If you listen carefully he is actually saying that we have no idea the shape of the universe. The universe is so much bigger than what we can see. What we do see looks flat to us.

"It's like taking a square mile of the earth and trying to determine its shape. It just looks flat to us."

When we travel out 13 billion years and look at the universe its geometry looks the same. It looks the same from every point in space. We measure the flat universe the same was we once measured the circular earth. with triangles.

You pick a point in space 13 billion light years away, and then you pick another point 13 billion light years away. You draw a line connecting the points and then two lines back to you and you measure the angles. On a flat plain you get 180 degrees. If the universe were curved you would get a non-Euclidean triangle with degrees greater than one hundred eighty.

He is not saying that the universe is not three dimensional. He is only asserting that it is best to view it on a two dimensional plain. That is what the measurements suggest. Remember, we are only measuring a very small amount of the known universe. We don't know how big it is. The idea of universe stacked on universe stacked on universe is a allusion to string theory. It's not that important in understanding the flat universe (at this point).

He is using Einstein's theory of gravity. All objects warp the space around them, like bowling balls on a flat trampoline. Remember this is a model of space. It is what scientists can observe. As our comprehension increases and as we learn more. our theories will change. It is also said that space is expanding like the surface of a large balloon.

This might help. The expansion of space is not the things in space but space itself. It is not three dimensional objects that are expanding but one dimensional space. The surface of the balloon.

https://www.youtube.com/watch?v=i1UC6HpxY28

I am not an astrophysicist and this is the way I grasp it. Someone else might do a better job. I am only an atheist. My background is psychology.

It isn't a references to dimensions (although it's easy to see how it could be mistaken for that). It means that space is approximately Euclidean. Which is just a fancy way of saying it obeys the rules you learned in (Euclidean) Geometry class. That triangles contain 180 degrees, that spheres have a volume of 4/3*pi*r^3, etc. It is a reference to a LACK of curvature.

They can only put limits on the universes' curvature. Or in other words, no measurement has revealed any detectable curvature; so if it's curved, that curvature must be very small at the scale of the observable universe. That is why they say it is "flat".

Thank you for the explanation guys. When I attempt to learn about such topics, I run across concepts that can take me a while to get my head around. For example, when I first encountered the concept of "nothing", that not even time would be present, wow, that was a tough one to comprehend.

Somehow it is more palatable to deal with that than visualize Tin-Man activating his optional wet-vac.

Nyarlathotep: Sorry
I wrote my response before reading yours. I should have just given you an agree.

Others have already commented here on the fact that the word "flat" is used as a synonym for "Euclidean", regardless of how many dimensions the space in question has, but it's worth expanding on this point.

Basically, mathematicians have devised a number of different coordinate systems, that can be used to represent a given space. The most familiar coordinate system is the Cartesian coordinate system, which is used to represent Euclidean spaces, but other coordinate systems exist, such as cylindrical polar coordinates, spherical polar coordinates, prolate spherical coordinates, oblate spherical coordinates, ellipsoidal coordinates, and there's even a bipolar coordinate system (for more details on these, see Vector Analysis, With An Introduction To Tensor Analysis by Murray R. Spiegel (Schaum's Outline Series, ISBN 0 07 084378 3)).

However, all the other coordinate systems differ from Cartesian coordinates, in that they possess a curvature. The coordinate curves and surfaces in these systems include curved surfaces and curved lines, which define the geodesics in these spaces, namely, the paths between two points in space that cover the shortest distance. It's tempting to think that a geodesic between two points is always a straight line segment between those points, but in general coordinate systems, this is not the case. The geodesics in general coordinate systems, consist of coordinate curves passing between the two points. Admittedly, this is a counter-intuitive concept when first encountered, but, this concept applies in a robust manner, when applying these coordinate systems to the modelling of real spaces. For example, the shortest distance between two points on the Earth's surface, if one is restricted to travelling on that surface, are great circles passing through the two points, and of course, those great circles possess a curvature.

Which brings us neatly to the concept of orthogonal curvilinear coordinates, also covered in detail in the reference above (which was a standard textbook for undergraduate mathematicians back in the 1980s). These are usually defined in terms of transformations from Euclidean coordinates. So, for example, if a 3D space has points referenced by coordinates (u, v, w) in a general curvilinear system, then u, v and w can themselves be represented as functions of x, y and z where x, y and z are the familiar Euclidean coordinates. For example, in the particular case of spherical polar coordinates (r, θ, φ), the relationships are as follows:

x = r cos θ cos φ
y = r cos θ sin φ
z = r sin θ

and inverse relationships for r, θ and φ in terms of x, y and z can also be written (though the board facilities make it difficult for me to reproduce these here).

Now, each coordinate system has associated with it, a quantity called a metric tensor, which in the case of a 3D space, can be written as a 3×3 matrix. In the case of a 3D Euclidean coordinate system, the metric tensor for that coordinate system is the 3×3 identity matrix, with values of 1 in the diagonal entries, 0 in other entries. The metric tensor for a spherical coordinate system, on the other hand, is rather more complicated, and its entries include variable entries involving functions of θ and φ, some of these being off-diagonal in the matrix. Spaces represented by general curvilinear coordinate systems, with metric tensors of this form (off-diagonal and variable components), all possess intrinsic curvature. Unfortunately the board doesn't make it easy to post the actual matrix representations thereof, but you can find the details in numerous textbooks.

The same principle can be generalised to spaces of N dimensions, where N is any number you care to choose, at which point the metric tensor for a space of this sort becomes an N×N square matrix. For the four-dimensional space-time representation of general relativity, matters become complicated further by the need to distinguish between local spaces, which may have a local curvature dominated by gravitating objects in the local space, and global space, which on the large scale may have a curvature that is negligible, and indistinguishable observationally from the zero curvature of a true Euclidean space. It turns out that there are interesting connections between the global curvature and the amount of matter in the observable universe, but this is starting to run deep into cosmological territory that isn't necessary here.

In short, as others have commented, "flat" is used as a synonym for "Euclidean", i.e., zero global curvature of space.

Have you heard of the Pythagorean Theorem? Did you know there could be infinite number of them? One for the (x, y) 2d plane, and the (x, y, z) 3D space. This can also be used for 4D universe, 5D universe, etc., etc.

Funny thing is I thought I had invented the 3D Pythagorean Theorem when I was only 8 years old. A year later, I found out I had not invented it, only derived as others had already done about 25 centuries before. What a let-down. However, from a certain point of view, I DID invent the 3D Pythagorean Theorem since I did derive it from mine own thoughts while star gazing one night.

Attachement: PNG of the two theorems.

rmfr