails me on a curved surface we will not measure 60 degree angles at all corners. The only answer to us Flatlanders is we live on a curved surface. We would not be able to see or experience the third dimension directly but we can see its effects.
In short, the globe is three dimensional even if the Flatlanders on the surface only perceive the two dimensional.
What’s a donut? Ooooh, it’s raining again!
Yes, the surface of a globe is a 2D surface, but in order to be able to recognize that it is “curved” (and compute its curvature), we need for it to be plunged into 3-space. That’s what Mole is saying.
So what you’re saying is that the warping of space occurs because space-time has four dimensions, without the need for a fourth spatial dimension. Am I correct? (I am actually a mathematician, but I know very little about physics.)
Four dimensional space-time is distinct from four dimensional space I believe.
The idea being in space/time I can use 4 data points to define where and when I am. E.g. Starting at the Statue of Liberty I will meet you 10 miles west, 4 miles north and 200 feet up at 3pm on July 17.
For a fourth spatial dimension we need something that works at a right angle to length/wight/height which is more than a little mind bending to picture.
Maybe my brain is asleep today, but I don’t understand what you mean. It seems to me that a single point in space-time will give your location in space and time, if you choose it to be the one where you actually are. (Of course, there is the problem that you cannot stop the flow of time, and therefore you cannot stop moving.) I must be missing something, so can you explain your idea in different words?
Maybe it’s harder to picture, but it’s not more complicated, mathematically speaking.
To specify a point in 3d space you use three coordinates (length, width, height). However you can also specify a fourth coordinate…the time coordinate. So not only the where of the point but the when of it (e.g. the spatial coordinate of the earth in space at some future moment…the coordinates of where we are today probably wouldn’t get you to the earth).
Near as I can tell mathematically is why they use this. Regardless of how the universe is constructed it is still useful to use higher dimensions in mathematics which seems well understood.
Being able to specify “when” doesn’t make it a dimension, necessarily, does it? I hear this all the time, that time is a dimension, but it’s not in the sense of geometry at all.
It’s a property, of course, one that’s very important but why should it be considered in the same breath as “length, width, & height”. Every three dimensional object has length, width, & height. It doesn’t necessarily have a time/duration component, does it?
I would think everything in the universe has a time/duration component. There are significant differences between a sheet of paper that was manufactured yesterday and a sheet of paper bound into a book that was published in 1884. The thing about time as a dimension is that it’s unidirectional. We can traverse the other three dimensions freely. We’re straitjacketed to traverse time both at a pace and in a direction outside of our control.
There are all kinds of interesting thought experiments that go with this as well. Does the past actually exist? Does the future actually exist? What if time is nothing but our awareness traveling through an infinite succession of static femtoseconds that exist eternally frozen in their individual existences?
Just because a space is curved, does not imply the existence of extra dimensions. Curvature, as it’s discussed by physicists, is a strictly intrinsic property, not extrinsic. That is to say, we define the curvature of the space we live in in terms of nothing more than the measurements we can make within the space. That curvature could be due to the existence of other dimensions, but so long as those other dimensions don’t make any difference to what we can observe, we might as well not assume that they exist.
One analogy that Einstein used: If we’re measuring the surface of a flat tabletop, using identical little metal rods, we’ll find that we can arrange all of the rods in a perfect square grid, with all of the angles 90 degrees. Based on this, we could say that the tabletop is flat. But suppose that part of the table were heated, such that rods on that part of the table are slightly longer: Now, we won’t be able to lay all of our rods out in that regular grid, any more. If we’re using those metal rods as the definition of our measurement standard, then we could say that the tabletop is no longer flat. But this does not in any way imply anything about any dimensions of space beyond the two dimensions of the tabletop.