The doughnut's surface looks curvature, but it's mathematically flat.

No, it is curved.

It has both principle and Gaussian curvature.

It is not flat in any way.

The simple way to demonstrate this is with a triangle.

For a "flat" surface (that is one with 0 Gaussian curvature), the angle sum will always be 180 degrees. For a surface with constant positive Gaussian curvature, it will be more than 180 degrees. For a surface with constant negative Gaussian curvature it will be less than 180 degrees. For a surface where the Gaussian curvature varies (such as a simple torus) it will vary depending upon the curvature of the area the triangle spans.

So now lets make a triangle.

The "equator" corresponds to a straight line. So that can be one line.

Now we pick a point on this and draw another line at a right angle.

We will continue this up to the top of the torus, and then put in another right angle.

Then we follow this line back down, and get back to the equator, at some angle.

We can easily show that this some angle is not 0, as that would mean the top has to be at the equator.

This means our angle sum will be 90+90+some angle. This is over 180 degrees and thus the surface is not flat.

The other way of showing it is measuring the distance between parallel lines.

The simple lines to show this with are those which go perpendicular to the equator.

For simplicity rather than considering a bunch, I will just add up the angles between them all to get a complete revolution.

For the outside of the torus, this sum will be 2*pi*R, where R is the outer radius of the torus.

For the inside of the torus this will will be 2*pi*r, where r is the inner radius of the torus.

This means that in order for them to be the same, the torus must have the same inner and outer radius.

In our space that corresponds to a single line, not a torus.

A "torus" which is flat is a flat torus.

This cannot be embedded in our 3D space, as it is physically impossible to do so.

This is the space of packman.

If you go off the right side, you appear at the left side. If you go off the top you appear at the bottom.

If you lived in the appropriate space you could even construct this from a sheet of paper.

The first step can be done in our space, by joining the 2 sides of the page together into a cylinder.

If you would like a "flat" shape that can be embedded in our space, the surface of an infinite cylinder is one.

This now has principle curvature, which depends upon the space it is put in, but it has 0 Gaussian curvature.

To complete the flat torus, you need to join the 2 ends of the cylinder together, without bending it, which simply cannot be done in our space.

Either way, WHO CARES?

This is not the shape of the surface of Earth.

First, an important point, just like the flat paper representation, all triangles drawn on this surface will have an angle sum of 180 degrees.

This does not apply for Earth.

For Earth, a triangle with an angle sum of 270 degrees can easily be constructed (or 180+x).

Start at the equator and go due north to the north pole. Then turn 90 degrees (or x), and head due south back to the equator.

Then turn 90 degrees at the equator and travel along until you reach your original line which is going off at 90 degrees.

That alone means Earth isn't flat, and thus can't be a flat torus.

But more importantly, Earth isn't a torus.

Again, the simple way to focus on it is the flat paper representation.

When you go off the right (or east), you come back in from the left (or west). This part does match Earth.

But then when you go off the top (or north), you come back in from the bottom (or south). This doesn't match Earth.

Instead when you go up north, you came back from the north at a different location.

So Earth isn't any kind of torus. (As this applies for all, not just a flat torus).

Earth is topologically a sphere. And while there is a flat torus, there is no flat sphere.

In other words, Earth's surface is curved, not flat.

It is a non-flat surface mathematically with curvature, and not just principle curvature which would allow it to be a flat surface, but Gaussian curvature which prevents it from being so.

I already pointed this out at the start of the thread.

So why wait so long just to bring up this BS again?